Properties

Label 1150.2.a.o.1.2
Level $1150$
Weight $2$
Character 1150.1
Self dual yes
Analytic conductor $9.183$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.18279623245\)
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 \(\chi\) \(=\) 1150.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.79129 q^{3} +1.00000 q^{4} +2.79129 q^{6} +1.79129 q^{7} +1.00000 q^{8} +4.79129 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.79129 q^{3} +1.00000 q^{4} +2.79129 q^{6} +1.79129 q^{7} +1.00000 q^{8} +4.79129 q^{9} -0.791288 q^{11} +2.79129 q^{12} -5.79129 q^{13} +1.79129 q^{14} +1.00000 q^{16} -0.791288 q^{17} +4.79129 q^{18} +5.79129 q^{19} +5.00000 q^{21} -0.791288 q^{22} -1.00000 q^{23} +2.79129 q^{24} -5.79129 q^{26} +5.00000 q^{27} +1.79129 q^{28} +7.58258 q^{29} -3.37386 q^{31} +1.00000 q^{32} -2.20871 q^{33} -0.791288 q^{34} +4.79129 q^{36} +4.00000 q^{37} +5.79129 q^{38} -16.1652 q^{39} -6.79129 q^{41} +5.00000 q^{42} -11.1652 q^{43} -0.791288 q^{44} -1.00000 q^{46} +4.41742 q^{47} +2.79129 q^{48} -3.79129 q^{49} -2.20871 q^{51} -5.79129 q^{52} -6.00000 q^{53} +5.00000 q^{54} +1.79129 q^{56} +16.1652 q^{57} +7.58258 q^{58} -13.5826 q^{59} +10.3739 q^{61} -3.37386 q^{62} +8.58258 q^{63} +1.00000 q^{64} -2.20871 q^{66} -11.1652 q^{67} -0.791288 q^{68} -2.79129 q^{69} +8.37386 q^{71} +4.79129 q^{72} -12.7477 q^{73} +4.00000 q^{74} +5.79129 q^{76} -1.41742 q^{77} -16.1652 q^{78} +8.00000 q^{79} -0.417424 q^{81} -6.79129 q^{82} +6.00000 q^{83} +5.00000 q^{84} -11.1652 q^{86} +21.1652 q^{87} -0.791288 q^{88} +15.1652 q^{89} -10.3739 q^{91} -1.00000 q^{92} -9.41742 q^{93} +4.41742 q^{94} +2.79129 q^{96} +7.95644 q^{97} -3.79129 q^{98} -3.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + q^{3} + 2q^{4} + q^{6} - q^{7} + 2q^{8} + 5q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + q^{3} + 2q^{4} + q^{6} - q^{7} + 2q^{8} + 5q^{9} + 3q^{11} + q^{12} - 7q^{13} - q^{14} + 2q^{16} + 3q^{17} + 5q^{18} + 7q^{19} + 10q^{21} + 3q^{22} - 2q^{23} + q^{24} - 7q^{26} + 10q^{27} - q^{28} + 6q^{29} + 7q^{31} + 2q^{32} - 9q^{33} + 3q^{34} + 5q^{36} + 8q^{37} + 7q^{38} - 14q^{39} - 9q^{41} + 10q^{42} - 4q^{43} + 3q^{44} - 2q^{46} + 18q^{47} + q^{48} - 3q^{49} - 9q^{51} - 7q^{52} - 12q^{53} + 10q^{54} - q^{56} + 14q^{57} + 6q^{58} - 18q^{59} + 7q^{61} + 7q^{62} + 8q^{63} + 2q^{64} - 9q^{66} - 4q^{67} + 3q^{68} - q^{69} + 3q^{71} + 5q^{72} + 2q^{73} + 8q^{74} + 7q^{76} - 12q^{77} - 14q^{78} + 16q^{79} - 10q^{81} - 9q^{82} + 12q^{83} + 10q^{84} - 4q^{86} + 24q^{87} + 3q^{88} + 12q^{89} - 7q^{91} - 2q^{92} - 28q^{93} + 18q^{94} + q^{96} - 7q^{97} - 3q^{98} - 3q^{99} + O(q^{100}) \)

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 \(-0.201749\pi\)
0.805775 + 0.592221i \(0.201749\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 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
\(10\) 0 0
\(11\) −0.791288 −0.238582 −0.119291 0.992859i \(-0.538062\pi\)
−0.119291 + 0.992859i \(0.538062\pi\)
\(12\) 2.79129 0.805775
\(13\) −5.79129 −1.60621 −0.803107 0.595835i \(-0.796821\pi\)
−0.803107 + 0.595835i \(0.796821\pi\)
\(14\) 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 \(0.530591\pi\)
\(18\) 4.79129 1.12932
\(19\) 5.79129 1.32861 0.664306 0.747460i \(-0.268727\pi\)
0.664306 + 0.747460i \(0.268727\pi\)
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) −0.791288 −0.168703
\(23\) −1.00000 −0.208514
\(24\) 2.79129 0.569769
\(25\) 0 0
\(26\) −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 \(0.251385\pi\)
\(30\) 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
\(33\) −2.20871 −0.384487
