Properties

Label 1840.2.a.n.1.2
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \(x^{2} - x - 5\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 1.2
Root \(2.79129\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.79129 q^{3} -1.00000 q^{5} +1.79129 q^{7} +4.79129 q^{9} +O(q^{10})\) \(q+2.79129 q^{3} -1.00000 q^{5} +1.79129 q^{7} +4.79129 q^{9} +0.791288 q^{11} +5.79129 q^{13} -2.79129 q^{15} +0.791288 q^{17} -5.79129 q^{19} +5.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} +7.58258 q^{29} +3.37386 q^{31} +2.20871 q^{33} -1.79129 q^{35} -4.00000 q^{37} +16.1652 q^{39} -6.79129 q^{41} -11.1652 q^{43} -4.79129 q^{45} +4.41742 q^{47} -3.79129 q^{49} +2.20871 q^{51} +6.00000 q^{53} -0.791288 q^{55} -16.1652 q^{57} +13.5826 q^{59} +10.3739 q^{61} +8.58258 q^{63} -5.79129 q^{65} -11.1652 q^{67} -2.79129 q^{69} -8.37386 q^{71} +12.7477 q^{73} +2.79129 q^{75} +1.41742 q^{77} -8.00000 q^{79} -0.417424 q^{81} +6.00000 q^{83} -0.791288 q^{85} +21.1652 q^{87} +15.1652 q^{89} +10.3739 q^{91} +9.41742 q^{93} +5.79129 q^{95} -7.95644 q^{97} +3.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 2q^{5} - q^{7} + 5q^{9} + O(q^{10}) \) \( 2q + q^{3} - 2q^{5} - q^{7} + 5q^{9} - 3q^{11} + 7q^{13} - q^{15} - 3q^{17} - 7q^{19} + 10q^{21} - 2q^{23} + 2q^{25} + 10q^{27} + 6q^{29} - 7q^{31} + 9q^{33} + q^{35} - 8q^{37} + 14q^{39} - 9q^{41} - 4q^{43} - 5q^{45} + 18q^{47} - 3q^{49} + 9q^{51} + 12q^{53} + 3q^{55} - 14q^{57} + 18q^{59} + 7q^{61} + 8q^{63} - 7q^{65} - 4q^{67} - q^{69} - 3q^{71} - 2q^{73} + q^{75} + 12q^{77} - 16q^{79} - 10q^{81} + 12q^{83} + 3q^{85} + 24q^{87} + 12q^{89} + 7q^{91} + 28q^{93} + 7q^{95} + 7q^{97} + 3q^{99} + O(q^{100}) \)

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 0
\(3\) 2.79129 1.61155 0.805775 0.592221i \(-0.201749\pi\)
0.805775 + 0.592221i \(0.201749\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.79129 0.677043 0.338522 0.940959i \(-0.390073\pi\)
0.338522 + 0.940959i \(0.390073\pi\)
\(8\) 0 0
\(9\) 4.79129 1.59710
\(10\) 0 0
\(11\) 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 \(0.203179\pi\)
\(14\) 0 0
\(15\) −2.79129 −0.720707
\(16\) 0 0
\(17\) 0.791288 0.191915 0.0959577 0.995385i \(-0.469409\pi\)
0.0959577 + 0.995385i \(0.469409\pi\)
\(18\) 0 0
\(19\) −5.79129 −1.32861 −0.664306 0.747460i \(-0.731273\pi\)
−0.664306 + 0.747460i \(0.731273\pi\)
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 7.58258 1.40805 0.704024 0.710176i \(-0.251385\pi\)
0.704024 + 0.710176i \(0.251385\pi\)
\(30\) 0 0
\(31\) 3.37386 0.605964 0.302982 0.952996i \(-0.402018\pi\)
0.302982 + 0.952996i \(0.402018\pi\)
\(32\) 0 0
\(33\) 2.20871 0.384487
\(34\) 0 0
\(35\) −1.79129 −0.302783
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 16.1652 2.58850
\(40\) 0 0
\(41\) −6.79129 −1.06062 −0.530310 0.847804i \(-0.677925\pi\)
−0.530310 + 0.847804i \(0.677925\pi\)
\(42\) 0 0
\(43\) −11.1652 −1.70267 −0.851335 0.524623i \(-0.824206\pi\)
−0.851335 + 0.524623i \(0.824206\pi\)
\(44\) 0 0
\(45\) −4.79129 −0.714243
\(46\) 0 0
\(47\) 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
\(50\) 0 0
\(51\) 2.20871 0.309282
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −0.791288 −0.106697
\(56\) 0 0
\(57\) −16.1652 −2.14113
\(58\) 0 0
\(59\) 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 \(0.268807\pi\)
\(62\) 0 0
\(63\) 8.58258 1.08130
\(64\) 0 0
\(65\) −5.79129 −0.718321
\(66\) 0 0
\(67\) −11.1652 −1.36404 −0.682020 0.731333i \(-0.738898\pi\)
−0.682020 + 0.731333i \(0.738898\pi\)
\(68\) 0 0
\(69\) −2.79129 −0.336032
\(70\) 0 0
