Properties

Label 6030.2.a.bu.1.3
Level $6030$
Weight $2$
Character 6030.1
Self dual yes
Analytic conductor $48.150$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6030.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1497924188\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.70292.1
Defining polynomial: \(x^{4} - x^{3} - 8 x^{2} + 10 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2010)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.46640\) of defining polynomial
Character \(\chi\) \(=\) 6030.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.08313 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.08313 q^{7} +1.00000 q^{8} +1.00000 q^{10} -2.20503 q^{11} -4.28816 q^{13} +2.08313 q^{14} +1.00000 q^{16} +4.93280 q^{17} +4.93280 q^{19} +1.00000 q^{20} -2.20503 q^{22} +3.35536 q^{23} +1.00000 q^{25} -4.28816 q^{26} +2.08313 q^{28} +2.81090 q^{29} -2.28816 q^{31} +1.00000 q^{32} +4.93280 q^{34} +2.08313 q^{35} -0.727768 q^{37} +4.93280 q^{38} +1.00000 q^{40} +1.47726 q^{41} +4.41006 q^{43} -2.20503 q^{44} +3.35536 q^{46} -9.38722 q^{47} -2.66057 q^{49} +1.00000 q^{50} -4.28816 q^{52} +6.16626 q^{53} -2.20503 q^{55} +2.08313 q^{56} +2.81090 q^{58} +1.47726 q^{59} +4.08313 q^{61} -2.28816 q^{62} +1.00000 q^{64} -4.28816 q^{65} -1.00000 q^{67} +4.93280 q^{68} +2.08313 q^{70} +4.84967 q^{71} -0.544464 q^{73} -0.727768 q^{74} +4.93280 q^{76} -4.59337 q^{77} +4.83263 q^{79} +1.00000 q^{80} +1.47726 q^{82} +16.5934 q^{83} +4.93280 q^{85} +4.41006 q^{86} -2.20503 q^{88} -2.18330 q^{89} -8.93280 q^{91} +3.35536 q^{92} -9.38722 q^{94} +4.93280 q^{95} +2.20503 q^{97} -2.66057 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{4} + 4q^{5} + q^{7} + 4q^{8} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{4} + 4q^{5} + q^{7} + 4q^{8} + 4q^{10} + 3q^{11} + 2q^{13} + q^{14} + 4q^{16} + 2q^{17} + 2q^{19} + 4q^{20} + 3q^{22} + 12q^{23} + 4q^{25} + 2q^{26} + q^{28} - 2q^{29} + 10q^{31} + 4q^{32} + 2q^{34} + q^{35} + 3q^{37} + 2q^{38} + 4q^{40} - 6q^{43} + 3q^{44} + 12q^{46} + 14q^{47} + 13q^{49} + 4q^{50} + 2q^{52} + 10q^{53} + 3q^{55} + q^{56} - 2q^{58} + 9q^{61} + 10q^{62} + 4q^{64} + 2q^{65} - 4q^{67} + 2q^{68} + q^{70} + 9q^{71} - 14q^{73} + 3q^{74} + 2q^{76} + 23q^{77} + 12q^{79} + 4q^{80} + 25q^{83} + 2q^{85} - 6q^{86} + 3q^{88} + 9q^{89} - 18q^{91} + 12q^{92} + 14q^{94} + 2q^{95} - 3q^{97} + 13q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.08313 0.787349 0.393675 0.919250i \(-0.371204\pi\)
0.393675 + 0.919250i \(0.371204\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −2.20503 −0.664842 −0.332421 0.943131i \(-0.607866\pi\)
−0.332421 + 0.943131i \(0.607866\pi\)
\(12\) 0 0
\(13\) −4.28816 −1.18932 −0.594661 0.803976i \(-0.702714\pi\)
−0.594661 + 0.803976i \(0.702714\pi\)
\(14\) 2.08313 0.556740
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.93280 1.19638 0.598190 0.801354i \(-0.295887\pi\)
0.598190 + 0.801354i \(0.295887\pi\)
\(18\) 0 0
\(19\) 4.93280 1.13166 0.565831 0.824521i \(-0.308555\pi\)
0.565831 + 0.824521i \(0.308555\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −2.20503 −0.470114
\(23\) 3.35536 0.699641 0.349821 0.936817i \(-0.386243\pi\)
0.349821 + 0.936817i \(0.386243\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.28816 −0.840978
\(27\) 0 0
\(28\) 2.08313 0.393675
\(29\) 2.81090 0.521971 0.260985 0.965343i \(-0.415953\pi\)
0.260985 + 0.965343i \(0.415953\pi\)
\(30\) 0 0
\(31\) −2.28816 −0.410966 −0.205483 0.978661i \(-0.565877\pi\)
−0.205483 + 0.978661i \(0.565877\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.93280 0.845968
\(35\) 2.08313 0.352113
\(36\) 0 0
\(37\) −0.727768 −0.119644 −0.0598222 0.998209i \(-0.519053\pi\)
−0.0598222 + 0.998209i \(0.519053\pi\)
\(38\) 4.93280 0.800206
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 1.47726 0.230710 0.115355 0.993324i \(-0.463199\pi\)
0.115355 + 0.993324i \(0.463199\pi\)
\(42\) 0 0
\(43\) 4.41006 0.672529 0.336264 0.941768i \(-0.390836\pi\)
0.336264 + 0.941768i \(0.390836\pi\)
\(44\) −2.20503 −0.332421
\(45\) 0 0
\(46\) 3.35536 0.494721
\(47\) −9.38722 −1.36927 −0.684634 0.728887i \(-0.740038\pi\)
−0.684634 + 0.728887i \(0.740038\pi\)
\(48\) 0 0
\(49\) −2.66057 −0.380081
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.28816 −0.594661
\(53\) 6.16626 0.847001 0.423500 0.905896i \(-0.360801\pi\)
0.423500 + 0.905896i \(0.360801\pi\)
\(54\) 0 0
\(55\) −2.20503 −0.297326
\(56\) 2.08313 0.278370
\(57\) 0 0
\(58\) 2.81090 0.369089
\(59\) 1.47726 0.192323 0.0961617 0.995366i \(-0.469343\pi\)