\(34\) −0.791288 −0.135705
\(35\) 0 0
\(36\) 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
\(40\) 0 0
\(41\) −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
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 4.41742 0.644348 0.322174 0.946681i \(-0.395586\pi\)
0.322174 + 0.946681i \(0.395586\pi\)
\(48\) 2.79129 0.402888
\(49\) −3.79129 −0.541613
\(50\) 0 0
\(51\) −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 0
\(56\) 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 \(0.845262\pi\)
\(60\) 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
\(63\) 8.58258 1.08130
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −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
\(69\) −2.79129 −0.336032
\(70\) 0 0
\(71\) 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 \(-0.768030\pi\)
−0.746004 + 0.665941i \(0.768030\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 5.79129 0.664306
\(77\) −1.41742 −0.161530
\(78\) −16.1652 −1.83034
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −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 0
\(86\) −11.1652 −1.20397
\(87\) 21.1652 2.26914
\(88\) −0.791288 −0.0843516
\(89\) 15.1652 1.60750 0.803751 0.594965i \(-0.202834\pi\)
0.803751 + 0.594965i \(0.202834\pi\)
\(90\) 0 0
\(91\) −10.3739 −1.08748
\(92\) −1.00000 −0.104257
\(93\) −9.41742 −0.976541
\(94\) 4.41742 0.455623
\(95\) 0 0
\(96\) 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
\(100\) 0 0
\(101\) 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
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −4.41742 −0.427049 −0.213524 0.976938i \(-0.568494\pi\)
−0.213524 + 0.976938i \(0.568494\pi\)
\(108\) 5.00000 0.481125
\(109\) −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
\(112\) 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
\(120\) 0 0
\(121\) −10.3739 −0.943079
\(122\) 10.3739 0.939205
\(123\) −18.9564 −1.70924
\(124\) −3.37386 −0.302982
\(125\) 0 0
\(126\) 8.58258 0.764597
\(127\) −12.7477 −1.13118 −0.565589 0.824687i \(-0.691351\pi\)
−0.565589 + 0.824687i \(0.691351\pi\)
\(128\) 1.00000 0.0883883
\(129\) −31.1652 −2.74394
\(130\) 0 0
\(131\) −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
\(135\) 0 0
\(136\) −0.791288 −0.0678524
\(137\) −3.79129 −0.323912 −0.161956 0.986798i \(-0.551780\pi\)
−0.161956 + 0.986798i \(0.551780\pi\)
\(138\) −2.79129 −0.237610
\(139\) 12.7477 1.08125 0.540624 0.841264i \(-0.318188\pi\)
0.540624 + 0.841264i \(0.318188\pi\)
\(140\) 0 0
\(141\) 12.3303 1.03840
\(142\) 8.37386 0.702719
\(143\) 4.58258 0.383214
\(144\) 4.79129 0.399274
\(145\) 0 0
\(146\) −12.7477 −1.05501
\(147\) −10.5826 −0.872836
\(148\) 4.00000 0.328798
\(149\) 8.20871 0.672484 0.336242 0.941776i \(-0.390844\pi\)
0.336242 + 0.941776i \(0.390844\pi\)
\(150\) 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
\(153\) −3.79129 −0.306507
\(154\) −1.41742 −0.114219
\(155\) 0 0
\(156\) −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
\(160\) 0 0
\(161\) −1.79129 −0.141173
\(162\) −0.417424 −0.0327960
\(163\) −8.62614 −0.675651 −0.337826 0.941209i \(-0.609691\pi\)
−0.337826 + 0.941209i \(0.609691\pi\)
\(164\) −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 \(0.750953\pi\)
\(168\) 5.00000 0.385758
\(169\) 20.5390 1.57992
\(170\) 0 0
\(171\) 27.7477 2.12192
\(172\) −11.1652 −0.851335
\(173\) 18.7913 1.42868 0.714338 0.699801i \(-0.246728\pi\)
0.714338 + 0.699801i \(0.246728\pi\)
\(174\) 21.1652 1.60453
\(175\) 0 0
\(176\) −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 \(0.368433\pi\)
\(180\) 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