\(71\) −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 \(0.231970\pi\)
\(74\) 0 0
\(75\) 2.79129 0.322310
\(76\) 0 0
\(77\) 1.41742 0.161530
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −0.417424 −0.0463805
\(82\) 0 0
\(83\) 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
\(86\) 0 0
\(87\) 21.1652 2.26914
\(88\) 0 0
\(89\) 15.1652 1.60750 0.803751 0.594965i \(-0.202834\pi\)
0.803751 + 0.594965i \(0.202834\pi\)
\(90\) 0 0
\(91\) 10.3739 1.08748
\(92\) 0 0
\(93\) 9.41742 0.976541
\(94\) 0 0
\(95\) 5.79129 0.594174
\(96\) 0 0
\(97\) −7.95644 −0.807854 −0.403927 0.914791i \(-0.632355\pi\)
−0.403927 + 0.914791i \(0.632355\pi\)
\(98\) 0 0
\(99\) 3.79129 0.381039
\(100\) 0 0
\(101\) 4.41742 0.439550 0.219775 0.975551i \(-0.429468\pi\)
0.219775 + 0.975551i \(0.429468\pi\)
\(102\) 0 0
\(103\) 6.37386 0.628035 0.314018 0.949417i \(-0.398325\pi\)
0.314018 + 0.949417i \(0.398325\pi\)
\(104\) 0 0
\(105\) −5.00000 −0.487950
\(106\) 0 0
\(107\) −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 0
\(111\) −11.1652 −1.05975
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 27.7477 2.56528
\(118\) 0 0
\(119\) 1.41742 0.129935
\(120\) 0 0
\(121\) −10.3739 −0.943079
\(122\) 0 0
\(123\) −18.9564 −1.70924
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.7477 −1.13118 −0.565589 0.824687i \(-0.691351\pi\)
−0.565589 + 0.824687i \(0.691351\pi\)
\(128\) 0 0
\(129\) −31.1652 −2.74394
\(130\) 0 0
\(131\) 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
\(134\) 0 0
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) 3.79129 0.323912 0.161956 0.986798i \(-0.448220\pi\)
0.161956 + 0.986798i \(0.448220\pi\)
\(138\) 0 0
\(139\) −12.7477 −1.08125 −0.540624 0.841264i \(-0.681812\pi\)
−0.540624 + 0.841264i \(0.681812\pi\)
\(140\) 0 0
\(141\) 12.3303 1.03840
\(142\) 0 0
\(143\) 4.58258 0.383214
\(144\) 0 0
\(145\) −7.58258 −0.629699
\(146\) 0 0
\(147\) −10.5826 −0.872836
\(148\) 0 0
\(149\) 8.20871 0.672484 0.336242 0.941776i \(-0.390844\pi\)
0.336242 + 0.941776i \(0.390844\pi\)
\(150\) 0 0
\(151\) 10.7913 0.878183 0.439091 0.898442i \(-0.355300\pi\)
0.439091 + 0.898442i \(0.355300\pi\)
\(152\) 0 0
\(153\) 3.79129 0.306507
\(154\) 0 0
\(155\) −3.37386 −0.270995
\(156\) 0 0
\(157\) −14.7477 −1.17700 −0.588498 0.808498i \(-0.700281\pi\)
−0.588498 + 0.808498i \(0.700281\pi\)
\(158\) 0 0
\(159\) 16.7477 1.32818
\(160\) 0 0
\(161\) −1.79129 −0.141173
\(162\) 0 0
\(163\) −8.62614 −0.675651 −0.337826 0.941209i \(-0.609691\pi\)
−0.337826 + 0.941209i \(0.609691\pi\)
\(164\) 0 0
\(165\) −2.20871 −0.171948
\(166\) 0 0
\(167\) −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 0
\(171\) −27.7477 −2.12192
\(172\) 0 0
\(173\) −18.7913 −1.42868 −0.714338 0.699801i \(-0.753272\pi\)
−0.714338 + 0.699801i \(0.753272\pi\)
\(174\) 0 0
\(175\) 1.79129 0.135409
\(176\) 0 0
\(177\) 37.9129 2.84971
\(178\) 0 0
\(179\) −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 \(0.741949\pi\)
\(182\) 0 0
\(183\) 28.9564 2.14052
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 0.626136 0.0457876
\(188\) 0 0
\(189\) 8.95644 0.651485
\(190\) 0 0
\(191\) −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 \(0.768376\pi\)
\(194\) 0 0
\(195\) −16.1652 −1.15761
\(196\) 0 0
\(197\) −11.5390 −0.822121 −0.411060 0.911608i \(-0.634841\pi\)
−0.411060 + 0.911608i \(0.634841\pi\)
\(198\) 0 0
\(199\) 16.3303 1.15762 0.578812 0.815461i \(-0.303516\pi\)
0.578812 + 0.815461i \(0.303516\pi\)
\(200\) 0 0
\(201\) −31.1652 −2.19822
\(202\) 0 0
\(203\) 13.5826 0.953310
\(204\) 0 0
