Properties

Label 9702.2.a.dz.1.2
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.4708600410\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3234)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.74912\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.47363 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -2.47363 q^{5} -1.00000 q^{8} +2.47363 q^{10} -1.00000 q^{11} -3.03127 q^{13} +1.00000 q^{16} -6.64167 q^{17} -0.557647 q^{19} -2.47363 q^{20} +1.00000 q^{22} +2.66981 q^{23} +1.11882 q^{25} +3.03127 q^{26} -3.77568 q^{29} -2.00666 q^{31} -1.00000 q^{32} +6.64167 q^{34} -0.158619 q^{37} +0.557647 q^{38} +2.47363 q^{40} -5.93207 q^{41} +4.32666 q^{43} -1.00000 q^{44} -2.66981 q^{46} -7.38607 q^{47} -1.11882 q^{50} -3.03127 q^{52} -2.83785 q^{53} +2.47363 q^{55} +3.77568 q^{58} -1.48881 q^{59} -3.19932 q^{61} +2.00666 q^{62} +1.00000 q^{64} +7.49824 q^{65} -11.2739 q^{67} -6.64167 q^{68} +11.0098 q^{71} -3.18323 q^{73} +0.158619 q^{74} -0.557647 q^{76} -6.95668 q^{79} -2.47363 q^{80} +5.93207 q^{82} +4.59744 q^{83} +16.4290 q^{85} -4.32666 q^{86} +1.00000 q^{88} -6.62558 q^{89} +2.66981 q^{92} +7.38607 q^{94} +1.37941 q^{95} +1.17656 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8} + O(q^{10}) \) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8} + 4 q^{10} - 4 q^{11} + 8 q^{13} + 4 q^{16} - 4 q^{17} + 12 q^{19} - 4 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 8 q^{26} + 8 q^{29} + 4 q^{31} - 4 q^{32} + 4 q^{34} + 8 q^{37} - 12 q^{38} + 4 q^{40} - 12 q^{41} - 8 q^{43} - 4 q^{44} - 8 q^{46} - 4 q^{47} - 4 q^{50} + 8 q^{52} + 8 q^{53} + 4 q^{55} - 8 q^{58} + 24 q^{61} - 4 q^{62} + 4 q^{64} + 16 q^{65} - 8 q^{67} - 4 q^{68} - 8 q^{71} + 4 q^{73} - 8 q^{74} + 12 q^{76} - 8 q^{79} - 4 q^{80} + 12 q^{82} - 4 q^{83} - 8 q^{85} + 8 q^{86} + 4 q^{88} - 24 q^{89} + 8 q^{92} + 4 q^{94} - 8 q^{95} + O(q^{100}) \)

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
\(5\) −2.47363 −1.10624 −0.553120 0.833102i \(-0.686563\pi\)
−0.553120 + 0.833102i \(0.686563\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 2.47363 0.782229
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.03127 −0.840724 −0.420362 0.907357i \(-0.638097\pi\)
−0.420362 + 0.907357i \(0.638097\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.64167 −1.61084 −0.805421 0.592704i \(-0.798061\pi\)
−0.805421 + 0.592704i \(0.798061\pi\)
\(18\) 0 0
\(19\) −0.557647 −0.127933 −0.0639665 0.997952i \(-0.520375\pi\)
−0.0639665 + 0.997952i \(0.520375\pi\)
\(20\) −2.47363 −0.553120
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 2.66981 0.556693 0.278347 0.960481i \(-0.410214\pi\)
0.278347 + 0.960481i \(0.410214\pi\)
\(24\) 0 0
\(25\) 1.11882 0.223765
\(26\) 3.03127 0.594482
\(27\) 0 0
\(28\) 0 0
\(29\) −3.77568 −0.701126 −0.350563 0.936539i \(-0.614010\pi\)
−0.350563 + 0.936539i \(0.614010\pi\)
\(30\) 0 0
\(31\) −2.00666 −0.360407 −0.180204 0.983629i \(-0.557676\pi\)
−0.180204 + 0.983629i \(0.557676\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.64167 1.13904
\(35\) 0 0
\(36\) 0 0
\(37\) −0.158619 −0.0260768 −0.0130384 0.999915i \(-0.504150\pi\)
−0.0130384 + 0.999915i \(0.504150\pi\)
\(38\) 0.557647 0.0904623
\(39\) 0 0
\(40\) 2.47363 0.391115
\(41\) −5.93207 −0.926433 −0.463217 0.886245i \(-0.653305\pi\)
−0.463217 + 0.886245i \(0.653305\pi\)
\(42\) 0 0
\(43\) 4.32666 0.659810 0.329905 0.944014i \(-0.392983\pi\)
0.329905 + 0.944014i \(0.392983\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −2.66981 −0.393642
\(47\) −7.38607 −1.07737 −0.538685 0.842507i \(-0.681079\pi\)
−0.538685 + 0.842507i \(0.681079\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.11882 −0.158226
\(51\) 0 0
\(52\) −3.03127 −0.420362
\(53\) −2.83785 −0.389809 −0.194904 0.980822i \(-0.562440\pi\)
−0.194904 + 0.980822i \(0.562440\pi\)
\(54\) 0 0
\(55\) 2.47363 0.333544
\(56\) 0 0
\(57\) 0 0
\(58\) 3.77568 0.495771
\(59\) −1.48881 −0.193827 −0.0969134 0.995293i \(-0.530897\pi\)
−0.0969134 + 0.995293i \(0.530897\pi\)
\(60\) 0 0
\(61\) −3.19932 −0.409630 −0.204815 0.978801i \(-0.565659\pi\)
