Properties

Label 2070.2.a.x.1.1
Level $2070$
Weight $2$
Character 2070.1
Self dual yes
Analytic conductor $16.529$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \(x^{2} - x - 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.79129\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.79129 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.79129 q^{7} +1.00000 q^{8} +1.00000 q^{10} +0.791288 q^{11} +5.79129 q^{13} -1.79129 q^{14} +1.00000 q^{16} -0.791288 q^{17} +5.79129 q^{19} +1.00000 q^{20} +0.791288 q^{22} -1.00000 q^{23} +1.00000 q^{25} +5.79129 q^{26} -1.79129 q^{28} -7.58258 q^{29} -3.37386 q^{31} +1.00000 q^{32} -0.791288 q^{34} -1.79129 q^{35} -4.00000 q^{37} +5.79129 q^{38} +1.00000 q^{40} +6.79129 q^{41} +11.1652 q^{43} +0.791288 q^{44} -1.00000 q^{46} +4.41742 q^{47} -3.79129 q^{49} +1.00000 q^{50} +5.79129 q^{52} -6.00000 q^{53} +0.791288 q^{55} -1.79129 q^{56} -7.58258 q^{58} +13.5826 q^{59} +10.3739 q^{61} -3.37386 q^{62} +1.00000 q^{64} +5.79129 q^{65} +11.1652 q^{67} -0.791288 q^{68} -1.79129 q^{70} -8.37386 q^{71} +12.7477 q^{73} -4.00000 q^{74} +5.79129 q^{76} -1.41742 q^{77} +8.00000 q^{79} +1.00000 q^{80} +6.79129 q^{82} +6.00000 q^{83} -0.791288 q^{85} +11.1652 q^{86} +0.791288 q^{88} -15.1652 q^{89} -10.3739 q^{91} -1.00000 q^{92} +4.41742 q^{94} +5.79129 q^{95} -7.95644 q^{97} -3.79129 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + q^{7} + 2q^{8} + 2q^{10} - 3q^{11} + 7q^{13} + q^{14} + 2q^{16} + 3q^{17} + 7q^{19} + 2q^{20} - 3q^{22} - 2q^{23} + 2q^{25} + 7q^{26} + q^{28} - 6q^{29} + 7q^{31} + 2q^{32} + 3q^{34} + q^{35} - 8q^{37} + 7q^{38} + 2q^{40} + 9q^{41} + 4q^{43} - 3q^{44} - 2q^{46} + 18q^{47} - 3q^{49} + 2q^{50} + 7q^{52} - 12q^{53} - 3q^{55} + q^{56} - 6q^{58} + 18q^{59} + 7q^{61} + 7q^{62} + 2q^{64} + 7q^{65} + 4q^{67} + 3q^{68} + q^{70} - 3q^{71} - 2q^{73} - 8q^{74} + 7q^{76} - 12q^{77} + 16q^{79} + 2q^{80} + 9q^{82} + 12q^{83} + 3q^{85} + 4q^{86} - 3q^{88} - 12q^{89} - 7q^{91} - 2q^{92} + 18q^{94} + 7q^{95} + 7q^{97} - 3q^{98} + O(q^{100}) \)

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
\(7\) −1.79129 −0.677043 −0.338522 0.940959i \(-0.609927\pi\)
−0.338522 + 0.940959i \(0.609927\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 0.791288 0.238582 0.119291 0.992859i \(-0.461938\pi\)
0.119291 + 0.992859i \(0.461938\pi\)
\(12\) 0 0
\(13\) 5.79129 1.60621 0.803107 0.595835i \(-0.203179\pi\)
0.803107 + 0.595835i \(0.203179\pi\)
\(14\) −1.79129 −0.478742
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.791288 −0.191915 −0.0959577 0.995385i \(-0.530591\pi\)
−0.0959577 + 0.995385i \(0.530591\pi\)
\(18\) 0 0
\(19\) 5.79129 1.32861 0.664306 0.747460i \(-0.268727\pi\)
0.664306 + 0.747460i \(0.268727\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 0.791288 0.168703
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 5.79129 1.13576
\(27\) 0 0
\(28\) −1.79129 −0.338522
\(29\) −7.58258 −1.40805 −0.704024 0.710176i \(-0.748615\pi\)
−0.704024 + 0.710176i \(0.748615\pi\)
\(30\) 0 0
\(31\) −3.37386 −0.605964 −0.302982 0.952996i \(-0.597982\pi\)
−0.302982 + 0.952996i \(0.597982\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.791288 −0.135705
\(35\) −1.79129 −0.302783
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 5.79129 0.939471
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 6.79129 1.06062 0.530310 0.847804i \(-0.322075\pi\)
0.530310 + 0.847804i \(0.322075\pi\)
\(42\) 0 0
\(43\) 11.1652 1.70267 0.851335 0.524623i \(-0.175794\pi\)
0.851335 + 0.524623i \(0.175794\pi\)
\(44\) 0.791288 0.119291
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 4.41742 0.644348 0.322174 0.946681i \(-0.395586\pi\)
0.322174 + 0.946681i \(0.395586\pi\)
\(48\) 0 0
\(49\) −3.79129 −0.541613
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 5.79129 0.803107
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0.791288 0.106697
\(56\) −1.79129 −0.239371
\(57\) 0 0
\(58\) −7.58258 −0.995641
\(59\) 13.5826 1.76830 0.884150 0.467202i \(-0.154738\pi\)
0.884150 + 0.467202i \(0.154738\pi\)
\(60\) 0 0
\(61\) 10.3739 1.32824 0.664119 0.747627i \(-0.268807\pi\)
0.664119 + 0.747627i \(0.268807\pi\)
\(62\) −3.37386 −0.428481
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 5.79129 0.718321
\(66\) 0 0
\(67\) 11.1652 1.36404 0.682020 0.731333i \(-0.261102\pi\)
0.682020 + 0.731333i \(0.261102\pi\)
\(68\) −0.791288 −0.0959577
\(69\) 0 0