\(183\) 28.9564 2.14052
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −9.41742 −0.690519
\(187\) 0.626136 0.0457876
\(188\) 4.41742 0.322174
\(189\) 8.95644 0.651485
\(190\) 0 0
\(191\) 25.5826 1.85109 0.925545 0.378637i \(-0.123607\pi\)
0.925545 + 0.378637i \(0.123607\pi\)
\(192\) 2.79129 0.201444
\(193\) 20.7477 1.49345 0.746727 0.665131i \(-0.231624\pi\)
0.746727 + 0.665131i \(0.231624\pi\)
\(194\) 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 \(0.365159\pi\)
\(198\) −3.79129 −0.269435
\(199\) −16.3303 −1.15762 −0.578812 0.815461i \(-0.696484\pi\)
−0.578812 + 0.815461i \(0.696484\pi\)
\(200\) 0 0
\(201\) −31.1652 −2.19822
\(202\) 4.41742 0.310809
\(203\) 13.5826 0.953310
\(204\) −2.20871 −0.154641
\(205\) 0 0
\(206\) 6.37386 0.444088
\(207\) −4.79129 −0.333018
\(208\) −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
\(213\) 23.3739 1.60155
\(214\) −4.41742 −0.301969
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) −6.04356 −0.410264
\(218\) −3.37386 −0.228507
\(219\) −35.5826 −2.40445
\(220\) 0 0
\(221\) 4.58258 0.308257
\(222\) 11.1652 0.749356
\(223\) 7.16515 0.479814 0.239907 0.970796i \(-0.422883\pi\)
0.239907 + 0.970796i \(0.422883\pi\)
\(224\) 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 \(0.227681\pi\)
\(228\) 16.1652 1.07056
\(229\) 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
\(232\) 7.58258 0.497820
\(233\) 1.58258 0.103678 0.0518390 0.998655i \(-0.483492\pi\)
0.0518390 + 0.998655i \(0.483492\pi\)
\(234\) −27.7477 −1.81393
\(235\) 0 0
\(236\) −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 \(0.663177\pi\)
\(240\) 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
\(243\) −16.1652 −1.03699
\(244\) 10.3739 0.664119
\(245\) 0 0
\(246\) −18.9564 −1.20862
\(247\) −33.5390 −2.13404
\(248\) −3.37386 −0.214241
\(249\) 16.7477 1.06134
\(250\) 0 0
\(251\) 26.2087 1.65428 0.827140 0.561996i \(-0.189967\pi\)
0.827140 + 0.561996i \(0.189967\pi\)
\(252\) 8.58258 0.540651
\(253\) 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 \(0.452691\pi\)
\(258\) −31.1652 −1.94026
\(259\) 7.16515 0.445221
\(260\) 0 0
\(261\) 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
\(265\) 0 0
\(266\) 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 \(0.606257\pi\)
\(270\) 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
\(273\) −28.9564 −1.75252
\(274\) −3.79129 −0.229040
\(275\) 0 0
\(276\) −2.79129 −0.168016
\(277\) −17.1652 −1.03135 −0.515677 0.856783i \(-0.672460\pi\)
−0.515677 + 0.856783i \(0.672460\pi\)
\(278\) 12.7477 0.764558
\(279\) −16.1652 −0.967782
\(280\) 0 0
\(281\) 10.7477 0.641156 0.320578 0.947222i \(-0.396123\pi\)
0.320578 + 0.947222i \(0.396123\pi\)
\(282\) 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
\(285\) 0 0
\(286\) 4.58258 0.270973
\(287\) −12.1652 −0.718086
\(288\) 4.79129 0.282329
\(289\) −16.3739 −0.963168
\(290\) 0 0
\(291\) 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
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) −3.95644 −0.229576
\(298\) 8.20871 0.475518
\(299\) 5.79129 0.334919
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) −10.7913 −0.620969
\(303\) 12.3303 0.708357
\(304\) 5.79129 0.332153
\(305\) 0 0
\(306\) −3.79129 −0.216734
\(307\) 15.5390 0.886858 0.443429 0.896309i \(-0.353762\pi\)
0.443429 + 0.896309i \(0.353762\pi\)
\(308\) −1.41742 −0.0807652
\(309\) 17.7913 1.01211
\(310\) 0 0
\(311\) 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
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −9.79129 −0.549934 −0.274967 0.961454i \(-0.588667\pi\)
−0.274967 + 0.961454i \(0.588667\pi\)
