Properties

Label 5290.2.a.e.1.1
Level $5290$
Weight $2$
Character 5290.1
Self dual yes
Analytic conductor $42.241$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5290 = 2 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5290.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.2408626693\)
Analytic rank: \(1\)
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 1.1
Root \(2.79129\) of defining polynomial
Character \(\chi\) \(=\) 5290.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.79129 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.79129 q^{6} +1.79129 q^{7} -1.00000 q^{8} +4.79129 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.79129 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.79129 q^{6} +1.79129 q^{7} -1.00000 q^{8} +4.79129 q^{9} -1.00000 q^{10} +0.791288 q^{11} -2.79129 q^{12} +5.79129 q^{13} -1.79129 q^{14} -2.79129 q^{15} +1.00000 q^{16} -0.791288 q^{17} -4.79129 q^{18} -5.79129 q^{19} +1.00000 q^{20} -5.00000 q^{21} -0.791288 q^{22} +2.79129 q^{24} +1.00000 q^{25} -5.79129 q^{26} -5.00000 q^{27} +1.79129 q^{28} +7.58258 q^{29} +2.79129 q^{30} -3.37386 q^{31} -1.00000 q^{32} -2.20871 q^{33} +0.791288 q^{34} +1.79129 q^{35} +4.79129 q^{36} +4.00000 q^{37} +5.79129 q^{38} -16.1652 q^{39} -1.00000 q^{40} -6.79129 q^{41} +5.00000 q^{42} -11.1652 q^{43} +0.791288 q^{44} +4.79129 q^{45} -4.41742 q^{47} -2.79129 q^{48} -3.79129 q^{49} -1.00000 q^{50} +2.20871 q^{51} +5.79129 q^{52} -6.00000 q^{53} +5.00000 q^{54} +0.791288 q^{55} -1.79129 q^{56} +16.1652 q^{57} -7.58258 q^{58} -13.5826 q^{59} -2.79129 q^{60} -10.3739 q^{61} +3.37386 q^{62} +8.58258 q^{63} +1.00000 q^{64} +5.79129 q^{65} +2.20871 q^{66} -11.1652 q^{67} -0.791288 q^{68} -1.79129 q^{70} +8.37386 q^{71} -4.79129 q^{72} +12.7477 q^{73} -4.00000 q^{74} -2.79129 q^{75} -5.79129 q^{76} +1.41742 q^{77} +16.1652 q^{78} -8.00000 q^{79} +1.00000 q^{80} -0.417424 q^{81} +6.79129 q^{82} +6.00000 q^{83} -5.00000 q^{84} -0.791288 q^{85} +11.1652 q^{86} -21.1652 q^{87} -0.791288 q^{88} -15.1652 q^{89} -4.79129 q^{90} +10.3739 q^{91} +9.41742 q^{93} +4.41742 q^{94} -5.79129 q^{95} +2.79129 q^{96} +7.95644 q^{97} +3.79129 q^{98} +3.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - q^{3} + 2q^{4} + 2q^{5} + q^{6} - q^{7} - 2q^{8} + 5q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - q^{3} + 2q^{4} + 2q^{5} + q^{6} - q^{7} - 2q^{8} + 5q^{9} - 2q^{10} - 3q^{11} - q^{12} + 7q^{13} + q^{14} - q^{15} + 2q^{16} + 3q^{17} - 5q^{18} - 7q^{19} + 2q^{20} - 10q^{21} + 3q^{22} + q^{24} + 2q^{25} - 7q^{26} - 10q^{27} - q^{28} + 6q^{29} + q^{30} + 7q^{31} - 2q^{32} - 9q^{33} - 3q^{34} - q^{35} + 5q^{36} + 8q^{37} + 7q^{38} - 14q^{39} - 2q^{40} - 9q^{41} + 10q^{42} - 4q^{43} - 3q^{44} + 5q^{45} - 18q^{47} - q^{48} - 3q^{49} - 2q^{50} + 9q^{51} + 7q^{52} - 12q^{53} + 10q^{54} - 3q^{55} + q^{56} + 14q^{57} - 6q^{58} - 18q^{59} - q^{60} - 7q^{61} - 7q^{62} + 8q^{63} + 2q^{64} + 7q^{65} + 9q^{66} - 4q^{67} + 3q^{68} + q^{70} + 3q^{71} - 5q^{72} - 2q^{73} - 8q^{74} - q^{75} - 7q^{76} + 12q^{77} + 14q^{78} - 16q^{79} + 2q^{80} - 10q^{81} + 9q^{82} + 12q^{83} - 10q^{84} + 3q^{85} + 4q^{86} - 24q^{87} + 3q^{88} - 12q^{89} - 5q^{90} + 7q^{91} + 28q^{93} + 18q^{94} - 7q^{95} + q^{96} - 7q^{97} + 3q^{98} + 3q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.79129 −1.61155 −0.805775 0.592221i \(-0.798251\pi\)
−0.805775 + 0.592221i \(0.798251\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.79129 1.13954
\(7\) 1.79129 0.677043 0.338522 0.940959i \(-0.390073\pi\)
0.338522 + 0.940959i \(0.390073\pi\)
\(8\) −1.00000 −0.353553
\(9\) 4.79129 1.59710
\(10\) −1.00000 −0.316228
\(11\) 0.791288 0.238582 0.119291 0.992859i \(-0.461938\pi\)
0.119291 + 0.992859i \(0.461938\pi\)
\(12\) −2.79129 −0.805775
\(13\) 5.79129 1.60621 0.803107 0.595835i \(-0.203179\pi\)
0.803107 + 0.595835i \(0.203179\pi\)
\(14\) −1.79129 −0.478742
\(15\) −2.79129 −0.720707
\(16\) 1.00000 0.250000
\(17\) −0.791288 −0.191915 −0.0959577 0.995385i \(-0.530591\pi\)
−0.0959577 + 0.995385i \(0.530591\pi\)
\(18\) −4.79129 −1.12932
\(19\) −5.79129 −1.32861 −0.664306 0.747460i \(-0.731273\pi\)
−0.664306 + 0.747460i \(0.731273\pi\)
\(20\) 1.00000 0.223607
\(21\) −5.00000 −1.09109
\(22\) −0.791288 −0.168703
\(23\) 0 0
\(24\) 2.79129 0.569769
\(25\) 1.00000 0.200000
\(26\) −5.79129 −1.13576
\(27\) −5.00000 −0.962250
\(28\) 1.79129 0.338522
\(29\) 7.58258 1.40805 0.704024 0.710176i \(-0.251385\pi\)
0.704024 + 0.710176i \(0.251385\pi\)
\(30\) 2.79129 0.509617
\(31\) −3.37386 −0.605964 −0.302982 0.952996i \(-0.597982\pi\)
−0.302982 + 0.952996i \(0.597982\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.20871 −0.384487
\(34\) 0.791288 0.135705
\(35\) 1.79129 0.302783
\(36\) 4.79129 0.798548
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 5.79129 0.939471
\(39\) −16.1652 −2.58850
\(40\) −1.00000 −0.158114
\(41\) −6.79129 −1.06062 −0.530310 0.847804i \(-0.677925\pi\)
−0.530310 + 0.847804i \(0.677925\pi\)