0.0961617 + 0.995366i \(0.469343\pi\)
\(60\) 0 0
\(61\) 4.08313 0.522791 0.261396 0.965232i \(-0.415817\pi\)
0.261396 + 0.965232i \(0.415817\pi\)
\(62\) −2.28816 −0.290597
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.28816 −0.531881
\(66\) 0 0
\(67\) −1.00000 −0.122169
\(68\) 4.93280 0.598190
\(69\) 0 0
\(70\) 2.08313 0.248982
\(71\) 4.84967 0.575550 0.287775 0.957698i \(-0.407084\pi\)
0.287775 + 0.957698i \(0.407084\pi\)
\(72\) 0 0
\(73\) −0.544464 −0.0637246 −0.0318623 0.999492i \(-0.510144\pi\)
−0.0318623 + 0.999492i \(0.510144\pi\)
\(74\) −0.727768 −0.0846013
\(75\) 0 0
\(76\) 4.93280 0.565831
\(77\) −4.59337 −0.523463
\(78\) 0 0
\(79\) 4.83263 0.543713 0.271856 0.962338i \(-0.412362\pi\)
0.271856 + 0.962338i \(0.412362\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 1.47726 0.163137
\(83\) 16.5934 1.82136 0.910679 0.413114i \(-0.135559\pi\)
0.910679 + 0.413114i \(0.135559\pi\)
\(84\) 0 0
\(85\) 4.93280 0.535037
\(86\) 4.41006 0.475549
\(87\) 0 0
\(88\) −2.20503 −0.235057
\(89\) −2.18330 −0.231430 −0.115715 0.993282i \(-0.536916\pi\)
−0.115715 + 0.993282i \(0.536916\pi\)
\(90\) 0 0
\(91\) −8.93280 −0.936412
\(92\) 3.35536 0.349821
\(93\) 0 0
\(94\) −9.38722 −0.968218
\(95\) 4.93280 0.506095
\(96\) 0 0
\(97\) 2.20503 0.223887 0.111944 0.993715i \(-0.464292\pi\)
0.111944 + 0.993715i \(0.464292\pi\)
\(98\) −2.66057 −0.268758
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 9.13783 0.909248 0.454624 0.890683i \(-0.349774\pi\)
0.454624 + 0.890683i \(0.349774\pi\)
\(102\) 0 0
\(103\) 0.410065 0.0404049 0.0202024 0.999796i \(-0.493569\pi\)
0.0202024 + 0.999796i \(0.493569\pi\)
\(104\) −4.28816 −0.420489
\(105\) 0 0
\(106\) 6.16626 0.598920
\(107\) −0.576325 −0.0557154 −0.0278577 0.999612i \(-0.508869\pi\)
−0.0278577 + 0.999612i \(0.508869\pi\)
\(108\) 0 0
\(109\) 5.78247 0.553860 0.276930 0.960890i \(-0.410683\pi\)
0.276930 + 0.960890i \(0.410683\pi\)
\(110\) −2.20503 −0.210242
\(111\) 0 0
\(112\) 2.08313 0.196837
\(113\) 13.8268 1.30072 0.650359 0.759627i \(-0.274618\pi\)
0.650359 + 0.759627i \(0.274618\pi\)
\(114\) 0 0
\(115\) 3.35536 0.312889
\(116\) 2.81090 0.260985
\(117\) 0 0
\(118\) 1.47726 0.135993
\(119\) 10.2757 0.941969
\(120\) 0 0
\(121\) −6.13783 −0.557985
\(122\) 4.08313 0.369669
\(123\) 0 0
\(124\) −2.28816 −0.205483
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −11.1208 −0.986810 −0.493405 0.869800i \(-0.664248\pi\)
−0.493405 + 0.869800i \(0.664248\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −4.28816 −0.376097
\(131\) 17.6435 1.54152 0.770761 0.637124i \(-0.219876\pi\)
0.770761 + 0.637124i \(0.219876\pi\)
\(132\) 0 0
\(133\) 10.2757 0.891013
\(134\) −1.00000 −0.0863868
\(135\) 0 0
\(136\) 4.93280 0.422984
\(137\) 4.72777 0.403920 0.201960 0.979394i \(-0.435269\pi\)
0.201960 + 0.979394i \(0.435269\pi\)
\(138\) 0 0
\(139\) −20.0855 −1.70363 −0.851813 0.523846i \(-0.824497\pi\)
−0.851813 + 0.523846i \(0.824497\pi\)
\(140\) 2.08313 0.176057
\(141\) 0 0
\(142\) 4.84967 0.406975
\(143\) 9.45554 0.790712
\(144\) 0 0
\(145\) 2.81090 0.233432
\(146\) −0.544464 −0.0450601
\(147\) 0 0
\(148\) −0.727768 −0.0598222
\(149\) 2.81090 0.230278 0.115139 0.993349i \(-0.463269\pi\)
0.115139 + 0.993349i \(0.463269\pi\)
\(150\) 0 0
\(151\) 11.7142 0.953285 0.476642 0.879097i \(-0.341854\pi\)
0.476642 + 0.879097i \(0.341854\pi\)
\(152\) 4.93280 0.400103
\(153\) 0 0
\(154\) −4.59337 −0.370144
\(155\) −2.28816 −0.183790
\(156\) 0 0
\(157\) −20.1981 −1.61199 −0.805993 0.591925i \(-0.798368\pi\)
−0.805993 + 0.591925i \(0.798368\pi\)
\(158\) 4.83263 0.384463
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 6.98966 0.550862
\(162\) 0 0
\(163\) −10.7278 −0.840264 −0.420132 0.907463i \(-0.638016\pi\)
−0.420132 + 0.907463i \(0.638016\pi\)
\(164\) 1.47726 0.115355
\(165\) 0 0
\(166\) 16.5934 1.28790
\(167\) −3.93169 −0.304243 −0.152122 0.988362i \(-0.548611\pi\)
−0.152122 + 0.988362i \(0.548611\pi\)
\(168\) 0 0
\(169\) 5.38834 0.414487
\(170\) 4.93280 0.378329
\(171\) 0 0
\(172\) 4.41006 0.336264
\(173\) −9.04779 −0.687891 −0.343945 0.938990i \(-0.611764\pi\)
−0.343945 + 0.938990i \(0.611764\pi\)
\(174\) 0 0
\(175\) 2.08313 0.157470
\(176\) −2.20503 −0.166211
\(177\) 0 0
\(178\) −2.18330 −0.163646
\(179\) −13.6754 −1.02215 −0.511073 0.859537i \(-0.670752\pi\)
−0.511073 + 0.859537i \(0.670752\pi\)
\(180\) 0 0
\(181\) 0.876984 0.0651857 0.0325929 0.999469i \(-0.489624\pi\)
0.0325929 + 0.999469i \(0.489624\pi\)
\(182\) −8.93280 −0.662143
\(183\) 0 0
\(184\) 3.35536 0.247361