\(318\) −16.7477 −0.939166
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) −12.3303 −0.688210
\(322\) −1.79129 −0.0998246
\(323\) −4.58258 −0.254981
\(324\) −0.417424 −0.0231902
\(325\) 0 0
\(326\) −8.62614 −0.477758
\(327\) −9.41742 −0.520785
\(328\) −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
\(333\) 19.1652 1.05024
\(334\) −18.3303 −1.00299
\(335\) 0 0
\(336\) 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 0
\(341\) 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 \(-0.544649\pi\)
−0.139809 + 0.990178i \(0.544649\pi\)
\(348\) 21.1652 1.13457
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −28.9564 −1.54558
\(352\) −0.791288 −0.0421758
\(353\) −3.16515 −0.168464 −0.0842320 0.996446i \(-0.526844\pi\)
−0.0842320 + 0.996446i \(0.526844\pi\)
\(354\) −37.9129 −2.01505
\(355\) 0 0
\(356\) 15.1652 0.803751
\(357\) −3.95644 −0.209397
\(358\) 10.7477 0.568035
\(359\) 9.16515 0.483718 0.241859 0.970311i \(-0.422243\pi\)
0.241859 + 0.970311i \(0.422243\pi\)
\(360\) 0 0
\(361\) 14.5390 0.765211
\(362\) −18.5390 −0.974389
\(363\) −28.9564 −1.51982
\(364\) −10.3739 −0.543738
\(365\) 0 0
\(366\) 28.9564 1.51358
\(367\) 19.1652 1.00041 0.500206 0.865906i \(-0.333257\pi\)
0.500206 + 0.865906i \(0.333257\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −32.5390 −1.69391
\(370\) 0 0
\(371\) −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
\(375\) 0 0
\(376\) 4.41742 0.227811
\(377\) −43.9129 −2.26163
\(378\) 8.95644 0.460670
\(379\) −6.37386 −0.327403 −0.163702 0.986510i \(-0.552343\pi\)
−0.163702 + 0.986510i \(0.552343\pi\)
\(380\) 0 0
\(381\) −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
\(385\) 0 0
\(386\) 20.7477 1.05603
\(387\) −53.4955 −2.71933
\(388\) 7.95644 0.403927
\(389\) −20.7042 −1.04974 −0.524871 0.851182i \(-0.675887\pi\)
−0.524871 + 0.851182i \(0.675887\pi\)
\(390\) 0 0
\(391\) 0.791288 0.0400171
\(392\) −3.79129 −0.191489
\(393\) −25.5826 −1.29047
\(394\) 11.5390 0.581327
\(395\) 0 0
\(396\) −3.79129 −0.190519
\(397\) 15.5390 0.779881 0.389940 0.920840i \(-0.372496\pi\)
0.389940 + 0.920840i \(0.372496\pi\)
\(398\) −16.3303 −0.818564
\(399\) 28.9564 1.44964
\(400\) 0 0
\(401\) −4.74773 −0.237090 −0.118545 0.992949i \(-0.537823\pi\)
−0.118545 + 0.992949i \(0.537823\pi\)
\(402\) −31.1652 −1.55438
\(403\) 19.5390 0.973308
\(404\) 4.41742 0.219775
\(405\) 0 0
\(406\) 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 \(0.648641\pi\)
\(410\) 0 0
\(411\) −10.5826 −0.522000
\(412\) 6.37386 0.314018
\(413\) −24.3303 −1.19722
\(414\) −4.79129 −0.235479
\(415\) 0 0
\(416\) −5.79129 −0.283941
\(417\) 35.5826 1.74249
\(418\) −4.58258 −0.224141
\(419\) 20.8348 1.01785 0.508924 0.860811i \(-0.330043\pi\)
0.508924 + 0.860811i \(0.330043\pi\)
\(420\) 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
\(423\) 21.1652 1.02908
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 23.3739 1.13247
\(427\) 18.5826 0.899274
\(428\) −4.41742 −0.213524
\(429\) 12.7913 0.617569
\(430\) 0 0
\(431\) 25.9129 1.24818 0.624090 0.781353i \(-0.285470\pi\)
0.624090 + 0.781353i \(0.285470\pi\)
\(432\) 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
\(435\) 0 0
\(436\) −3.37386 −0.161579
\(437\) −5.79129 −0.277035
\(438\) −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 0
\(441\) −18.1652 −0.865007
\(442\) 4.58258 0.217971
\(443\) 39.7913 1.89054 0.945271 0.326288i \(-0.105798\pi\)
0.945271 + 0.326288i \(0.105798\pi\)
\(444\) 11.1652 0.529875
\(445\) 0 0
\(446\) 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 \(0.624218\pi\)
\(450\) 0 0
\(451\) 5.37386 0.253045