\(42\) 5.00000 0.771517
\(43\) −11.1652 −1.70267 −0.851335 0.524623i \(-0.824206\pi\)
−0.851335 + 0.524623i \(0.824206\pi\)
\(44\) 0.791288 0.119291
\(45\) 4.79129 0.714243
\(46\) 0 0
\(47\) −4.41742 −0.644348 −0.322174 0.946681i \(-0.604414\pi\)
−0.322174 + 0.946681i \(0.604414\pi\)
\(48\) −2.79129 −0.402888
\(49\) −3.79129 −0.541613
\(50\) −1.00000 −0.141421
\(51\) 2.20871 0.309282
\(52\) 5.79129 0.803107
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 5.00000 0.680414
\(55\) 0.791288 0.106697
\(56\) −1.79129 −0.239371
\(57\) 16.1652 2.14113
\(58\) −7.58258 −0.995641
\(59\) −13.5826 −1.76830 −0.884150 0.467202i \(-0.845262\pi\)
−0.884150 + 0.467202i \(0.845262\pi\)
\(60\) −2.79129 −0.360354
\(61\) −10.3739 −1.32824 −0.664119 0.747627i \(-0.731193\pi\)
−0.664119 + 0.747627i \(0.731193\pi\)
\(62\) 3.37386 0.428481
\(63\) 8.58258 1.08130
\(64\) 1.00000 0.125000
\(65\) 5.79129 0.718321
\(66\) 2.20871 0.271874
\(67\) −11.1652 −1.36404 −0.682020 0.731333i \(-0.738898\pi\)
−0.682020 + 0.731333i \(0.738898\pi\)
\(68\) −0.791288 −0.0959577
\(69\) 0 0
\(70\) −1.79129 −0.214100
\(71\) 8.37386 0.993795 0.496897 0.867809i \(-0.334473\pi\)
0.496897 + 0.867809i \(0.334473\pi\)
\(72\) −4.79129 −0.564659
\(73\) 12.7477 1.49201 0.746004 0.665941i \(-0.231970\pi\)
0.746004 + 0.665941i \(0.231970\pi\)
\(74\) −4.00000 −0.464991
\(75\) −2.79129 −0.322310
\(76\) −5.79129 −0.664306
\(77\) 1.41742 0.161530
\(78\) 16.1652 1.83034
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) −0.417424 −0.0463805
\(82\) 6.79129 0.749972
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −5.00000 −0.545545
\(85\) −0.791288 −0.0858272
\(86\) 11.1652 1.20397
\(87\) −21.1652 −2.26914
\(88\) −0.791288 −0.0843516
\(89\) −15.1652 −1.60750 −0.803751 0.594965i \(-0.797166\pi\)
−0.803751 + 0.594965i \(0.797166\pi\)
\(90\) −4.79129 −0.505046
\(91\) 10.3739 1.08748
\(92\) 0 0
\(93\) 9.41742 0.976541
\(94\) 4.41742 0.455623
\(95\) −5.79129 −0.594174
\(96\) 2.79129 0.284885
\(97\) 7.95644 0.807854 0.403927 0.914791i \(-0.367645\pi\)
0.403927 + 0.914791i \(0.367645\pi\)
\(98\) 3.79129 0.382978
\(99\) 3.79129 0.381039
\(100\) 1.00000 0.100000
\(101\) 4.41742 0.439550 0.219775 0.975551i \(-0.429468\pi\)
0.219775 + 0.975551i \(0.429468\pi\)
\(102\) −2.20871 −0.218695
\(103\) 6.37386 0.628035 0.314018 0.949417i \(-0.398325\pi\)
0.314018 + 0.949417i \(0.398325\pi\)
\(104\) −5.79129 −0.567882
\(105\) −5.00000 −0.487950
\(106\) 6.00000 0.582772
\(107\) −4.41742 −0.427049 −0.213524 0.976938i \(-0.568494\pi\)
−0.213524 + 0.976938i \(0.568494\pi\)
\(108\) −5.00000 −0.481125
\(109\) 3.37386 0.323158 0.161579 0.986860i \(-0.448341\pi\)
0.161579 + 0.986860i \(0.448341\pi\)
\(110\) −0.791288 −0.0754463
\(111\) −11.1652 −1.05975
\(112\) 1.79129 0.169261
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −16.1652 −1.51401
\(115\) 0 0
\(116\) 7.58258 0.704024
\(117\) 27.7477 2.56528
\(118\) 13.5826 1.25038
\(119\) −1.41742 −0.129935
\(120\) 2.79129 0.254809
\(121\) −10.3739 −0.943079
\(122\) 10.3739 0.939205
\(123\) 18.9564 1.70924
\(124\) −3.37386 −0.302982
\(125\) 1.00000 0.0894427
\(126\) −8.58258 −0.764597
\(127\) 12.7477 1.13118 0.565589 0.824687i \(-0.308649\pi\)
0.565589 + 0.824687i \(0.308649\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 31.1652 2.74394
\(130\) −5.79129 −0.507930
\(131\) −9.16515 −0.800763 −0.400381 0.916349i \(-0.631122\pi\)
−0.400381 + 0.916349i \(0.631122\pi\)
\(132\) −2.20871 −0.192244
\(133\) −10.3739 −0.899528
\(134\) 11.1652 0.964522
\(135\) −5.00000 −0.430331
\(136\) 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
\(141\) 12.3303 1.03840
\(142\) −8.37386 −0.702719
\(143\) 4.58258 0.383214
\(144\) 4.79129 0.399274
\(145\) 7.58258 0.629699
\(146\) −12.7477 −1.05501
\(147\) 10.5826 0.872836
\(148\) 4.00000 0.328798
\(149\) −8.20871 −0.672484 −0.336242 0.941776i \(-0.609156\pi\)
−0.336242 + 0.941776i \(0.609156\pi\)
\(150\) 2.79129 0.227908
\(151\) −10.7913 −0.878183 −0.439091 0.898442i \(-0.644700\pi\)
−0.439091 + 0.898442i \(0.644700\pi\)
\(152\) 5.79129 0.469735
\(153\) −3.79129 −0.306507
\(154\) −1.41742 −0.114219
\(155\) −3.37386 −0.270995
\(156\) −16.1652 −1.29425
\(157\) 14.7477 1.17700 0.588498 0.808498i \(-0.299719\pi\)
0.588498 + 0.808498i \(0.299719\pi\)
\(158\) 8.00000 0.636446
\(159\) 16.7477 1.32818
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0.417424 0.0327960
\(163\) 8.62614 0.675651 0.337826 0.941209i \(-0.390309\pi\)
0.337826 + 0.941209i \(0.390309\pi\)
\(164\) −6.79129 −0.530310
\(165\) −2.20871 −0.171948
\(166\) −6.00000 −0.465690
\(167\) 18.3303 1.41844 0.709221 0.704987i \(-0.249047\pi\)
0.709221 + 0.704987i \(0.249047\pi\)
\(168\) 5.00000 0.385758
\(169\) 20.5390 1.57992
\(170\) 0.791288 0.0606890
\(171\) −27.7477 −2.12192
\(172\) −11.1652 −0.851335
\(173\) −18.7913 −1.42868 −0.714338 0.699801i \(-0.753272\pi\)
−0.714338 + 0.699801i \(0.753272\pi\)
\(174\) 21.1652 1.60453
\(175\) 1.79129 0.135409
\(176\) 0.791288 0.0596456