\(185\) −0.727768 −0.0535066
\(186\) 0 0
\(187\) −10.8770 −0.795404
\(188\) −9.38722 −0.684634
\(189\) 0 0
\(190\) 4.93280 0.357863
\(191\) 16.4101 1.18739 0.593695 0.804690i \(-0.297668\pi\)
0.593695 + 0.804690i \(0.297668\pi\)
\(192\) 0 0
\(193\) −3.17660 −0.228657 −0.114329 0.993443i \(-0.536472\pi\)
−0.114329 + 0.993443i \(0.536472\pi\)
\(194\) 2.20503 0.158312
\(195\) 0 0
\(196\) −2.66057 −0.190041
\(197\) 10.5763 0.753532 0.376766 0.926308i \(-0.377036\pi\)
0.376766 + 0.926308i \(0.377036\pi\)
\(198\) 0 0
\(199\) 8.59337 0.609168 0.304584 0.952486i \(-0.401483\pi\)
0.304584 + 0.952486i \(0.401483\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 9.13783 0.642936
\(203\) 5.85547 0.410973
\(204\) 0 0
\(205\) 1.47726 0.103177
\(206\) 0.410065 0.0285705
\(207\) 0 0
\(208\) −4.28816 −0.297331
\(209\) −10.8770 −0.752377
\(210\) 0 0
\(211\) −12.8553 −0.884992 −0.442496 0.896770i \(-0.645907\pi\)
−0.442496 + 0.896770i \(0.645907\pi\)
\(212\) 6.16626 0.423500
\(213\) 0 0
\(214\) −0.576325 −0.0393968
\(215\) 4.41006 0.300764
\(216\) 0 0
\(217\) −4.76654 −0.323574
\(218\) 5.78247 0.391638
\(219\) 0 0
\(220\) −2.20503 −0.148663
\(221\) −21.1526 −1.42288
\(222\) 0 0
\(223\) 18.8520 1.26242 0.631211 0.775611i \(-0.282558\pi\)
0.631211 + 0.775611i \(0.282558\pi\)
\(224\) 2.08313 0.139185
\(225\) 0 0
\(226\) 13.8268 0.919747
\(227\) 22.1094 1.46745 0.733726 0.679445i \(-0.237779\pi\)
0.733726 + 0.679445i \(0.237779\pi\)
\(228\) 0 0
\(229\) 0.326934 0.0216044 0.0108022 0.999942i \(-0.496561\pi\)
0.0108022 + 0.999942i \(0.496561\pi\)
\(230\) 3.35536 0.221246
\(231\) 0 0
\(232\) 2.81090 0.184545
\(233\) 2.14922 0.140800 0.0703999 0.997519i \(-0.477572\pi\)
0.0703999 + 0.997519i \(0.477572\pi\)
\(234\) 0 0
\(235\) −9.38722 −0.612355
\(236\) 1.47726 0.0961617
\(237\) 0 0
\(238\) 10.2757 0.666072
\(239\) −11.1208 −0.719344 −0.359672 0.933079i \(-0.617111\pi\)
−0.359672 + 0.933079i \(0.617111\pi\)
\(240\) 0 0
\(241\) −7.15956 −0.461188 −0.230594 0.973050i \(-0.574067\pi\)
−0.230594 + 0.973050i \(0.574067\pi\)
\(242\) −6.13783 −0.394555
\(243\) 0 0
\(244\) 4.08313 0.261396
\(245\) −2.66057 −0.169978
\(246\) 0 0
\(247\) −21.1526 −1.34591
\(248\) −2.28816 −0.145298
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 10.7278 0.677131 0.338565 0.940943i \(-0.390058\pi\)
0.338565 + 0.940943i \(0.390058\pi\)
\(252\) 0 0
\(253\) −7.39868 −0.465151
\(254\) −11.1208 −0.697780
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 20.4202 1.27378 0.636888 0.770956i \(-0.280221\pi\)
0.636888 + 0.770956i \(0.280221\pi\)
\(258\) 0 0
\(259\) −1.51604 −0.0942019
\(260\) −4.28816 −0.265941
\(261\) 0 0
\(262\) 17.6435 1.09002
\(263\) 3.59917 0.221934 0.110967 0.993824i \(-0.464605\pi\)
0.110967 + 0.993824i \(0.464605\pi\)
\(264\) 0 0
\(265\) 6.16626 0.378790
\(266\) 10.2757 0.630041
\(267\) 0 0
\(268\) −1.00000 −0.0610847
\(269\) −8.94530 −0.545404 −0.272702 0.962098i \(-0.587917\pi\)
−0.272702 + 0.962098i \(0.587917\pi\)
\(270\) 0 0
\(271\) −7.41118 −0.450197 −0.225099 0.974336i \(-0.572270\pi\)
−0.225099 + 0.974336i \(0.572270\pi\)
\(272\) 4.93280 0.299095
\(273\) 0 0
\(274\) 4.72777 0.285615
\(275\) −2.20503 −0.132968
\(276\) 0 0
\(277\) 16.2586 0.976886 0.488443 0.872596i \(-0.337565\pi\)
0.488443 + 0.872596i \(0.337565\pi\)
\(278\) −20.0855 −1.20465
\(279\) 0 0
\(280\) 2.08313 0.124491
\(281\) −19.1526 −1.14255 −0.571276 0.820758i \(-0.693551\pi\)
−0.571276 + 0.820758i \(0.693551\pi\)
\(282\) 0 0
\(283\) −24.2905 −1.44392 −0.721960 0.691935i \(-0.756758\pi\)
−0.721960 + 0.691935i \(0.756758\pi\)
\(284\) 4.84967 0.287775
\(285\) 0 0
\(286\) 9.45554 0.559118
\(287\) 3.07733 0.181649
\(288\) 0 0
\(289\) 7.33252 0.431325
\(290\) 2.81090 0.165062
\(291\) 0 0
\(292\) −0.544464 −0.0318623
\(293\) −13.9705 −0.816163 −0.408081 0.912946i \(-0.633802\pi\)
−0.408081 + 0.912946i \(0.633802\pi\)
\(294\) 0 0
\(295\) 1.47726 0.0860096
\(296\) −0.727768 −0.0423007
\(297\) 0 0
\(298\) 2.81090 0.162831
\(299\) −14.3883 −0.832099
\(300\) 0 0
\(301\) 9.18674 0.529515
\(302\) 11.7142 0.674074
\(303\) 0 0
\(304\) 4.93280 0.282916
\(305\) 4.08313 0.233799
\(306\) 0 0
\(307\) −20.3144 −1.15941 −0.579703 0.814828i \(-0.696831\pi\)
−0.579703 + 0.814828i \(0.696831\pi\)
\(308\) −4.59337 −0.261732
\(309\) 0 0
\(310\) −2.28816 −0.129959
\(311\) 7.75620 0.439814 0.219907 0.975521i \(-0.429425\pi\)
0.219907 + 0.975521i \(0.429425\pi\)
\(312\) 0 0
\(313\) −20.8132 −1.17643 −0.588216 0.808704i \(-0.700170\pi\)
−0.588216 + 0.808704i \(0.700170\pi\)
\(314\) −20.1981 −1.13985