−0.204815 + 0.978801i \(0.565659\pi\)
\(62\) 2.00666 0.254847
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.49824 0.930042
\(66\) 0 0
\(67\) −11.2739 −1.37733 −0.688664 0.725081i \(-0.741802\pi\)
−0.688664 + 0.725081i \(0.741802\pi\)
\(68\) −6.64167 −0.805421
\(69\) 0 0
\(70\) 0 0
\(71\) 11.0098 1.30662 0.653311 0.757089i \(-0.273379\pi\)
0.653311 + 0.757089i \(0.273379\pi\)
\(72\) 0 0
\(73\) −3.18323 −0.372569 −0.186284 0.982496i \(-0.559645\pi\)
−0.186284 + 0.982496i \(0.559645\pi\)
\(74\) 0.158619 0.0184391
\(75\) 0 0
\(76\) −0.557647 −0.0639665
\(77\) 0 0
\(78\) 0 0
\(79\) −6.95668 −0.782687 −0.391344 0.920245i \(-0.627990\pi\)
−0.391344 + 0.920245i \(0.627990\pi\)
\(80\) −2.47363 −0.276560
\(81\) 0 0
\(82\) 5.93207 0.655087
\(83\) 4.59744 0.504635 0.252317 0.967645i \(-0.418807\pi\)
0.252317 + 0.967645i \(0.418807\pi\)
\(84\) 0 0
\(85\) 16.4290 1.78198
\(86\) −4.32666 −0.466556
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −6.62558 −0.702310 −0.351155 0.936317i \(-0.614211\pi\)
−0.351155 + 0.936317i \(0.614211\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.66981 0.278347
\(93\) 0 0
\(94\) 7.38607 0.761815
\(95\) 1.37941 0.141525
\(96\) 0 0
\(97\) 1.17656 0.119462 0.0597310 0.998215i \(-0.480976\pi\)
0.0597310 + 0.998215i \(0.480976\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.11882 0.111882
\(101\) −14.5295 −1.44574 −0.722870 0.690984i \(-0.757177\pi\)
−0.722870 + 0.690984i \(0.757177\pi\)
\(102\) 0 0
\(103\) 12.2239 1.20446 0.602230 0.798323i \(-0.294279\pi\)
0.602230 + 0.798323i \(0.294279\pi\)
\(104\) 3.03127 0.297241
\(105\) 0 0
\(106\) 2.83785 0.275636
\(107\) 2.81900 0.272523 0.136262 0.990673i \(-0.456491\pi\)
0.136262 + 0.990673i \(0.456491\pi\)
\(108\) 0 0
\(109\) −20.5478 −1.96812 −0.984062 0.177823i \(-0.943095\pi\)
−0.984062 + 0.177823i \(0.943095\pi\)
\(110\) −2.47363 −0.235851
\(111\) 0 0
\(112\) 0 0
\(113\) −11.7194 −1.10247 −0.551234 0.834351i \(-0.685843\pi\)
−0.551234 + 0.834351i \(0.685843\pi\)
\(114\) 0 0
\(115\) −6.60411 −0.615836
\(116\) −3.77568 −0.350563
\(117\) 0 0
\(118\) 1.48881 0.137056
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 3.19932 0.289652
\(123\) 0 0
\(124\) −2.00666 −0.180204
\(125\) 9.60058 0.858702
\(126\) 0 0
\(127\) 0.0397948 0.00353121 0.00176561 0.999998i \(-0.499438\pi\)
0.00176561 + 0.999998i \(0.499438\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −7.49824 −0.657639
\(131\) −9.60097 −0.838841 −0.419420 0.907792i \(-0.637767\pi\)
−0.419420 + 0.907792i \(0.637767\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 11.2739 0.973918
\(135\) 0 0
\(136\) 6.64167 0.569518
\(137\) 18.6667 1.59480 0.797400 0.603451i \(-0.206208\pi\)
0.797400 + 0.603451i \(0.206208\pi\)
\(138\) 0 0
\(139\) 8.33333 0.706823 0.353412 0.935468i \(-0.385021\pi\)
0.353412 + 0.935468i \(0.385021\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −11.0098 −0.923922
\(143\) 3.03127 0.253488
\(144\) 0 0
\(145\) 9.33962 0.775613
\(146\) 3.18323 0.263446
\(147\) 0 0
\(148\) −0.158619 −0.0130384
\(149\) 15.2341 1.24803 0.624014 0.781413i \(-0.285501\pi\)
0.624014 + 0.781413i \(0.285501\pi\)
\(150\) 0 0
\(151\) −8.31724 −0.676847 −0.338424 0.940994i \(-0.609894\pi\)
−0.338424 + 0.940994i \(0.609894\pi\)
\(152\) 0.557647 0.0452312
\(153\) 0 0
\(154\) 0 0
\(155\) 4.96374 0.398697
\(156\) 0 0
\(157\) −9.13401 −0.728973 −0.364487 0.931209i \(-0.618756\pi\)
−0.364487 + 0.931209i \(0.618756\pi\)
\(158\) 6.95668 0.553443
\(159\) 0 0
\(160\) 2.47363 0.195557
\(161\) 0 0
\(162\) 0 0
\(163\) 8.33609 0.652933 0.326466 0.945209i \(-0.394142\pi\)
0.326466 + 0.945209i \(0.394142\pi\)
\(164\) −5.93207 −0.463217
\(165\) 0 0
\(166\) −4.59744 −0.356831
\(167\) 22.1610 1.71487 0.857434 0.514594i \(-0.172057\pi\)
0.857434 + 0.514594i \(0.172057\pi\)
\(168\) 0 0
\(169\) −3.81138 −0.293183
\(170\) −16.4290 −1.26005
\(171\) 0 0
\(172\) 4.32666 0.329905
\(173\) −20.9656 −1.59398 −0.796991 0.603991i \(-0.793576\pi\)
−0.796991 + 0.603991i \(0.793576\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 6.62558 0.496608
\(179\) 12.3759 0.925017 0.462508 0.886615i \(-0.346950\pi\)