\(70\) −1.79129 −0.214100
\(71\) −8.37386 −0.993795 −0.496897 0.867809i \(-0.665527\pi\)
−0.496897 + 0.867809i \(0.665527\pi\)
\(72\) 0 0
\(73\) 12.7477 1.49201 0.746004 0.665941i \(-0.231970\pi\)
0.746004 + 0.665941i \(0.231970\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 5.79129 0.664306
\(77\) −1.41742 −0.161530
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 6.79129 0.749972
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −0.791288 −0.0858272
\(86\) 11.1652 1.20397
\(87\) 0 0
\(88\) 0.791288 0.0843516
\(89\) −15.1652 −1.60750 −0.803751 0.594965i \(-0.797166\pi\)
−0.803751 + 0.594965i \(0.797166\pi\)
\(90\) 0 0
\(91\) −10.3739 −1.08748
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 4.41742 0.455623
\(95\) 5.79129 0.594174
\(96\) 0 0
\(97\) −7.95644 −0.807854 −0.403927 0.914791i \(-0.632355\pi\)
−0.403927 + 0.914791i \(0.632355\pi\)
\(98\) −3.79129 −0.382978
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −4.41742 −0.439550 −0.219775 0.975551i \(-0.570532\pi\)
−0.219775 + 0.975551i \(0.570532\pi\)
\(102\) 0 0
\(103\) −6.37386 −0.628035 −0.314018 0.949417i \(-0.601675\pi\)
−0.314018 + 0.949417i \(0.601675\pi\)
\(104\) 5.79129 0.567882
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −4.41742 −0.427049 −0.213524 0.976938i \(-0.568494\pi\)
−0.213524 + 0.976938i \(0.568494\pi\)
\(108\) 0 0
\(109\) −3.37386 −0.323158 −0.161579 0.986860i \(-0.551659\pi\)
−0.161579 + 0.986860i \(0.551659\pi\)
\(110\) 0.791288 0.0754463
\(111\) 0 0
\(112\) −1.79129 −0.169261
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) −7.58258 −0.704024
\(117\) 0 0
\(118\) 13.5826 1.25038
\(119\) 1.41742 0.129935
\(120\) 0 0
\(121\) −10.3739 −0.943079
\(122\) 10.3739 0.939205
\(123\) 0 0
\(124\) −3.37386 −0.302982
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 12.7477 1.13118 0.565589 0.824687i \(-0.308649\pi\)
0.565589 + 0.824687i \(0.308649\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 5.79129 0.507930
\(131\) 9.16515 0.800763 0.400381 0.916349i \(-0.368878\pi\)
0.400381 + 0.916349i \(0.368878\pi\)
\(132\) 0 0
\(133\) −10.3739 −0.899528
\(134\) 11.1652 0.964522
\(135\) 0 0
\(136\) −0.791288 −0.0678524
\(137\) −3.79129 −0.323912 −0.161956 0.986798i \(-0.551780\pi\)
−0.161956 + 0.986798i \(0.551780\pi\)
\(138\) 0 0
\(139\) 12.7477 1.08125 0.540624 0.841264i \(-0.318188\pi\)
0.540624 + 0.841264i \(0.318188\pi\)
\(140\) −1.79129 −0.151391
\(141\) 0 0
\(142\) −8.37386 −0.702719
\(143\) 4.58258 0.383214
\(144\) 0 0
\(145\) −7.58258 −0.629699
\(146\) 12.7477 1.05501
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) −8.20871 −0.672484 −0.336242 0.941776i \(-0.609156\pi\)
−0.336242 + 0.941776i \(0.609156\pi\)
\(150\) 0 0
\(151\) −10.7913 −0.878183 −0.439091 0.898442i \(-0.644700\pi\)
−0.439091 + 0.898442i \(0.644700\pi\)
\(152\) 5.79129 0.469735
\(153\) 0 0
\(154\) −1.41742 −0.114219
\(155\) −3.37386 −0.270995
\(156\) 0 0
\(157\) −14.7477 −1.17700 −0.588498 0.808498i \(-0.700281\pi\)
−0.588498 + 0.808498i \(0.700281\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 1.79129 0.141173
\(162\) 0 0
\(163\) 8.62614 0.675651 0.337826 0.941209i \(-0.390309\pi\)
0.337826 + 0.941209i \(0.390309\pi\)
\(164\) 6.79129 0.530310
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −18.3303 −1.41844 −0.709221 0.704987i \(-0.750953\pi\)
−0.709221 + 0.704987i \(0.750953\pi\)
\(168\) 0 0
\(169\) 20.5390 1.57992
\(170\) −0.791288 −0.0606890
\(171\) 0 0
\(172\) 11.1652 0.851335
\(173\) 18.7913 1.42868 0.714338 0.699801i \(-0.246728\pi\)
0.714338 + 0.699801i \(0.246728\pi\)
\(174\) 0 0
\(175\) −1.79129 −0.135409
\(176\) 0.791288 0.0596456
\(177\) 0 0
\(178\) −15.1652 −1.13668
\(179\) −10.7477 −0.803323 −0.401661 0.915788i \(-0.631567\pi\)
−0.401661 + 0.915788i \(0.631567\pi\)
\(180\) 0 0
\(181\) −18.5390 −1.37799 −0.688997 0.724764i \(-0.741949\pi\)
−0.688997 + 0.724764i \(0.741949\pi\)
\(182\) −10.3739 −0.768962
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) −0.626136 −0.0457876
\(188\) 4.41742 0.322174
\(189\) 0 0
\(190\) 5.79129 0.420144
\(191\) −25.5826 −1.85109 −0.925545 0.378637i \(-0.876393\pi\)
−0.925545 + 0.378637i \(0.876393\pi\)
\(192\) 0 0
\(193\) −20.7477 −1.49345 −0.746727 0.665131i \(-0.768376\pi\)
−0.746727 + 0.665131i \(0.768376\pi\)
\(194\) −7.95644 −0.571239
\(195\) 0 0
\(196\) −3.79129 −0.270806
\(197\) 11.5390 0.822121 0.411060 0.911608i \(-0.365159\pi\)
0.411060 + 0.911608i \(0.365159\pi\)