\(315\) 0 0
\(316\) 4.83263 0.271856
\(317\) −0.488511 −0.0274375 −0.0137188 0.999906i \(-0.504367\pi\)
−0.0137188 + 0.999906i \(0.504367\pi\)
\(318\) 0 0
\(319\) −6.19812 −0.347028
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 6.98966 0.389518
\(323\) 24.3325 1.35390
\(324\) 0 0
\(325\) −4.28816 −0.237864
\(326\) −10.7278 −0.594156
\(327\) 0 0
\(328\) 1.47726 0.0815683
\(329\) −19.5548 −1.07809
\(330\) 0 0
\(331\) −23.0570 −1.26733 −0.633664 0.773608i \(-0.718450\pi\)
−0.633664 + 0.773608i \(0.718450\pi\)
\(332\) 16.5934 0.910679
\(333\) 0 0
\(334\) −3.93169 −0.215132
\(335\) −1.00000 −0.0546358
\(336\) 0 0
\(337\) 30.3293 1.65214 0.826070 0.563568i \(-0.190572\pi\)
0.826070 + 0.563568i \(0.190572\pi\)
\(338\) 5.38834 0.293087
\(339\) 0 0
\(340\) 4.93280 0.267519
\(341\) 5.04547 0.273228
\(342\) 0 0
\(343\) −20.1242 −1.08661
\(344\) 4.41006 0.237775
\(345\) 0 0
\(346\) −9.04779 −0.486412
\(347\) −17.5432 −0.941769 −0.470885 0.882195i \(-0.656065\pi\)
−0.470885 + 0.882195i \(0.656065\pi\)
\(348\) 0 0
\(349\) 14.8201 0.793303 0.396652 0.917969i \(-0.370172\pi\)
0.396652 + 0.917969i \(0.370172\pi\)
\(350\) 2.08313 0.111348
\(351\) 0 0
\(352\) −2.20503 −0.117529
\(353\) −18.7665 −0.998842 −0.499421 0.866359i \(-0.666454\pi\)
−0.499421 + 0.866359i \(0.666454\pi\)
\(354\) 0 0
\(355\) 4.84967 0.257394
\(356\) −2.18330 −0.115715
\(357\) 0 0
\(358\) −13.6754 −0.722767
\(359\) 6.43738 0.339752 0.169876 0.985465i \(-0.445663\pi\)
0.169876 + 0.985465i \(0.445663\pi\)
\(360\) 0 0
\(361\) 5.33252 0.280659
\(362\) 0.876984 0.0460933
\(363\) 0 0
\(364\) −8.93280 −0.468206
\(365\) −0.544464 −0.0284985
\(366\) 0 0
\(367\) −18.1298 −0.946368 −0.473184 0.880964i \(-0.656895\pi\)
−0.473184 + 0.880964i \(0.656895\pi\)
\(368\) 3.35536 0.174910
\(369\) 0 0
\(370\) −0.727768 −0.0378349
\(371\) 12.8451 0.666886
\(372\) 0 0
\(373\) 12.6207 0.653474 0.326737 0.945115i \(-0.394051\pi\)
0.326737 + 0.945115i \(0.394051\pi\)
\(374\) −10.8770 −0.562435
\(375\) 0 0
\(376\) −9.38722 −0.484109
\(377\) −12.0536 −0.620791
\(378\) 0 0
\(379\) 28.4180 1.45973 0.729867 0.683590i \(-0.239582\pi\)
0.729867 + 0.683590i \(0.239582\pi\)
\(380\) 4.93280 0.253047
\(381\) 0 0
\(382\) 16.4101 0.839612
\(383\) −13.6048 −0.695170 −0.347585 0.937648i \(-0.612998\pi\)
−0.347585 + 0.937648i \(0.612998\pi\)
\(384\) 0 0
\(385\) −4.59337 −0.234100
\(386\) −3.17660 −0.161685
\(387\) 0 0
\(388\) 2.20503 0.111944
\(389\) 23.6948 1.20137 0.600686 0.799485i \(-0.294894\pi\)
0.600686 + 0.799485i \(0.294894\pi\)
\(390\) 0 0
\(391\) 16.5513 0.837037
\(392\) −2.66057 −0.134379
\(393\) 0 0
\(394\) 10.5763 0.532828
\(395\) 4.83263 0.243156
\(396\) 0 0
\(397\) −4.65022 −0.233388 −0.116694 0.993168i \(-0.537230\pi\)
−0.116694 + 0.993168i \(0.537230\pi\)
\(398\) 8.59337 0.430747
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −23.5193 −1.17450 −0.587248 0.809407i \(-0.699789\pi\)
−0.587248 + 0.809407i \(0.699789\pi\)
\(402\) 0 0
\(403\) 9.81201 0.488771
\(404\) 9.13783 0.454624
\(405\) 0 0
\(406\) 5.85547 0.290602
\(407\) 1.60475 0.0795446
\(408\) 0 0
\(409\) 18.5763 0.918540 0.459270 0.888297i \(-0.348111\pi\)
0.459270 + 0.888297i \(0.348111\pi\)
\(410\) 1.47726 0.0729569
\(411\) 0 0
\(412\) 0.410065 0.0202024
\(413\) 3.07733 0.151426
\(414\) 0 0
\(415\) 16.5934 0.814536
\(416\) −4.28816 −0.210244
\(417\) 0 0
\(418\) −10.8770 −0.532011
\(419\) 1.97605 0.0965361 0.0482681 0.998834i \(-0.484630\pi\)
0.0482681 + 0.998834i \(0.484630\pi\)
\(420\) 0 0
\(421\) 9.92246 0.483591 0.241795 0.970327i \(-0.422264\pi\)
0.241795 + 0.970327i \(0.422264\pi\)
\(422\) −12.8553 −0.625784
\(423\) 0 0
\(424\) 6.16626 0.299460
\(425\) 4.93280 0.239276
\(426\) 0 0
\(427\) 8.50569 0.411619
\(428\) −0.576325 −0.0278577
\(429\) 0 0
\(430\) 4.41006 0.212672
\(431\) −2.38275 −0.114773 −0.0573865 0.998352i \(-0.518277\pi\)
−0.0573865 + 0.998352i \(0.518277\pi\)
\(432\) 0 0
\(433\) −20.9965 −1.00903 −0.504514 0.863403i \(-0.668328\pi\)
−0.504514 + 0.863403i \(0.668328\pi\)
\(434\) −4.76654 −0.228801
\(435\) 0 0
\(436\) 5.78247 0.276930
\(437\) 16.5513 0.791758
\(438\) 0 0
\(439\) 17.0034 0.811530 0.405765 0.913978i \(-0.367005\pi\)
0.405765 + 0.913978i \(0.367005\pi\)
\(440\) −2.20503 −0.105121
\(441\) 0 0
\(442\) −21.1526 −1.00613
\(443\) 13.8656 0.658775 0.329387 0.944195i \(-0.393158\pi\)
0.329387 + 0.944195i \(0.393158\pi\)
\(444\) 0 0
\(445\) −2.18330 −0.103499
\(446\) 18.8520 0.892668
\(447\) 0 0
\(448\) 2.08313 0.0984186
\(449\) −39.0922 −1.84487 −0.922436 0.386149i \(-0.873805\pi\)