\(205\) 6.79129 0.474324
\(206\) 0 0
\(207\) −4.79129 −0.333018
\(208\) 0 0
\(209\) −4.58258 −0.316983
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 0 0
\(213\) −23.3739 −1.60155
\(214\) 0 0
\(215\) 11.1652 0.761457
\(216\) 0 0
\(217\) 6.04356 0.410264
\(218\) 0 0
\(219\) 35.5826 2.40445
\(220\) 0 0
\(221\) 4.58258 0.308257
\(222\) 0 0
\(223\) 7.16515 0.479814 0.239907 0.970796i \(-0.422883\pi\)
0.239907 + 0.970796i \(0.422883\pi\)
\(224\) 0 0
\(225\) 4.79129 0.319419
\(226\) 0 0
\(227\) 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 \(0.265551\pi\)
\(230\) 0 0
\(231\) 3.95644 0.260315
\(232\) 0 0
\(233\) −1.58258 −0.103678 −0.0518390 0.998655i \(-0.516508\pi\)
−0.0518390 + 0.998655i \(0.516508\pi\)
\(234\) 0 0
\(235\) −4.41742 −0.288161
\(236\) 0 0
\(237\) −22.3303 −1.45051
\(238\) 0 0
\(239\) 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 \(0.857768\pi\)
\(242\) 0 0
\(243\) −16.1652 −1.03699
\(244\) 0 0
\(245\) 3.79129 0.242216
\(246\) 0 0
\(247\) −33.5390 −2.13404
\(248\) 0 0
\(249\) 16.7477 1.06134
\(250\) 0 0
\(251\) −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
\(254\) 0 0
\(255\) −2.20871 −0.138315
\(256\) 0 0
\(257\) −4.74773 −0.296155 −0.148078 0.988976i \(-0.547309\pi\)
−0.148078 + 0.988976i \(0.547309\pi\)
\(258\) 0 0
\(259\) −7.16515 −0.445221
\(260\) 0 0
\(261\) 36.3303 2.24879
\(262\) 0 0
\(263\) −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
\(266\) 0 0
\(267\) 42.3303 2.59057
\(268\) 0 0
\(269\) −10.7477 −0.655300 −0.327650 0.944799i \(-0.606257\pi\)
−0.327650 + 0.944799i \(0.606257\pi\)
\(270\) 0 0
\(271\) −18.1216 −1.10081 −0.550404 0.834898i \(-0.685526\pi\)
−0.550404 + 0.834898i \(0.685526\pi\)
\(272\) 0 0
\(273\) 28.9564 1.75252
\(274\) 0 0
\(275\) 0.791288 0.0477165
\(276\) 0 0
\(277\) 17.1652 1.03135 0.515677 0.856783i \(-0.327540\pi\)
0.515677 + 0.856783i \(0.327540\pi\)
\(278\) 0 0
\(279\) 16.1652 0.967782
\(280\) 0 0
\(281\) 10.7477 0.641156 0.320578 0.947222i \(-0.396123\pi\)
0.320578 + 0.947222i \(0.396123\pi\)
\(282\) 0 0
\(283\) −8.33030 −0.495185 −0.247593 0.968864i \(-0.579639\pi\)
−0.247593 + 0.968864i \(0.579639\pi\)
\(284\) 0 0
\(285\) 16.1652 0.957541
\(286\) 0 0
\(287\) −12.1652 −0.718086
\(288\) 0 0
\(289\) −16.3739 −0.963168
\(290\) 0 0
\(291\) −22.2087 −1.30190
\(292\) 0 0
\(293\) 27.4955 1.60630 0.803151 0.595776i \(-0.203155\pi\)
0.803151 + 0.595776i \(0.203155\pi\)
\(294\) 0 0
\(295\) −13.5826 −0.790808
\(296\) 0 0
\(297\) 3.95644 0.229576
\(298\) 0 0
\(299\) −5.79129 −0.334919
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) 0 0
\(303\) 12.3303 0.708357
\(304\) 0 0
\(305\) −10.3739 −0.594006
\(306\) 0 0
\(307\) 15.5390 0.886858 0.443429 0.896309i \(-0.353762\pi\)
0.443429 + 0.896309i \(0.353762\pi\)
\(308\) 0 0
\(309\) 17.7913 1.01211
\(310\) 0 0
\(311\) −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 \(0.541736\pi\)
\(314\) 0 0
\(315\) −8.58258 −0.483573
\(316\) 0 0
\(317\) 9.79129 0.549934 0.274967 0.961454i \(-0.411333\pi\)
0.274967 + 0.961454i \(0.411333\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) −12.3303 −0.688210
\(322\) 0 0
\(323\) −4.58258 −0.254981
\(324\) 0 0
\(325\) 5.79129 0.321243
\(326\) 0 0
\(327\) −9.41742 −0.520785
\(328\) 0 0
\(329\) 7.91288 0.436251
\(330\) 0 0
\(331\) 20.7477 1.14040 0.570199 0.821507i \(-0.306866\pi\)
0.570199 + 0.821507i \(0.306866\pi\)
\(332\) 0 0
\(333\) −19.1652 −1.05024
\(334\) 0 0
\(335\) 11.1652 0.610017
\(336\) 0 0
\(337\) −12.2087 −0.665051 −0.332525 0.943094i \(-0.607901\pi\)
−0.332525 + 0.943094i \(0.607901\pi\)