\(452\) −6.00000 −0.282216
\(453\) −30.1216 −1.41524
\(454\) 22.7477 1.06760
\(455\) 0 0
\(456\) 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 \(-0.425352\pi\)
0.232370 + 0.972628i \(0.425352\pi\)
\(464\) 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 \(0.347588\pi\)
\(468\) −27.7477 −1.28264
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) 41.1652 1.89679
\(472\) −13.5826 −0.625189
\(473\) 8.83485 0.406227
\(474\) 22.3303 1.02566
\(475\) 0 0
\(476\) −1.41742 −0.0649675
\(477\) −28.7477 −1.31627
\(478\) −15.1652 −0.693638
\(479\) 15.4955 0.708005 0.354003 0.935244i \(-0.384820\pi\)
0.354003 + 0.935244i \(0.384820\pi\)
\(480\) 0 0
\(481\) −23.1652 −1.05624
\(482\) −28.0000 −1.27537
\(483\) −5.00000 −0.227508
\(484\) −10.3739 −0.471539
\(485\) 0 0
\(486\) −16.1652 −0.733266
\(487\) −6.41742 −0.290801 −0.145401 0.989373i \(-0.546447\pi\)
−0.145401 + 0.989373i \(0.546447\pi\)
\(488\) 10.3739 0.469603
\(489\) −24.0780 −1.08885
\(490\) 0 0
\(491\) 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
\(495\) 0 0
\(496\) −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 \(0.465485\pi\)
\(500\) 0 0
\(501\) −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
\(505\) 0 0
\(506\) 0.791288 0.0351770
\(507\) 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 \(0.220215\pi\)
\(510\) 0 0
\(511\) −22.8348 −1.01015
\(512\) 1.00000 0.0441942
\(513\) 28.9564 1.27846
\(514\) 4.74773 0.209413
\(515\) 0 0
\(516\) −31.1652 −1.37197
\(517\) −3.49545 −0.153730
\(518\) 7.16515 0.314819
\(519\) 52.4519 2.30238
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 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
\(525\) 0 0
\(526\) −11.2087 −0.488723
\(527\) 2.66970 0.116294
\(528\) −2.20871 −0.0961219
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −65.0780 −2.82415
\(532\) 10.3739 0.449764
\(533\) 39.3303 1.70358
\(534\) 42.3303 1.83181
\(535\) 0 0
\(536\) −11.1652 −0.482261
\(537\) 30.0000 1.29460
\(538\) −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
\(543\) −51.7477 −2.22071
\(544\) −0.791288 −0.0339262
\(545\) 0 0
\(546\) −28.9564 −1.23922
\(547\) 26.1216 1.11688 0.558439 0.829545i \(-0.311400\pi\)
0.558439 + 0.829545i \(0.311400\pi\)
\(548\) −3.79129 −0.161956
\(549\) 49.7042 2.12132
\(550\) 0 0
\(551\) 43.9129 1.87075
\(552\) −2.79129 −0.118805
\(553\) 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 \(0.457182\pi\)
\(558\) −16.1652 −0.684325
\(559\) 64.6606 2.73485
\(560\) 0 0
\(561\) 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
\(565\) 0 0
\(566\) −8.33030 −0.350149
\(567\) −0.747727 −0.0314016
\(568\) 8.37386 0.351360
\(569\) −39.4955 −1.65574 −0.827868 0.560923i \(-0.810446\pi\)
−0.827868 + 0.560923i \(0.810446\pi\)
\(570\) 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
\(573\) 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 \(-0.827590\pi\)
−0.856864 + 0.515543i \(0.827590\pi\)
\(578\) −16.3739 −0.681063
\(579\) 57.9129 2.40678
\(580\) 0 0
\(581\) 10.7477 0.445891
\(582\) 22.2087 0.920581
\(583\) 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 \(0.280816\pi\)
\(588\) −10.5826 −0.436418
\(589\) −19.5390 −0.805091
\(590\) 0 0
\(591\) 32.2087 1.32489
\(592\) 4.00000 0.164399
\(593\) 31.9129 1.31050 0.655252 0.755410i \(-0.272562\pi\)
0.655252 + 0.755410i \(0.272562\pi\)
\(594\) −3.95644 −0.162335
\(595\) 0 0
\(596\) 8.20871 0.336242
\(597\) −45.5826 −1.86557
\(598\) 5.79129 0.236823
\(599\) 1.12159 0.0458270 0.0229135 0.999737i \(-0.492706\pi\)
0.0229135 + 0.999737i \(0.492706\pi\)
\(600\) 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