−0.922436 + 0.386149i \(0.873805\pi\)
\(450\) 0 0
\(451\) −3.25741 −0.153386
\(452\) 13.8268 0.650359
\(453\) 0 0
\(454\) 22.1094 1.03765
\(455\) −8.93280 −0.418776
\(456\) 0 0
\(457\) −6.68900 −0.312898 −0.156449 0.987686i \(-0.550005\pi\)
−0.156449 + 0.987686i \(0.550005\pi\)
\(458\) 0.326934 0.0152766
\(459\) 0 0
\(460\) 3.35536 0.156445
\(461\) 22.0545 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(462\) 0 0
\(463\) 39.4191 1.83196 0.915980 0.401224i \(-0.131415\pi\)
0.915980 + 0.401224i \(0.131415\pi\)
\(464\) 2.81090 0.130493
\(465\) 0 0
\(466\) 2.14922 0.0995605
\(467\) −10.0148 −0.463430 −0.231715 0.972784i \(-0.574434\pi\)
−0.231715 + 0.972784i \(0.574434\pi\)
\(468\) 0 0
\(469\) −2.08313 −0.0961900
\(470\) −9.38722 −0.433000
\(471\) 0 0
\(472\) 1.47726 0.0679966
\(473\) −9.72433 −0.447125
\(474\) 0 0
\(475\) 4.93280 0.226332
\(476\) 10.2757 0.470984
\(477\) 0 0
\(478\) −11.1208 −0.508653
\(479\) −22.8474 −1.04393 −0.521963 0.852968i \(-0.674800\pi\)
−0.521963 + 0.852968i \(0.674800\pi\)
\(480\) 0 0
\(481\) 3.12079 0.142296
\(482\) −7.15956 −0.326109
\(483\) 0 0
\(484\) −6.13783 −0.278992
\(485\) 2.20503 0.100125
\(486\) 0 0
\(487\) 35.8896 1.62632 0.813158 0.582044i \(-0.197747\pi\)
0.813158 + 0.582044i \(0.197747\pi\)
\(488\) 4.08313 0.184835
\(489\) 0 0
\(490\) −2.66057 −0.120192
\(491\) 35.2403 1.59037 0.795187 0.606364i \(-0.207373\pi\)
0.795187 + 0.606364i \(0.207373\pi\)
\(492\) 0 0
\(493\) 13.8656 0.624475
\(494\) −21.1526 −0.951703
\(495\) 0 0
\(496\) −2.28816 −0.102742
\(497\) 10.1025 0.453159
\(498\) 0 0
\(499\) 8.17094 0.365782 0.182891 0.983133i \(-0.441455\pi\)
0.182891 + 0.983133i \(0.441455\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 10.7278 0.478804
\(503\) 0.470564 0.0209814 0.0104907 0.999945i \(-0.496661\pi\)
0.0104907 + 0.999945i \(0.496661\pi\)
\(504\) 0 0
\(505\) 9.13783 0.406628
\(506\) −7.39868 −0.328912
\(507\) 0 0
\(508\) −11.1208 −0.493405
\(509\) −1.02284 −0.0453366 −0.0226683 0.999743i \(-0.507216\pi\)
−0.0226683 + 0.999743i \(0.507216\pi\)
\(510\) 0 0
\(511\) −1.13419 −0.0501735
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 20.4202 0.900696
\(515\) 0.410065 0.0180696
\(516\) 0 0
\(517\) 20.6991 0.910347
\(518\) −1.51604 −0.0666108
\(519\) 0 0
\(520\) −4.28816 −0.188048
\(521\) 0.841857 0.0368824 0.0184412 0.999830i \(-0.494130\pi\)
0.0184412 + 0.999830i \(0.494130\pi\)
\(522\) 0 0
\(523\) 38.8987 1.70092 0.850460 0.526040i \(-0.176324\pi\)
0.850460 + 0.526040i \(0.176324\pi\)
\(524\) 17.6435 0.770761
\(525\) 0 0
\(526\) 3.59917 0.156931
\(527\) −11.2870 −0.491672
\(528\) 0 0
\(529\) −11.7415 −0.510502
\(530\) 6.16626 0.267845
\(531\) 0 0
\(532\) 10.2757 0.445507
\(533\) −6.33475 −0.274388
\(534\) 0 0
\(535\) −0.576325 −0.0249167
\(536\) −1.00000 −0.0431934
\(537\) 0 0
\(538\) −8.94530 −0.385659
\(539\) 5.86664 0.252694
\(540\) 0 0
\(541\) −11.2644 −0.484295 −0.242148 0.970239i \(-0.577852\pi\)
−0.242148 + 0.970239i \(0.577852\pi\)
\(542\) −7.41118 −0.318337
\(543\) 0 0
\(544\) 4.93280 0.211492
\(545\) 5.78247 0.247694
\(546\) 0 0
\(547\) 33.1092 1.41565 0.707823 0.706389i \(-0.249677\pi\)
0.707823 + 0.706389i \(0.249677\pi\)
\(548\) 4.72777 0.201960
\(549\) 0 0
\(550\) −2.20503 −0.0940229
\(551\) 13.8656 0.590694
\(552\) 0 0
\(553\) 10.0670 0.428092
\(554\) 16.2586 0.690763
\(555\) 0 0
\(556\) −20.0855 −0.851813
\(557\) −14.1103 −0.597873 −0.298936 0.954273i \(-0.596632\pi\)
−0.298936 + 0.954273i \(0.596632\pi\)
\(558\) 0 0
\(559\) −18.9111 −0.799853
\(560\) 2.08313 0.0880283
\(561\) 0 0
\(562\) −19.1526 −0.807906
\(563\) 16.4101 0.691602 0.345801 0.938308i \(-0.387607\pi\)
0.345801 + 0.938308i \(0.387607\pi\)
\(564\) 0 0
\(565\) 13.8268 0.581699
\(566\) −24.2905 −1.02101
\(567\) 0 0
\(568\) 4.84967 0.203488
\(569\) −30.7255 −1.28808 −0.644041 0.764991i \(-0.722743\pi\)
−0.644041 + 0.764991i \(0.722743\pi\)
\(570\) 0 0
\(571\) −9.86560 −0.412863 −0.206431 0.978461i \(-0.566185\pi\)
−0.206431 + 0.978461i \(0.566185\pi\)
\(572\) 9.45554 0.395356
\(573\) 0 0
\(574\) 3.07733 0.128445
\(575\) 3.35536 0.139928
\(576\) 0 0
\(577\) −2.86685 −0.119349 −0.0596743 0.998218i \(-0.519006\pi\)
−0.0596743 + 0.998218i \(0.519006\pi\)
\(578\) 7.33252 0.304993
\(579\) 0 0
\(580\) 2.81090 0.116716
\(581\) 34.5661 1.43405
\(582\) 0 0
\(583\) −13.5968 −0.563122
\(584\) −0.544464 −0.0225301
\(585\) 0 0
\(586\) −13.9705 −0.577114
\(587\) 6.26553 0.258606 0.129303 0.991605i \(-0.458726\pi\)
0.129303 + 0.991605i \(0.458726\pi\)
\(588\) 0 0
\(589\) −11.2870 −0.465075