0.462508 + 0.886615i \(0.346950\pi\)
\(180\) 0 0
\(181\) 20.5595 1.52817 0.764087 0.645113i \(-0.223190\pi\)
0.764087 + 0.645113i \(0.223190\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2.66981 −0.196821
\(185\) 0.392364 0.0288472
\(186\) 0 0
\(187\) 6.64167 0.485687
\(188\) −7.38607 −0.538685
\(189\) 0 0
\(190\) −1.37941 −0.100073
\(191\) 20.3267 1.47079 0.735393 0.677641i \(-0.236998\pi\)
0.735393 + 0.677641i \(0.236998\pi\)
\(192\) 0 0
\(193\) −13.0761 −0.941235 −0.470618 0.882337i \(-0.655969\pi\)
−0.470618 + 0.882337i \(0.655969\pi\)
\(194\) −1.17656 −0.0844724
\(195\) 0 0
\(196\) 0 0
\(197\) 3.80159 0.270852 0.135426 0.990787i \(-0.456760\pi\)
0.135426 + 0.990787i \(0.456760\pi\)
\(198\) 0 0
\(199\) −6.87098 −0.487071 −0.243536 0.969892i \(-0.578307\pi\)
−0.243536 + 0.969892i \(0.578307\pi\)
\(200\) −1.11882 −0.0791128
\(201\) 0 0
\(202\) 14.5295 1.02229
\(203\) 0 0
\(204\) 0 0
\(205\) 14.6737 1.02486
\(206\) −12.2239 −0.851681
\(207\) 0 0
\(208\) −3.03127 −0.210181
\(209\) 0.557647 0.0385733
\(210\) 0 0
\(211\) 11.7984 0.812237 0.406119 0.913820i \(-0.366882\pi\)
0.406119 + 0.913820i \(0.366882\pi\)
\(212\) −2.83785 −0.194904
\(213\) 0 0
\(214\) −2.81900 −0.192703
\(215\) −10.7025 −0.729907
\(216\) 0 0
\(217\) 0 0
\(218\) 20.5478 1.39167
\(219\) 0 0
\(220\) 2.47363 0.166772
\(221\) 20.1327 1.35427
\(222\) 0 0
\(223\) −21.0031 −1.40647 −0.703237 0.710956i \(-0.748263\pi\)
−0.703237 + 0.710956i \(0.748263\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 11.7194 0.779563
\(227\) 29.4884 1.95721 0.978607 0.205736i \(-0.0659590\pi\)
0.978607 + 0.205736i \(0.0659590\pi\)
\(228\) 0 0
\(229\) −17.3254 −1.14489 −0.572446 0.819942i \(-0.694005\pi\)
−0.572446 + 0.819942i \(0.694005\pi\)
\(230\) 6.60411 0.435462
\(231\) 0 0
\(232\) 3.77568 0.247885
\(233\) −29.4388 −1.92860 −0.964300 0.264812i \(-0.914690\pi\)
−0.964300 + 0.264812i \(0.914690\pi\)
\(234\) 0 0
\(235\) 18.2704 1.19183
\(236\) −1.48881 −0.0969134
\(237\) 0 0
\(238\) 0 0
\(239\) −24.2047 −1.56567 −0.782835 0.622229i \(-0.786227\pi\)
−0.782835 + 0.622229i \(0.786227\pi\)
\(240\) 0 0
\(241\) −15.3083 −0.986096 −0.493048 0.870002i \(-0.664117\pi\)
−0.493048 + 0.870002i \(0.664117\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −3.19932 −0.204815
\(245\) 0 0
\(246\) 0 0
\(247\) 1.69038 0.107556
\(248\) 2.00666 0.127423
\(249\) 0 0
\(250\) −9.60058 −0.607194
\(251\) −7.79453 −0.491986 −0.245993 0.969272i \(-0.579114\pi\)
−0.245993 + 0.969272i \(0.579114\pi\)
\(252\) 0 0
\(253\) −2.66981 −0.167849
\(254\) −0.0397948 −0.00249695
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0973 −0.879368 −0.439684 0.898152i \(-0.644910\pi\)
−0.439684 + 0.898152i \(0.644910\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 7.49824 0.465021
\(261\) 0 0
\(262\) 9.60097 0.593150
\(263\) −4.61353 −0.284482 −0.142241 0.989832i \(-0.545431\pi\)
−0.142241 + 0.989832i \(0.545431\pi\)
\(264\) 0 0
\(265\) 7.01978 0.431222
\(266\) 0 0
\(267\) 0 0
\(268\) −11.2739 −0.688664
\(269\) 3.18323 0.194085 0.0970424 0.995280i \(-0.469062\pi\)
0.0970424 + 0.995280i \(0.469062\pi\)
\(270\) 0 0
\(271\) 0.722930 0.0439149 0.0219574 0.999759i \(-0.493010\pi\)
0.0219574 + 0.999759i \(0.493010\pi\)
\(272\) −6.64167 −0.402710
\(273\) 0 0
\(274\) −18.6667 −1.12769
\(275\) −1.11882 −0.0674676
\(276\) 0 0
\(277\) 1.18490 0.0711938 0.0355969 0.999366i \(-0.488667\pi\)
0.0355969 + 0.999366i \(0.488667\pi\)
\(278\) −8.33333 −0.499800
\(279\) 0 0
\(280\) 0 0
\(281\) −2.89097 −0.172461 −0.0862305 0.996275i \(-0.527482\pi\)
−0.0862305 + 0.996275i \(0.527482\pi\)
\(282\) 0 0
\(283\) −5.32353 −0.316451 −0.158225 0.987403i \(-0.550577\pi\)
−0.158225 + 0.987403i \(0.550577\pi\)
\(284\) 11.0098 0.653311
\(285\) 0 0
\(286\) −3.03127 −0.179243
\(287\) 0 0
\(288\) 0 0
\(289\) 27.1118 1.59481
\(290\) −9.33962 −0.548441
\(291\) 0 0
\(292\) −3.18323 −0.186284
\(293\) −10.6413 −0.621670 −0.310835 0.950464i \(-0.600609\pi\)
−0.310835 + 0.950464i \(0.600609\pi\)
\(294\) 0 0
\(295\) 3.68276 0.214419
\(296\) 0.158619 0.00921955
\(297\) 0 0
\(298\) −15.2341 −0.882489
