Properties

Label 1134.2.a.p.1.2
Level $1134$
Weight $2$
Character 1134.1
Self dual yes
Analytic conductor $9.055$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 1134.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.44949 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.44949 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.44949 q^{10} +2.00000 q^{11} -4.89898 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +7.44949 q^{19} +3.44949 q^{20} +2.00000 q^{22} -1.00000 q^{23} +6.89898 q^{25} -4.89898 q^{26} -1.00000 q^{28} +2.89898 q^{29} +6.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} -3.44949 q^{35} -7.79796 q^{37} +7.44949 q^{38} +3.44949 q^{40} -9.79796 q^{41} -2.89898 q^{43} +2.00000 q^{44} -1.00000 q^{46} -9.79796 q^{47} +1.00000 q^{49} +6.89898 q^{50} -4.89898 q^{52} -1.10102 q^{53} +6.89898 q^{55} -1.00000 q^{56} +2.89898 q^{58} -2.00000 q^{59} +11.4495 q^{61} +6.00000 q^{62} +1.00000 q^{64} -16.8990 q^{65} -3.10102 q^{67} +2.00000 q^{68} -3.44949 q^{70} +9.89898 q^{71} +2.89898 q^{73} -7.79796 q^{74} +7.44949 q^{76} -2.00000 q^{77} +7.89898 q^{79} +3.44949 q^{80} -9.79796 q^{82} +2.00000 q^{83} +6.89898 q^{85} -2.89898 q^{86} +2.00000 q^{88} -7.10102 q^{89} +4.89898 q^{91} -1.00000 q^{92} -9.79796 q^{94} +25.6969 q^{95} -6.89898 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 2q^{8} + 2q^{10} + 4q^{11} - 2q^{14} + 2q^{16} + 4q^{17} + 10q^{19} + 2q^{20} + 4q^{22} - 2q^{23} + 4q^{25} - 2q^{28} - 4q^{29} + 12q^{31} + 2q^{32} + 4q^{34} - 2q^{35} + 4q^{37} + 10q^{38} + 2q^{40} + 4q^{43} + 4q^{44} - 2q^{46} + 2q^{49} + 4q^{50} - 12q^{53} + 4q^{55} - 2q^{56} - 4q^{58} - 4q^{59} + 18q^{61} + 12q^{62} + 2q^{64} - 24q^{65} - 16q^{67} + 4q^{68} - 2q^{70} + 10q^{71} - 4q^{73} + 4q^{74} + 10q^{76} - 4q^{77} + 6q^{79} + 2q^{80} + 4q^{83} + 4q^{85} + 4q^{86} + 4q^{88} - 24q^{89} - 2q^{92} + 22q^{95} - 4q^{97} + 2q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.44949 1.54266 0.771329 0.636436i \(-0.219592\pi\)
0.771329 + 0.636436i \(0.219592\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.44949 1.09082
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −4.89898 −1.35873 −0.679366 0.733799i \(-0.737745\pi\)
−0.679366 + 0.733799i \(0.737745\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 7.44949 1.70903 0.854515 0.519427i \(-0.173854\pi\)
0.854515 + 0.519427i \(0.173854\pi\)
\(20\) 3.44949 0.771329
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) 6.89898 1.37980
\(26\) −4.89898 −0.960769
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 2.89898 0.538327 0.269163 0.963095i \(-0.413253\pi\)
0.269163 + 0.963095i \(0.413253\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −3.44949 −0.583070
\(36\) 0 0
\(37\) −7.79796 −1.28198 −0.640988 0.767551i \(-0.721475\pi\)
−0.640988 + 0.767551i \(0.721475\pi\)
\(38\) 7.44949 1.20847
\(39\) 0 0
\(40\) 3.44949 0.545412
\(41\) −9.79796 −1.53018 −0.765092 0.643921i \(-0.777307\pi\)
−0.765092 + 0.643921i \(0.777307\pi\)
\(42\) 0 0
\(43\) −2.89898 −0.442090 −0.221045 0.975264i \(-0.570947\pi\)
−0.221045 + 0.975264i \(0.570947\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −9.79796 −1.42918 −0.714590 0.699544i \(-0.753387\pi\)
−0.714590 + 0.699544i \(0.753387\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 6.89898 0.975663
\(51\) 0 0
\(52\) −4.89898 −0.679366
\(53\) −1.10102 −0.151237 −0.0756184 0.997137i \(-0.524093\pi\)
−0.0756184 + 0.997137i \(0.524093\pi\)
\(54\) 0 0
\(55\) 6.89898 0.930258
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 2.89898 0.380655
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) 11.4495 1.46596 0.732978 0.680252i \(-0.238130\pi\)
0.732978 + 0.680252i \(0.238130\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −16.8990 −2.09606
\(66\) 0 0
\(67\) −3.10102 −0.378850 −0.189425 0.981895i \(-0.560662\pi\)
−0.189425 + 0.981895i \(0.560662\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −3.44949 −0.412293
\(71\) 9.89898 1.17479 0.587396 0.809299i \(-0.300153\pi\)
0.587396 + 0.809299i \(0.300153\pi\)
\(72\) 0 0
\(73\) 2.89898 0.339300 0.169650 0.985504i \(-0.445736\pi\)
0.169650 + 0.985504i \(0.445736\pi\)
\(74\) −7.79796 −0.906494
\(75\) 0 0
\(76\) 7.44949 0.854515
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) 7.89898 0.888705 0.444352 0.895852i \(-0.353434\pi\)
0.444352 + 0.895852i \(0.353434\pi\)
\(80\) 3.44949 0.385665
\(81\) 0 0
\(82\) −9.79796 −1.08200
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 6.89898 0.748299
\(86\) −2.89898 −0.312605
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) −7.10102 −0.752707 −0.376353 0.926476i \(-0.622822\pi\)