\(338\) 0 0
\(339\) 16.7477 0.909612
\(340\) 0 0
\(341\) 2.66970 0.144572
\(342\) 0 0
\(343\) −19.3303 −1.04374
\(344\) 0 0
\(345\) 2.79129 0.150278
\(346\) 0 0
\(347\) −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 \(0.255017\pi\)
\(350\) 0 0
\(351\) 28.9564 1.54558
\(352\) 0 0
\(353\) 3.16515 0.168464 0.0842320 0.996446i \(-0.473156\pi\)
0.0842320 + 0.996446i \(0.473156\pi\)
\(354\) 0 0
\(355\) 8.37386 0.444439
\(356\) 0 0
\(357\) 3.95644 0.209397
\(358\) 0 0
\(359\) −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
\(362\) 0 0
\(363\) −28.9564 −1.51982
\(364\) 0 0
\(365\) −12.7477 −0.667247
\(366\) 0 0
\(367\) 19.1652 1.00041 0.500206 0.865906i \(-0.333257\pi\)
0.500206 + 0.865906i \(0.333257\pi\)
\(368\) 0 0
\(369\) −32.5390 −1.69391
\(370\) 0 0
\(371\) 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 0
\(375\) −2.79129 −0.144141
\(376\) 0 0
\(377\) 43.9129 2.26163
\(378\) 0 0
\(379\) 6.37386 0.327403 0.163702 0.986510i \(-0.447657\pi\)
0.163702 + 0.986510i \(0.447657\pi\)
\(380\) 0 0
\(381\) −35.5826 −1.82295
\(382\) 0 0
\(383\) 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
\(386\) 0 0
\(387\) −53.4955 −2.71933
\(388\) 0 0
\(389\) −20.7042 −1.04974 −0.524871 0.851182i \(-0.675887\pi\)
−0.524871 + 0.851182i \(0.675887\pi\)
\(390\) 0 0
\(391\) −0.791288 −0.0400171
\(392\) 0 0
\(393\) 25.5826 1.29047
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −15.5390 −0.779881 −0.389940 0.920840i \(-0.627504\pi\)
−0.389940 + 0.920840i \(0.627504\pi\)
\(398\) 0 0
\(399\) −28.9564 −1.44964
\(400\) 0 0
\(401\) −4.74773 −0.237090 −0.118545 0.992949i \(-0.537823\pi\)
−0.118545 + 0.992949i \(0.537823\pi\)
\(402\) 0 0
\(403\) 19.5390 0.973308
\(404\) 0 0
\(405\) 0.417424 0.0207420
\(406\) 0 0
\(407\) −3.16515 −0.156891
\(408\) 0 0
\(409\) −18.2087 −0.900363 −0.450181 0.892937i \(-0.648641\pi\)
−0.450181 + 0.892937i \(0.648641\pi\)
\(410\) 0 0
\(411\) 10.5826 0.522000
\(412\) 0 0
\(413\) 24.3303 1.19722
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) −35.5826 −1.74249
\(418\) 0 0
\(419\) −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 \(0.354412\pi\)
\(422\) 0 0
\(423\) 21.1652 1.02908
\(424\) 0 0
\(425\) 0.791288 0.0383831
\(426\) 0 0
\(427\) 18.5826 0.899274
\(428\) 0 0
\(429\) 12.7913 0.617569
\(430\) 0 0
\(431\) −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 \(0.762258\pi\)
\(434\) 0 0
\(435\) −21.1652 −1.01479
\(436\) 0 0
\(437\) 5.79129 0.277035
\(438\) 0 0
\(439\) 6.53901 0.312090 0.156045 0.987750i \(-0.450125\pi\)
0.156045 + 0.987750i \(0.450125\pi\)
\(440\) 0 0
\(441\) −18.1652 −0.865007
\(442\) 0 0
\(443\) 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
\(446\) 0 0
\(447\) 22.9129 1.08374
\(448\) 0 0
\(449\) −16.1216 −0.760825 −0.380412 0.924817i \(-0.624218\pi\)
−0.380412 + 0.924817i \(0.624218\pi\)
\(450\) 0 0
\(451\) −5.37386 −0.253045
\(452\) 0 0
\(453\) 30.1216 1.41524
\(454\) 0 0
\(455\) −10.3739 −0.486334
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) 3.95644 0.184671
\(460\) 0 0
\(461\) −28.7477 −1.33892 −0.669458 0.742850i \(-0.733473\pi\)
−0.669458 + 0.742850i \(0.733473\pi\)
\(462\) 0 0
\(463\) 10.0000 0.464739 0.232370 0.972628i \(-0.425352\pi\)
0.232370 + 0.972628i \(0.425352\pi\)
\(464\) 0 0
\(465\) −9.41742 −0.436723
\(466\) 0 0
\(467\) 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
\(470\) 0 0
\(471\) −41.1652 −1.89679
\(472\) 0 0
\(473\) −8.83485 −0.406227
\(474\) 0 0
\(475\) −5.79129 −0.265723
\(476\) 0 0
\(477\) 28.7477 1.31627
\(478\) 0 0
\(479\) −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