\(590\) 1.47726 0.0608180
\(591\) 0 0
\(592\) −0.727768 −0.0299111
\(593\) 9.77324 0.401339 0.200669 0.979659i \(-0.435688\pi\)
0.200669 + 0.979659i \(0.435688\pi\)
\(594\) 0 0
\(595\) 10.2757 0.421261
\(596\) 2.81090 0.115139
\(597\) 0 0
\(598\) −14.3883 −0.588383
\(599\) 15.1845 0.620422 0.310211 0.950668i \(-0.399600\pi\)
0.310211 + 0.950668i \(0.399600\pi\)
\(600\) 0 0
\(601\) −20.2688 −0.826780 −0.413390 0.910554i \(-0.635655\pi\)
−0.413390 + 0.910554i \(0.635655\pi\)
\(602\) 9.18674 0.374424
\(603\) 0 0
\(604\) 11.7142 0.476642
\(605\) −6.13783 −0.249538
\(606\) 0 0
\(607\) −5.31891 −0.215888 −0.107944 0.994157i \(-0.534427\pi\)
−0.107944 + 0.994157i \(0.534427\pi\)
\(608\) 4.93280 0.200051
\(609\) 0 0
\(610\) 4.08313 0.165321
\(611\) 40.2539 1.62850
\(612\) 0 0
\(613\) −10.6650 −0.430757 −0.215378 0.976531i \(-0.569099\pi\)
−0.215378 + 0.976531i \(0.569099\pi\)
\(614\) −20.3144 −0.819824
\(615\) 0 0
\(616\) −4.59337 −0.185072
\(617\) −37.1628 −1.49612 −0.748059 0.663633i \(-0.769014\pi\)
−0.748059 + 0.663633i \(0.769014\pi\)
\(618\) 0 0
\(619\) −15.0672 −0.605602 −0.302801 0.953054i \(-0.597922\pi\)
−0.302801 + 0.953054i \(0.597922\pi\)
\(620\) −2.28816 −0.0918948
\(621\) 0 0
\(622\) 7.75620 0.310995
\(623\) −4.54811 −0.182216
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −20.8132 −0.831864
\(627\) 0 0
\(628\) −20.1981 −0.805993
\(629\) −3.58994 −0.143140
\(630\) 0 0
\(631\) −12.4226 −0.494534 −0.247267 0.968947i \(-0.579533\pi\)
−0.247267 + 0.968947i \(0.579533\pi\)
\(632\) 4.83263 0.192232
\(633\) 0 0
\(634\) −0.488511 −0.0194013
\(635\) −11.1208 −0.441315
\(636\) 0 0
\(637\) 11.4090 0.452039
\(638\) −6.19812 −0.245386
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −31.8177 −1.25672 −0.628362 0.777921i \(-0.716274\pi\)
−0.628362 + 0.777921i \(0.716274\pi\)
\(642\) 0 0
\(643\) −40.3427 −1.59096 −0.795479 0.605981i \(-0.792781\pi\)
−0.795479 + 0.605981i \(0.792781\pi\)
\(644\) 6.98966 0.275431
\(645\) 0 0
\(646\) 24.3325 0.957350
\(647\) −43.6662 −1.71670 −0.858349 0.513067i \(-0.828509\pi\)
−0.858349 + 0.513067i \(0.828509\pi\)
\(648\) 0 0
\(649\) −3.25741 −0.127865
\(650\) −4.28816 −0.168196
\(651\) 0 0
\(652\) −10.7278 −0.420132
\(653\) −33.1411 −1.29691 −0.648455 0.761253i \(-0.724584\pi\)
−0.648455 + 0.761253i \(0.724584\pi\)
\(654\) 0 0
\(655\) 17.6435 0.689390
\(656\) 1.47726 0.0576775
\(657\) 0 0
\(658\) −19.5548 −0.762326
\(659\) −34.6640 −1.35032 −0.675159 0.737672i \(-0.735925\pi\)
−0.675159 + 0.737672i \(0.735925\pi\)
\(660\) 0 0
\(661\) −10.8884 −0.423511 −0.211756 0.977323i \(-0.567918\pi\)
−0.211756 + 0.977323i \(0.567918\pi\)
\(662\) −23.0570 −0.896137
\(663\) 0 0
\(664\) 16.5934 0.643948
\(665\) 10.2757 0.398473
\(666\) 0 0
\(667\) 9.43158 0.365192
\(668\) −3.93169 −0.152122
\(669\) 0 0
\(670\) −1.00000 −0.0386334
\(671\) −9.00343 −0.347574
\(672\) 0 0
\(673\) −8.48865 −0.327213 −0.163607 0.986526i \(-0.552313\pi\)
−0.163607 + 0.986526i \(0.552313\pi\)
\(674\) 30.3293 1.16824
\(675\) 0 0
\(676\) 5.38834 0.207244
\(677\) 15.1526 0.582364 0.291182 0.956668i \(-0.405952\pi\)
0.291182 + 0.956668i \(0.405952\pi\)
\(678\) 0 0
\(679\) 4.59337 0.176277
\(680\) 4.93280 0.189164
\(681\) 0 0
\(682\) 5.04547 0.193201
\(683\) −5.01138 −0.191755 −0.0958776 0.995393i \(-0.530566\pi\)
−0.0958776 + 0.995393i \(0.530566\pi\)
\(684\) 0 0
\(685\) 4.72777 0.180639
\(686\) −20.1242 −0.768346
\(687\) 0 0
\(688\) 4.41006 0.168132
\(689\) −26.4419 −1.00736
\(690\) 0 0
\(691\) −9.89969 −0.376602 −0.188301 0.982111i \(-0.560298\pi\)
−0.188301 + 0.982111i \(0.560298\pi\)
\(692\) −9.04779 −0.343945
\(693\) 0 0
\(694\) −17.5432 −0.665931
\(695\) −20.0855 −0.761885
\(696\) 0 0
\(697\) 7.28705 0.276017
\(698\) 14.8201 0.560950
\(699\) 0 0
\(700\) 2.08313 0.0787349
\(701\) −11.3609 −0.429097 −0.214549 0.976713i \(-0.568828\pi\)
−0.214549 + 0.976713i \(0.568828\pi\)
\(702\) 0 0
\(703\) −3.58994 −0.135397
\(704\) −2.20503 −0.0831053
\(705\) 0 0
\(706\) −18.7665 −0.706288
\(707\) 19.0353 0.715896
\(708\) 0 0
\(709\) −49.5170 −1.85965 −0.929826 0.368001i \(-0.880042\pi\)
−0.929826 + 0.368001i \(0.880042\pi\)
\(710\) 4.84967 0.182005
\(711\) 0 0
\(712\) −2.18330 −0.0818228
\(713\) −7.67761 −0.287529
\(714\) 0 0
\(715\) 9.45554 0.353617
\(716\) −13.6754 −0.511073
\(717\) 0 0
\(718\) 6.43738 0.240241
\(719\) 28.4863 1.06236 0.531180 0.847259i \(-0.321749\pi\)
0.531180 + 0.847259i \(0.321749\pi\)
\(720\) 0 0
\(721\) 0.854218 0.0318127
\(722\) 5.33252 0.198456
\(723\) 0 0
\(724\) 0.876984 0.0325929