\(198\) 0 0
\(199\) −16.3303 −1.15762 −0.578812 0.815461i \(-0.696484\pi\)
−0.578812 + 0.815461i \(0.696484\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −4.41742 −0.310809
\(203\) 13.5826 0.953310
\(204\) 0 0
\(205\) 6.79129 0.474324
\(206\) −6.37386 −0.444088
\(207\) 0 0
\(208\) 5.79129 0.401554
\(209\) 4.58258 0.316983
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −4.41742 −0.301969
\(215\) 11.1652 0.761457
\(216\) 0 0
\(217\) 6.04356 0.410264
\(218\) −3.37386 −0.228507
\(219\) 0 0
\(220\) 0.791288 0.0533486
\(221\) −4.58258 −0.308257
\(222\) 0 0
\(223\) −7.16515 −0.479814 −0.239907 0.970796i \(-0.577117\pi\)
−0.239907 + 0.970796i \(0.577117\pi\)
\(224\) −1.79129 −0.119685
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 22.7477 1.50982 0.754910 0.655829i \(-0.227681\pi\)
0.754910 + 0.655829i \(0.227681\pi\)
\(228\) 0 0
\(229\) 20.3303 1.34346 0.671732 0.740794i \(-0.265551\pi\)
0.671732 + 0.740794i \(0.265551\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −7.58258 −0.497820
\(233\) 1.58258 0.103678 0.0518390 0.998655i \(-0.483492\pi\)
0.0518390 + 0.998655i \(0.483492\pi\)
\(234\) 0 0
\(235\) 4.41742 0.288161
\(236\) 13.5826 0.884150
\(237\) 0 0
\(238\) 1.41742 0.0918780
\(239\) 15.1652 0.980952 0.490476 0.871455i \(-0.336823\pi\)
0.490476 + 0.871455i \(0.336823\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) −10.3739 −0.666857
\(243\) 0 0
\(244\) 10.3739 0.664119
\(245\) −3.79129 −0.242216
\(246\) 0 0
\(247\) 33.5390 2.13404
\(248\) −3.37386 −0.214241
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −26.2087 −1.65428 −0.827140 0.561996i \(-0.810033\pi\)
−0.827140 + 0.561996i \(0.810033\pi\)
\(252\) 0 0
\(253\) −0.791288 −0.0497478
\(254\) 12.7477 0.799864
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.74773 0.296155 0.148078 0.988976i \(-0.452691\pi\)
0.148078 + 0.988976i \(0.452691\pi\)
\(258\) 0 0
\(259\) 7.16515 0.445221
\(260\) 5.79129 0.359160
\(261\) 0 0
\(262\) 9.16515 0.566225
\(263\) −11.2087 −0.691159 −0.345579 0.938390i \(-0.612318\pi\)
−0.345579 + 0.938390i \(0.612318\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −10.3739 −0.636062
\(267\) 0 0
\(268\) 11.1652 0.682020
\(269\) 10.7477 0.655300 0.327650 0.944799i \(-0.393743\pi\)
0.327650 + 0.944799i \(0.393743\pi\)
\(270\) 0 0
\(271\) 18.1216 1.10081 0.550404 0.834898i \(-0.314474\pi\)
0.550404 + 0.834898i \(0.314474\pi\)
\(272\) −0.791288 −0.0479789
\(273\) 0 0
\(274\) −3.79129 −0.229040
\(275\) 0.791288 0.0477165
\(276\) 0 0
\(277\) 17.1652 1.03135 0.515677 0.856783i \(-0.327540\pi\)
0.515677 + 0.856783i \(0.327540\pi\)
\(278\) 12.7477 0.764558
\(279\) 0 0
\(280\) −1.79129 −0.107050
\(281\) −10.7477 −0.641156 −0.320578 0.947222i \(-0.603877\pi\)
−0.320578 + 0.947222i \(0.603877\pi\)
\(282\) 0 0
\(283\) 8.33030 0.495185 0.247593 0.968864i \(-0.420361\pi\)
0.247593 + 0.968864i \(0.420361\pi\)
\(284\) −8.37386 −0.496897
\(285\) 0 0
\(286\) 4.58258 0.270973
\(287\) −12.1652 −0.718086
\(288\) 0 0
\(289\) −16.3739 −0.963168
\(290\) −7.58258 −0.445264
\(291\) 0 0
\(292\) 12.7477 0.746004
\(293\) −27.4955 −1.60630 −0.803151 0.595776i \(-0.796845\pi\)
−0.803151 + 0.595776i \(0.796845\pi\)
\(294\) 0 0
\(295\) 13.5826 0.790808
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) −8.20871 −0.475518
\(299\) −5.79129 −0.334919
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) −10.7913 −0.620969
\(303\) 0 0
\(304\) 5.79129 0.332153
\(305\) 10.3739 0.594006
\(306\) 0 0
\(307\) −15.5390 −0.886858 −0.443429 0.896309i \(-0.646238\pi\)
−0.443429 + 0.896309i \(0.646238\pi\)
\(308\) −1.41742 −0.0807652
\(309\) 0 0
\(310\) −3.37386 −0.191623
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −4.62614 −0.261485 −0.130742 0.991416i \(-0.541736\pi\)
−0.130742 + 0.991416i \(0.541736\pi\)
\(314\) −14.7477 −0.832262
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −9.79129 −0.549934 −0.274967 0.961454i \(-0.588667\pi\)
−0.274967 + 0.961454i \(0.588667\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 1.79129 0.0998246
\(323\) −4.58258 −0.254981
\(324\) 0 0
\(325\) 5.79129 0.321243
\(326\) 8.62614 0.477758
\(327\) 0 0
\(328\) 6.79129 0.374986
\(329\) −7.91288 −0.436251
\(330\) 0 0
\(331\) −20.7477 −1.14040 −0.570199 0.821507i \(-0.693134\pi\)
−0.570199 + 0.821507i \(0.693134\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) −18.3303 −1.00299
\(335\) 11.1652 0.610017
\(336\) 0 0