\(299\) −8.09292 −0.468025
\(300\) 0 0
\(301\) 0 0
\(302\) 8.31724 0.478603
\(303\) 0 0
\(304\) −0.557647 −0.0319833
\(305\) 7.91391 0.453149
\(306\) 0 0
\(307\) 28.9733 1.65359 0.826797 0.562500i \(-0.190160\pi\)
0.826797 + 0.562500i \(0.190160\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.96374 −0.281921
\(311\) −2.62019 −0.148578 −0.0742888 0.997237i \(-0.523669\pi\)
−0.0742888 + 0.997237i \(0.523669\pi\)
\(312\) 0 0
\(313\) −31.7566 −1.79499 −0.897494 0.441027i \(-0.854614\pi\)
−0.897494 + 0.441027i \(0.854614\pi\)
\(314\) 9.13401 0.515462
\(315\) 0 0
\(316\) −6.95668 −0.391344
\(317\) 24.8308 1.39464 0.697318 0.716762i \(-0.254377\pi\)
0.697318 + 0.716762i \(0.254377\pi\)
\(318\) 0 0
\(319\) 3.77568 0.211397
\(320\) −2.47363 −0.138280
\(321\) 0 0
\(322\) 0 0
\(323\) 3.70371 0.206080
\(324\) 0 0
\(325\) −3.39146 −0.188124
\(326\) −8.33609 −0.461693
\(327\) 0 0
\(328\) 5.93207 0.327544
\(329\) 0 0
\(330\) 0 0
\(331\) 21.2669 1.16893 0.584466 0.811418i \(-0.301304\pi\)
0.584466 + 0.811418i \(0.301304\pi\)
\(332\) 4.59744 0.252317
\(333\) 0 0
\(334\) −22.1610 −1.21260
\(335\) 27.8874 1.52365
\(336\) 0 0
\(337\) 23.3927 1.27428 0.637142 0.770747i \(-0.280117\pi\)
0.637142 + 0.770747i \(0.280117\pi\)
\(338\) 3.81138 0.207312
\(339\) 0 0
\(340\) 16.4290 0.890988
\(341\) 2.00666 0.108667
\(342\) 0 0
\(343\) 0 0
\(344\) −4.32666 −0.233278
\(345\) 0 0
\(346\) 20.9656 1.12712
\(347\) −6.60020 −0.354317 −0.177159 0.984182i \(-0.556691\pi\)
−0.177159 + 0.984182i \(0.556691\pi\)
\(348\) 0 0
\(349\) 5.06164 0.270944 0.135472 0.990781i \(-0.456745\pi\)
0.135472 + 0.990781i \(0.456745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 11.5166 0.612964 0.306482 0.951876i \(-0.400848\pi\)
0.306482 + 0.951876i \(0.400848\pi\)
\(354\) 0 0
\(355\) −27.2341 −1.44544
\(356\) −6.62558 −0.351155
\(357\) 0 0
\(358\) −12.3759 −0.654086
\(359\) 10.5080 0.554593 0.277296 0.960784i \(-0.410562\pi\)
0.277296 + 0.960784i \(0.410562\pi\)
\(360\) 0 0
\(361\) −18.6890 −0.983633
\(362\) −20.5595 −1.08058
\(363\) 0 0
\(364\) 0 0
\(365\) 7.87412 0.412150
\(366\) 0 0
\(367\) 10.8976 0.568852 0.284426 0.958698i \(-0.408197\pi\)
0.284426 + 0.958698i \(0.408197\pi\)
\(368\) 2.66981 0.139173
\(369\) 0 0
\(370\) −0.392364 −0.0203981
\(371\) 0 0
\(372\) 0 0
\(373\) 21.3459 1.10525 0.552624 0.833431i \(-0.313627\pi\)
0.552624 + 0.833431i \(0.313627\pi\)
\(374\) −6.64167 −0.343433
\(375\) 0 0
\(376\) 7.38607 0.380908
\(377\) 11.4451 0.589453
\(378\) 0 0
\(379\) −21.3066 −1.09445 −0.547225 0.836986i \(-0.684316\pi\)
−0.547225 + 0.836986i \(0.684316\pi\)
\(380\) 1.37941 0.0707623
\(381\) 0 0
\(382\) −20.3267 −1.04000
\(383\) 34.6202 1.76901 0.884505 0.466531i \(-0.154497\pi\)
0.884505 + 0.466531i \(0.154497\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 13.0761 0.665554
\(387\) 0 0
\(388\) 1.17656 0.0597310
\(389\) 9.68276 0.490936 0.245468 0.969405i \(-0.421058\pi\)
0.245468 + 0.969405i \(0.421058\pi\)
\(390\) 0 0
\(391\) −17.7320 −0.896745
\(392\) 0 0
\(393\) 0 0
\(394\) −3.80159 −0.191521
\(395\) 17.2082 0.865839
\(396\) 0 0
\(397\) −11.8928 −0.596884 −0.298442 0.954428i \(-0.596467\pi\)
−0.298442 + 0.954428i \(0.596467\pi\)
\(398\) 6.87098 0.344411
\(399\) 0 0
\(400\) 1.11882 0.0559412
\(401\) −33.9804 −1.69690 −0.848449 0.529277i \(-0.822463\pi\)
−0.848449 + 0.529277i \(0.822463\pi\)
\(402\) 0 0
\(403\) 6.08275 0.303003
\(404\) −14.5295 −0.722870
\(405\) 0 0
\(406\) 0 0
\(407\) 0.158619 0.00786246
\(408\) 0 0
\(409\) 9.51305 0.470390 0.235195 0.971948i \(-0.424427\pi\)
0.235195 + 0.971948i \(0.424427\pi\)
\(410\) −14.6737 −0.724683
\(411\) 0 0
\(412\) 12.2239 0.602230
\(413\) 0 0
\(414\) 0 0
\(415\) −11.3724 −0.558247
\(416\) 3.03127 0.148620
\(417\) 0 0
\(418\) −0.557647 −0.0272754
\(419\) 15.9249 0.777981 0.388990 0.921242i \(-0.372824\pi\)
0.388990 + 0.921242i \(0.372824\pi\)
\(420\) 0 0
\(421\) 3.55842 0.173427 0.0867133 0.996233i \(-0.472364\pi\)
0.0867133 + 0.996233i \(0.472364\pi\)
\(422\) −11.7984 −0.574339
\(423\) 0 0
\(424\) 2.83785 0.137818
\(425\) −7.43086 −0.360450
\(426\) 0 0
\(427\) 0 0