−0.376353 + 0.926476i \(0.622822\pi\)
\(90\) 0 0
\(91\) 4.89898 0.513553
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −9.79796 −1.01058
\(95\) 25.6969 2.63645
\(96\) 0 0
\(97\) −6.89898 −0.700485 −0.350243 0.936659i \(-0.613901\pi\)
−0.350243 + 0.936659i \(0.613901\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 6.89898 0.689898
\(101\) 7.24745 0.721148 0.360574 0.932731i \(-0.382581\pi\)
0.360574 + 0.932731i \(0.382581\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −4.89898 −0.480384
\(105\) 0 0
\(106\) −1.10102 −0.106941
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −16.6969 −1.59928 −0.799638 0.600482i \(-0.794975\pi\)
−0.799638 + 0.600482i \(0.794975\pi\)
\(110\) 6.89898 0.657792
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −15.8990 −1.49565 −0.747825 0.663896i \(-0.768902\pi\)
−0.747825 + 0.663896i \(0.768902\pi\)
\(114\) 0 0
\(115\) −3.44949 −0.321667
\(116\) 2.89898 0.269163
\(117\) 0 0
\(118\) −2.00000 −0.184115
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 11.4495 1.03659
\(123\) 0 0
\(124\) 6.00000 0.538816
\(125\) 6.55051 0.585895
\(126\) 0 0
\(127\) −3.00000 −0.266207 −0.133103 0.991102i \(-0.542494\pi\)
−0.133103 + 0.991102i \(0.542494\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −16.8990 −1.48214
\(131\) −13.4495 −1.17509 −0.587544 0.809192i \(-0.699905\pi\)
−0.587544 + 0.809192i \(0.699905\pi\)
\(132\) 0 0
\(133\) −7.44949 −0.645953
\(134\) −3.10102 −0.267887
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −11.7980 −1.00797 −0.503984 0.863713i \(-0.668133\pi\)
−0.503984 + 0.863713i \(0.668133\pi\)
\(138\) 0 0
\(139\) 9.44949 0.801495 0.400748 0.916188i \(-0.368750\pi\)
0.400748 + 0.916188i \(0.368750\pi\)
\(140\) −3.44949 −0.291535
\(141\) 0 0
\(142\) 9.89898 0.830704
\(143\) −9.79796 −0.819346
\(144\) 0 0
\(145\) 10.0000 0.830455
\(146\) 2.89898 0.239921
\(147\) 0 0
\(148\) −7.79796 −0.640988
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 7.44949 0.604233
\(153\) 0 0
\(154\) −2.00000 −0.161165
\(155\) 20.6969 1.66242
\(156\) 0 0
\(157\) 6.34847 0.506663 0.253332 0.967380i \(-0.418474\pi\)
0.253332 + 0.967380i \(0.418474\pi\)
\(158\) 7.89898 0.628409
\(159\) 0 0
\(160\) 3.44949 0.272706
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −0.202041 −0.0158251 −0.00791254 0.999969i \(-0.502519\pi\)
−0.00791254 + 0.999969i \(0.502519\pi\)
\(164\) −9.79796 −0.765092
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) 18.6969 1.44681 0.723406 0.690423i \(-0.242575\pi\)
0.723406 + 0.690423i \(0.242575\pi\)
\(168\) 0 0
\(169\) 11.0000 0.846154
\(170\) 6.89898 0.529128
\(171\) 0 0
\(172\) −2.89898 −0.221045
\(173\) −12.8990 −0.980691 −0.490346 0.871528i \(-0.663129\pi\)
−0.490346 + 0.871528i \(0.663129\pi\)
\(174\) 0 0
\(175\) −6.89898 −0.521514
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) −7.10102 −0.532244
\(179\) −8.69694 −0.650040 −0.325020 0.945707i \(-0.605371\pi\)
−0.325020 + 0.945707i \(0.605371\pi\)
\(180\) 0 0
\(181\) 4.34847 0.323219 0.161610 0.986855i \(-0.448331\pi\)
0.161610 + 0.986855i \(0.448331\pi\)
\(182\) 4.89898 0.363137
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −26.8990 −1.97765
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) −9.79796 −0.714590
\(189\) 0 0
\(190\) 25.6969 1.86425
\(191\) 13.8990 1.00569 0.502847 0.864375i \(-0.332286\pi\)
0.502847 + 0.864375i \(0.332286\pi\)
\(192\) 0 0
\(193\) −8.10102 −0.583124 −0.291562 0.956552i \(-0.594175\pi\)
−0.291562 + 0.956552i \(0.594175\pi\)
\(194\) −6.89898 −0.495318
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −12.6969 −0.904619 −0.452310 0.891861i \(-0.649400\pi\)
−0.452310 + 0.891861i \(0.649400\pi\)
\(198\) 0 0
\(199\) 6.89898 0.489056 0.244528 0.969642i \(-0.421367\pi\)
0.244528 + 0.969642i \(0.421367\pi\)
\(200\) 6.89898 0.487832
\(201\) 0 0
\(202\) 7.24745 0.509929
\(203\) −2.89898 −0.203468
\(204\) 0 0
\(205\) −33.7980 −2.36055
\(206\) 14.0000 0.975426
\(207\) 0 0
\(208\) −4.89898 −0.339683
\(209\) 14.8990 1.03058
\(210\) 0 0
\(211\) 3.10102 0.213483 0.106742 0.994287i \(-0.465958\pi\)
0.106742 + 0.994287i \(0.465958\pi\)
\(212\) −1.10102 −0.0756184
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −10.0000 −0.681994
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) −16.6969 −1.13086
\(219\) 0 0
\(220\) 6.89898 0.465129
\(221\) −9.79796 −0.659082
\(222\) 0 0
\(223\) −20.8990 −1.39950 −0.699750 0.714388i \(-0.746705\pi\)
−0.699750 + 0.714388i \(0.746705\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −15.8990 −1.05758