\(337\) −12.2087 −0.665051 −0.332525 0.943094i \(-0.607901\pi\)
−0.332525 + 0.943094i \(0.607901\pi\)
\(338\) 20.5390 1.11718
\(339\) 0 0
\(340\) −0.791288 −0.0429136
\(341\) −2.66970 −0.144572
\(342\) 0 0
\(343\) 19.3303 1.04374
\(344\) 11.1652 0.601985
\(345\) 0 0
\(346\) 18.7913 1.01023
\(347\) −5.20871 −0.279618 −0.139809 0.990178i \(-0.544649\pi\)
−0.139809 + 0.990178i \(0.544649\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) −1.79129 −0.0957484
\(351\) 0 0
\(352\) 0.791288 0.0421758
\(353\) −3.16515 −0.168464 −0.0842320 0.996446i \(-0.526844\pi\)
−0.0842320 + 0.996446i \(0.526844\pi\)
\(354\) 0 0
\(355\) −8.37386 −0.444439
\(356\) −15.1652 −0.803751
\(357\) 0 0
\(358\) −10.7477 −0.568035
\(359\) −9.16515 −0.483718 −0.241859 0.970311i \(-0.577757\pi\)
−0.241859 + 0.970311i \(0.577757\pi\)
\(360\) 0 0
\(361\) 14.5390 0.765211
\(362\) −18.5390 −0.974389
\(363\) 0 0
\(364\) −10.3739 −0.543738
\(365\) 12.7477 0.667247
\(366\) 0 0
\(367\) −19.1652 −1.00041 −0.500206 0.865906i \(-0.666743\pi\)
−0.500206 + 0.865906i \(0.666743\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −4.00000 −0.207950
\(371\) 10.7477 0.557994
\(372\) 0 0
\(373\) 12.7477 0.660052 0.330026 0.943972i \(-0.392942\pi\)
0.330026 + 0.943972i \(0.392942\pi\)
\(374\) −0.626136 −0.0323767
\(375\) 0 0
\(376\) 4.41742 0.227811
\(377\) −43.9129 −2.26163
\(378\) 0 0
\(379\) −6.37386 −0.327403 −0.163702 0.986510i \(-0.552343\pi\)
−0.163702 + 0.986510i \(0.552343\pi\)
\(380\) 5.79129 0.297087
\(381\) 0 0
\(382\) −25.5826 −1.30892
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) −1.41742 −0.0722386
\(386\) −20.7477 −1.05603
\(387\) 0 0
\(388\) −7.95644 −0.403927
\(389\) 20.7042 1.04974 0.524871 0.851182i \(-0.324113\pi\)
0.524871 + 0.851182i \(0.324113\pi\)
\(390\) 0 0
\(391\) 0.791288 0.0400171
\(392\) −3.79129 −0.191489
\(393\) 0 0
\(394\) 11.5390 0.581327
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −15.5390 −0.779881 −0.389940 0.920840i \(-0.627504\pi\)
−0.389940 + 0.920840i \(0.627504\pi\)
\(398\) −16.3303 −0.818564
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 4.74773 0.237090 0.118545 0.992949i \(-0.462177\pi\)
0.118545 + 0.992949i \(0.462177\pi\)
\(402\) 0 0
\(403\) −19.5390 −0.973308
\(404\) −4.41742 −0.219775
\(405\) 0 0
\(406\) 13.5826 0.674092
\(407\) −3.16515 −0.156891
\(408\) 0 0
\(409\) −18.2087 −0.900363 −0.450181 0.892937i \(-0.648641\pi\)
−0.450181 + 0.892937i \(0.648641\pi\)
\(410\) 6.79129 0.335398
\(411\) 0 0
\(412\) −6.37386 −0.314018
\(413\) −24.3303 −1.19722
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 5.79129 0.283941
\(417\) 0 0
\(418\) 4.58258 0.224141
\(419\) −20.8348 −1.01785 −0.508924 0.860811i \(-0.669957\pi\)
−0.508924 + 0.860811i \(0.669957\pi\)
\(420\) 0 0
\(421\) 18.1216 0.883192 0.441596 0.897214i \(-0.354412\pi\)
0.441596 + 0.897214i \(0.354412\pi\)
\(422\) −10.0000 −0.486792
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −0.791288 −0.0383831
\(426\) 0 0
\(427\) −18.5826 −0.899274
\(428\) −4.41742 −0.213524
\(429\) 0 0
\(430\) 11.1652 0.538431
\(431\) −25.9129 −1.24818 −0.624090 0.781353i \(-0.714530\pi\)
−0.624090 + 0.781353i \(0.714530\pi\)
\(432\) 0 0
\(433\) −30.5390 −1.46761 −0.733806 0.679359i \(-0.762258\pi\)
−0.733806 + 0.679359i \(0.762258\pi\)
\(434\) 6.04356 0.290100
\(435\) 0 0
\(436\) −3.37386 −0.161579
\(437\) −5.79129 −0.277035
\(438\) 0 0
\(439\) −6.53901 −0.312090 −0.156045 0.987750i \(-0.549875\pi\)
−0.156045 + 0.987750i \(0.549875\pi\)
\(440\) 0.791288 0.0377232
\(441\) 0 0
\(442\) −4.58258 −0.217971
\(443\) 39.7913 1.89054 0.945271 0.326288i \(-0.105798\pi\)
0.945271 + 0.326288i \(0.105798\pi\)
\(444\) 0 0
\(445\) −15.1652 −0.718897
\(446\) −7.16515 −0.339280
\(447\) 0 0
\(448\) −1.79129 −0.0846304
\(449\) 16.1216 0.760825 0.380412 0.924817i \(-0.375782\pi\)
0.380412 + 0.924817i \(0.375782\pi\)
\(450\) 0 0
\(451\) 5.37386 0.253045
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 22.7477 1.06760
\(455\) −10.3739 −0.486334
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 20.3303 0.949973
\(459\) 0 0
\(460\) −1.00000 −0.0466252
\(461\) 28.7477 1.33892 0.669458 0.742850i \(-0.266527\pi\)
0.669458 + 0.742850i \(0.266527\pi\)
\(462\) 0 0
\(463\) −10.0000 −0.464739 −0.232370 0.972628i \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) −7.58258 −0.352012
\(465\) 0 0
\(466\) 1.58258 0.0733114
\(467\) 19.9129 0.921458 0.460729 0.887541i \(-0.347588\pi\)