\(603\) −53.4955 −2.17850
\(604\) −10.7913 −0.439091
\(605\) 0 0
\(606\) 12.3303 0.500884
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 5.79129 0.234868
\(609\) 37.9129 1.53631
\(610\) 0 0
\(611\) −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
\(615\) 0 0
\(616\) −1.41742 −0.0571097
\(617\) 23.8693 0.960943 0.480471 0.877010i \(-0.340466\pi\)
0.480471 + 0.877010i \(0.340466\pi\)
\(618\) 17.7913 0.715671
\(619\) −1.79129 −0.0719979 −0.0359990 0.999352i \(-0.511461\pi\)
−0.0359990 + 0.999352i \(0.511461\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 12.0000 0.481156
\(623\) 27.1652 1.08835
\(624\) −16.1652 −0.647124
\(625\) 0 0
\(626\) 4.62614 0.184898
\(627\) −12.7913 −0.510835
\(628\) 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
\(633\) −27.9129 −1.10944
\(634\) −9.79129 −0.388862
\(635\) 0 0
\(636\) −16.7477 −0.664091
\(637\) 21.9564 0.869946
\(638\) −6.00000 −0.237542
\(639\) 40.1216 1.58719
\(640\) 0 0
\(641\) 15.1652 0.598987 0.299494 0.954098i \(-0.403182\pi\)
0.299494 + 0.954098i \(0.403182\pi\)
\(642\) −12.3303 −0.486638
\(643\) −6.74773 −0.266104 −0.133052 0.991109i \(-0.542478\pi\)
−0.133052 + 0.991109i \(0.542478\pi\)
\(644\) −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 \(0.636584\pi\)
\(648\) −0.417424 −0.0163980
\(649\) 10.7477 0.421885
\(650\) 0 0
\(651\) −16.8693 −0.661161
\(652\) −8.62614 −0.337826
\(653\) −3.46099 −0.135439 −0.0677194 0.997704i \(-0.521572\pi\)
−0.0677194 + 0.997704i \(0.521572\pi\)
\(654\) −9.41742 −0.368250
\(655\) 0 0
\(656\) −6.79129 −0.265155
\(657\) −61.0780 −2.38288
\(658\) 7.91288 0.308476
\(659\) −8.83485 −0.344157 −0.172078 0.985083i \(-0.555048\pi\)
−0.172078 + 0.985083i \(0.555048\pi\)
\(660\) 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
\(663\) 12.7913 0.496772
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 19.1652 0.742635
\(667\) −7.58258 −0.293599
\(668\) −18.3303 −0.709221
\(669\) 20.0000 0.773245
\(670\) 0 0
\(671\) −8.20871 −0.316894
\(672\) 5.00000 0.192879
\(673\) −38.0000 −1.46479 −0.732396 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) 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 \(0.805910\pi\)
\(678\) −16.7477 −0.643193
\(679\) 14.2523 0.546952
\(680\) 0 0
\(681\) 63.4955 2.43315
\(682\) 2.66970 0.102228
\(683\) 11.3739 0.435209 0.217604 0.976037i \(-0.430176\pi\)
0.217604 + 0.976037i \(0.430176\pi\)
\(684\) 27.7477 1.06096
\(685\) 0 0
\(686\) −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
\(695\) 0 0
\(696\) 21.1652 0.802263
\(697\) 5.37386 0.203550
\(698\) 26.0000 0.984115
\(699\) 4.41742 0.167082
\(700\) 0 0
\(701\) 23.3739 0.882819 0.441409 0.897306i \(-0.354479\pi\)
0.441409 + 0.897306i \(0.354479\pi\)
\(702\) −28.9564 −1.09289
\(703\) 23.1652 0.873690
\(704\) −0.791288 −0.0298228
\(705\) 0 0
\(706\) −3.16515 −0.119122
\(707\) 7.91288 0.297594
\(708\) −37.9129 −1.42485
\(709\) 2.46099 0.0924242 0.0462121 0.998932i \(-0.485285\pi\)
0.0462121 + 0.998932i \(0.485285\pi\)
\(710\) 0 0
\(711\) 38.3303 1.43750
\(712\) 15.1652 0.568338
\(713\) 3.37386 0.126352
\(714\) −3.95644 −0.148066
\(715\) 0 0
\(716\) 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 \(0.484924\pi\)
\(720\) 0 0
\(721\) 11.4174 0.425207
\(722\) 14.5390 0.541086
\(723\) −78.1561 −2.90666
\(724\) −18.5390 −0.688997
\(725\) 0 0
\(726\) −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
\(730\) 0 0
\(731\) 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
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 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 \(0.452992\pi\)
\(740\) 0 0