\(177\) 37.9129 2.84971
\(178\) 15.1652 1.13668
\(179\) 10.7477 0.803323 0.401661 0.915788i \(-0.368433\pi\)
0.401661 + 0.915788i \(0.368433\pi\)
\(180\) 4.79129 0.357122
\(181\) 18.5390 1.37799 0.688997 0.724764i \(-0.258051\pi\)
0.688997 + 0.724764i \(0.258051\pi\)
\(182\) −10.3739 −0.768962
\(183\) 28.9564 2.14052
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) −9.41742 −0.690519
\(187\) −0.626136 −0.0457876
\(188\) −4.41742 −0.322174
\(189\) −8.95644 −0.651485
\(190\) 5.79129 0.420144
\(191\) −25.5826 −1.85109 −0.925545 0.378637i \(-0.876393\pi\)
−0.925545 + 0.378637i \(0.876393\pi\)
\(192\) −2.79129 −0.201444
\(193\) −20.7477 −1.49345 −0.746727 0.665131i \(-0.768376\pi\)
−0.746727 + 0.665131i \(0.768376\pi\)
\(194\) −7.95644 −0.571239
\(195\) −16.1652 −1.15761
\(196\) −3.79129 −0.270806
\(197\) −11.5390 −0.822121 −0.411060 0.911608i \(-0.634841\pi\)
−0.411060 + 0.911608i \(0.634841\pi\)
\(198\) −3.79129 −0.269435
\(199\) 16.3303 1.15762 0.578812 0.815461i \(-0.303516\pi\)
0.578812 + 0.815461i \(0.303516\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 31.1652 2.19822
\(202\) −4.41742 −0.310809
\(203\) 13.5826 0.953310
\(204\) 2.20871 0.154641
\(205\) −6.79129 −0.474324
\(206\) −6.37386 −0.444088
\(207\) 0 0
\(208\) 5.79129 0.401554
\(209\) −4.58258 −0.316983
\(210\) 5.00000 0.345033
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) −6.00000 −0.412082
\(213\) −23.3739 −1.60155
\(214\) 4.41742 0.301969
\(215\) −11.1652 −0.761457
\(216\) 5.00000 0.340207
\(217\) −6.04356 −0.410264
\(218\) −3.37386 −0.228507
\(219\) −35.5826 −2.40445
\(220\) 0.791288 0.0533486
\(221\) −4.58258 −0.308257
\(222\) 11.1652 0.749356
\(223\) −7.16515 −0.479814 −0.239907 0.970796i \(-0.577117\pi\)
−0.239907 + 0.970796i \(0.577117\pi\)
\(224\) −1.79129 −0.119685
\(225\) 4.79129 0.319419
\(226\) 6.00000 0.399114
\(227\) 22.7477 1.50982 0.754910 0.655829i \(-0.227681\pi\)
0.754910 + 0.655829i \(0.227681\pi\)
\(228\) 16.1652 1.07056
\(229\) −20.3303 −1.34346 −0.671732 0.740794i \(-0.734449\pi\)
−0.671732 + 0.740794i \(0.734449\pi\)
\(230\) 0 0
\(231\) −3.95644 −0.260315
\(232\) −7.58258 −0.497820
\(233\) −1.58258 −0.103678 −0.0518390 0.998655i \(-0.516508\pi\)
−0.0518390 + 0.998655i \(0.516508\pi\)
\(234\) −27.7477 −1.81393
\(235\) −4.41742 −0.288161
\(236\) −13.5826 −0.884150
\(237\) 22.3303 1.45051
\(238\) 1.41742 0.0918780
\(239\) −15.1652 −0.980952 −0.490476 0.871455i \(-0.663177\pi\)
−0.490476 + 0.871455i \(0.663177\pi\)
\(240\) −2.79129 −0.180177
\(241\) 28.0000 1.80364 0.901819 0.432113i \(-0.142232\pi\)
0.901819 + 0.432113i \(0.142232\pi\)
\(242\) 10.3739 0.666857
\(243\) 16.1652 1.03699
\(244\) −10.3739 −0.664119
\(245\) −3.79129 −0.242216
\(246\) −18.9564 −1.20862
\(247\) −33.5390 −2.13404
\(248\) 3.37386 0.214241
\(249\) −16.7477 −1.06134
\(250\) −1.00000 −0.0632456
\(251\) −26.2087 −1.65428 −0.827140 0.561996i \(-0.810033\pi\)
−0.827140 + 0.561996i \(0.810033\pi\)
\(252\) 8.58258 0.540651
\(253\) 0 0
\(254\) −12.7477 −0.799864
\(255\) 2.20871 0.138315
\(256\) 1.00000 0.0625000
\(257\) −4.74773 −0.296155 −0.148078 0.988976i \(-0.547309\pi\)
−0.148078 + 0.988976i \(0.547309\pi\)
\(258\) −31.1652 −1.94026
\(259\) 7.16515 0.445221
\(260\) 5.79129 0.359160
\(261\) 36.3303 2.24879
\(262\) 9.16515 0.566225
\(263\) −11.2087 −0.691159 −0.345579 0.938390i \(-0.612318\pi\)
−0.345579 + 0.938390i \(0.612318\pi\)
\(264\) 2.20871 0.135937
\(265\) −6.00000 −0.368577
\(266\) 10.3739 0.636062
\(267\) 42.3303 2.59057
\(268\) −11.1652 −0.682020
\(269\) −10.7477 −0.655300 −0.327650 0.944799i \(-0.606257\pi\)
−0.327650 + 0.944799i \(0.606257\pi\)
\(270\) 5.00000 0.304290
\(271\) 18.1216 1.10081 0.550404 0.834898i \(-0.314474\pi\)
0.550404 + 0.834898i \(0.314474\pi\)
\(272\) −0.791288 −0.0479789
\(273\) −28.9564 −1.75252
\(274\) 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
\(279\) −16.1652 −0.967782
\(280\) −1.79129 −0.107050
\(281\) −10.7477 −0.641156 −0.320578 0.947222i \(-0.603877\pi\)
−0.320578 + 0.947222i \(0.603877\pi\)
\(282\) −12.3303 −0.734259
\(283\) −8.33030 −0.495185 −0.247593 0.968864i \(-0.579639\pi\)
−0.247593 + 0.968864i \(0.579639\pi\)
\(284\) 8.37386 0.496897
\(285\) 16.1652 0.957541
\(286\) −4.58258 −0.270973
\(287\) −12.1652 −0.718086
\(288\) −4.79129 −0.282329
\(289\) −16.3739 −0.963168
\(290\) −7.58258 −0.445264
\(291\) −22.2087 −1.30190
\(292\) 12.7477 0.746004
\(293\) −27.4955 −1.60630 −0.803151 0.595776i \(-0.796845\pi\)
−0.803151 + 0.595776i \(0.796845\pi\)
\(294\) −10.5826 −0.617188
\(295\) −13.5826 −0.790808
\(296\) −4.00000 −0.232495
\(297\) −3.95644 −0.229576
\(298\) 8.20871 0.475518
\(299\) 0 0
\(300\) −2.79129 −0.161155
\(301\) −20.0000 −1.15278
\(302\) 10.7913 0.620969
\(303\) −12.3303 −0.708357
\(304\) −5.79129 −0.332153
\(305\) −10.3739 −0.594006
\(306\) 3.79129 0.216734
\(307\) −15.5390 −0.886858 −0.443429 0.896309i \(-0.646238\pi\)
−0.443429 + 0.896309i \(0.646238\pi\)
\(308\) 1.41742 0.0807652
\(309\) −17.7913 −1.01211