0.460729 + 0.887541i \(0.347588\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 4.41742 0.203761
\(471\) 0 0
\(472\) 13.5826 0.625189
\(473\) 8.83485 0.406227
\(474\) 0 0
\(475\) 5.79129 0.265723
\(476\) 1.41742 0.0649675
\(477\) 0 0
\(478\) 15.1652 0.693638
\(479\) −15.4955 −0.708005 −0.354003 0.935244i \(-0.615180\pi\)
−0.354003 + 0.935244i \(0.615180\pi\)
\(480\) 0 0
\(481\) −23.1652 −1.05624
\(482\) −28.0000 −1.27537
\(483\) 0 0
\(484\) −10.3739 −0.471539
\(485\) −7.95644 −0.361283
\(486\) 0 0
\(487\) 6.41742 0.290801 0.145401 0.989373i \(-0.453553\pi\)
0.145401 + 0.989373i \(0.453553\pi\)
\(488\) 10.3739 0.469603
\(489\) 0 0
\(490\) −3.79129 −0.171273
\(491\) −10.7477 −0.485038 −0.242519 0.970147i \(-0.577974\pi\)
−0.242519 + 0.970147i \(0.577974\pi\)
\(492\) 0 0
\(493\) 6.00000 0.270226
\(494\) 33.5390 1.50899
\(495\) 0 0
\(496\) −3.37386 −0.151491
\(497\) 15.0000 0.672842
\(498\) 0 0
\(499\) 4.83485 0.216438 0.108219 0.994127i \(-0.465485\pi\)
0.108219 + 0.994127i \(0.465485\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −26.2087 −1.16975
\(503\) −14.2087 −0.633535 −0.316768 0.948503i \(-0.602598\pi\)
−0.316768 + 0.948503i \(0.602598\pi\)
\(504\) 0 0
\(505\) −4.41742 −0.196573
\(506\) −0.791288 −0.0351770
\(507\) 0 0
\(508\) 12.7477 0.565589
\(509\) −34.7477 −1.54017 −0.770083 0.637944i \(-0.779785\pi\)
−0.770083 + 0.637944i \(0.779785\pi\)
\(510\) 0 0
\(511\) −22.8348 −1.01015
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 4.74773 0.209413
\(515\) −6.37386 −0.280866
\(516\) 0 0
\(517\) 3.49545 0.153730
\(518\) 7.16515 0.314819
\(519\) 0 0
\(520\) 5.79129 0.253965
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 17.1652 0.750580 0.375290 0.926908i \(-0.377543\pi\)
0.375290 + 0.926908i \(0.377543\pi\)
\(524\) 9.16515 0.400381
\(525\) 0 0
\(526\) −11.2087 −0.488723
\(527\) 2.66970 0.116294
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) −10.3739 −0.449764
\(533\) 39.3303 1.70358
\(534\) 0 0
\(535\) −4.41742 −0.190982
\(536\) 11.1652 0.482261
\(537\) 0 0
\(538\) 10.7477 0.463367
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 1.66970 0.0717859 0.0358929 0.999356i \(-0.488572\pi\)
0.0358929 + 0.999356i \(0.488572\pi\)
\(542\) 18.1216 0.778389
\(543\) 0 0
\(544\) −0.791288 −0.0339262
\(545\) −3.37386 −0.144520
\(546\) 0 0
\(547\) −26.1216 −1.11688 −0.558439 0.829545i \(-0.688600\pi\)
−0.558439 + 0.829545i \(0.688600\pi\)
\(548\) −3.79129 −0.161956
\(549\) 0 0
\(550\) 0.791288 0.0337406
\(551\) −43.9129 −1.87075
\(552\) 0 0
\(553\) −14.3303 −0.609386
\(554\) 17.1652 0.729277
\(555\) 0 0
\(556\) 12.7477 0.540624
\(557\) 6.33030 0.268224 0.134112 0.990966i \(-0.457182\pi\)
0.134112 + 0.990966i \(0.457182\pi\)
\(558\) 0 0
\(559\) 64.6606 2.73485
\(560\) −1.79129 −0.0756957
\(561\) 0 0
\(562\) −10.7477 −0.453366
\(563\) 15.1652 0.639135 0.319567 0.947564i \(-0.396462\pi\)
0.319567 + 0.947564i \(0.396462\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 8.33030 0.350149
\(567\) 0 0
\(568\) −8.37386 −0.351360
\(569\) 39.4955 1.65574 0.827868 0.560923i \(-0.189554\pi\)
0.827868 + 0.560923i \(0.189554\pi\)
\(570\) 0 0
\(571\) −11.1216 −0.465424 −0.232712 0.972546i \(-0.574760\pi\)
−0.232712 + 0.972546i \(0.574760\pi\)
\(572\) 4.58258 0.191607
\(573\) 0 0
\(574\) −12.1652 −0.507764
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 41.1652 1.71373 0.856864 0.515543i \(-0.172410\pi\)
0.856864 + 0.515543i \(0.172410\pi\)
\(578\) −16.3739 −0.681063
\(579\) 0 0
\(580\) −7.58258 −0.314849
\(581\) −10.7477 −0.445891
\(582\) 0 0
\(583\) −4.74773 −0.196631
\(584\) 12.7477 0.527505
\(585\) 0 0
\(586\) −27.4955 −1.13583
\(587\) 30.7913 1.27089 0.635446 0.772145i \(-0.280816\pi\)
0.635446 + 0.772145i \(0.280816\pi\)
\(588\) 0 0
\(589\) −19.5390 −0.805091
\(590\) 13.5826 0.559186
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) 31.9129 1.31050 0.655252 0.755410i \(-0.272562\pi\)
0.655252 + 0.755410i \(0.272562\pi\)
\(594\) 0 0
\(595\) 1.41742 0.0581087
\(596\) −8.20871 −0.336242
\(597\) 0 0
\(598\) −5.79129 −0.236823
\(599\) −1.12159 −0.0458270 −0.0229135 0.999737i \(-0.507294\pi\)
−0.0229135 + 0.999737i \(0.507294\pi\)
\(600\) 0 0
\(601\) −18.2087 −0.742749 −0.371374 0.928483i \(-0.621113\pi\)
−0.371374 + 0.928483i \(0.621113\pi\)
\(602\) −20.0000 −0.815139
\(603\) 0 0
\(604\) −10.7913 −0.439091
\(605\) −10.3739 −0.421758
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 5.79129 0.234868