\(227\) −0.550510 −0.0365386 −0.0182693 0.999833i \(-0.505816\pi\)
−0.0182693 + 0.999833i \(0.505816\pi\)
\(228\) 0 0
\(229\) −23.2474 −1.53623 −0.768117 0.640309i \(-0.778806\pi\)
−0.768117 + 0.640309i \(0.778806\pi\)
\(230\) −3.44949 −0.227453
\(231\) 0 0
\(232\) 2.89898 0.190327
\(233\) −7.00000 −0.458585 −0.229293 0.973358i \(-0.573641\pi\)
−0.229293 + 0.973358i \(0.573641\pi\)
\(234\) 0 0
\(235\) −33.7980 −2.20474
\(236\) −2.00000 −0.130189
\(237\) 0 0
\(238\) −2.00000 −0.129641
\(239\) −12.7980 −0.827831 −0.413916 0.910315i \(-0.635839\pi\)
−0.413916 + 0.910315i \(0.635839\pi\)
\(240\) 0 0
\(241\) −8.89898 −0.573234 −0.286617 0.958045i \(-0.592531\pi\)
−0.286617 + 0.958045i \(0.592531\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) 11.4495 0.732978
\(245\) 3.44949 0.220380
\(246\) 0 0
\(247\) −36.4949 −2.32211
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) 6.55051 0.414291
\(251\) 12.5505 0.792181 0.396091 0.918211i \(-0.370367\pi\)
0.396091 + 0.918211i \(0.370367\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) −3.00000 −0.188237
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 27.7980 1.73399 0.866995 0.498318i \(-0.166049\pi\)
0.866995 + 0.498318i \(0.166049\pi\)
\(258\) 0 0
\(259\) 7.79796 0.484542
\(260\) −16.8990 −1.04803
\(261\) 0 0
\(262\) −13.4495 −0.830912
\(263\) 16.1010 0.992831 0.496416 0.868085i \(-0.334649\pi\)
0.496416 + 0.868085i \(0.334649\pi\)
\(264\) 0 0
\(265\) −3.79796 −0.233307
\(266\) −7.44949 −0.456758
\(267\) 0 0
\(268\) −3.10102 −0.189425
\(269\) −3.65153 −0.222638 −0.111319 0.993785i \(-0.535507\pi\)
−0.111319 + 0.993785i \(0.535507\pi\)
\(270\) 0 0
\(271\) 16.8990 1.02654 0.513270 0.858227i \(-0.328434\pi\)
0.513270 + 0.858227i \(0.328434\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −11.7980 −0.712741
\(275\) 13.7980 0.832048
\(276\) 0 0
\(277\) 10.6969 0.642717 0.321358 0.946958i \(-0.395861\pi\)
0.321358 + 0.946958i \(0.395861\pi\)
\(278\) 9.44949 0.566743
\(279\) 0 0
\(280\) −3.44949 −0.206146
\(281\) −19.0000 −1.13344 −0.566722 0.823909i \(-0.691789\pi\)
−0.566722 + 0.823909i \(0.691789\pi\)
\(282\) 0 0
\(283\) −20.5505 −1.22160 −0.610801 0.791785i \(-0.709152\pi\)
−0.610801 + 0.791785i \(0.709152\pi\)
\(284\) 9.89898 0.587396
\(285\) 0 0
\(286\) −9.79796 −0.579365
\(287\) 9.79796 0.578355
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 10.0000 0.587220
\(291\) 0 0
\(292\) 2.89898 0.169650
\(293\) 27.2474 1.59181 0.795906 0.605420i \(-0.206995\pi\)
0.795906 + 0.605420i \(0.206995\pi\)
\(294\) 0 0
\(295\) −6.89898 −0.401674
\(296\) −7.79796 −0.453247
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 4.89898 0.283315
\(300\) 0 0
\(301\) 2.89898 0.167094
\(302\) −5.00000 −0.287718
\(303\) 0 0
\(304\) 7.44949 0.427258
\(305\) 39.4949 2.26147
\(306\) 0 0
\(307\) 0.752551 0.0429504 0.0214752 0.999769i \(-0.493164\pi\)
0.0214752 + 0.999769i \(0.493164\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) 20.6969 1.17551
\(311\) 1.30306 0.0738898 0.0369449 0.999317i \(-0.488237\pi\)
0.0369449 + 0.999317i \(0.488237\pi\)
\(312\) 0 0
\(313\) 24.6969 1.39595 0.697977 0.716120i \(-0.254084\pi\)
0.697977 + 0.716120i \(0.254084\pi\)
\(314\) 6.34847 0.358265
\(315\) 0 0
\(316\) 7.89898 0.444352
\(317\) −8.69694 −0.488469 −0.244234 0.969716i \(-0.578537\pi\)
−0.244234 + 0.969716i \(0.578537\pi\)
\(318\) 0 0
\(319\) 5.79796 0.324623
\(320\) 3.44949 0.192832
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) 14.8990 0.829001
\(324\) 0 0
\(325\) −33.7980 −1.87477
\(326\) −0.202041 −0.0111900
\(327\) 0 0
\(328\) −9.79796 −0.541002
\(329\) 9.79796 0.540179
\(330\) 0 0
\(331\) −24.6969 −1.35747 −0.678733 0.734385i \(-0.737471\pi\)
−0.678733 + 0.734385i \(0.737471\pi\)
\(332\) 2.00000 0.109764
\(333\) 0 0
\(334\) 18.6969 1.02305
\(335\) −10.6969 −0.584436
\(336\) 0 0
\(337\) 35.3939 1.92803 0.964014 0.265853i \(-0.0856535\pi\)
0.964014 + 0.265853i \(0.0856535\pi\)
\(338\) 11.0000 0.598321
\(339\) 0 0
\(340\) 6.89898 0.374150
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −2.89898 −0.156302
\(345\) 0 0
\(346\) −12.8990 −0.693453
\(347\) −19.5959 −1.05196 −0.525982 0.850496i \(-0.676302\pi\)
−0.525982 + 0.850496i \(0.676302\pi\)
\(348\) 0 0
\(349\) 20.8990 1.11870 0.559348 0.828933i \(-0.311051\pi\)
0.559348 + 0.828933i \(0.311051\pi\)
\(350\) −6.89898 −0.368766
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 34.1464 1.81230
\(356\) −7.10102 −0.376353
\(357\) 0 0
\(358\) −8.69694 −0.459647
\(359\) 10.7980 0.569894 0.284947 0.958543i \(-0.408024\pi\)
0.284947 + 0.958543i \(0.408024\pi\)
\(360\) 0 0
\(361\) 36.4949 1.92078