\(428\) 2.81900 0.136262
\(429\) 0 0
\(430\) 10.7025 0.516122
\(431\) −20.2774 −0.976730 −0.488365 0.872639i \(-0.662407\pi\)
−0.488365 + 0.872639i \(0.662407\pi\)
\(432\) 0 0
\(433\) 27.7144 1.33187 0.665935 0.746010i \(-0.268033\pi\)
0.665935 + 0.746010i \(0.268033\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −20.5478 −0.984062
\(437\) −1.48881 −0.0712195
\(438\) 0 0
\(439\) 4.85433 0.231685 0.115842 0.993268i \(-0.463043\pi\)
0.115842 + 0.993268i \(0.463043\pi\)
\(440\) −2.47363 −0.117925
\(441\) 0 0
\(442\) −20.1327 −0.957615
\(443\) 30.5269 1.45038 0.725188 0.688551i \(-0.241753\pi\)
0.725188 + 0.688551i \(0.241753\pi\)
\(444\) 0 0
\(445\) 16.3892 0.776923
\(446\) 21.0031 0.994527
\(447\) 0 0
\(448\) 0 0
\(449\) 30.3986 1.43460 0.717300 0.696764i \(-0.245378\pi\)
0.717300 + 0.696764i \(0.245378\pi\)
\(450\) 0 0
\(451\) 5.93207 0.279330
\(452\) −11.7194 −0.551234
\(453\) 0 0
\(454\) −29.4884 −1.38396
\(455\) 0 0
\(456\) 0 0
\(457\) 11.1020 0.519328 0.259664 0.965699i \(-0.416388\pi\)
0.259664 + 0.965699i \(0.416388\pi\)
\(458\) 17.3254 0.809561
\(459\) 0 0
\(460\) −6.60411 −0.307918
\(461\) −38.1693 −1.77772 −0.888861 0.458177i \(-0.848503\pi\)
−0.888861 + 0.458177i \(0.848503\pi\)
\(462\) 0 0
\(463\) −31.4388 −1.46108 −0.730542 0.682867i \(-0.760733\pi\)
−0.730542 + 0.682867i \(0.760733\pi\)
\(464\) −3.77568 −0.175281
\(465\) 0 0
\(466\) 29.4388 1.36373
\(467\) 11.9204 0.551611 0.275805 0.961213i \(-0.411055\pi\)
0.275805 + 0.961213i \(0.411055\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −18.2704 −0.842750
\(471\) 0 0
\(472\) 1.48881 0.0685281
\(473\) −4.32666 −0.198940
\(474\) 0 0
\(475\) −0.623909 −0.0286269
\(476\) 0 0
\(477\) 0 0
\(478\) 24.2047 1.10710
\(479\) 39.4155 1.80094 0.900470 0.434918i \(-0.143223\pi\)
0.900470 + 0.434918i \(0.143223\pi\)
\(480\) 0 0
\(481\) 0.480818 0.0219234
\(482\) 15.3083 0.697275
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −2.91038 −0.132154
\(486\) 0 0
\(487\) 23.1551 1.04926 0.524629 0.851331i \(-0.324204\pi\)
0.524629 + 0.851331i \(0.324204\pi\)
\(488\) 3.19932 0.144826
\(489\) 0 0
\(490\) 0 0
\(491\) 4.47530 0.201967 0.100984 0.994888i \(-0.467801\pi\)
0.100984 + 0.994888i \(0.467801\pi\)
\(492\) 0 0
\(493\) 25.0768 1.12940
\(494\) −1.69038 −0.0760538
\(495\) 0 0
\(496\) −2.00666 −0.0901019
\(497\) 0 0
\(498\) 0 0
\(499\) −16.7727 −0.750850 −0.375425 0.926853i \(-0.622503\pi\)
−0.375425 + 0.926853i \(0.622503\pi\)
\(500\) 9.60058 0.429351
\(501\) 0 0
\(502\) 7.79453 0.347887
\(503\) 41.1653 1.83547 0.917734 0.397195i \(-0.130016\pi\)
0.917734 + 0.397195i \(0.130016\pi\)
\(504\) 0 0
\(505\) 35.9406 1.59933
\(506\) 2.66981 0.118687
\(507\) 0 0
\(508\) 0.0397948 0.00176561
\(509\) −14.5987 −0.647077 −0.323538 0.946215i \(-0.604872\pi\)
−0.323538 + 0.946215i \(0.604872\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 14.0973 0.621807
\(515\) −30.2374 −1.33242
\(516\) 0 0
\(517\) 7.38607 0.324839
\(518\) 0 0
\(519\) 0 0
\(520\) −7.49824 −0.328819
\(521\) −17.0054 −0.745019 −0.372509 0.928028i \(-0.621503\pi\)
−0.372509 + 0.928028i \(0.621503\pi\)
\(522\) 0 0
\(523\) 32.8874 1.43807 0.719033 0.694976i \(-0.244585\pi\)
0.719033 + 0.694976i \(0.244585\pi\)
\(524\) −9.60097 −0.419420
\(525\) 0 0
\(526\) 4.61353 0.201159
\(527\) 13.3276 0.580559
\(528\) 0 0
\(529\) −15.8721 −0.690092
\(530\) −7.01978 −0.304920
\(531\) 0 0
\(532\) 0 0
\(533\) 17.9817 0.778874
\(534\) 0 0
\(535\) −6.97316 −0.301476
\(536\) 11.2739 0.486959
\(537\) 0 0
\(538\) −3.18323 −0.137239
\(539\) 0 0
\(540\) 0 0
\(541\) −3.11529 −0.133937 −0.0669685 0.997755i \(-0.521333\pi\)
−0.0669685 + 0.997755i \(0.521333\pi\)
\(542\) −0.722930 −0.0310525
\(543\) 0 0
\(544\) 6.64167 0.284759
\(545\) 50.8276 2.17722
\(546\) 0 0
\(547\) −4.58706 −0.196129 −0.0980643 0.995180i \(-0.531265\pi\)
−0.0980643 + 0.995180i \(0.531265\pi\)
\(548\) 18.6667 0.797400
\(549\) 0 0
\(550\) 1.11882 0.0477068
\(551\) 2.10550 0.0896972
\(552\) 0 0
\(553\) 0 0
\(554\) −1.18490 −0.0503416
\(555\) 0 0
\(556\) 8.33333 0.353412
\(557\) −18.4290 −0.780862 −0.390431 0.920632i \(-0.627674\pi\)