\(609\) 0 0
\(610\) 10.3739 0.420025
\(611\) 25.5826 1.03496
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −15.5390 −0.627104
\(615\) 0 0
\(616\) −1.41742 −0.0571097
\(617\) 23.8693 0.960943 0.480471 0.877010i \(-0.340466\pi\)
0.480471 + 0.877010i \(0.340466\pi\)
\(618\) 0 0
\(619\) −1.79129 −0.0719979 −0.0359990 0.999352i \(-0.511461\pi\)
−0.0359990 + 0.999352i \(0.511461\pi\)
\(620\) −3.37386 −0.135498
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 27.1652 1.08835
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −4.62614 −0.184898
\(627\) 0 0
\(628\) −14.7477 −0.588498
\(629\) 3.16515 0.126203
\(630\) 0 0
\(631\) 27.9129 1.11119 0.555597 0.831452i \(-0.312490\pi\)
0.555597 + 0.831452i \(0.312490\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) −9.79129 −0.388862
\(635\) 12.7477 0.505878
\(636\) 0 0
\(637\) −21.9564 −0.869946
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −15.1652 −0.598987 −0.299494 0.954098i \(-0.596818\pi\)
−0.299494 + 0.954098i \(0.596818\pi\)
\(642\) 0 0
\(643\) 6.74773 0.266104 0.133052 0.991109i \(-0.457522\pi\)
0.133052 + 0.991109i \(0.457522\pi\)
\(644\) 1.79129 0.0705866
\(645\) 0 0
\(646\) −4.58258 −0.180299
\(647\) −21.1652 −0.832088 −0.416044 0.909344i \(-0.636584\pi\)
−0.416044 + 0.909344i \(0.636584\pi\)
\(648\) 0 0
\(649\) 10.7477 0.421885
\(650\) 5.79129 0.227153
\(651\) 0 0
\(652\) 8.62614 0.337826
\(653\) −3.46099 −0.135439 −0.0677194 0.997704i \(-0.521572\pi\)
−0.0677194 + 0.997704i \(0.521572\pi\)
\(654\) 0 0
\(655\) 9.16515 0.358112
\(656\) 6.79129 0.265155
\(657\) 0 0
\(658\) −7.91288 −0.308476
\(659\) 8.83485 0.344157 0.172078 0.985083i \(-0.444952\pi\)
0.172078 + 0.985083i \(0.444952\pi\)
\(660\) 0 0
\(661\) −25.6261 −0.996741 −0.498371 0.866964i \(-0.666068\pi\)
−0.498371 + 0.866964i \(0.666068\pi\)
\(662\) −20.7477 −0.806383
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) −10.3739 −0.402281
\(666\) 0 0
\(667\) 7.58258 0.293599
\(668\) −18.3303 −0.709221
\(669\) 0 0
\(670\) 11.1652 0.431347
\(671\) 8.20871 0.316894
\(672\) 0 0
\(673\) 38.0000 1.46479 0.732396 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(674\) −12.2087 −0.470262
\(675\) 0 0
\(676\) 20.5390 0.789962
\(677\) −42.6606 −1.63958 −0.819790 0.572664i \(-0.805910\pi\)
−0.819790 + 0.572664i \(0.805910\pi\)
\(678\) 0 0
\(679\) 14.2523 0.546952
\(680\) −0.791288 −0.0303445
\(681\) 0 0
\(682\) −2.66970 −0.102228
\(683\) 11.3739 0.435209 0.217604 0.976037i \(-0.430176\pi\)
0.217604 + 0.976037i \(0.430176\pi\)
\(684\) 0 0
\(685\) −3.79129 −0.144858
\(686\) 19.3303 0.738034
\(687\) 0 0
\(688\) 11.1652 0.425667
\(689\) −34.7477 −1.32378
\(690\) 0 0
\(691\) 42.7477 1.62620 0.813100 0.582124i \(-0.197778\pi\)
0.813100 + 0.582124i \(0.197778\pi\)
\(692\) 18.7913 0.714338
\(693\) 0 0
\(694\) −5.20871 −0.197720
\(695\) 12.7477 0.483549
\(696\) 0 0
\(697\) −5.37386 −0.203550
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) −1.79129 −0.0677043
\(701\) −23.3739 −0.882819 −0.441409 0.897306i \(-0.645521\pi\)
−0.441409 + 0.897306i \(0.645521\pi\)
\(702\) 0 0
\(703\) −23.1652 −0.873690
\(704\) 0.791288 0.0298228
\(705\) 0 0
\(706\) −3.16515 −0.119122
\(707\) 7.91288 0.297594
\(708\) 0 0
\(709\) 2.46099 0.0924242 0.0462121 0.998932i \(-0.485285\pi\)
0.0462121 + 0.998932i \(0.485285\pi\)
\(710\) −8.37386 −0.314265
\(711\) 0 0
\(712\) −15.1652 −0.568338
\(713\) 3.37386 0.126352
\(714\) 0 0
\(715\) 4.58258 0.171379
\(716\) −10.7477 −0.401661
\(717\) 0 0
\(718\) −9.16515 −0.342040
\(719\) −2.53901 −0.0946893 −0.0473446 0.998879i \(-0.515076\pi\)
−0.0473446 + 0.998879i \(0.515076\pi\)
\(720\) 0 0
\(721\) 11.4174 0.425207
\(722\) 14.5390 0.541086
\(723\) 0 0
\(724\) −18.5390 −0.688997
\(725\) −7.58258 −0.281610
\(726\) 0 0
\(727\) 39.1216 1.45094 0.725470 0.688254i \(-0.241623\pi\)
0.725470 + 0.688254i \(0.241623\pi\)
\(728\) −10.3739 −0.384481
\(729\) 0 0
\(730\) 12.7477 0.471815
\(731\) −8.83485 −0.326769
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) −19.1652 −0.707399
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 8.83485 0.325436
\(738\) 0 0
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 10.7477 0.394561
\(743\) 12.9564 0.475326 0.237663 0.971348i \(-0.423619\pi\)
0.237663 + 0.971348i \(0.423619\pi\)
\(744\) 0 0
\(745\) −8.20871 −0.300744
\(746\) 12.7477 0.466727
\(747\) 0 0
\(748\) −0.626136 −0.0228938
\(749\) 7.91288 0.289130