\(362\) 4.34847 0.228550
\(363\) 0 0
\(364\) 4.89898 0.256776
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) 5.79796 0.302651 0.151325 0.988484i \(-0.451646\pi\)
0.151325 + 0.988484i \(0.451646\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −26.8990 −1.39841
\(371\) 1.10102 0.0571621
\(372\) 0 0
\(373\) 2.89898 0.150103 0.0750517 0.997180i \(-0.476088\pi\)
0.0750517 + 0.997180i \(0.476088\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −9.79796 −0.505291
\(377\) −14.2020 −0.731442
\(378\) 0 0
\(379\) −26.4949 −1.36095 −0.680476 0.732771i \(-0.738227\pi\)
−0.680476 + 0.732771i \(0.738227\pi\)
\(380\) 25.6969 1.31823
\(381\) 0 0
\(382\) 13.8990 0.711134
\(383\) 6.89898 0.352521 0.176261 0.984344i \(-0.443600\pi\)
0.176261 + 0.984344i \(0.443600\pi\)
\(384\) 0 0
\(385\) −6.89898 −0.351605
\(386\) −8.10102 −0.412331
\(387\) 0 0
\(388\) −6.89898 −0.350243
\(389\) −15.1010 −0.765652 −0.382826 0.923820i \(-0.625049\pi\)
−0.382826 + 0.923820i \(0.625049\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −12.6969 −0.639663
\(395\) 27.2474 1.37097
\(396\) 0 0
\(397\) 9.30306 0.466907 0.233454 0.972368i \(-0.424997\pi\)
0.233454 + 0.972368i \(0.424997\pi\)
\(398\) 6.89898 0.345815
\(399\) 0 0
\(400\) 6.89898 0.344949
\(401\) −10.1010 −0.504421 −0.252210 0.967672i \(-0.581158\pi\)
−0.252210 + 0.967672i \(0.581158\pi\)
\(402\) 0 0
\(403\) −29.3939 −1.46421
\(404\) 7.24745 0.360574
\(405\) 0 0
\(406\) −2.89898 −0.143874
\(407\) −15.5959 −0.773061
\(408\) 0 0
\(409\) 5.79796 0.286691 0.143345 0.989673i \(-0.454214\pi\)
0.143345 + 0.989673i \(0.454214\pi\)
\(410\) −33.7980 −1.66916
\(411\) 0 0
\(412\) 14.0000 0.689730
\(413\) 2.00000 0.0984136
\(414\) 0 0
\(415\) 6.89898 0.338658
\(416\) −4.89898 −0.240192
\(417\) 0 0
\(418\) 14.8990 0.728733
\(419\) 24.5505 1.19937 0.599685 0.800236i \(-0.295292\pi\)
0.599685 + 0.800236i \(0.295292\pi\)
\(420\) 0 0
\(421\) 13.1010 0.638505 0.319252 0.947670i \(-0.396568\pi\)
0.319252 + 0.947670i \(0.396568\pi\)
\(422\) 3.10102 0.150955
\(423\) 0 0
\(424\) −1.10102 −0.0534703
\(425\) 13.7980 0.669299
\(426\) 0 0
\(427\) −11.4495 −0.554080
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −10.0000 −0.482243
\(431\) −7.59592 −0.365882 −0.182941 0.983124i \(-0.558562\pi\)
−0.182941 + 0.983124i \(0.558562\pi\)
\(432\) 0 0
\(433\) 11.7980 0.566974 0.283487 0.958976i \(-0.408509\pi\)
0.283487 + 0.958976i \(0.408509\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) −16.6969 −0.799638
\(437\) −7.44949 −0.356357
\(438\) 0 0
\(439\) 21.7980 1.04036 0.520180 0.854057i \(-0.325865\pi\)
0.520180 + 0.854057i \(0.325865\pi\)
\(440\) 6.89898 0.328896
\(441\) 0 0
\(442\) −9.79796 −0.466041
\(443\) −5.10102 −0.242357 −0.121178 0.992631i \(-0.538667\pi\)
−0.121178 + 0.992631i \(0.538667\pi\)
\(444\) 0 0
\(445\) −24.4949 −1.16117
\(446\) −20.8990 −0.989595
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −18.5959 −0.877596 −0.438798 0.898586i \(-0.644596\pi\)
−0.438798 + 0.898586i \(0.644596\pi\)
\(450\) 0 0
\(451\) −19.5959 −0.922736
\(452\) −15.8990 −0.747825
\(453\) 0 0
\(454\) −0.550510 −0.0258367
\(455\) 16.8990 0.792236
\(456\) 0 0
\(457\) 31.4949 1.47327 0.736635 0.676291i \(-0.236414\pi\)
0.736635 + 0.676291i \(0.236414\pi\)
\(458\) −23.2474 −1.08628
\(459\) 0 0
\(460\) −3.44949 −0.160833
\(461\) 20.3485 0.947723 0.473861 0.880599i \(-0.342860\pi\)
0.473861 + 0.880599i \(0.342860\pi\)
\(462\) 0 0
\(463\) −25.6969 −1.19424 −0.597119 0.802153i \(-0.703688\pi\)
−0.597119 + 0.802153i \(0.703688\pi\)
\(464\) 2.89898 0.134582
\(465\) 0 0
\(466\) −7.00000 −0.324269
\(467\) 10.0000 0.462745 0.231372 0.972865i \(-0.425678\pi\)
0.231372 + 0.972865i \(0.425678\pi\)
\(468\) 0 0
\(469\) 3.10102 0.143192
\(470\) −33.7980 −1.55898
\(471\) 0 0
\(472\) −2.00000 −0.0920575
\(473\) −5.79796 −0.266590
\(474\) 0 0
\(475\) 51.3939 2.35811
\(476\) −2.00000 −0.0916698
\(477\) 0 0
\(478\) −12.7980 −0.585365
\(479\) −29.5959 −1.35227 −0.676136 0.736777i \(-0.736347\pi\)
−0.676136 + 0.736777i \(0.736347\pi\)
\(480\) 0 0
\(481\) 38.2020 1.74186
\(482\) −8.89898 −0.405337
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −23.7980 −1.08061
\(486\) 0 0
\(487\) 22.3939 1.01476 0.507382 0.861721i \(-0.330613\pi\)
0.507382 + 0.861721i \(0.330613\pi\)
\(488\) 11.4495 0.518294
\(489\) 0 0
\(490\) 3.44949 0.155832
\(491\) −3.79796 −0.171399 −0.0856997 0.996321i \(-0.527313\pi\)
−0.0856997 + 0.996321i \(0.527313\pi\)
\(492\) 0 0
\(493\) 5.79796 0.261127
\(494\) −36.4949 −1.64198
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) −9.89898 −0.444030
\(498\) 0 0