\(310\) 3.37386 0.191623
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 16.1652 0.915171
\(313\) 4.62614 0.261485 0.130742 0.991416i \(-0.458264\pi\)
0.130742 + 0.991416i \(0.458264\pi\)
\(314\) −14.7477 −0.832262
\(315\) 8.58258 0.483573
\(316\) −8.00000 −0.450035
\(317\) 9.79129 0.549934 0.274967 0.961454i \(-0.411333\pi\)
0.274967 + 0.961454i \(0.411333\pi\)
\(318\) −16.7477 −0.939166
\(319\) 6.00000 0.335936
\(320\) 1.00000 0.0559017
\(321\) 12.3303 0.688210
\(322\) 0 0
\(323\) 4.58258 0.254981
\(324\) −0.417424 −0.0231902
\(325\) 5.79129 0.321243
\(326\) −8.62614 −0.477758
\(327\) −9.41742 −0.520785
\(328\) 6.79129 0.374986
\(329\) −7.91288 −0.436251
\(330\) 2.20871 0.121586
\(331\) −20.7477 −1.14040 −0.570199 0.821507i \(-0.693134\pi\)
−0.570199 + 0.821507i \(0.693134\pi\)
\(332\) 6.00000 0.329293
\(333\) 19.1652 1.05024
\(334\) −18.3303 −1.00299
\(335\) −11.1652 −0.610017
\(336\) −5.00000 −0.272772
\(337\) 12.2087 0.665051 0.332525 0.943094i \(-0.392099\pi\)
0.332525 + 0.943094i \(0.392099\pi\)
\(338\) −20.5390 −1.11718
\(339\) 16.7477 0.909612
\(340\) −0.791288 −0.0429136
\(341\) −2.66970 −0.144572
\(342\) 27.7477 1.50043
\(343\) −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 \(-0.455351\pi\)
0.139809 + 0.990178i \(0.455351\pi\)
\(348\) −21.1652 −1.13457
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) −1.79129 −0.0957484
\(351\) −28.9564 −1.54558
\(352\) −0.791288 −0.0421758
\(353\) 3.16515 0.168464 0.0842320 0.996446i \(-0.473156\pi\)
0.0842320 + 0.996446i \(0.473156\pi\)
\(354\) −37.9129 −2.01505
\(355\) 8.37386 0.444439
\(356\) −15.1652 −0.803751
\(357\) 3.95644 0.209397
\(358\) −10.7477 −0.568035
\(359\) −9.16515 −0.483718 −0.241859 0.970311i \(-0.577757\pi\)
−0.241859 + 0.970311i \(0.577757\pi\)
\(360\) −4.79129 −0.252523
\(361\) 14.5390 0.765211
\(362\) −18.5390 −0.974389
\(363\) 28.9564 1.51982
\(364\) 10.3739 0.543738
\(365\) 12.7477 0.667247
\(366\) −28.9564 −1.51358
\(367\) 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
\(370\) −4.00000 −0.207950
\(371\) −10.7477 −0.557994
\(372\) 9.41742 0.488271
\(373\) −12.7477 −0.660052 −0.330026 0.943972i \(-0.607058\pi\)
−0.330026 + 0.943972i \(0.607058\pi\)
\(374\) 0.626136 0.0323767
\(375\) −2.79129 −0.144141
\(376\) 4.41742 0.227811
\(377\) 43.9129 2.26163
\(378\) 8.95644 0.460670
\(379\) 6.37386 0.327403 0.163702 0.986510i \(-0.447657\pi\)
0.163702 + 0.986510i \(0.447657\pi\)
\(380\) −5.79129 −0.297087
\(381\) −35.5826 −1.82295
\(382\) 25.5826 1.30892
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 2.79129 0.142442
\(385\) 1.41742 0.0722386
\(386\) 20.7477 1.05603
\(387\) −53.4955 −2.71933
\(388\) 7.95644 0.403927
\(389\) 20.7042 1.04974 0.524871 0.851182i \(-0.324113\pi\)
0.524871 + 0.851182i \(0.324113\pi\)
\(390\) 16.1652 0.818554
\(391\) 0 0
\(392\) 3.79129 0.191489
\(393\) 25.5826 1.29047
\(394\) 11.5390 0.581327
\(395\) −8.00000 −0.402524
\(396\) 3.79129 0.190519
\(397\) −15.5390 −0.779881 −0.389940 0.920840i \(-0.627504\pi\)
−0.389940 + 0.920840i \(0.627504\pi\)
\(398\) −16.3303 −0.818564
\(399\) 28.9564 1.44964
\(400\) 1.00000 0.0500000
\(401\) 4.74773 0.237090 0.118545 0.992949i \(-0.462177\pi\)
0.118545 + 0.992949i \(0.462177\pi\)
\(402\) −31.1652 −1.55438
\(403\) −19.5390 −0.973308
\(404\) 4.41742 0.219775
\(405\) −0.417424 −0.0207420
\(406\) −13.5826 −0.674092
\(407\) 3.16515 0.156891
\(408\) −2.20871 −0.109348
\(409\) −18.2087 −0.900363 −0.450181 0.892937i \(-0.648641\pi\)
−0.450181 + 0.892937i \(0.648641\pi\)
\(410\) 6.79129 0.335398
\(411\) 10.5826 0.522000
\(412\) 6.37386 0.314018
\(413\) −24.3303 −1.19722
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) −5.79129 −0.283941
\(417\) −35.5826 −1.74249
\(418\) 4.58258 0.224141
\(419\) −20.8348 −1.01785 −0.508924 0.860811i \(-0.669957\pi\)
−0.508924 + 0.860811i \(0.669957\pi\)
\(420\) −5.00000 −0.243975
\(421\) −18.1216 −0.883192 −0.441596 0.897214i \(-0.645588\pi\)
−0.441596 + 0.897214i \(0.645588\pi\)
\(422\) 10.0000 0.486792
\(423\) −21.1652 −1.02908
\(424\) 6.00000 0.291386
\(425\) −0.791288 −0.0383831
\(426\) 23.3739 1.13247
\(427\) −18.5826 −0.899274
\(428\) −4.41742 −0.213524
\(429\) −12.7913 −0.617569
\(430\) 11.1652 0.538431
\(431\) −25.9129 −1.24818 −0.624090 0.781353i \(-0.714530\pi\)
−0.624090 + 0.781353i \(0.714530\pi\)
\(432\) −5.00000 −0.240563
\(433\) 30.5390 1.46761 0.733806 0.679359i \(-0.237742\pi\)
0.733806 + 0.679359i \(0.237742\pi\)
\(434\) 6.04356 0.290100
\(435\) −21.1652 −1.01479
\(436\) 3.37386 0.161579
\(437\) 0 0
\(438\) 35.5826 1.70020
\(439\) −6.53901 −0.312090 −0.156045 0.987750i \(-0.549875\pi\)
−0.156045 + 0.987750i \(0.549875\pi\)
\(440\) −0.791288 −0.0377232
\(441\) −18.1652 −0.865007
\(442\) 4.58258 0.217971
\(443\) −39.7913 −1.89054 −0.945271 0.326288i \(-0.894202\pi\)
−0.945271 + 0.326288i \(0.894202\pi\)
\(444\) −11.1652 −0.529875
\(445\) −15.1652 −0.718897
\(446\) 7.16515 0.339280
\(447\) 22.9129 1.08374
\(448\) 1.79129 0.0846304