\(750\) 0 0
\(751\) −8.74773 −0.319209 −0.159605 0.987181i \(-0.551022\pi\)
−0.159605 + 0.987181i \(0.551022\pi\)
\(752\) 4.41742 0.161087
\(753\) 0 0
\(754\) −43.9129 −1.59921
\(755\) −10.7913 −0.392735
\(756\) 0 0
\(757\) −10.3303 −0.375461 −0.187731 0.982221i \(-0.560113\pi\)
−0.187731 + 0.982221i \(0.560113\pi\)
\(758\) −6.37386 −0.231509
\(759\) 0 0
\(760\) 5.79129 0.210072
\(761\) 11.0436 0.400329 0.200164 0.979762i \(-0.435852\pi\)
0.200164 + 0.979762i \(0.435852\pi\)
\(762\) 0 0
\(763\) 6.04356 0.218792
\(764\) −25.5826 −0.925545
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 78.6606 2.84027
\(768\) 0 0
\(769\) −40.3303 −1.45435 −0.727174 0.686453i \(-0.759167\pi\)
−0.727174 + 0.686453i \(0.759167\pi\)
\(770\) −1.41742 −0.0510804
\(771\) 0 0
\(772\) −20.7477 −0.746727
\(773\) −33.4955 −1.20475 −0.602374 0.798214i \(-0.705778\pi\)
−0.602374 + 0.798214i \(0.705778\pi\)
\(774\) 0 0
\(775\) −3.37386 −0.121193
\(776\) −7.95644 −0.285620
\(777\) 0 0
\(778\) 20.7042 0.742280
\(779\) 39.3303 1.40915
\(780\) 0 0
\(781\) −6.62614 −0.237102
\(782\) 0.791288 0.0282964
\(783\) 0 0
\(784\) −3.79129 −0.135403
\(785\) −14.7477 −0.526369
\(786\) 0 0
\(787\) −17.5826 −0.626751 −0.313376 0.949629i \(-0.601460\pi\)
−0.313376 + 0.949629i \(0.601460\pi\)
\(788\) 11.5390 0.411060
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) 10.7477 0.382145
\(792\) 0 0
\(793\) 60.0780 2.13343
\(794\) −15.5390 −0.551459
\(795\) 0 0
\(796\) −16.3303 −0.578812
\(797\) 4.08712 0.144773 0.0723866 0.997377i \(-0.476938\pi\)
0.0723866 + 0.997377i \(0.476938\pi\)
\(798\) 0 0
\(799\) −3.49545 −0.123660
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 4.74773 0.167648
\(803\) 10.0871 0.355967
\(804\) 0 0
\(805\) 1.79129 0.0631346
\(806\) −19.5390 −0.688232
\(807\) 0 0
\(808\) −4.41742 −0.155404
\(809\) −33.9564 −1.19384 −0.596922 0.802299i \(-0.703610\pi\)
−0.596922 + 0.802299i \(0.703610\pi\)
\(810\) 0 0
\(811\) −2.08712 −0.0732887 −0.0366444 0.999328i \(-0.511667\pi\)
−0.0366444 + 0.999328i \(0.511667\pi\)
\(812\) 13.5826 0.476655
\(813\) 0 0
\(814\) −3.16515 −0.110938
\(815\) 8.62614 0.302160
\(816\) 0 0
\(817\) 64.6606 2.26219
\(818\) −18.2087 −0.636653
\(819\) 0 0
\(820\) 6.79129 0.237162
\(821\) −21.1652 −0.738669 −0.369334 0.929297i \(-0.620414\pi\)
−0.369334 + 0.929297i \(0.620414\pi\)
\(822\) 0 0
\(823\) 22.8348 0.795973 0.397986 0.917391i \(-0.369709\pi\)
0.397986 + 0.917391i \(0.369709\pi\)
\(824\) −6.37386 −0.222044
\(825\) 0 0
\(826\) −24.3303 −0.846560
\(827\) 23.0780 0.802502 0.401251 0.915968i \(-0.368576\pi\)
0.401251 + 0.915968i \(0.368576\pi\)
\(828\) 0 0
\(829\) 23.4955 0.816031 0.408015 0.912975i \(-0.366221\pi\)
0.408015 + 0.912975i \(0.366221\pi\)
\(830\) 6.00000 0.208263
\(831\) 0 0
\(832\) 5.79129 0.200777
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) −18.3303 −0.634346
\(836\) 4.58258 0.158492
\(837\) 0 0
\(838\) −20.8348 −0.719728
\(839\) 31.5826 1.09035 0.545176 0.838322i \(-0.316463\pi\)
0.545176 + 0.838322i \(0.316463\pi\)
\(840\) 0 0
\(841\) 28.4955 0.982602
\(842\) 18.1216 0.624511
\(843\) 0 0
\(844\) −10.0000 −0.344214
\(845\) 20.5390 0.706564
\(846\) 0 0
\(847\) 18.5826 0.638505
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −0.791288 −0.0271409
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) 40.5390 1.38803 0.694015 0.719961i \(-0.255840\pi\)
0.694015 + 0.719961i \(0.255840\pi\)
\(854\) −18.5826 −0.635883
\(855\) 0 0
\(856\) −4.41742 −0.150984
\(857\) 9.16515 0.313076 0.156538 0.987672i \(-0.449967\pi\)
0.156538 + 0.987672i \(0.449967\pi\)
\(858\) 0 0
\(859\) −26.7477 −0.912621 −0.456310 0.889821i \(-0.650829\pi\)
−0.456310 + 0.889821i \(0.650829\pi\)
\(860\) 11.1652 0.380729
\(861\) 0 0
\(862\) −25.9129 −0.882596
\(863\) 22.4174 0.763098 0.381549 0.924349i \(-0.375391\pi\)
0.381549 + 0.924349i \(0.375391\pi\)
\(864\) 0 0
\(865\) 18.7913 0.638923
\(866\) −30.5390 −1.03776
\(867\) 0 0
\(868\) 6.04356 0.205132
\(869\) 6.33030 0.214741
\(870\) 0 0
\(871\) 64.6606 2.19094
\(872\) −3.37386 −0.114253
\(873\) 0 0
\(874\) −5.79129 −0.195893
\(875\) −1.79129 −0.0605566
\(876\) 0 0
\(877\) −42.7042 −1.44202 −0.721009 0.692926i \(-0.756321\pi\)
−0.721009 + 0.692926i \(0.756321\pi\)
\(878\) −6.53901 −0.220681
\(879\) 0 0
\(880\) 0.791288 0.0266743
\(881\) −30.3303 −1.02185 −0.510927 0.859624i \(-0.670698\pi\)
−0.510927 + 0.859624i \(0.670698\pi\)
\(882\) 0 0