−0.390431 + 0.920632i \(0.627674\pi\)
\(558\) 0 0
\(559\) −13.1153 −0.554718
\(560\) 0 0
\(561\) 0 0
\(562\) 2.89097 0.121948
\(563\) −7.03903 −0.296660 −0.148330 0.988938i \(-0.547390\pi\)
−0.148330 + 0.988938i \(0.547390\pi\)
\(564\) 0 0
\(565\) 28.9894 1.21959
\(566\) 5.32353 0.223765
\(567\) 0 0
\(568\) −11.0098 −0.461961
\(569\) 20.2913 0.850657 0.425328 0.905039i \(-0.360159\pi\)
0.425328 + 0.905039i \(0.360159\pi\)
\(570\) 0 0
\(571\) 10.8517 0.454131 0.227066 0.973879i \(-0.427087\pi\)
0.227066 + 0.973879i \(0.427087\pi\)
\(572\) 3.03127 0.126744
\(573\) 0 0
\(574\) 0 0
\(575\) 2.98705 0.124568
\(576\) 0 0
\(577\) −15.0944 −0.628387 −0.314194 0.949359i \(-0.601734\pi\)
−0.314194 + 0.949359i \(0.601734\pi\)
\(578\) −27.1118 −1.12770
\(579\) 0 0
\(580\) 9.33962 0.387806
\(581\) 0 0
\(582\) 0 0
\(583\) 2.83785 0.117532
\(584\) 3.18323 0.131723
\(585\) 0 0
\(586\) 10.6413 0.439587
\(587\) −8.46197 −0.349263 −0.174631 0.984634i \(-0.555873\pi\)
−0.174631 + 0.984634i \(0.555873\pi\)
\(588\) 0 0
\(589\) 1.11901 0.0461080
\(590\) −3.68276 −0.151617
\(591\) 0 0
\(592\) −0.158619 −0.00651921
\(593\) −28.2234 −1.15900 −0.579498 0.814974i \(-0.696751\pi\)
−0.579498 + 0.814974i \(0.696751\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.2341 0.624014
\(597\) 0 0
\(598\) 8.09292 0.330944
\(599\) −15.4396 −0.630845 −0.315422 0.948951i \(-0.602146\pi\)
−0.315422 + 0.948951i \(0.602146\pi\)
\(600\) 0 0
\(601\) −10.8464 −0.442432 −0.221216 0.975225i \(-0.571003\pi\)
−0.221216 + 0.975225i \(0.571003\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −8.31724 −0.338424
\(605\) −2.47363 −0.100567
\(606\) 0 0
\(607\) −13.9134 −0.564725 −0.282363 0.959308i \(-0.591118\pi\)
−0.282363 + 0.959308i \(0.591118\pi\)
\(608\) 0.557647 0.0226156
\(609\) 0 0
\(610\) −7.91391 −0.320425
\(611\) 22.3892 0.905770
\(612\) 0 0
\(613\) 6.82843 0.275798 0.137899 0.990446i \(-0.455965\pi\)
0.137899 + 0.990446i \(0.455965\pi\)
\(614\) −28.9733 −1.16927
\(615\) 0 0
\(616\) 0 0
\(617\) 32.0688 1.29104 0.645521 0.763743i \(-0.276640\pi\)
0.645521 + 0.763743i \(0.276640\pi\)
\(618\) 0 0
\(619\) −18.0492 −0.725459 −0.362730 0.931894i \(-0.618155\pi\)
−0.362730 + 0.931894i \(0.618155\pi\)
\(620\) 4.96374 0.199348
\(621\) 0 0
\(622\) 2.62019 0.105060
\(623\) 0 0
\(624\) 0 0
\(625\) −29.3424 −1.17369
\(626\) 31.7566 1.26925
\(627\) 0 0
\(628\) −9.13401 −0.364487
\(629\) 1.05350 0.0420056
\(630\) 0 0
\(631\) 9.28724 0.369719 0.184860 0.982765i \(-0.440817\pi\)
0.184860 + 0.982765i \(0.440817\pi\)
\(632\) 6.95668 0.276722
\(633\) 0 0
\(634\) −24.8308 −0.986157
\(635\) −0.0984373 −0.00390637
\(636\) 0 0
\(637\) 0 0
\(638\) −3.77568 −0.149481
\(639\) 0 0
\(640\) 2.47363 0.0977786
\(641\) 15.3700 0.607078 0.303539 0.952819i \(-0.401832\pi\)
0.303539 + 0.952819i \(0.401832\pi\)
\(642\) 0 0
\(643\) −11.3833 −0.448914 −0.224457 0.974484i \(-0.572061\pi\)
−0.224457 + 0.974484i \(0.572061\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −3.70371 −0.145720
\(647\) 20.1464 0.792038 0.396019 0.918242i \(-0.370391\pi\)
0.396019 + 0.918242i \(0.370391\pi\)
\(648\) 0 0
\(649\) 1.48881 0.0584410
\(650\) 3.39146 0.133024
\(651\) 0 0
\(652\) 8.33609 0.326466
\(653\) 38.1062 1.49121 0.745606 0.666387i \(-0.232160\pi\)
0.745606 + 0.666387i \(0.232160\pi\)
\(654\) 0 0
\(655\) 23.7492 0.927958
\(656\) −5.93207 −0.231608
\(657\) 0 0
\(658\) 0 0
\(659\) −21.4653 −0.836168 −0.418084 0.908408i \(-0.637298\pi\)
−0.418084 + 0.908408i \(0.637298\pi\)
\(660\) 0 0
\(661\) 6.01166 0.233826 0.116913 0.993142i \(-0.462700\pi\)
0.116913 + 0.993142i \(0.462700\pi\)
\(662\) −21.2669 −0.826560
\(663\) 0 0
\(664\) −4.59744 −0.178415
\(665\) 0 0
\(666\) 0 0
\(667\) −10.0803 −0.390312
\(668\) 22.1610 0.857434
\(669\) 0 0
\(670\) −27.8874 −1.07739
\(671\) 3.19932 0.123508
\(672\) 0 0
\(673\) 0.415116 0.0160015 0.00800077 0.999968i \(-0.497453\pi\)
0.00800077 + 0.999968i \(0.497453\pi\)
\(674\) −23.3927 −0.901055
\(675\) 0 0
\(676\) −3.81138 −0.146592
\(677\) 18.2067 0.699742 0.349871 0.936798i \(-0.386225\pi\)
0.349871 + 0.936798i \(0.386225\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −16.4290 −0.630024