\(741\) −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
\(745\) 0 0
\(746\) −12.7477 −0.466727
\(747\) 28.7477 1.05182
\(748\) 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
\(753\) 73.1561 2.66596
\(754\) −43.9129 −1.59921
\(755\) 0 0
\(756\) 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
\(759\) 2.20871 0.0801712
\(760\) 0 0
\(761\) −11.0436 −0.400329 −0.200164 0.979762i \(-0.564148\pi\)
−0.200164 + 0.979762i \(0.564148\pi\)
\(762\) −35.5826 −1.28902
\(763\) −6.04356 −0.218792
\(764\) 25.5826 0.925545
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 78.6606 2.84027
\(768\) 2.79129 0.100722
\(769\) −40.3303 −1.45435 −0.727174 0.686453i \(-0.759167\pi\)
−0.727174 + 0.686453i \(0.759167\pi\)
\(770\) 0 0
\(771\) 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
\(775\) 0 0
\(776\) 7.95644 0.285620
\(777\) 20.0000 0.717496
\(778\) −20.7042 −0.742280
\(779\) −39.3303 −1.40915
\(780\) 0 0
\(781\) −6.62614 −0.237102
\(782\) 0.791288 0.0282964
\(783\) 37.9129 1.35490
\(784\) −3.79129 −0.135403
\(785\) 0 0
\(786\) −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
\(790\) 0 0
\(791\) −10.7477 −0.382145
\(792\) −3.79129 −0.134718
\(793\) −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 \(0.476938\pi\)
\(798\) 28.9564 1.02505
\(799\) −3.49545 −0.123660
\(800\) 0 0
\(801\) 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 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
\(813\) 50.5826 1.77401
\(814\) −3.16515 −0.110938
\(815\) 0 0
\(816\) −2.20871 −0.0773204
\(817\) −64.6606 −2.26219
\(818\) −18.2087 −0.636653
\(819\) −49.7042 −1.73680
\(820\) 0 0
\(821\) 21.1652 0.738669 0.369334 0.929297i \(-0.379586\pi\)
0.369334 + 0.929297i \(0.379586\pi\)
\(822\) −10.5826 −0.369110
\(823\) −22.8348 −0.795973 −0.397986 0.917391i \(-0.630291\pi\)
−0.397986 + 0.917391i \(0.630291\pi\)
\(824\) 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 \(0.368576\pi\)
\(828\) −4.79129 −0.166509
\(829\) 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
\(832\) −5.79129 −0.200777
\(833\) 3.00000 0.103944
\(834\) 35.5826 1.23212
\(835\) 0 0
\(836\) −4.58258 −0.158492
\(837\) −16.8693 −0.583089
\(838\) 20.8348 0.719728
\(839\) −31.5826 −1.09035 −0.545176 0.838322i \(-0.683537\pi\)
−0.545176 + 0.838322i \(0.683537\pi\)
\(840\) 0 0
\(841\) 28.4955 0.982602
\(842\) 18.1216 0.624511
\(843\) 30.0000 1.03325
\(844\) −10.0000 −0.344214
\(845\) 0 0
\(846\) 21.1652 0.727673
\(847\) −18.5826 −0.638505
\(848\) −6.00000 −0.206041
\(849\) −23.2523 −0.798016
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) 23.3739 0.800775
\(853\) −40.5390 −1.38803 −0.694015 0.719961i \(-0.744160\pi\)
−0.694015 + 0.719961i \(0.744160\pi\)
\(854\) 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 \(0.449967\pi\)
\(858\) 12.7913 0.436687
\(859\) −26.7477 −0.912621 −0.456310 0.889821i \(-0.650829\pi\)
−0.456310 + 0.889821i \(0.650829\pi\)
\(860\) 0 0
\(861\) −33.9564 −1.15723
\(862\) 25.9129 0.882596
\(863\) 22.4174 0.763098 0.381549 0.924349i \(-0.375391\pi\)
0.381549 + 0.924349i \(0.375391\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) 30.5390 1.03776
\(867\) −45.7042 −1.55219
\(868\) −6.04356 −0.205132
\(869\) −6.33030 −0.214741
\(870\) 0 0
\(871\) 64.6606 2.19094
\(872\) −3.37386 −0.114253
\(873\) 38.1216 1.29022
\(874\) −5.79129 −0.195893
\(875\) 0 0
\(876\) −35.5826 −1.20222
\(877\) 42.7042 1.44202 0.721009 0.692926i \(-0.243679\pi\)
0.721009 + 0.692926i \(0.243679\pi\)
\(878\) −6.53901 −0.220681
\(879\) −76.7477 −2.58864
\(880\) 0 0
\(881\) 30.3303 1.02185 0.510927 0.859624i \(-0.329302\pi\)