\(449\) −16.1216 −0.760825 −0.380412 0.924817i \(-0.624218\pi\)
−0.380412 + 0.924817i \(0.624218\pi\)
\(450\) −4.79129 −0.225863
\(451\) −5.37386 −0.253045
\(452\) −6.00000 −0.282216
\(453\) 30.1216 1.41524
\(454\) −22.7477 −1.06760
\(455\) 10.3739 0.486334
\(456\) −16.1652 −0.757003
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 20.3303 0.949973
\(459\) 3.95644 0.184671
\(460\) 0 0
\(461\) −28.7477 −1.33892 −0.669458 0.742850i \(-0.733473\pi\)
−0.669458 + 0.742850i \(0.733473\pi\)
\(462\) 3.95644 0.184070
\(463\) −10.0000 −0.464739 −0.232370 0.972628i \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) 7.58258 0.352012
\(465\) 9.41742 0.436723
\(466\) 1.58258 0.0733114
\(467\) 19.9129 0.921458 0.460729 0.887541i \(-0.347588\pi\)
0.460729 + 0.887541i \(0.347588\pi\)
\(468\) 27.7477 1.28264
\(469\) −20.0000 −0.923514
\(470\) 4.41742 0.203761
\(471\) −41.1652 −1.89679
\(472\) 13.5826 0.625189
\(473\) −8.83485 −0.406227
\(474\) −22.3303 −1.02566
\(475\) −5.79129 −0.265723
\(476\) −1.41742 −0.0649675
\(477\) −28.7477 −1.31627
\(478\) 15.1652 0.693638
\(479\) −15.4955 −0.708005 −0.354003 0.935244i \(-0.615180\pi\)
−0.354003 + 0.935244i \(0.615180\pi\)
\(480\) 2.79129 0.127404
\(481\) 23.1652 1.05624
\(482\) −28.0000 −1.27537
\(483\) 0 0
\(484\) −10.3739 −0.471539
\(485\) 7.95644 0.361283
\(486\) −16.1652 −0.733266
\(487\) 6.41742 0.290801 0.145401 0.989373i \(-0.453553\pi\)
0.145401 + 0.989373i \(0.453553\pi\)
\(488\) 10.3739 0.469603
\(489\) −24.0780 −1.08885
\(490\) 3.79129 0.171273
\(491\) 10.7477 0.485038 0.242519 0.970147i \(-0.422026\pi\)
0.242519 + 0.970147i \(0.422026\pi\)
\(492\) 18.9564 0.854622
\(493\) −6.00000 −0.270226
\(494\) 33.5390 1.50899
\(495\) 3.79129 0.170406
\(496\) −3.37386 −0.151491
\(497\) 15.0000 0.672842
\(498\) 16.7477 0.750484
\(499\) 4.83485 0.216438 0.108219 0.994127i \(-0.465485\pi\)
0.108219 + 0.994127i \(0.465485\pi\)
\(500\) 1.00000 0.0447214
\(501\) −51.1652 −2.28589
\(502\) 26.2087 1.16975
\(503\) −14.2087 −0.633535 −0.316768 0.948503i \(-0.602598\pi\)
−0.316768 + 0.948503i \(0.602598\pi\)
\(504\) −8.58258 −0.382298
\(505\) 4.41742 0.196573
\(506\) 0 0
\(507\) −57.3303 −2.54613
\(508\) 12.7477 0.565589
\(509\) 34.7477 1.54017 0.770083 0.637944i \(-0.220215\pi\)
0.770083 + 0.637944i \(0.220215\pi\)
\(510\) −2.20871 −0.0978034
\(511\) 22.8348 1.01015
\(512\) −1.00000 −0.0441942
\(513\) 28.9564 1.27846
\(514\) 4.74773 0.209413
\(515\) 6.37386 0.280866
\(516\) 31.1652 1.37197
\(517\) −3.49545 −0.153730
\(518\) −7.16515 −0.314819
\(519\) 52.4519 2.30238
\(520\) −5.79129 −0.253965
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −36.3303 −1.59013
\(523\) −17.1652 −0.750580 −0.375290 0.926908i \(-0.622457\pi\)
−0.375290 + 0.926908i \(0.622457\pi\)
\(524\) −9.16515 −0.400381
\(525\) −5.00000 −0.218218
\(526\) 11.2087 0.488723
\(527\) 2.66970 0.116294
\(528\) −2.20871 −0.0961219
\(529\) 0 0
\(530\) 6.00000 0.260623
\(531\) −65.0780 −2.82415
\(532\) −10.3739 −0.449764
\(533\) −39.3303 −1.70358
\(534\) −42.3303 −1.83181
\(535\) −4.41742 −0.190982
\(536\) 11.1652 0.482261
\(537\) −30.0000 −1.29460
\(538\) 10.7477 0.463367
\(539\) −3.00000 −0.129219
\(540\) −5.00000 −0.215166
\(541\) 1.66970 0.0717859 0.0358929 0.999356i \(-0.488572\pi\)
0.0358929 + 0.999356i \(0.488572\pi\)
\(542\) −18.1216 −0.778389
\(543\) −51.7477 −2.22071
\(544\) 0.791288 0.0339262
\(545\) 3.37386 0.144520
\(546\) 28.9564 1.23922
\(547\) −26.1216 −1.11688 −0.558439 0.829545i \(-0.688600\pi\)
−0.558439 + 0.829545i \(0.688600\pi\)
\(548\) −3.79129 −0.161956
\(549\) −49.7042 −2.12132
\(550\) −0.791288 −0.0337406
\(551\) −43.9129 −1.87075
\(552\) 0 0
\(553\) −14.3303 −0.609386
\(554\) −17.1652 −0.729277
\(555\) −11.1652 −0.473934
\(556\) 12.7477 0.540624
\(557\) 6.33030 0.268224 0.134112 0.990966i \(-0.457182\pi\)
0.134112 + 0.990966i \(0.457182\pi\)
\(558\) 16.1652 0.684325
\(559\) −64.6606 −2.73485
\(560\) 1.79129 0.0756957
\(561\) 1.74773 0.0737891
\(562\) 10.7477 0.453366
\(563\) 15.1652 0.639135 0.319567 0.947564i \(-0.396462\pi\)
0.319567 + 0.947564i \(0.396462\pi\)
\(564\) 12.3303 0.519199
\(565\) −6.00000 −0.252422
\(566\) 8.33030 0.350149
\(567\) −0.747727 −0.0314016
\(568\) −8.37386 −0.351360
\(569\) 39.4955 1.65574 0.827868 0.560923i \(-0.189554\pi\)
0.827868 + 0.560923i \(0.189554\pi\)
\(570\) −16.1652 −0.677084
\(571\) 11.1216 0.465424 0.232712 0.972546i \(-0.425240\pi\)
0.232712 + 0.972546i \(0.425240\pi\)
\(572\) 4.58258 0.191607
\(573\) 71.4083 2.98313
\(574\) 12.1652 0.507764
\(575\) 0 0
\(576\) 4.79129 0.199637
\(577\) 41.1652 1.71373 0.856864 0.515543i \(-0.172410\pi\)
0.856864 + 0.515543i \(0.172410\pi\)
\(578\) 16.3739 0.681063
\(579\) 57.9129 2.40678
\(580\) 7.58258 0.314849
\(581\) 10.7477 0.445891
\(582\) 22.2087 0.920581
\(583\) −4.74773 −0.196631
\(584\) −12.7477 −0.527505
\(585\) 27.7477 1.14723
\(586\) 27.4955 1.13583
\(587\) −30.7913 −1.27089 −0.635446 0.772145i \(-0.719184\pi\)
−0.635446 + 0.772145i \(0.719184\pi\)