\(681\) 0 0
\(682\) −2.00666 −0.0768391
\(683\) 44.9763 1.72097 0.860485 0.509476i \(-0.170161\pi\)
0.860485 + 0.509476i \(0.170161\pi\)
\(684\) 0 0
\(685\) −46.1743 −1.76423
\(686\) 0 0
\(687\) 0 0
\(688\) 4.32666 0.164952
\(689\) 8.60230 0.327722
\(690\) 0 0
\(691\) −30.7910 −1.17134 −0.585672 0.810548i \(-0.699169\pi\)
−0.585672 + 0.810548i \(0.699169\pi\)
\(692\) −20.9656 −0.796991
\(693\) 0 0
\(694\) 6.60020 0.250540
\(695\) −20.6135 −0.781916
\(696\) 0 0
\(697\) 39.3988 1.49234
\(698\) −5.06164 −0.191586
\(699\) 0 0
\(700\) 0 0
\(701\) 7.83376 0.295877 0.147939 0.988997i \(-0.452736\pi\)
0.147939 + 0.988997i \(0.452736\pi\)
\(702\) 0 0
\(703\) 0.0884535 0.00333609
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −11.5166 −0.433431
\(707\) 0 0
\(708\) 0 0
\(709\) 44.2507 1.66187 0.830936 0.556368i \(-0.187806\pi\)
0.830936 + 0.556368i \(0.187806\pi\)
\(710\) 27.2341 1.02208
\(711\) 0 0
\(712\) 6.62558 0.248304
\(713\) −5.35741 −0.200636
\(714\) 0 0
\(715\) −7.49824 −0.280418
\(716\) 12.3759 0.462508
\(717\) 0 0
\(718\) −10.5080 −0.392156
\(719\) 6.50843 0.242723 0.121362 0.992608i \(-0.461274\pi\)
0.121362 + 0.992608i \(0.461274\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 18.6890 0.695534
\(723\) 0 0
\(724\) 20.5595 0.764087
\(725\) −4.22432 −0.156887
\(726\) 0 0
\(727\) 18.9110 0.701369 0.350684 0.936494i \(-0.385949\pi\)
0.350684 + 0.936494i \(0.385949\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −7.87412 −0.291434
\(731\) −28.7363 −1.06285
\(732\) 0 0
\(733\) −10.2146 −0.377286 −0.188643 0.982046i \(-0.560409\pi\)
−0.188643 + 0.982046i \(0.560409\pi\)
\(734\) −10.8976 −0.402239
\(735\) 0 0
\(736\) −2.66981 −0.0984104
\(737\) 11.2739 0.415280
\(738\) 0 0
\(739\) 7.08939 0.260787 0.130394 0.991462i \(-0.458376\pi\)
0.130394 + 0.991462i \(0.458376\pi\)
\(740\) 0.392364 0.0144236
\(741\) 0 0
\(742\) 0 0
\(743\) 32.5408 1.19380 0.596902 0.802314i \(-0.296398\pi\)
0.596902 + 0.802314i \(0.296398\pi\)
\(744\) 0 0
\(745\) −37.6835 −1.38062
\(746\) −21.3459 −0.781528
\(747\) 0 0
\(748\) 6.64167 0.242843
\(749\) 0 0
\(750\) 0 0
\(751\) −29.0496 −1.06003 −0.530017 0.847987i \(-0.677815\pi\)
−0.530017 + 0.847987i \(0.677815\pi\)
\(752\) −7.38607 −0.269342
\(753\) 0 0
\(754\) −11.4451 −0.416806
\(755\) 20.5737 0.748755
\(756\) 0 0
\(757\) 28.4151 1.03276 0.516382 0.856358i \(-0.327278\pi\)
0.516382 + 0.856358i \(0.327278\pi\)
\(758\) 21.3066 0.773892
\(759\) 0 0
\(760\) −1.37941 −0.0500365
\(761\) 25.7999 0.935245 0.467622 0.883928i \(-0.345111\pi\)
0.467622 + 0.883928i \(0.345111\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 20.3267 0.735393
\(765\) 0 0
\(766\) −34.6202 −1.25088
\(767\) 4.51299 0.162955
\(768\) 0 0
\(769\) 27.1932 0.980612 0.490306 0.871550i \(-0.336885\pi\)
0.490306 + 0.871550i \(0.336885\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −13.0761 −0.470618
\(773\) 24.5032 0.881319 0.440660 0.897674i \(-0.354745\pi\)
0.440660 + 0.897674i \(0.354745\pi\)
\(774\) 0 0
\(775\) −2.24510 −0.0806465
\(776\) −1.17656 −0.0422362
\(777\) 0 0
\(778\) −9.68276 −0.347144
\(779\) 3.30800 0.118521
\(780\) 0 0
\(781\) −11.0098 −0.393962
\(782\) 17.7320 0.634094
\(783\) 0 0
\(784\) 0 0
\(785\) 22.5941 0.806419
\(786\) 0 0
\(787\) −52.8741 −1.88476 −0.942379 0.334547i \(-0.891417\pi\)
−0.942379 + 0.334547i \(0.891417\pi\)
\(788\) 3.80159 0.135426
\(789\) 0 0
\(790\) −17.2082 −0.612241
\(791\) 0 0
\(792\) 0 0
\(793\) 9.69800 0.344386
\(794\) 11.8928 0.422061
\(795\) 0 0
\(796\) −6.87098 −0.243536
\(797\) −27.1028 −0.960032 −0.480016 0.877260i \(-0.659369\pi\)
−0.480016 + 0.877260i \(0.659369\pi\)
\(798\) 0 0
\(799\) 49.0559 1.73547
\(800\) −1.11882 −0.0395564
\(801\) 0 0
\(802\) 33.9804 1.19989
\(803\) 3.18323 0.112334
\(804\) 0 0
\(805\) 0 0
\(806\) −6.08275 −0.214256
\(807\) 0 0
\(808\) 14.5295 0.511146
\(809\) 3.58488 0.126038 0.0630189 0.998012i \(-0.479927\pi\)
0.0630189 + 0.998012i \(0.479927\pi\)
\(810\) 0 0
\(811\) 50.2933 1.76604 0.883018 0.469339i \(-0.155508\pi\)