\(482\) 0 0
\(483\) −5.00000 −0.227508
\(484\) 0 0
\(485\) 7.95644 0.361283
\(486\) 0 0
\(487\) −6.41742 −0.290801 −0.145401 0.989373i \(-0.546447\pi\)
−0.145401 + 0.989373i \(0.546447\pi\)
\(488\) 0 0
\(489\) −24.0780 −1.08885
\(490\) 0 0
\(491\) −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
\(494\) 0 0
\(495\) −3.79129 −0.170406
\(496\) 0 0
\(497\) −15.0000 −0.672842
\(498\) 0 0
\(499\) −4.83485 −0.216438 −0.108219 0.994127i \(-0.534515\pi\)
−0.108219 + 0.994127i \(0.534515\pi\)
\(500\) 0 0
\(501\) −51.1652 −2.28589
\(502\) 0 0
\(503\) −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 0
\(507\) 57.3303 2.54613
\(508\) 0 0
\(509\) 34.7477 1.54017 0.770083 0.637944i \(-0.220215\pi\)
0.770083 + 0.637944i \(0.220215\pi\)
\(510\) 0 0
\(511\) 22.8348 1.01015
\(512\) 0 0
\(513\) −28.9564 −1.27846
\(514\) 0 0
\(515\) −6.37386 −0.280866
\(516\) 0 0
\(517\) 3.49545 0.153730
\(518\) 0 0
\(519\) −52.4519 −2.30238
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) −17.1652 −0.750580 −0.375290 0.926908i \(-0.622457\pi\)
−0.375290 + 0.926908i \(0.622457\pi\)
\(524\) 0 0
\(525\) 5.00000 0.218218
\(526\) 0 0
\(527\) 2.66970 0.116294
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 65.0780 2.82415
\(532\) 0 0
\(533\) −39.3303 −1.70358
\(534\) 0 0
\(535\) 4.41742 0.190982
\(536\) 0 0
\(537\) −30.0000 −1.29460
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 1.66970 0.0717859 0.0358929 0.999356i \(-0.488572\pi\)
0.0358929 + 0.999356i \(0.488572\pi\)
\(542\) 0 0
\(543\) −51.7477 −2.22071
\(544\) 0 0
\(545\) 3.37386 0.144520
\(546\) 0 0
\(547\) 26.1216 1.11688 0.558439 0.829545i \(-0.311400\pi\)
0.558439 + 0.829545i \(0.311400\pi\)
\(548\) 0 0
\(549\) 49.7042 2.12132
\(550\) 0 0
\(551\) −43.9129 −1.87075
\(552\) 0 0
\(553\) −14.3303 −0.609386
\(554\) 0 0
\(555\) 11.1652 0.473934
\(556\) 0 0
\(557\) −6.33030 −0.268224 −0.134112 0.990966i \(-0.542818\pi\)
−0.134112 + 0.990966i \(0.542818\pi\)
\(558\) 0 0
\(559\) −64.6606 −2.73485
\(560\) 0 0
\(561\) 1.74773 0.0737891
\(562\) 0 0
\(563\) 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
\(566\) 0 0
\(567\) −0.747727 −0.0314016
\(568\) 0 0
\(569\) −39.4955 −1.65574 −0.827868 0.560923i \(-0.810446\pi\)
−0.827868 + 0.560923i \(0.810446\pi\)
\(570\) 0 0
\(571\) 11.1216 0.465424 0.232712 0.972546i \(-0.425240\pi\)
0.232712 + 0.972546i \(0.425240\pi\)
\(572\) 0 0
\(573\) −71.4083 −2.98313
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 41.1652 1.71373 0.856864 0.515543i \(-0.172410\pi\)
0.856864 + 0.515543i \(0.172410\pi\)
\(578\) 0 0
\(579\) −57.9129 −2.40678
\(580\) 0 0
\(581\) 10.7477 0.445891
\(582\) 0 0
\(583\) 4.74773 0.196631
\(584\) 0 0
\(585\) −27.7477 −1.14723
\(586\) 0 0
\(587\) 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
\(590\) 0 0
\(591\) −32.2087 −1.32489
\(592\) 0 0
\(593\) −31.9129 −1.31050 −0.655252 0.755410i \(-0.727438\pi\)
−0.655252 + 0.755410i \(0.727438\pi\)
\(594\) 0 0
\(595\) −1.41742 −0.0581087
\(596\) 0 0
\(597\) 45.5826 1.86557
\(598\) 0 0
\(599\) −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 \(0.621113\pi\)
\(602\) 0 0
\(603\) −53.4955 −2.17850
\(604\) 0 0
\(605\) 10.3739 0.421758
\(606\) 0 0
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 0 0
\(609\) 37.9129 1.53631
\(610\) 0 0
\(611\) 25.5826 1.03496
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 0 0
\(615\) 18.9564 0.764397
\(616\) 0 0
\(617\) −23.8693 −0.960943 −0.480471 0.877010i \(-0.659534\pi\)
−0.480471 + 0.877010i \(0.659534\pi\)
\(618\) 0 0