\(588\) 10.5826 0.436418
\(589\) 19.5390 0.805091
\(590\) 13.5826 0.559186
\(591\) 32.2087 1.32489
\(592\) 4.00000 0.164399
\(593\) −31.9129 −1.31050 −0.655252 0.755410i \(-0.727438\pi\)
−0.655252 + 0.755410i \(0.727438\pi\)
\(594\) 3.95644 0.162335
\(595\) −1.41742 −0.0581087
\(596\) −8.20871 −0.336242
\(597\) −45.5826 −1.86557
\(598\) 0 0
\(599\) 1.12159 0.0458270 0.0229135 0.999737i \(-0.492706\pi\)
0.0229135 + 0.999737i \(0.492706\pi\)
\(600\) 2.79129 0.113954
\(601\) −18.2087 −0.742749 −0.371374 0.928483i \(-0.621113\pi\)
−0.371374 + 0.928483i \(0.621113\pi\)
\(602\) 20.0000 0.815139
\(603\) −53.4955 −2.17850
\(604\) −10.7913 −0.439091
\(605\) −10.3739 −0.421758
\(606\) 12.3303 0.500884
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 5.79129 0.234868
\(609\) −37.9129 −1.53631
\(610\) 10.3739 0.420025
\(611\) −25.5826 −1.03496
\(612\) −3.79129 −0.153254
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) 15.5390 0.627104
\(615\) 18.9564 0.764397
\(616\) −1.41742 −0.0571097
\(617\) 23.8693 0.960943 0.480471 0.877010i \(-0.340466\pi\)
0.480471 + 0.877010i \(0.340466\pi\)
\(618\) 17.7913 0.715671
\(619\) 1.79129 0.0719979 0.0359990 0.999352i \(-0.488539\pi\)
0.0359990 + 0.999352i \(0.488539\pi\)
\(620\) −3.37386 −0.135498
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) −27.1652 −1.08835
\(624\) −16.1652 −0.647124
\(625\) 1.00000 0.0400000
\(626\) −4.62614 −0.184898
\(627\) 12.7913 0.510835
\(628\) 14.7477 0.588498
\(629\) −3.16515 −0.126203
\(630\) −8.58258 −0.341938
\(631\) −27.9129 −1.11119 −0.555597 0.831452i \(-0.687510\pi\)
−0.555597 + 0.831452i \(0.687510\pi\)
\(632\) 8.00000 0.318223
\(633\) 27.9129 1.10944
\(634\) −9.79129 −0.388862
\(635\) 12.7477 0.505878
\(636\) 16.7477 0.664091
\(637\) −21.9564 −0.869946
\(638\) −6.00000 −0.237542
\(639\) 40.1216 1.58719
\(640\) −1.00000 −0.0395285
\(641\) −15.1652 −0.598987 −0.299494 0.954098i \(-0.596818\pi\)
−0.299494 + 0.954098i \(0.596818\pi\)
\(642\) −12.3303 −0.486638
\(643\) −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
\(646\) −4.58258 −0.180299
\(647\) 21.1652 0.832088 0.416044 0.909344i \(-0.363416\pi\)
0.416044 + 0.909344i \(0.363416\pi\)
\(648\) 0.417424 0.0163980
\(649\) −10.7477 −0.421885
\(650\) −5.79129 −0.227153
\(651\) 16.8693 0.661161
\(652\) 8.62614 0.337826
\(653\) 3.46099 0.135439 0.0677194 0.997704i \(-0.478428\pi\)
0.0677194 + 0.997704i \(0.478428\pi\)
\(654\) 9.41742 0.368250
\(655\) −9.16515 −0.358112
\(656\) −6.79129 −0.265155
\(657\) 61.0780 2.38288
\(658\) 7.91288 0.308476
\(659\) 8.83485 0.344157 0.172078 0.985083i \(-0.444952\pi\)
0.172078 + 0.985083i \(0.444952\pi\)
\(660\) −2.20871 −0.0859740
\(661\) 25.6261 0.996741 0.498371 0.866964i \(-0.333932\pi\)
0.498371 + 0.866964i \(0.333932\pi\)
\(662\) 20.7477 0.806383
\(663\) 12.7913 0.496772
\(664\) −6.00000 −0.232845
\(665\) −10.3739 −0.402281
\(666\) −19.1652 −0.742635
\(667\) 0 0
\(668\) 18.3303 0.709221
\(669\) 20.0000 0.773245
\(670\) 11.1652 0.431347
\(671\) −8.20871 −0.316894
\(672\) 5.00000 0.192879
\(673\) 38.0000 1.46479 0.732396 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(674\) −12.2087 −0.470262
\(675\) −5.00000 −0.192450
\(676\) 20.5390 0.789962
\(677\) −42.6606 −1.63958 −0.819790 0.572664i \(-0.805910\pi\)
−0.819790 + 0.572664i \(0.805910\pi\)
\(678\) −16.7477 −0.643193
\(679\) 14.2523 0.546952
\(680\) 0.791288 0.0303445
\(681\) −63.4955 −2.43315
\(682\) 2.66970 0.102228
\(683\) −11.3739 −0.435209 −0.217604 0.976037i \(-0.569824\pi\)
−0.217604 + 0.976037i \(0.569824\pi\)
\(684\) −27.7477 −1.06096
\(685\) −3.79129 −0.144858
\(686\) 19.3303 0.738034
\(687\) 56.7477 2.16506
\(688\) −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
\(693\) 6.79129 0.257980
\(694\) −5.20871 −0.197720
\(695\) 12.7477 0.483549
\(696\) 21.1652 0.802263
\(697\) 5.37386 0.203550
\(698\) −26.0000 −0.984115
\(699\) 4.41742 0.167082
\(700\) 1.79129 0.0677043
\(701\) −23.3739 −0.882819 −0.441409 0.897306i \(-0.645521\pi\)
−0.441409 + 0.897306i \(0.645521\pi\)
\(702\) 28.9564 1.09289
\(703\) −23.1652 −0.873690
\(704\) 0.791288 0.0298228
\(705\) 12.3303 0.464386
\(706\) −3.16515 −0.119122
\(707\) 7.91288 0.297594
\(708\) 37.9129 1.42485
\(709\) −2.46099 −0.0924242 −0.0462121 0.998932i \(-0.514715\pi\)
−0.0462121 + 0.998932i \(0.514715\pi\)
\(710\) −8.37386 −0.314265
\(711\) −38.3303 −1.43750
\(712\) 15.1652 0.568338
\(713\) 0 0
\(714\) −3.95644 −0.148066
\(715\) 4.58258 0.171379
\(716\) 10.7477 0.401661
\(717\) 42.3303 1.58085
\(718\) 9.16515 0.342040
\(719\) 2.53901 0.0946893 0.0473446 0.998879i \(-0.484924\pi\)
0.0473446 + 0.998879i \(0.484924\pi\)
\(720\) 4.79129 0.178561
\(721\) 11.4174 0.425207
\(722\) −14.5390 −0.541086
\(723\) −78.1561 −2.90666
\(724\) 18.5390 0.688997
\(725\) 7.58258 0.281610
\(726\) −28.9564 −1.07467
\(727\) −39.1216 −1.45094 −0.725470 0.688254i \(-0.758377\pi\)
−0.725470 + 0.688254i \(0.758377\pi\)
\(728\) −10.3739 −0.384481
\(729\) −43.8693 −1.62479
\(730\) −12.7477 −0.471815