\(725\) 2.81090 0.104394
\(726\) 0 0
\(727\) −19.4191 −0.720214 −0.360107 0.932911i \(-0.617260\pi\)
−0.360107 + 0.932911i \(0.617260\pi\)
\(728\) −8.93280 −0.331072
\(729\) 0 0
\(730\) −0.544464 −0.0201515
\(731\) 21.7540 0.804600
\(732\) 0 0
\(733\) 20.0628 0.741037 0.370519 0.928825i \(-0.379180\pi\)
0.370519 + 0.928825i \(0.379180\pi\)
\(734\) −18.1298 −0.669183
\(735\) 0 0
\(736\) 3.35536 0.123680
\(737\) 2.20503 0.0812234
\(738\) 0 0
\(739\) 6.21985 0.228801 0.114400 0.993435i \(-0.463505\pi\)
0.114400 + 0.993435i \(0.463505\pi\)
\(740\) −0.727768 −0.0267533
\(741\) 0 0
\(742\) 12.8451 0.471559
\(743\) −51.6925 −1.89641 −0.948207 0.317652i \(-0.897106\pi\)
−0.948207 + 0.317652i \(0.897106\pi\)
\(744\) 0 0
\(745\) 2.81090 0.102983
\(746\) 12.6207 0.462076
\(747\) 0 0
\(748\) −10.8770 −0.397702
\(749\) −1.20056 −0.0438675
\(750\) 0 0
\(751\) −12.8201 −0.467813 −0.233907 0.972259i \(-0.575151\pi\)
−0.233907 + 0.972259i \(0.575151\pi\)
\(752\) −9.38722 −0.342317
\(753\) 0 0
\(754\) −12.0536 −0.438966
\(755\) 11.7142 0.426322
\(756\) 0 0
\(757\) −30.4613 −1.10713 −0.553567 0.832805i \(-0.686734\pi\)
−0.553567 + 0.832805i \(0.686734\pi\)
\(758\) 28.4180 1.03219
\(759\) 0 0
\(760\) 4.93280 0.178931
\(761\) −0.937484 −0.0339838 −0.0169919 0.999856i \(-0.505409\pi\)
−0.0169919 + 0.999856i \(0.505409\pi\)
\(762\) 0 0
\(763\) 12.0456 0.436081
\(764\) 16.4101 0.593695
\(765\) 0 0
\(766\) −13.6048 −0.491560
\(767\) −6.33475 −0.228734
\(768\) 0 0
\(769\) −30.2734 −1.09169 −0.545844 0.837887i \(-0.683791\pi\)
−0.545844 + 0.837887i \(0.683791\pi\)
\(770\) −4.59337 −0.165534
\(771\) 0 0
\(772\) −3.17660 −0.114329
\(773\) 26.8949 0.967343 0.483672 0.875249i \(-0.339303\pi\)
0.483672 + 0.875249i \(0.339303\pi\)
\(774\) 0 0
\(775\) −2.28816 −0.0821932
\(776\) 2.20503 0.0791560
\(777\) 0 0
\(778\) 23.6948 0.849498
\(779\) 7.28705 0.261086
\(780\) 0 0
\(781\) −10.6937 −0.382650
\(782\) 16.5513 0.591874
\(783\) 0 0
\(784\) −2.66057 −0.0950203
\(785\) −20.1981 −0.720902
\(786\) 0 0
\(787\) 33.0936 1.17966 0.589829 0.807528i \(-0.299195\pi\)
0.589829 + 0.807528i \(0.299195\pi\)
\(788\) 10.5763 0.376766
\(789\) 0 0
\(790\) 4.83263 0.171937
\(791\) 28.8031 1.02412
\(792\) 0 0
\(793\) −17.5091 −0.621767
\(794\) −4.65022 −0.165030
\(795\) 0 0
\(796\) 8.59337 0.304584
\(797\) −22.6243 −0.801395 −0.400697 0.916210i \(-0.631232\pi\)
−0.400697 + 0.916210i \(0.631232\pi\)
\(798\) 0 0
\(799\) −46.3053 −1.63816
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −23.5193 −0.830494
\(803\) 1.20056 0.0423668
\(804\) 0 0
\(805\) 6.98966 0.246353
\(806\) 9.81201 0.345613
\(807\) 0 0
\(808\) 9.13783 0.321468
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) −43.5217 −1.52825 −0.764127 0.645066i \(-0.776830\pi\)
−0.764127 + 0.645066i \(0.776830\pi\)
\(812\) 5.85547 0.205487
\(813\) 0 0
\(814\) 1.60475 0.0562465
\(815\) −10.7278 −0.375777
\(816\) 0 0
\(817\) 21.7540 0.761075
\(818\) 18.5763 0.649506
\(819\) 0 0
\(820\) 1.47726 0.0515883
\(821\) −10.9431 −0.381916 −0.190958 0.981598i \(-0.561159\pi\)
−0.190958 + 0.981598i \(0.561159\pi\)
\(822\) 0 0
\(823\) 8.63541 0.301011 0.150506 0.988609i \(-0.451910\pi\)
0.150506 + 0.988609i \(0.451910\pi\)
\(824\) 0.410065 0.0142853
\(825\) 0 0
\(826\) 3.07733 0.107074
\(827\) 23.9395 0.832458 0.416229 0.909260i \(-0.363351\pi\)
0.416229 + 0.909260i \(0.363351\pi\)
\(828\) 0 0
\(829\) −15.6993 −0.545261 −0.272630 0.962119i \(-0.587894\pi\)
−0.272630 + 0.962119i \(0.587894\pi\)
\(830\) 16.5934 0.575964
\(831\) 0 0
\(832\) −4.28816 −0.148665
\(833\) −13.1241 −0.454722
\(834\) 0 0
\(835\) −3.93169 −0.136062
\(836\) −10.8770 −0.376188
\(837\) 0 0
\(838\) 1.97605 0.0682613
\(839\) −25.1742 −0.869111 −0.434556 0.900645i \(-0.643095\pi\)
−0.434556 + 0.900645i \(0.643095\pi\)
\(840\) 0 0
\(841\) −21.0989 −0.727547
\(842\) 9.92246 0.341950
\(843\) 0 0
\(844\) −12.8553 −0.442496
\(845\) 5.38834 0.185364
\(846\) 0 0
\(847\) −12.7859 −0.439329
\(848\) 6.16626 0.211750
\(849\) 0 0
\(850\) 4.93280 0.169194
\(851\) −2.44193 −0.0837081
\(852\) 0 0
\(853\) 8.33029 0.285224 0.142612 0.989779i \(-0.454450\pi\)
0.142612 + 0.989779i \(0.454450\pi\)
\(854\) 8.50569 0.291059
\(855\) 0 0
\(856\) −0.576325 −0.0196984
\(857\) 35.7562 1.22141 0.610703 0.791859i \(-0.290887\pi\)
0.610703 + 0.791859i \(0.290887\pi\)
\(858\) 0 0
\(859\) −51.5410 −1.75856 −0.879278 0.476309i \(-0.841974\pi\)
−0.879278 + 0.476309i \(0.841974\pi\)
\(860\) 4.41006 0.150382
\(861\) 0 0
\(862\) −2.38275 −0.0811568
\(863\) 3.65825 0.124528 0.0622641 0.998060i \(-0.480168\pi\)