0.510927 + 0.859624i \(0.329302\pi\)
\(882\) −18.1652 −0.611652
\(883\) 34.9564 1.17638 0.588189 0.808724i \(-0.299841\pi\)
0.588189 + 0.808724i \(0.299841\pi\)
\(884\) 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 \(0.581943\pi\)
\(888\) 11.1652 0.374678
\(889\) −22.8348 −0.765856
\(890\) 0 0
\(891\) 0.330303 0.0110656
\(892\) 7.16515 0.239907
\(893\) 25.5826 0.856088
\(894\) 22.9129 0.766321
\(895\) 0 0
\(896\) 1.79129 0.0598427
\(897\) 16.1652 0.539739
\(898\) −16.1216 −0.537984
\(899\) −25.5826 −0.853227
\(900\) 0 0
\(901\) 4.74773 0.158170
\(902\) 5.37386 0.178930
\(903\) −55.8258 −1.85776
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −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
\(910\) 0 0
\(911\) 4.41742 0.146356 0.0731779 0.997319i \(-0.476686\pi\)
0.0731779 + 0.997319i \(0.476686\pi\)
\(912\) 16.1652 0.535282
\(913\) −4.74773 −0.157127
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 20.3303 0.671732
\(917\) −16.4174 −0.542151
\(918\) −3.95644 −0.130582
\(919\) −55.1652 −1.81973 −0.909865 0.414904i \(-0.863815\pi\)
−0.909865 + 0.414904i \(0.863815\pi\)
\(920\) 0 0
\(921\) 43.3739 1.42922
\(922\) −28.7477 −0.946756
\(923\) −48.4955 −1.59625
\(924\) −3.95644 −0.130157
\(925\) 0 0
\(926\) 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 \(0.418190\pi\)
\(930\) 0 0
\(931\) −21.9564 −0.719593
\(932\) 1.58258 0.0518390
\(933\) 33.4955 1.09659
\(934\) 19.9129 0.651569
\(935\) 0 0
\(936\) −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
\(940\) 0 0
\(941\) 32.0436 1.04459 0.522295 0.852765i \(-0.325076\pi\)
0.522295 + 0.852765i \(0.325076\pi\)
\(942\) 41.1652 1.34123
\(943\) 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 \(0.513135\pi\)
\(948\) 22.3303 0.725255
\(949\) 73.8258 2.39649
\(950\) 0 0
\(951\) −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
\(955\) 0 0
\(956\) −15.1652 −0.490476
\(957\) −16.7477 −0.541377
\(958\) 15.4955 0.500635
\(959\) −6.79129 −0.219302
\(960\) 0 0
\(961\) −19.6170 −0.632808
\(962\) −23.1652 −0.746874
\(963\) −21.1652 −0.682037
\(964\) −28.0000 −0.901819
\(965\) 0 0
\(966\) −5.00000 −0.160872
\(967\) 32.7477 1.05310 0.526548 0.850145i \(-0.323486\pi\)
0.526548 + 0.850145i \(0.323486\pi\)
\(968\) −10.3739 −0.333429
\(969\) −12.7913 −0.410915
\(970\) 0 0
\(971\) 15.9564 0.512067 0.256033 0.966668i \(-0.417584\pi\)
0.256033 + 0.966668i \(0.417584\pi\)
\(972\) −16.1652 −0.518497
\(973\) 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 \(0.316217\pi\)
\(978\) −24.0780 −0.769930
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) −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
\(985\) 0 0
\(986\) −6.00000 −0.191079
\(987\) 22.0871 0.703041
\(988\) −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
\(993\) −57.9129 −1.83781
\(994\) 15.0000 0.475771
\(995\) 0 0
\(996\) 16.7477 0.530672
\(997\) 43.4955 1.37751 0.688757 0.724992i \(-0.258157\pi\)
0.688757 + 0.724992i \(0.258157\pi\)
\(998\) 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 1150.2.a.o.1.2 2
4.3 odd 2 9200.2.a.bs.1.1 2
5.2 odd 4 1150.2.b.g.599.3 4
5.3 odd 4 1150.2.b.g.599.2 4
5.4 even 2 230.2.a.a.1.1 2
15.14 odd 2 2070.2.a.x.1.1 2
20.19 odd 2 1840.2.a.n.1.2 2
40.19 odd 2 7360.2.a.bk.1.1 2
40.29 even 2 7360.2.a.bq.1.2 2
115.114 odd 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 5.4 even 2
1150.2.a.o.1.2 2 1.1 even 1 trivial
1150.2.b.g.599.2 4 5.3 odd 4
1150.2.b.g.599.3 4 5.2 odd 4
1840.2.a.n.1.2 2 20.19 odd 2
2070.2.a.x.1.1 2 15.14 odd 2
5290.2.a.e.1.1 2 115.114 odd 2
7360.2.a.bk.1.1 2 40.19 odd 2
7360.2.a.bq.1.2 2 40.29 even 2
9200.2.a.bs.1.1 2 4.3 odd 2