\(883\) −34.9564 −1.17638 −0.588189 0.808724i \(-0.700159\pi\)
−0.588189 + 0.808724i \(0.700159\pi\)
\(884\) −4.58258 −0.154129
\(885\) 0 0
\(886\) 39.7913 1.33681
\(887\) −15.1652 −0.509196 −0.254598 0.967047i \(-0.581943\pi\)
−0.254598 + 0.967047i \(0.581943\pi\)
\(888\) 0 0
\(889\) −22.8348 −0.765856
\(890\) −15.1652 −0.508337
\(891\) 0 0
\(892\) −7.16515 −0.239907
\(893\) 25.5826 0.856088
\(894\) 0 0
\(895\) −10.7477 −0.359257
\(896\) −1.79129 −0.0598427
\(897\) 0 0
\(898\) 16.1216 0.537984
\(899\) 25.5826 0.853227
\(900\) 0 0
\(901\) 4.74773 0.158170
\(902\) 5.37386 0.178930
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −18.5390 −0.616258
\(906\) 0 0
\(907\) 6.74773 0.224055 0.112027 0.993705i \(-0.464266\pi\)
0.112027 + 0.993705i \(0.464266\pi\)
\(908\) 22.7477 0.754910
\(909\) 0 0
\(910\) −10.3739 −0.343890
\(911\) −4.41742 −0.146356 −0.0731779 0.997319i \(-0.523314\pi\)
−0.0731779 + 0.997319i \(0.523314\pi\)
\(912\) 0 0
\(913\) 4.74773 0.157127
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 20.3303 0.671732
\(917\) −16.4174 −0.542151
\(918\) 0 0
\(919\) −55.1652 −1.81973 −0.909865 0.414904i \(-0.863815\pi\)
−0.909865 + 0.414904i \(0.863815\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 0 0
\(922\) 28.7477 0.946756
\(923\) −48.4955 −1.59625
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −10.0000 −0.328620
\(927\) 0 0
\(928\) −7.58258 −0.248910
\(929\) −15.4955 −0.508389 −0.254195 0.967153i \(-0.581810\pi\)
−0.254195 + 0.967153i \(0.581810\pi\)
\(930\) 0 0
\(931\) −21.9564 −0.719593
\(932\) 1.58258 0.0518390
\(933\) 0 0
\(934\) 19.9129 0.651569
\(935\) −0.626136 −0.0204769
\(936\) 0 0
\(937\) 44.6261 1.45787 0.728936 0.684582i \(-0.240015\pi\)
0.728936 + 0.684582i \(0.240015\pi\)
\(938\) −20.0000 −0.653023
\(939\) 0 0
\(940\) 4.41742 0.144080
\(941\) −32.0436 −1.04459 −0.522295 0.852765i \(-0.674924\pi\)
−0.522295 + 0.852765i \(0.674924\pi\)
\(942\) 0 0
\(943\) −6.79129 −0.221155
\(944\) 13.5826 0.442075
\(945\) 0 0
\(946\) 8.83485 0.287246
\(947\) −2.53901 −0.0825069 −0.0412534 0.999149i \(-0.513135\pi\)
−0.0412534 + 0.999149i \(0.513135\pi\)
\(948\) 0 0
\(949\) 73.8258 2.39649
\(950\) 5.79129 0.187894
\(951\) 0 0
\(952\) 1.41742 0.0459390
\(953\) −5.53901 −0.179426 −0.0897131 0.995968i \(-0.528595\pi\)
−0.0897131 + 0.995968i \(0.528595\pi\)
\(954\) 0 0
\(955\) −25.5826 −0.827833
\(956\) 15.1652 0.490476
\(957\) 0 0
\(958\) −15.4955 −0.500635
\(959\) 6.79129 0.219302
\(960\) 0 0
\(961\) −19.6170 −0.632808
\(962\) −23.1652 −0.746874
\(963\) 0 0
\(964\) −28.0000 −0.901819
\(965\) −20.7477 −0.667893
\(966\) 0 0
\(967\) −32.7477 −1.05310 −0.526548 0.850145i \(-0.676514\pi\)
−0.526548 + 0.850145i \(0.676514\pi\)
\(968\) −10.3739 −0.333429
\(969\) 0 0
\(970\) −7.95644 −0.255466
\(971\) −15.9564 −0.512067 −0.256033 0.966668i \(-0.582416\pi\)
−0.256033 + 0.966668i \(0.582416\pi\)
\(972\) 0 0
\(973\) −22.8348 −0.732052
\(974\) 6.41742 0.205628
\(975\) 0 0
\(976\) 10.3739 0.332059
\(977\) 34.1216 1.09165 0.545823 0.837900i \(-0.316217\pi\)
0.545823 + 0.837900i \(0.316217\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) −3.79129 −0.121108
\(981\) 0 0
\(982\) −10.7477 −0.342974
\(983\) 14.3739 0.458455 0.229228 0.973373i \(-0.426380\pi\)
0.229228 + 0.973373i \(0.426380\pi\)
\(984\) 0 0
\(985\) 11.5390 0.367664
\(986\) 6.00000 0.191079
\(987\) 0 0
\(988\) 33.5390 1.06702
\(989\) −11.1652 −0.355031
\(990\) 0 0
\(991\) −33.2087 −1.05491 −0.527455 0.849583i \(-0.676854\pi\)
−0.527455 + 0.849583i \(0.676854\pi\)
\(992\) −3.37386 −0.107120
\(993\) 0 0
\(994\) 15.0000 0.475771
\(995\) −16.3303 −0.517705
\(996\) 0 0
\(997\) −43.4955 −1.37751 −0.688757 0.724992i \(-0.741843\pi\)
−0.688757 + 0.724992i \(0.741843\pi\)
\(998\) 4.83485 0.153044
\(999\) 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 2070.2.a.x.1.1 2
3.2 odd 2 230.2.a.a.1.1 2
12.11 even 2 1840.2.a.n.1.2 2
15.2 even 4 1150.2.b.g.599.2 4
15.8 even 4 1150.2.b.g.599.3 4
15.14 odd 2 1150.2.a.o.1.2 2
24.5 odd 2 7360.2.a.bq.1.2 2
24.11 even 2 7360.2.a.bk.1.1 2
60.59 even 2 9200.2.a.bs.1.1 2
69.68 even 2 5290.2.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.1 2 3.2 odd 2
1150.2.a.o.1.2 2 15.14 odd 2
1150.2.b.g.599.2 4 15.2 even 4
1150.2.b.g.599.3 4 15.8 even 4
1840.2.a.n.1.2 2 12.11 even 2
2070.2.a.x.1.1 2 1.1 even 1 trivial
5290.2.a.e.1.1 2 69.68 even 2
7360.2.a.bk.1.1 2 24.11 even 2
7360.2.a.bq.1.2 2 24.5 odd 2
9200.2.a.bs.1.1 2 60.59 even 2