\(731\) 8.83485 0.326769
\(732\) 28.9564 1.07026
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) −19.1652 −0.707399
\(735\) 10.5826 0.390344
\(736\) 0 0
\(737\) −8.83485 −0.325436
\(738\) 32.5390 1.19778
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 4.00000 0.147043
\(741\) 93.6170 3.43911
\(742\) 10.7477 0.394561
\(743\) 12.9564 0.475326 0.237663 0.971348i \(-0.423619\pi\)
0.237663 + 0.971348i \(0.423619\pi\)
\(744\) −9.41742 −0.345260
\(745\) −8.20871 −0.300744
\(746\) 12.7477 0.466727
\(747\) 28.7477 1.05182
\(748\) −0.626136 −0.0228938
\(749\) −7.91288 −0.289130
\(750\) 2.79129 0.101923
\(751\) 8.74773 0.319209 0.159605 0.987181i \(-0.448978\pi\)
0.159605 + 0.987181i \(0.448978\pi\)
\(752\) −4.41742 −0.161087
\(753\) 73.1561 2.66596
\(754\) −43.9129 −1.59921
\(755\) −10.7913 −0.392735
\(756\) −8.95644 −0.325743
\(757\) 10.3303 0.375461 0.187731 0.982221i \(-0.439887\pi\)
0.187731 + 0.982221i \(0.439887\pi\)
\(758\) −6.37386 −0.231509
\(759\) 0 0
\(760\) 5.79129 0.210072
\(761\) −11.0436 −0.400329 −0.200164 0.979762i \(-0.564148\pi\)
−0.200164 + 0.979762i \(0.564148\pi\)
\(762\) 35.5826 1.28902
\(763\) 6.04356 0.218792
\(764\) −25.5826 −0.925545
\(765\) −3.79129 −0.137074
\(766\) −24.0000 −0.867155
\(767\) −78.6606 −2.84027
\(768\) −2.79129 −0.100722
\(769\) 40.3303 1.45435 0.727174 0.686453i \(-0.240833\pi\)
0.727174 + 0.686453i \(0.240833\pi\)
\(770\) −1.41742 −0.0510804
\(771\) 13.2523 0.477269
\(772\) −20.7477 −0.746727
\(773\) −33.4955 −1.20475 −0.602374 0.798214i \(-0.705778\pi\)
−0.602374 + 0.798214i \(0.705778\pi\)
\(774\) 53.4955 1.92285
\(775\) −3.37386 −0.121193
\(776\) −7.95644 −0.285620
\(777\) −20.0000 −0.717496
\(778\) −20.7042 −0.742280
\(779\) 39.3303 1.40915
\(780\) −16.1652 −0.578805
\(781\) 6.62614 0.237102
\(782\) 0 0
\(783\) −37.9129 −1.35490
\(784\) −3.79129 −0.135403
\(785\) 14.7477 0.526369
\(786\) −25.5826 −0.912500
\(787\) 17.5826 0.626751 0.313376 0.949629i \(-0.398540\pi\)
0.313376 + 0.949629i \(0.398540\pi\)
\(788\) −11.5390 −0.411060
\(789\) 31.2867 1.11384
\(790\) 8.00000 0.284627
\(791\) −10.7477 −0.382145
\(792\) −3.79129 −0.134718
\(793\) −60.0780 −2.13343
\(794\) 15.5390 0.551459
\(795\) 16.7477 0.593981
\(796\) 16.3303 0.578812
\(797\) 4.08712 0.144773 0.0723866 0.997377i \(-0.476938\pi\)
0.0723866 + 0.997377i \(0.476938\pi\)
\(798\) −28.9564 −1.02505
\(799\) 3.49545 0.123660
\(800\) −1.00000 −0.0353553
\(801\) −72.6606 −2.56734
\(802\) −4.74773 −0.167648
\(803\) 10.0871 0.355967
\(804\) 31.1652 1.09911
\(805\) 0 0
\(806\) 19.5390 0.688232
\(807\) 30.0000 1.05605
\(808\) −4.41742 −0.155404
\(809\) 33.9564 1.19384 0.596922 0.802299i \(-0.296390\pi\)
0.596922 + 0.802299i \(0.296390\pi\)
\(810\) 0.417424 0.0146668
\(811\) −2.08712 −0.0732887 −0.0366444 0.999328i \(-0.511667\pi\)
−0.0366444 + 0.999328i \(0.511667\pi\)
\(812\) 13.5826 0.476655
\(813\) −50.5826 −1.77401
\(814\) −3.16515 −0.110938
\(815\) 8.62614 0.302160
\(816\) 2.20871 0.0773204
\(817\) 64.6606 2.26219
\(818\) 18.2087 0.636653
\(819\) 49.7042 1.73680
\(820\) −6.79129 −0.237162
\(821\) 21.1652 0.738669 0.369334 0.929297i \(-0.379586\pi\)
0.369334 + 0.929297i \(0.379586\pi\)
\(822\) −10.5826 −0.369110
\(823\) 22.8348 0.795973 0.397986 0.917391i \(-0.369709\pi\)
0.397986 + 0.917391i \(0.369709\pi\)
\(824\) −6.37386 −0.222044
\(825\) −2.20871 −0.0768975
\(826\) 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
\(831\) −47.9129 −1.66208
\(832\) 5.79129 0.200777
\(833\) 3.00000 0.103944
\(834\) 35.5826 1.23212
\(835\) 18.3303 0.634346
\(836\) −4.58258 −0.158492
\(837\) 16.8693 0.583089
\(838\) 20.8348 0.719728
\(839\) 31.5826 1.09035 0.545176 0.838322i \(-0.316463\pi\)
0.545176 + 0.838322i \(0.316463\pi\)
\(840\) 5.00000 0.172516
\(841\) 28.4955 0.982602
\(842\) 18.1216 0.624511
\(843\) 30.0000 1.03325
\(844\) −10.0000 −0.344214
\(845\) 20.5390 0.706564
\(846\) 21.1652 0.727673
\(847\) −18.5826 −0.638505
\(848\) −6.00000 −0.206041
\(849\) 23.2523 0.798016
\(850\) 0.791288 0.0271409
\(851\) 0 0
\(852\) −23.3739 −0.800775
\(853\) 40.5390 1.38803 0.694015 0.719961i \(-0.255840\pi\)
0.694015 + 0.719961i \(0.255840\pi\)
\(854\) 18.5826 0.635883
\(855\) −27.7477 −0.948952
\(856\) 4.41742 0.150984
\(857\) −9.16515 −0.313076 −0.156538 0.987672i \(-0.550033\pi\)
−0.156538 + 0.987672i \(0.550033\pi\)
\(858\) 12.7913 0.436687
\(859\) −26.7477 −0.912621 −0.456310 0.889821i \(-0.650829\pi\)
−0.456310 + 0.889821i \(0.650829\pi\)
\(860\) −11.1652 −0.380729
\(861\) 33.9564 1.15723
\(862\) 25.9129 0.882596
\(863\) −22.4174 −0.763098 −0.381549 0.924349i \(-0.624609\pi\)
−0.381549 + 0.924349i \(0.624609\pi\)
\(864\) 5.00000 0.170103
\(865\) −18.7913 −0.638923
\(866\) −30.5390 −1.03776
\(867\) 45.7042 1.55219
\(868\) −6.04356 −0.205132
\(869\) −6.33030 −0.214741
\(870\) 21.1652 0.717566
\(871\) −64.6606 −2.19094
\(872\) −3.37386 −0.114253
\(873\) 38.1216 1.29022
\(874\) 0 0
\(875\) 1.79129 0.0605566
\(876\) −35.5826 −1.20222