\(619\) 1.79129 0.0719979 0.0359990 0.999352i \(-0.488539\pi\)
0.0359990 + 0.999352i \(0.488539\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) 27.1652 1.08835
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −12.7913 −0.510835
\(628\) 0 0
\(629\) −3.16515 −0.126203
\(630\) 0 0
\(631\) −27.9129 −1.11119 −0.555597 0.831452i \(-0.687510\pi\)
−0.555597 + 0.831452i \(0.687510\pi\)
\(632\) 0 0
\(633\) 27.9129 1.10944
\(634\) 0 0
\(635\) 12.7477 0.505878
\(636\) 0 0
\(637\) −21.9564 −0.869946
\(638\) 0 0
\(639\) −40.1216 −1.58719
\(640\) 0 0
\(641\) 15.1652 0.598987 0.299494 0.954098i \(-0.403182\pi\)
0.299494 + 0.954098i \(0.403182\pi\)
\(642\) 0 0
\(643\) −6.74773 −0.266104 −0.133052 0.991109i \(-0.542478\pi\)
−0.133052 + 0.991109i \(0.542478\pi\)
\(644\) 0 0
\(645\) 31.1652 1.22713
\(646\) 0 0
\(647\) −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
\(650\) 0 0
\(651\) 16.8693 0.661161
\(652\) 0 0
\(653\) 3.46099 0.135439 0.0677194 0.997704i \(-0.478428\pi\)
0.0677194 + 0.997704i \(0.478428\pi\)
\(654\) 0 0
\(655\) −9.16515 −0.358112
\(656\) 0 0
\(657\) 61.0780 2.38288
\(658\) 0 0
\(659\) 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 \(0.666068\pi\)
\(662\) 0 0
\(663\) 12.7913 0.496772
\(664\) 0 0
\(665\) 10.3739 0.402281
\(666\) 0 0
\(667\) −7.58258 −0.293599
\(668\) 0 0
\(669\) 20.0000 0.773245
\(670\) 0 0
\(671\) 8.20871 0.316894
\(672\) 0 0
\(673\) 38.0000 1.46479 0.732396 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) 42.6606 1.63958 0.819790 0.572664i \(-0.194090\pi\)
0.819790 + 0.572664i \(0.194090\pi\)
\(678\) 0 0
\(679\) −14.2523 −0.546952
\(680\) 0 0
\(681\) 63.4955 2.43315
\(682\) 0 0
\(683\) 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
\(686\) 0 0
\(687\) 56.7477 2.16506
\(688\) 0 0
\(689\) 34.7477 1.32378
\(690\) 0 0
\(691\) −42.7477 −1.62620 −0.813100 0.582124i \(-0.802222\pi\)
−0.813100 + 0.582124i \(0.802222\pi\)
\(692\) 0 0
\(693\) 6.79129 0.257980
\(694\) 0 0
\(695\) 12.7477 0.483549
\(696\) 0 0
\(697\) −5.37386 −0.203550
\(698\) 0 0
\(699\) −4.41742 −0.167082
\(700\) 0 0
\(701\) 23.3739 0.882819 0.441409 0.897306i \(-0.354479\pi\)
0.441409 + 0.897306i \(0.354479\pi\)
\(702\) 0 0
\(703\) 23.1652 0.873690
\(704\) 0 0
\(705\) −12.3303 −0.464386
\(706\) 0 0
\(707\) 7.91288 0.297594
\(708\) 0 0
\(709\) 2.46099 0.0924242 0.0462121 0.998932i \(-0.485285\pi\)
0.0462121 + 0.998932i \(0.485285\pi\)
\(710\) 0 0
\(711\) −38.3303 −1.43750
\(712\) 0 0
\(713\) −3.37386 −0.126352
\(714\) 0 0
\(715\) −4.58258 −0.171379
\(716\) 0 0
\(717\) 42.3303 1.58085
\(718\) 0 0
\(719\) −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
\(722\) 0 0
\(723\) −78.1561 −2.90666
\(724\) 0 0
\(725\) 7.58258 0.281610
\(726\) 0 0
\(727\) −39.1216 −1.45094 −0.725470 0.688254i \(-0.758377\pi\)
−0.725470 + 0.688254i \(0.758377\pi\)
\(728\) 0 0
\(729\) −43.8693 −1.62479
\(730\) 0 0
\(731\) −8.83485 −0.326769
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 0 0
\(735\) 10.5826 0.390344
\(736\) 0 0
\(737\) −8.83485 −0.325436
\(738\) 0 0
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 0 0
\(741\) −93.6170 −3.43911
\(742\) 0 0
\(743\) 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
\(746\) 0 0
\(747\) 28.7477 1.05182
\(748\) 0 0
\(749\) −7.91288 −0.289130
\(750\) 0 0
\(751\) 8.74773 0.319209 0.159605 0.987181i \(-0.448978\pi\)
0.159605 + 0.987181i \(0.448978\pi\)
\(752\) 0 0
\(753\) −73.1561 −2.66596
\(754\) 0 0
\(755\) −10.7913 −0.392735
\(756\) 0 0
\(757\) −10.3303 −0.375461 −0.187731 0.982221i \(-0.560113\pi\)
−0.187731 + 0.982221i \(0.560113\pi\)