0.0622641 + 0.998060i \(0.480168\pi\)
\(864\) 0 0
\(865\) −9.04779 −0.307634
\(866\) −20.9965 −0.713491
\(867\) 0 0
\(868\) −4.76654 −0.161787
\(869\) −10.6561 −0.361483
\(870\) 0 0
\(871\) 4.28816 0.145299
\(872\) 5.78247 0.195819
\(873\) 0 0
\(874\) 16.5513 0.559857
\(875\) 2.08313 0.0704227
\(876\) 0 0
\(877\) 10.0185 0.338299 0.169150 0.985590i \(-0.445898\pi\)
0.169150 + 0.985590i \(0.445898\pi\)
\(878\) 17.0034 0.573838
\(879\) 0 0
\(880\) −2.20503 −0.0743316
\(881\) −13.9145 −0.468792 −0.234396 0.972141i \(-0.575311\pi\)
−0.234396 + 0.972141i \(0.575311\pi\)
\(882\) 0 0
\(883\) −19.7562 −0.664849 −0.332424 0.943130i \(-0.607867\pi\)
−0.332424 + 0.943130i \(0.607867\pi\)
\(884\) −21.1526 −0.711441
\(885\) 0 0
\(886\) 13.8656 0.465824
\(887\) 8.75182 0.293857 0.146929 0.989147i \(-0.453061\pi\)
0.146929 + 0.989147i \(0.453061\pi\)
\(888\) 0 0
\(889\) −23.1660 −0.776964
\(890\) −2.18330 −0.0731845
\(891\) 0 0
\(892\) 18.8520 0.631211
\(893\) −46.3053 −1.54955
\(894\) 0 0
\(895\) −13.6754 −0.457118
\(896\) 2.08313 0.0695925
\(897\) 0 0
\(898\) −39.0922 −1.30452
\(899\) −6.43179 −0.214512
\(900\) 0 0
\(901\) 30.4169 1.01333
\(902\) −3.25741 −0.108460
\(903\) 0 0
\(904\) 13.8268 0.459873
\(905\) 0.876984 0.0291519
\(906\) 0 0
\(907\) −24.3920 −0.809922 −0.404961 0.914334i \(-0.632715\pi\)
−0.404961 + 0.914334i \(0.632715\pi\)
\(908\) 22.1094 0.733726
\(909\) 0 0
\(910\) −8.93280 −0.296119
\(911\) 6.94061 0.229953 0.114976 0.993368i \(-0.463321\pi\)
0.114976 + 0.993368i \(0.463321\pi\)
\(912\) 0 0
\(913\) −36.5889 −1.21092
\(914\) −6.68900 −0.221252
\(915\) 0 0
\(916\) 0.326934 0.0108022
\(917\) 36.7538 1.21372
\(918\) 0 0
\(919\) −17.0764 −0.563299 −0.281650 0.959517i \(-0.590882\pi\)
−0.281650 + 0.959517i \(0.590882\pi\)
\(920\) 3.35536 0.110623
\(921\) 0 0
\(922\) 22.0545 0.726326
\(923\) −20.7962 −0.684514
\(924\) 0 0
\(925\) −0.727768 −0.0239289
\(926\) 39.4191 1.29539
\(927\) 0 0
\(928\) 2.81090 0.0922723
\(929\) −16.6433 −0.546049 −0.273025 0.962007i \(-0.588024\pi\)
−0.273025 + 0.962007i \(0.588024\pi\)
\(930\) 0 0
\(931\) −13.1241 −0.430124
\(932\) 2.14922 0.0703999
\(933\) 0 0
\(934\) −10.0148 −0.327695
\(935\) −10.8770 −0.355715
\(936\) 0 0
\(937\) 9.72652 0.317752 0.158876 0.987299i \(-0.449213\pi\)
0.158876 + 0.987299i \(0.449213\pi\)
\(938\) −2.08313 −0.0680166
\(939\) 0 0
\(940\) −9.38722 −0.306177
\(941\) 7.20056 0.234732 0.117366 0.993089i \(-0.462555\pi\)
0.117366 + 0.993089i \(0.462555\pi\)
\(942\) 0 0
\(943\) 4.95676 0.161414
\(944\) 1.47726 0.0480808
\(945\) 0 0
\(946\) −9.72433 −0.316165
\(947\) −30.1561 −0.979941 −0.489971 0.871739i \(-0.662992\pi\)
−0.489971 + 0.871739i \(0.662992\pi\)
\(948\) 0 0
\(949\) 2.33475 0.0757891
\(950\) 4.93280 0.160041
\(951\) 0 0
\(952\) 10.2757 0.333036
\(953\) 19.0193 0.616095 0.308048 0.951371i \(-0.400324\pi\)
0.308048 + 0.951371i \(0.400324\pi\)
\(954\) 0 0
\(955\) 16.4101 0.531017
\(956\) −11.1208 −0.359672
\(957\) 0 0
\(958\) −22.8474 −0.738167
\(959\) 9.84856 0.318026
\(960\) 0 0
\(961\) −25.7643 −0.831107
\(962\) 3.12079 0.100618
\(963\) 0 0
\(964\) −7.15956 −0.230594
\(965\) −3.17660 −0.102259
\(966\) 0 0
\(967\) 40.1865 1.29231 0.646156 0.763206i \(-0.276376\pi\)
0.646156 + 0.763206i \(0.276376\pi\)
\(968\) −6.13783 −0.197277
\(969\) 0 0
\(970\) 2.20503 0.0707993
\(971\) 50.4387 1.61865 0.809327 0.587359i \(-0.199832\pi\)
0.809327 + 0.587359i \(0.199832\pi\)
\(972\) 0 0
\(973\) −41.8406 −1.34135
\(974\) 35.8896 1.14998
\(975\) 0 0
\(976\) 4.08313 0.130698
\(977\) −58.8142 −1.88163 −0.940817 0.338916i \(-0.889940\pi\)
−0.940817 + 0.338916i \(0.889940\pi\)
\(978\) 0 0
\(979\) 4.81426 0.153864
\(980\) −2.66057 −0.0849888
\(981\) 0 0
\(982\) 35.2403 1.12456
\(983\) −13.2643 −0.423065 −0.211532 0.977371i \(-0.567845\pi\)
−0.211532 + 0.977371i \(0.567845\pi\)
\(984\) 0 0
\(985\) 10.5763 0.336990
\(986\) 13.8656 0.441571
\(987\) 0 0
\(988\) −21.1526 −0.672955
\(989\) 14.7974 0.470529
\(990\) 0 0
\(991\) −15.6071 −0.495775 −0.247887 0.968789i \(-0.579736\pi\)
−0.247887 + 0.968789i \(0.579736\pi\)
\(992\) −2.28816 −0.0726492
\(993\) 0 0
\(994\) 10.1025 0.320432
\(995\) 8.59337 0.272428
\(996\) 0 0
\(997\) −20.5911 −0.652128 −0.326064 0.945348i \(-0.605722\pi\)
−0.326064 + 0.945348i \(0.605722\pi\)
\(998\) 8.17094 0.258647
\(999\) 0 0
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 6030.2.a.bu.1.3 4
3.2 odd 2 2010.2.a.r.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2010.2.a.r.1.3 4 3.2 odd 2
6030.2.a.bu.1.3 4 1.1 even 1 trivial