\(499\) 33.3939 1.49492 0.747458 0.664309i \(-0.231274\pi\)
0.747458 + 0.664309i \(0.231274\pi\)
\(500\) 6.55051 0.292948
\(501\) 0 0
\(502\) 12.5505 0.560157
\(503\) 24.4949 1.09217 0.546087 0.837729i \(-0.316117\pi\)
0.546087 + 0.837729i \(0.316117\pi\)
\(504\) 0 0
\(505\) 25.0000 1.11249
\(506\) −2.00000 −0.0889108
\(507\) 0 0
\(508\) −3.00000 −0.133103
\(509\) −16.8990 −0.749034 −0.374517 0.927220i \(-0.622191\pi\)
−0.374517 + 0.927220i \(0.622191\pi\)
\(510\) 0 0
\(511\) −2.89898 −0.128243
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 27.7980 1.22612
\(515\) 48.2929 2.12804
\(516\) 0 0
\(517\) −19.5959 −0.861827
\(518\) 7.79796 0.342623
\(519\) 0 0
\(520\) −16.8990 −0.741069
\(521\) −38.6969 −1.69534 −0.847672 0.530521i \(-0.821996\pi\)
−0.847672 + 0.530521i \(0.821996\pi\)
\(522\) 0 0
\(523\) 0.348469 0.0152375 0.00761875 0.999971i \(-0.497575\pi\)
0.00761875 + 0.999971i \(0.497575\pi\)
\(524\) −13.4495 −0.587544
\(525\) 0 0
\(526\) 16.1010 0.702038
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) −3.79796 −0.164973
\(531\) 0 0
\(532\) −7.44949 −0.322976
\(533\) 48.0000 2.07911
\(534\) 0 0
\(535\) −41.3939 −1.78961
\(536\) −3.10102 −0.133944
\(537\) 0 0
\(538\) −3.65153 −0.157429
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 30.4949 1.31108 0.655539 0.755161i \(-0.272441\pi\)
0.655539 + 0.755161i \(0.272441\pi\)
\(542\) 16.8990 0.725873
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) −57.5959 −2.46714
\(546\) 0 0
\(547\) 31.5959 1.35094 0.675472 0.737386i \(-0.263940\pi\)
0.675472 + 0.737386i \(0.263940\pi\)
\(548\) −11.7980 −0.503984
\(549\) 0 0
\(550\) 13.7980 0.588347
\(551\) 21.5959 0.920017
\(552\) 0 0
\(553\) −7.89898 −0.335899
\(554\) 10.6969 0.454469
\(555\) 0 0
\(556\) 9.44949 0.400748
\(557\) −3.10102 −0.131394 −0.0656972 0.997840i \(-0.520927\pi\)
−0.0656972 + 0.997840i \(0.520927\pi\)
\(558\) 0 0
\(559\) 14.2020 0.600682
\(560\) −3.44949 −0.145768
\(561\) 0 0
\(562\) −19.0000 −0.801467
\(563\) 13.9444 0.587686 0.293843 0.955854i \(-0.405066\pi\)
0.293843 + 0.955854i \(0.405066\pi\)
\(564\) 0 0
\(565\) −54.8434 −2.30728
\(566\) −20.5505 −0.863802
\(567\) 0 0
\(568\) 9.89898 0.415352
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 14.2020 0.594337 0.297168 0.954825i \(-0.403958\pi\)
0.297168 + 0.954825i \(0.403958\pi\)
\(572\) −9.79796 −0.409673
\(573\) 0 0
\(574\) 9.79796 0.408959
\(575\) −6.89898 −0.287707
\(576\) 0 0
\(577\) 23.5959 0.982311 0.491155 0.871072i \(-0.336575\pi\)
0.491155 + 0.871072i \(0.336575\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 10.0000 0.415227
\(581\) −2.00000 −0.0829740
\(582\) 0 0
\(583\) −2.20204 −0.0911992
\(584\) 2.89898 0.119961
\(585\) 0 0
\(586\) 27.2474 1.12558
\(587\) −18.1464 −0.748983 −0.374492 0.927230i \(-0.622183\pi\)
−0.374492 + 0.927230i \(0.622183\pi\)
\(588\) 0 0
\(589\) 44.6969 1.84171
\(590\) −6.89898 −0.284026
\(591\) 0 0
\(592\) −7.79796 −0.320494
\(593\) −14.6969 −0.603531 −0.301765 0.953382i \(-0.597576\pi\)
−0.301765 + 0.953382i \(0.597576\pi\)
\(594\) 0 0
\(595\) −6.89898 −0.282831
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 4.89898 0.200334
\(599\) −14.2020 −0.580280 −0.290140 0.956984i \(-0.593702\pi\)
−0.290140 + 0.956984i \(0.593702\pi\)
\(600\) 0 0
\(601\) −12.6969 −0.517919 −0.258959 0.965888i \(-0.583380\pi\)
−0.258959 + 0.965888i \(0.583380\pi\)
\(602\) 2.89898 0.118154
\(603\) 0 0
\(604\) −5.00000 −0.203447
\(605\) −24.1464 −0.981692
\(606\) 0 0
\(607\) −8.69694 −0.352998 −0.176499 0.984301i \(-0.556477\pi\)
−0.176499 + 0.984301i \(0.556477\pi\)
\(608\) 7.44949 0.302117
\(609\) 0 0
\(610\) 39.4949 1.59910
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) 14.6969 0.593604 0.296802 0.954939i \(-0.404080\pi\)
0.296802 + 0.954939i \(0.404080\pi\)
\(614\) 0.752551 0.0303705
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) 43.3939 1.74697 0.873486 0.486850i \(-0.161854\pi\)
0.873486 + 0.486850i \(0.161854\pi\)
\(618\) 0 0
\(619\) −4.14643 −0.166659 −0.0833295 0.996522i \(-0.526555\pi\)
−0.0833295 + 0.996522i \(0.526555\pi\)
\(620\) 20.6969 0.831209
\(621\) 0 0
\(622\) 1.30306 0.0522480
\(623\) 7.10102 0.284496
\(624\) 0 0
\(625\) −11.8990 −0.475959
\(626\) 24.6969 0.987088
\(627\) 0 0
\(628\) 6.34847 0.253332
\(629\) −15.5959 −0.621850
\(630\) 0 0
\(631\) 18.1010 0.720590 0.360295 0.932838i \(-0.382676\pi\)
0.360295 + 0.932838i \(0.382676\pi\)
\(632\) 7.89898 0.314205
\(633\) 0 0
\(634\) −8.69694 −0.345400
\(635\) −10.3485 −0.410666
\(636\) 0 0
\(637\) −4.89898 −0.194105
\(638\) 5.79796 0.229543
\(639\) 0 0
\(640\) 3.44949 0.136353