\(877\) −42.7042 −1.44202 −0.721009 0.692926i \(-0.756321\pi\)
−0.721009 + 0.692926i \(0.756321\pi\)
\(878\) 6.53901 0.220681
\(879\) 76.7477 2.58864
\(880\) 0.791288 0.0266743
\(881\) −30.3303 −1.02185 −0.510927 0.859624i \(-0.670698\pi\)
−0.510927 + 0.859624i \(0.670698\pi\)
\(882\) 18.1652 0.611652
\(883\) −34.9564 −1.17638 −0.588189 0.808724i \(-0.700159\pi\)
−0.588189 + 0.808724i \(0.700159\pi\)
\(884\) −4.58258 −0.154129
\(885\) 37.9129 1.27443
\(886\) 39.7913 1.33681
\(887\) 15.1652 0.509196 0.254598 0.967047i \(-0.418057\pi\)
0.254598 + 0.967047i \(0.418057\pi\)
\(888\) 11.1652 0.374678
\(889\) 22.8348 0.765856
\(890\) 15.1652 0.508337
\(891\) −0.330303 −0.0110656
\(892\) −7.16515 −0.239907
\(893\) 25.5826 0.856088
\(894\) −22.9129 −0.766321
\(895\) 10.7477 0.359257
\(896\) −1.79129 −0.0598427
\(897\) 0 0
\(898\) 16.1216 0.537984
\(899\) −25.5826 −0.853227
\(900\) 4.79129 0.159710
\(901\) 4.74773 0.158170
\(902\) 5.37386 0.178930
\(903\) 55.8258 1.85776
\(904\) 6.00000 0.199557
\(905\) 18.5390 0.616258
\(906\) −30.1216 −1.00072
\(907\) −6.74773 −0.224055 −0.112027 0.993705i \(-0.535734\pi\)
−0.112027 + 0.993705i \(0.535734\pi\)
\(908\) 22.7477 0.754910
\(909\) 21.1652 0.702004
\(910\) −10.3739 −0.343890
\(911\) −4.41742 −0.146356 −0.0731779 0.997319i \(-0.523314\pi\)
−0.0731779 + 0.997319i \(0.523314\pi\)
\(912\) 16.1652 0.535282
\(913\) 4.74773 0.157127
\(914\) −10.0000 −0.330771
\(915\) 28.9564 0.957270
\(916\) −20.3303 −0.671732
\(917\) −16.4174 −0.542151
\(918\) −3.95644 −0.130582
\(919\) 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
\(922\) 28.7477 0.946756
\(923\) 48.4955 1.59625
\(924\) −3.95644 −0.130157
\(925\) 4.00000 0.131519
\(926\) 10.0000 0.328620
\(927\) 30.5390 1.00303
\(928\) −7.58258 −0.248910
\(929\) 15.4955 0.508389 0.254195 0.967153i \(-0.418190\pi\)
0.254195 + 0.967153i \(0.418190\pi\)
\(930\) −9.41742 −0.308810
\(931\) 21.9564 0.719593
\(932\) −1.58258 −0.0518390
\(933\) −33.4955 −1.09659
\(934\) −19.9129 −0.651569
\(935\) −0.626136 −0.0204769
\(936\) −27.7477 −0.906963
\(937\) −44.6261 −1.45787 −0.728936 0.684582i \(-0.759985\pi\)
−0.728936 + 0.684582i \(0.759985\pi\)
\(938\) 20.0000 0.653023
\(939\) −12.9129 −0.421396
\(940\) −4.41742 −0.144080
\(941\) −32.0436 −1.04459 −0.522295 0.852765i \(-0.674924\pi\)
−0.522295 + 0.852765i \(0.674924\pi\)
\(942\) 41.1652 1.34123
\(943\) 0 0
\(944\) −13.5826 −0.442075
\(945\) −8.95644 −0.291353
\(946\) 8.83485 0.287246
\(947\) 2.53901 0.0825069 0.0412534 0.999149i \(-0.486865\pi\)
0.0412534 + 0.999149i \(0.486865\pi\)
\(948\) 22.3303 0.725255
\(949\) 73.8258 2.39649
\(950\) 5.79129 0.187894
\(951\) −27.3303 −0.886246
\(952\) 1.41742 0.0459390
\(953\) −5.53901 −0.179426 −0.0897131 0.995968i \(-0.528595\pi\)
−0.0897131 + 0.995968i \(0.528595\pi\)
\(954\) 28.7477 0.930742
\(955\) −25.5826 −0.827833
\(956\) −15.1652 −0.490476
\(957\) −16.7477 −0.541377
\(958\) 15.4955 0.500635
\(959\) −6.79129 −0.219302
\(960\) −2.79129 −0.0900884
\(961\) −19.6170 −0.632808
\(962\) −23.1652 −0.746874
\(963\) −21.1652 −0.682037
\(964\) 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
\(969\) −12.7913 −0.410915
\(970\) −7.95644 −0.255466
\(971\) −15.9564 −0.512067 −0.256033 0.966668i \(-0.582416\pi\)
−0.256033 + 0.966668i \(0.582416\pi\)
\(972\) 16.1652 0.518497
\(973\) 22.8348 0.732052
\(974\) −6.41742 −0.205628
\(975\) −16.1652 −0.517699
\(976\) −10.3739 −0.332059
\(977\) 34.1216 1.09165 0.545823 0.837900i \(-0.316217\pi\)
0.545823 + 0.837900i \(0.316217\pi\)
\(978\) 24.0780 0.769930
\(979\) −12.0000 −0.383522
\(980\) −3.79129 −0.121108
\(981\) 16.1652 0.516114
\(982\) −10.7477 −0.342974
\(983\) 14.3739 0.458455 0.229228 0.973373i \(-0.426380\pi\)
0.229228 + 0.973373i \(0.426380\pi\)
\(984\) −18.9564 −0.604309
\(985\) −11.5390 −0.367664
\(986\) 6.00000 0.191079
\(987\) 22.0871 0.703041
\(988\) −33.5390 −1.06702
\(989\) 0 0
\(990\) −3.79129 −0.120495
\(991\) −33.2087 −1.05491 −0.527455 0.849583i \(-0.676854\pi\)
−0.527455 + 0.849583i \(0.676854\pi\)
\(992\) 3.37386 0.107120
\(993\) 57.9129 1.83781
\(994\) −15.0000 −0.475771
\(995\) 16.3303 0.517705
\(996\) −16.7477 −0.530672
\(997\) −43.4955 −1.37751 −0.688757 0.724992i \(-0.741843\pi\)
−0.688757 + 0.724992i \(0.741843\pi\)
\(998\) −4.83485 −0.153044
\(999\) −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 5290.2.a.e.1.1 2
23.22 odd 2 230.2.a.a.1.1 2
69.68 even 2 2070.2.a.x.1.1 2
92.91 even 2 1840.2.a.n.1.2 2
115.22 even 4 1150.2.b.g.599.2 4
115.68 even 4 1150.2.b.g.599.3 4
115.114 odd 2 1150.2.a.o.1.2 2
184.45 odd 2 7360.2.a.bq.1.2 2
184.91 even 2 7360.2.a.bk.1.1 2
460.459 even 2 9200.2.a.bs.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.a.1.1 2 23.22 odd 2
1150.2.a.o.1.2 2 115.114 odd 2
1150.2.b.g.599.2 4 115.22 even 4
1150.2.b.g.599.3 4 115.68 even 4
1840.2.a.n.1.2 2 92.91 even 2
2070.2.a.x.1.1 2 69.68 even 2
5290.2.a.e.1.1 2 1.1 even 1 trivial
7360.2.a.bk.1.1 2 184.91 even 2
7360.2.a.bq.1.2 2 184.45 odd 2
9200.2.a.bs.1.1 2 460.459 even 2