\(758\) 0 0
\(759\) −2.20871 −0.0801712
\(760\) 0 0
\(761\) −11.0436 −0.400329 −0.200164 0.979762i \(-0.564148\pi\)
−0.200164 + 0.979762i \(0.564148\pi\)
\(762\) 0 0
\(763\) −6.04356 −0.218792
\(764\) 0 0
\(765\) −3.79129 −0.137074
\(766\) 0 0
\(767\) 78.6606 2.84027
\(768\) 0 0
\(769\) −40.3303 −1.45435 −0.727174 0.686453i \(-0.759167\pi\)
−0.727174 + 0.686453i \(0.759167\pi\)
\(770\) 0 0
\(771\) −13.2523 −0.477269
\(772\) 0 0
\(773\) 33.4955 1.20475 0.602374 0.798214i \(-0.294222\pi\)
0.602374 + 0.798214i \(0.294222\pi\)
\(774\) 0 0
\(775\) 3.37386 0.121193
\(776\) 0 0
\(777\) −20.0000 −0.717496
\(778\) 0 0
\(779\) 39.3303 1.40915
\(780\) 0 0
\(781\) −6.62614 −0.237102
\(782\) 0 0
\(783\) 37.9129 1.35490
\(784\) 0 0
\(785\) 14.7477 0.526369
\(786\) 0 0
\(787\) 17.5826 0.626751 0.313376 0.949629i \(-0.398540\pi\)
0.313376 + 0.949629i \(0.398540\pi\)
\(788\) 0 0
\(789\) −31.2867 −1.11384
\(790\) 0 0
\(791\) 10.7477 0.382145
\(792\) 0 0
\(793\) 60.0780 2.13343
\(794\) 0 0
\(795\) −16.7477 −0.593981
\(796\) 0 0
\(797\) −4.08712 −0.144773 −0.0723866 0.997377i \(-0.523062\pi\)
−0.0723866 + 0.997377i \(0.523062\pi\)
\(798\) 0 0
\(799\) 3.49545 0.123660
\(800\) 0 0
\(801\) 72.6606 2.56734
\(802\) 0 0
\(803\) 10.0871 0.355967
\(804\) 0 0
\(805\) 1.79129 0.0631346
\(806\) 0 0
\(807\) −30.0000 −1.05605
\(808\) 0 0
\(809\) 33.9564 1.19384 0.596922 0.802299i \(-0.296390\pi\)
0.596922 + 0.802299i \(0.296390\pi\)
\(810\) 0 0
\(811\) 2.08712 0.0732887 0.0366444 0.999328i \(-0.488333\pi\)
0.0366444 + 0.999328i \(0.488333\pi\)
\(812\) 0 0
\(813\) −50.5826 −1.77401
\(814\) 0 0
\(815\) 8.62614 0.302160
\(816\) 0 0
\(817\) 64.6606 2.26219
\(818\) 0 0
\(819\) 49.7042 1.73680
\(820\) 0 0
\(821\) 21.1652 0.738669 0.369334 0.929297i \(-0.379586\pi\)
0.369334 + 0.929297i \(0.379586\pi\)
\(822\) 0 0
\(823\) −22.8348 −0.795973 −0.397986 0.917391i \(-0.630291\pi\)
−0.397986 + 0.917391i \(0.630291\pi\)
\(824\) 0 0
\(825\) 2.20871 0.0768975
\(826\) 0 0
\(827\) 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 \(0.366221\pi\)
\(830\) 0 0
\(831\) 47.9129 1.66208
\(832\) 0 0
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) 18.3303 0.634346
\(836\) 0 0
\(837\) 16.8693 0.583089
\(838\) 0 0
\(839\) 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
\(842\) 0 0
\(843\) 30.0000 1.03325
\(844\) 0 0
\(845\) −20.5390 −0.706564
\(846\) 0 0
\(847\) −18.5826 −0.638505
\(848\) 0 0
\(849\) −23.2523 −0.798016
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) 40.5390 1.38803 0.694015 0.719961i \(-0.255840\pi\)
0.694015 + 0.719961i \(0.255840\pi\)
\(854\) 0 0
\(855\) 27.7477 0.948952
\(856\) 0 0
\(857\) −9.16515 −0.313076 −0.156538 0.987672i \(-0.550033\pi\)
−0.156538 + 0.987672i \(0.550033\pi\)
\(858\) 0 0
\(859\) 26.7477 0.912621 0.456310 0.889821i \(-0.349171\pi\)
0.456310 + 0.889821i \(0.349171\pi\)
\(860\) 0 0
\(861\) −33.9564 −1.15723
\(862\) 0 0
\(863\) 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
\(866\) 0 0
\(867\) −45.7042 −1.55219
\(868\) 0 0
\(869\) −6.33030 −0.214741
\(870\) 0 0
\(871\) −64.6606 −2.19094
\(872\) 0 0
\(873\) −38.1216 −1.29022
\(874\) 0 0
\(875\) −1.79129 −0.0605566
\(876\) 0 0
\(877\) −42.7042 −1.44202 −0.721009 0.692926i \(-0.756321\pi\)
−0.721009 + 0.692926i \(0.756321\pi\)
\(878\) 0 0
\(879\) 76.7477 2.58864
\(880\) 0 0
\(881\) 30.3303 1.02185 0.510927 0.859624i \(-0.329302\pi\)
0.510927 + 0.859624i \(0.329302\pi\)
\(882\) 0 0
\(883\) 34.9564 1.17638 0.588189 0.808724i \(-0.299841\pi\)
0.588189 + 0.808724i \(0.299841\pi\)
\(884\) 0 0
\(885\) −37.9129 −1.27443