\(641\) 41.4949 1.63895 0.819475 0.573115i \(-0.194265\pi\)
0.819475 + 0.573115i \(0.194265\pi\)
\(642\) 0 0
\(643\) 19.3939 0.764820 0.382410 0.923993i \(-0.375094\pi\)
0.382410 + 0.923993i \(0.375094\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 14.8990 0.586193
\(647\) −21.3031 −0.837510 −0.418755 0.908099i \(-0.637533\pi\)
−0.418755 + 0.908099i \(0.637533\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) −33.7980 −1.32567
\(651\) 0 0
\(652\) −0.202041 −0.00791254
\(653\) −9.79796 −0.383424 −0.191712 0.981451i \(-0.561404\pi\)
−0.191712 + 0.981451i \(0.561404\pi\)
\(654\) 0 0
\(655\) −46.3939 −1.81276
\(656\) −9.79796 −0.382546
\(657\) 0 0
\(658\) 9.79796 0.381964
\(659\) 4.69694 0.182967 0.0914834 0.995807i \(-0.470839\pi\)
0.0914834 + 0.995807i \(0.470839\pi\)
\(660\) 0 0
\(661\) 9.44949 0.367543 0.183771 0.982969i \(-0.441169\pi\)
0.183771 + 0.982969i \(0.441169\pi\)
\(662\) −24.6969 −0.959874
\(663\) 0 0
\(664\) 2.00000 0.0776151
\(665\) −25.6969 −0.996485
\(666\) 0 0
\(667\) −2.89898 −0.112249
\(668\) 18.6969 0.723406
\(669\) 0 0
\(670\) −10.6969 −0.413259
\(671\) 22.8990 0.884005
\(672\) 0 0
\(673\) 30.5959 1.17939 0.589693 0.807628i \(-0.299249\pi\)
0.589693 + 0.807628i \(0.299249\pi\)
\(674\) 35.3939 1.36332
\(675\) 0 0
\(676\) 11.0000 0.423077
\(677\) −14.6969 −0.564849 −0.282425 0.959289i \(-0.591139\pi\)
−0.282425 + 0.959289i \(0.591139\pi\)
\(678\) 0 0
\(679\) 6.89898 0.264759
\(680\) 6.89898 0.264564
\(681\) 0 0
\(682\) 12.0000 0.459504
\(683\) 32.2020 1.23218 0.616088 0.787677i \(-0.288716\pi\)
0.616088 + 0.787677i \(0.288716\pi\)
\(684\) 0 0
\(685\) −40.6969 −1.55495
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −2.89898 −0.110523
\(689\) 5.39388 0.205490
\(690\) 0 0
\(691\) 6.95459 0.264565 0.132283 0.991212i \(-0.457769\pi\)
0.132283 + 0.991212i \(0.457769\pi\)
\(692\) −12.8990 −0.490346
\(693\) 0 0
\(694\) −19.5959 −0.743851
\(695\) 32.5959 1.23643
\(696\) 0 0
\(697\) −19.5959 −0.742248
\(698\) 20.8990 0.791038
\(699\) 0 0
\(700\) −6.89898 −0.260757
\(701\) 51.3939 1.94112 0.970560 0.240860i \(-0.0774293\pi\)
0.970560 + 0.240860i \(0.0774293\pi\)
\(702\) 0 0
\(703\) −58.0908 −2.19094
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −7.24745 −0.272568
\(708\) 0 0
\(709\) −11.5959 −0.435494 −0.217747 0.976005i \(-0.569871\pi\)
−0.217747 + 0.976005i \(0.569871\pi\)
\(710\) 34.1464 1.28149
\(711\) 0 0
\(712\) −7.10102 −0.266122
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) −33.7980 −1.26397
\(716\) −8.69694 −0.325020
\(717\) 0 0
\(718\) 10.7980 0.402976
\(719\) −9.79796 −0.365402 −0.182701 0.983169i \(-0.558484\pi\)
−0.182701 + 0.983169i \(0.558484\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 36.4949 1.35820
\(723\) 0 0
\(724\) 4.34847 0.161610
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) 40.4949 1.50187 0.750936 0.660375i \(-0.229603\pi\)
0.750936 + 0.660375i \(0.229603\pi\)
\(728\) 4.89898 0.181568
\(729\) 0 0
\(730\) 10.0000 0.370117
\(731\) −5.79796 −0.214445
\(732\) 0 0
\(733\) −12.5505 −0.463564 −0.231782 0.972768i \(-0.574456\pi\)
−0.231782 + 0.972768i \(0.574456\pi\)
\(734\) 5.79796 0.214007
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −6.20204 −0.228455
\(738\) 0 0
\(739\) −25.5959 −0.941561 −0.470781 0.882250i \(-0.656028\pi\)
−0.470781 + 0.882250i \(0.656028\pi\)
\(740\) −26.8990 −0.988826
\(741\) 0 0
\(742\) 1.10102 0.0404197
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) −20.6969 −0.758277
\(746\) 2.89898 0.106139
\(747\) 0 0
\(748\) 4.00000 0.146254
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 40.5959 1.48137 0.740683 0.671855i \(-0.234502\pi\)
0.740683 + 0.671855i \(0.234502\pi\)
\(752\) −9.79796 −0.357295
\(753\) 0 0
\(754\) −14.2020 −0.517208
\(755\) −17.2474 −0.627699
\(756\) 0 0
\(757\) 23.3939 0.850265 0.425132 0.905131i \(-0.360228\pi\)
0.425132 + 0.905131i \(0.360228\pi\)
\(758\) −26.4949 −0.962338
\(759\) 0 0
\(760\) 25.6969 0.932126
\(761\) 2.00000 0.0724999 0.0362500 0.999343i \(-0.488459\pi\)
0.0362500 + 0.999343i \(0.488459\pi\)
\(762\) 0 0
\(763\) 16.6969 0.604470
\(764\) 13.8990 0.502847
\(765\) 0 0
\(766\) 6.89898 0.249270
\(767\) 9.79796 0.353784
\(768\) 0 0
\(769\) 54.0908 1.95056 0.975282 0.220962i \(-0.0709198\pi\)
0.975282 + 0.220962i \(0.0709198\pi\)
\(770\) −6.89898 −0.248622
\(771\) 0 0
\(772\) −8.10102 −0.291562
\(773\) −19.9444 −0.717350 −0.358675 0.933463i \(-0.616771\pi\)
−0.358675 + 0.933463i \(0.616771\pi\)
\(774\) 0 0
\(775\) 41.3939 1.48691
\(776\) −6.89898 −0.247659
\(777\) 0 0
\(778\) −15.1010 −0.541398
\(779\) −72.9898 −2.61513
\(780\) 0 0
\(781\) 19.7980 0.708427
\(782\) −2.00000 −0.0715199
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 21.8990 0.781608
\(786\) 0 0
\(787\) 47.3939 1.68941 0.844705 0.535233i \(-0.179776\pi\)
0.844705 + 0.535233i \(0.179776\pi\)
\(788\) −12.6969 −0.452310
\(789\) 0 0
\(790\) 27.2474 0.969421
\(791\) 15.8990 0.565303
\(792\) 0 0
\(793\) −56.0908 −1.99184
\(794\) 9.30306 0.330153
\(795\) 0 0
\(796\) 6.89898 0.244528
\(797\) 35.9444 1.27322 0.636608 0.771188i \(-0.280337\pi\)
0.636608 + 0.771188i \(0.280337\pi\)
\(798\) 0 0
\(799\) −19.5959 −0.693254
\(800\) 6.89898 0.243916
\(801\) 0 0
\(802\) −10.1010 −0.356679
\(803\) 5.79796 0.204606
\(804\) 0 0
\(805\) 3.44949 0.121579
\(806\) −29.3939 −1.03536
\(807\) 0 0
\(808\) 7.24745 0.254964
\(809\) −35.7980 −1.25859 −0.629295 0.777167i \(-0.716656\pi\)
−0.629295 + 0.777167i \(0.716656\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) −2.89898 −0.101734
\(813\) 0 0
\(814\) −15.5959 −0.546637
\(815\) −0.696938 −0.0244127
\(816\) 0 0
\(817\) −21.5959 −0.755546
\(818\) 5.79796 0.202721
\(819\) 0 0
\(820\) −33.7980 −1.18028
\(821\) 39.5959 1.38191 0.690954 0.722899i \(-0.257191\pi\)
0.690954 + 0.722899i \(0.257191\pi\)
\(822\) 0 0
\(823\) 45.3939 1.58233 0.791166 0.611602i \(-0.209475\pi\)
0.791166 + 0.611602i \(0.209475\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) 2.00000 0.0695889
\(827\) 12.4949 0.434490 0.217245 0.976117i \(-0.430293\pi\)
0.217245 + 0.976117i \(0.430293\pi\)
\(828\) 0 0
\(829\) 30.6969 1.06615 0.533074 0.846068i \(-0.321037\pi\)
0.533074 + 0.846068i \(0.321037\pi\)
\(830\) 6.89898 0.239467
\(831\) 0 0
\(832\) −4.89898 −0.169842
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 64.4949 2.23194
\(836\) 14.8990 0.515292
\(837\) 0 0
\(838\) 24.5505 0.848083
\(839\) −44.8990 −1.55008 −0.775042 0.631909i \(-0.782272\pi\)
−0.775042 + 0.631909i \(0.782272\pi\)
\(840\) 0 0
\(841\) −20.5959 −0.710204
\(842\) 13.1010 0.451491
\(843\) 0 0
\(844\) 3.10102 0.106742
\(845\) 37.9444 1.30533
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) −1.10102 −0.0378092
\(849\) 0 0
\(850\) 13.7980 0.473266
\(851\) 7.79796 0.267311
\(852\) 0 0
\(853\) −38.8434 −1.32997 −0.664986 0.746856i \(-0.731562\pi\)
−0.664986 + 0.746856i \(0.731562\pi\)
\(854\) −11.4495 −0.391793
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −25.1010 −0.857435 −0.428717 0.903439i \(-0.641034\pi\)
−0.428717 + 0.903439i \(0.641034\pi\)
\(858\) 0 0
\(859\) −10.0000 −0.341196 −0.170598 0.985341i \(-0.554570\pi\)
−0.170598 + 0.985341i \(0.554570\pi\)
\(860\) −10.0000 −0.340997
\(861\) 0 0
\(862\) −7.59592 −0.258718
\(863\) 2.10102 0.0715196 0.0357598 0.999360i \(-0.488615\pi\)
0.0357598 + 0.999360i \(0.488615\pi\)
\(864\) 0 0
\(865\) −44.4949 −1.51287
\(866\) 11.7980 0.400911
\(867\) 0 0
\(868\) −6.00000 −0.203653
\(869\) 15.7980 0.535909
\(870\) 0 0
\(871\) 15.1918 0.514756
\(872\) −16.6969 −0.565430
\(873\) 0 0
\(874\) −7.44949 −0.251983
\(875\) −6.55051 −0.221448
\(876\) 0 0
\(877\) −26.4949 −0.894669 −0.447335 0.894367i \(-0.647627\pi\)
−0.447335 + 0.894367i \(0.647627\pi\)
\(878\) 21.7980 0.735645
\(879\) 0 0
\(880\) 6.89898 0.232565
\(881\) −19.5959 −0.660203 −0.330102 0.943945i \(-0.607083\pi\)
−0.330102 + 0.943945i \(0.607083\pi\)
\(882\) 0 0
\(883\) −0.202041 −0.00679922 −0.00339961 0.999994i \(-0.501082\pi\)
−0.00339961 + 0.999994i \(0.501082\pi\)
\(884\) −9.79796 −0.329541
\(885\) 0 0
\(886\) −5.10102 −0.171372
\(887\) −33.7980 −1.13482 −0.567412 0.823434i \(-0.692055\pi\)
−0.567412 + 0.823434i \(0.692055\pi\)
\(888\) 0 0
\(889\) 3.00000 0.100617
\(890\) −24.4949 −0.821071
\(891\) 0 0
\(892\) −20.8990 −0.699750
\(893\) −72.9898 −2.44251
\(894\) 0 0
\(895\) −30.0000 −1.00279
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −18.5959 −0.620554
\(899\) 17.3939 0.580118
\(900\) 0 0
\(901\) −2.20204 −0.0733606
\(902\) −19.5959 −0.652473
\(903\) 0 0
\(904\) −15.8990 −0.528792
\(905\) 15.0000 0.498617
\(906\) 0 0
\(907\) −26.6969 −0.886457 −0.443229 0.896409i \(-0.646167\pi\)
−0.443229 + 0.896409i \(0.646167\pi\)
\(908\) −0.550510 −0.0182693
\(909\) 0 0
\(910\) 16.8990 0.560196
\(911\) −45.9898 −1.52371 −0.761855 0.647748i \(-0.775711\pi\)
−0.761855 + 0.647748i \(0.775711\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) 31.4949 1.04176
\(915\) 0 0
\(916\) −23.2474 −0.768117
\(917\) 13.4495 0.444141
\(918\) 0 0
\(919\) 3.69694 0.121951 0.0609754 0.998139i \(-0.480579\pi\)
0.0609754 + 0.998139i \(0.480579\pi\)
\(920\) −3.44949 −0.113726
\(921\) 0 0
\(922\) 20.3485 0.670141
\(923\) −48.4949 −1.59623
\(924\) 0 0
\(925\) −53.7980 −1.76887