\(886\) 0 0
\(887\) −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
\(890\) 0 0
\(891\) −0.330303 −0.0110656
\(892\) 0 0
\(893\) −25.5826 −0.856088
\(894\) 0 0
\(895\) 10.7477 0.359257
\(896\) 0 0
\(897\) −16.1652 −0.539739
\(898\) 0 0
\(899\) 25.5826 0.853227
\(900\) 0 0
\(901\) 4.74773 0.158170
\(902\) 0 0
\(903\) −55.8258 −1.85776
\(904\) 0 0
\(905\) 18.5390 0.616258
\(906\) 0 0
\(907\) −6.74773 −0.224055 −0.112027 0.993705i \(-0.535734\pi\)
−0.112027 + 0.993705i \(0.535734\pi\)
\(908\) 0 0
\(909\) 21.1652 0.702004
\(910\) 0 0
\(911\) −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
\(914\) 0 0
\(915\) −28.9564 −0.957270
\(916\) 0 0
\(917\) 16.4174 0.542151
\(918\) 0 0
\(919\) 55.1652 1.81973 0.909865 0.414904i \(-0.136185\pi\)
0.909865 + 0.414904i \(0.136185\pi\)
\(920\) 0 0
\(921\) 43.3739 1.42922
\(922\) 0 0
\(923\) −48.4955 −1.59625
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) 30.5390 1.00303
\(928\) 0 0
\(929\) 15.4955 0.508389 0.254195 0.967153i \(-0.418190\pi\)
0.254195 + 0.967153i \(0.418190\pi\)
\(930\) 0 0
\(931\) 21.9564 0.719593
\(932\) 0 0
\(933\) −33.4955 −1.09659
\(934\) 0 0
\(935\) −0.626136 −0.0204769
\(936\) 0 0
\(937\) 44.6261 1.45787 0.728936 0.684582i \(-0.240015\pi\)
0.728936 + 0.684582i \(0.240015\pi\)
\(938\) 0 0
\(939\) −12.9129 −0.421396
\(940\) 0 0
\(941\) 32.0436 1.04459 0.522295 0.852765i \(-0.325076\pi\)
0.522295 + 0.852765i \(0.325076\pi\)
\(942\) 0 0
\(943\) 6.79129 0.221155
\(944\) 0 0
\(945\) −8.95644 −0.291353
\(946\) 0 0
\(947\) −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
\(950\) 0 0
\(951\) 27.3303 0.886246
\(952\) 0 0
\(953\) 5.53901 0.179426 0.0897131 0.995968i \(-0.471405\pi\)
0.0897131 + 0.995968i \(0.471405\pi\)
\(954\) 0 0
\(955\) 25.5826 0.827833
\(956\) 0 0
\(957\) 16.7477 0.541377
\(958\) 0 0
\(959\) 6.79129 0.219302
\(960\) 0 0
\(961\) −19.6170 −0.632808
\(962\) 0 0
\(963\) −21.1652 −0.682037
\(964\) 0 0
\(965\) 20.7477 0.667893
\(966\) 0 0
\(967\) 32.7477 1.05310 0.526548 0.850145i \(-0.323486\pi\)
0.526548 + 0.850145i \(0.323486\pi\)
\(968\) 0 0
\(969\) −12.7913 −0.410915
\(970\) 0 0
\(971\) −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
\(974\) 0 0
\(975\) 16.1652 0.517699
\(976\) 0 0
\(977\) −34.1216 −1.09165 −0.545823 0.837900i \(-0.683783\pi\)
−0.545823 + 0.837900i \(0.683783\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) −16.1652 −0.516114
\(982\) 0 0
\(983\) 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
\(986\) 0 0
\(987\) 22.0871 0.703041
\(988\) 0 0
\(989\) 11.1652 0.355031
\(990\) 0 0
\(991\) 33.2087 1.05491 0.527455 0.849583i \(-0.323146\pi\)
0.527455 + 0.849583i \(0.323146\pi\)
\(992\) 0 0
\(993\) 57.9129 1.83781
\(994\) 0 0
\(995\) −16.3303 −0.517705
\(996\) 0 0
\(997\) −43.4955 −1.37751 −0.688757 0.724992i \(-0.741843\pi\)
−0.688757 + 0.724992i \(0.741843\pi\)
\(998\) 0 0
\(999\) −20.0000 −0.632772
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 1840.2.a.n.1.2 2
4.3 odd 2 230.2.a.a.1.1 2
5.4 even 2 9200.2.a.bs.1.1 2
8.3 odd 2 7360.2.a.bq.1.2 2
8.5 even 2 7360.2.a.bk.1.1 2
12.11 even 2 2070.2.a.x.1.1 2
20.3 even 4 1150.2.b.g.599.3 4
20.7 even 4 1150.2.b.g.599.2 4
20.19 odd 2 1150.2.a.o.1.2 2
92.91 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 4.3 odd 2
1150.2.a.o.1.2 2 20.19 odd 2
1150.2.b.g.599.2 4 20.7 even 4
1150.2.b.g.599.3 4 20.3 even 4
1840.2.a.n.1.2 2 1.1 even 1 trivial
2070.2.a.x.1.1 2 12.11 even 2
5290.2.a.e.1.1 2 92.91 even 2
7360.2.a.bk.1.1 2 8.5 even 2
7360.2.a.bq.1.2 2 8.3 odd 2
9200.2.a.bs.1.1 2 5.4 even 2