\(926\) −25.6969 −0.844454
\(927\) 0 0
\(928\) 2.89898 0.0951637
\(929\) −34.2929 −1.12511 −0.562556 0.826759i \(-0.690182\pi\)
−0.562556 + 0.826759i \(0.690182\pi\)
\(930\) 0 0
\(931\) 7.44949 0.244147
\(932\) −7.00000 −0.229293
\(933\) 0 0
\(934\) 10.0000 0.327210
\(935\) 13.7980 0.451242
\(936\) 0 0
\(937\) 6.40408 0.209212 0.104606 0.994514i \(-0.466642\pi\)
0.104606 + 0.994514i \(0.466642\pi\)
\(938\) 3.10102 0.101252
\(939\) 0 0
\(940\) −33.7980 −1.10237
\(941\) 3.44949 0.112450 0.0562251 0.998418i \(-0.482094\pi\)
0.0562251 + 0.998418i \(0.482094\pi\)
\(942\) 0 0
\(943\) 9.79796 0.319065
\(944\) −2.00000 −0.0650945
\(945\) 0 0
\(946\) −5.79796 −0.188508
\(947\) 3.50510 0.113901 0.0569503 0.998377i \(-0.481862\pi\)
0.0569503 + 0.998377i \(0.481862\pi\)
\(948\) 0 0
\(949\) −14.2020 −0.461018
\(950\) 51.3939 1.66744
\(951\) 0 0
\(952\) −2.00000 −0.0648204
\(953\) 55.3939 1.79438 0.897192 0.441641i \(-0.145604\pi\)
0.897192 + 0.441641i \(0.145604\pi\)
\(954\) 0 0
\(955\) 47.9444 1.55144
\(956\) −12.7980 −0.413916
\(957\) 0 0
\(958\) −29.5959 −0.956201
\(959\) 11.7980 0.380976
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 38.2020 1.23168
\(963\) 0 0
\(964\) −8.89898 −0.286617
\(965\) −27.9444 −0.899562
\(966\) 0 0
\(967\) −14.5959 −0.469373 −0.234687 0.972071i \(-0.575406\pi\)
−0.234687 + 0.972071i \(0.575406\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) −23.7980 −0.764106
\(971\) −53.9444 −1.73116 −0.865579 0.500773i \(-0.833049\pi\)
−0.865579 + 0.500773i \(0.833049\pi\)
\(972\) 0 0
\(973\) −9.44949 −0.302937
\(974\) 22.3939 0.717546
\(975\) 0 0
\(976\) 11.4495 0.366489
\(977\) 1.59592 0.0510579 0.0255290 0.999674i \(-0.491873\pi\)
0.0255290 + 0.999674i \(0.491873\pi\)
\(978\) 0 0
\(979\) −14.2020 −0.453899
\(980\) 3.44949 0.110190
\(981\) 0 0
\(982\) −3.79796 −0.121198
\(983\) 45.1918 1.44140 0.720698 0.693249i \(-0.243822\pi\)
0.720698 + 0.693249i \(0.243822\pi\)
\(984\) 0 0
\(985\) −43.7980 −1.39552
\(986\) 5.79796 0.184645
\(987\) 0 0
\(988\) −36.4949 −1.16106
\(989\) 2.89898 0.0921822
\(990\) 0 0
\(991\) 17.7980 0.565371 0.282685 0.959213i \(-0.408775\pi\)
0.282685 + 0.959213i \(0.408775\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) −9.89898 −0.313977
\(995\) 23.7980 0.754446
\(996\) 0 0
\(997\) 17.8536 0.565428 0.282714 0.959204i \(-0.408765\pi\)
0.282714 + 0.959204i \(0.408765\pi\)
\(998\) 33.3939 1.05706
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1134.2.a.p.1.2 2
3.2 odd 2 1134.2.a.i.1.1 2
4.3 odd 2 9072.2.a.bk.1.2 2
7.6 odd 2 7938.2.a.bn.1.1 2
9.2 odd 6 378.2.f.d.253.2 4
9.4 even 3 126.2.f.c.43.1 4
9.5 odd 6 378.2.f.d.127.2 4
9.7 even 3 126.2.f.c.85.2 yes 4
12.11 even 2 9072.2.a.bd.1.1 2
21.20 even 2 7938.2.a.bm.1.2 2
36.7 odd 6 1008.2.r.e.337.1 4
36.11 even 6 3024.2.r.e.1009.2 4
36.23 even 6 3024.2.r.e.2017.2 4
36.31 odd 6 1008.2.r.e.673.2 4
63.2 odd 6 2646.2.h.m.361.1 4
63.4 even 3 882.2.h.k.79.1 4
63.5 even 6 2646.2.e.k.2125.1 4
63.11 odd 6 2646.2.e.l.1549.2 4
63.13 odd 6 882.2.f.j.295.2 4
63.16 even 3 882.2.h.k.67.1 4
63.20 even 6 2646.2.f.k.1765.1 4
63.23 odd 6 2646.2.e.l.2125.2 4
63.25 even 3 882.2.e.m.373.2 4
63.31 odd 6 882.2.h.l.79.2 4
63.32 odd 6 2646.2.h.m.667.1 4
63.34 odd 6 882.2.f.j.589.1 4
63.38 even 6 2646.2.e.k.1549.1 4
63.40 odd 6 882.2.e.n.655.1 4
63.41 even 6 2646.2.f.k.883.1 4
63.47 even 6 2646.2.h.n.361.2 4
63.52 odd 6 882.2.e.n.373.1 4
63.58 even 3 882.2.e.m.655.2 4
63.59 even 6 2646.2.h.n.667.2 4
63.61 odd 6 882.2.h.l.67.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.f.c.43.1 4 9.4 even 3
126.2.f.c.85.2 yes 4 9.7 even 3
378.2.f.d.127.2 4 9.5 odd 6
378.2.f.d.253.2 4 9.2 odd 6
882.2.e.m.373.2 4 63.25 even 3
882.2.e.m.655.2 4 63.58 even 3
882.2.e.n.373.1 4 63.52 odd 6
882.2.e.n.655.1 4 63.40 odd 6
882.2.f.j.295.2 4 63.13 odd 6
882.2.f.j.589.1 4 63.34 odd 6
882.2.h.k.67.1 4 63.16 even 3
882.2.h.k.79.1 4 63.4 even 3
882.2.h.l.67.2 4 63.61 odd 6
882.2.h.l.79.2 4 63.31 odd 6
1008.2.r.e.337.1 4 36.7 odd 6
1008.2.r.e.673.2 4 36.31 odd 6
1134.2.a.i.1.1 2 3.2 odd 2
1134.2.a.p.1.2 2 1.1 even 1 trivial
2646.2.e.k.1549.1 4 63.38 even 6
2646.2.e.k.2125.1 4 63.5 even 6
2646.2.e.l.1549.2 4 63.11 odd 6
2646.2.e.l.2125.2 4 63.23 odd 6
2646.2.f.k.883.1 4 63.41 even 6
2646.2.f.k.1765.1 4 63.20 even 6
2646.2.h.m.361.1 4 63.2 odd 6
2646.2.h.m.667.1 4 63.32 odd 6
2646.2.h.n.361.2 4 63.47 even 6
2646.2.h.n.667.2 4 63.59 even 6
3024.2.r.e.1009.2 4 36.11 even 6
3024.2.r.e.2017.2 4 36.23 even 6
7938.2.a.bm.1.2 2 21.20 even 2
7938.2.a.bn.1.1 2 7.6 odd 2
9072.2.a.bd.1.1 2 12.11 even 2
9072.2.a.bk.1.2 2 4.3 odd 2