0.883018 + 0.469339i \(0.155508\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −0.158619 −0.00555960
\(815\) −20.6204 −0.722300
\(816\) 0 0
\(817\) −2.41275 −0.0844115
\(818\) −9.51305 −0.332616
\(819\) 0 0
\(820\) 14.6737 0.512428
\(821\) 11.1556 0.389335 0.194667 0.980869i \(-0.437637\pi\)
0.194667 + 0.980869i \(0.437637\pi\)
\(822\) 0 0
\(823\) 22.7860 0.794271 0.397136 0.917760i \(-0.370004\pi\)
0.397136 + 0.917760i \(0.370004\pi\)
\(824\) −12.2239 −0.425841
\(825\) 0 0
\(826\) 0 0
\(827\) −17.0490 −0.592853 −0.296426 0.955056i \(-0.595795\pi\)
−0.296426 + 0.955056i \(0.595795\pi\)
\(828\) 0 0
\(829\) −11.6952 −0.406190 −0.203095 0.979159i \(-0.565100\pi\)
−0.203095 + 0.979159i \(0.565100\pi\)
\(830\) 11.3724 0.394740
\(831\) 0 0
\(832\) −3.03127 −0.105090
\(833\) 0 0
\(834\) 0 0
\(835\) −54.8180 −1.89705
\(836\) 0.557647 0.0192866
\(837\) 0 0
\(838\) −15.9249 −0.550116
\(839\) −30.2833 −1.04550 −0.522748 0.852487i \(-0.675093\pi\)
−0.522748 + 0.852487i \(0.675093\pi\)
\(840\) 0 0
\(841\) −14.7443 −0.508422
\(842\) −3.55842 −0.122631
\(843\) 0 0
\(844\) 11.7984 0.406119
\(845\) 9.42794 0.324331
\(846\) 0 0
\(847\) 0 0
\(848\) −2.83785 −0.0974522
\(849\) 0 0
\(850\) 7.43086 0.254876
\(851\) −0.423483 −0.0145168
\(852\) 0 0
\(853\) −34.7318 −1.18920 −0.594598 0.804023i \(-0.702689\pi\)
−0.594598 + 0.804023i \(0.702689\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2.81900 −0.0963515
\(857\) −21.7344 −0.742433 −0.371216 0.928546i \(-0.621059\pi\)
−0.371216 + 0.928546i \(0.621059\pi\)
\(858\) 0 0
\(859\) −39.9241 −1.36219 −0.681096 0.732194i \(-0.738496\pi\)
−0.681096 + 0.732194i \(0.738496\pi\)
\(860\) −10.7025 −0.364954
\(861\) 0 0
\(862\) 20.2774 0.690652
\(863\) −57.8380 −1.96883 −0.984414 0.175866i \(-0.943727\pi\)
−0.984414 + 0.175866i \(0.943727\pi\)
\(864\) 0 0
\(865\) 51.8610 1.76333
\(866\) −27.7144 −0.941774
\(867\) 0 0
\(868\) 0 0
\(869\) 6.95668 0.235989
\(870\) 0 0
\(871\) 34.1743 1.15795
\(872\) 20.5478 0.695837
\(873\) 0 0
\(874\) 1.48881 0.0503598
\(875\) 0 0
\(876\) 0 0
\(877\) −38.1717 −1.28897 −0.644484 0.764618i \(-0.722928\pi\)
−0.644484 + 0.764618i \(0.722928\pi\)
\(878\) −4.85433 −0.163826
\(879\) 0 0
\(880\) 2.47363 0.0833859
\(881\) 50.5818 1.70415 0.852073 0.523423i \(-0.175345\pi\)
0.852073 + 0.523423i \(0.175345\pi\)
\(882\) 0 0
\(883\) −45.3333 −1.52559 −0.762794 0.646642i \(-0.776173\pi\)
−0.762794 + 0.646642i \(0.776173\pi\)
\(884\) 20.1327 0.677136
\(885\) 0 0
\(886\) −30.5269 −1.02557
\(887\) 14.6337 0.491353 0.245676 0.969352i \(-0.420990\pi\)
0.245676 + 0.969352i \(0.420990\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −16.3892 −0.549368
\(891\) 0 0
\(892\) −21.0031 −0.703237
\(893\) 4.11882 0.137831
\(894\) 0 0
\(895\) −30.6133 −1.02329
\(896\) 0 0
\(897\) 0 0
\(898\) −30.3986 −1.01442
\(899\) 7.57652 0.252691
\(900\) 0 0
\(901\) 18.8481 0.627920
\(902\) −5.93207 −0.197516
\(903\) 0 0
\(904\) 11.7194 0.389781
\(905\) −50.8565 −1.69053
\(906\) 0 0
\(907\) 51.4597 1.70869 0.854346 0.519704i \(-0.173958\pi\)
0.854346 + 0.519704i \(0.173958\pi\)
\(908\) 29.4884 0.978607
\(909\) 0 0
\(910\) 0 0
\(911\) −4.27373 −0.141595 −0.0707975 0.997491i \(-0.522554\pi\)
−0.0707975 + 0.997491i \(0.522554\pi\)
\(912\) 0 0
\(913\) −4.59744 −0.152153
\(914\) −11.1020 −0.367220
\(915\) 0 0
\(916\) −17.3254 −0.572446
\(917\) 0 0
\(918\) 0 0
\(919\) 46.0992 1.52067 0.760336 0.649530i \(-0.225034\pi\)
0.760336 + 0.649530i \(0.225034\pi\)
\(920\) 6.60411 0.217731
\(921\) 0 0
\(922\) 38.1693 1.25704
\(923\) −33.3737 −1.09851
\(924\) 0 0
\(925\) −0.177467 −0.00583508
\(926\) 31.4388 1.03314
\(927\) 0 0
\(928\) 3.77568 0.123943
\(929\) −46.3287 −1.51999 −0.759997 0.649926i \(-0.774800\pi\)
−0.759997 + 0.649926i \(0.774800\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −29.4388 −0.964300
\(933\) 0 0
\(934\) −11.9204 −0.390048
\(935\) −16.4290 −0.537286
\(936\) 0 0
\(937\) 50.7016 1.65635 0.828175 0.560470i \(-0.189379\pi\)
0.828175 + 0.560470i \(0.189379\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 18.2704 0.595914
\(941\) −23.8736 −0.778257 −0.389128 0.921184i \(-0.627224\pi\)
−0.389128 + 0.921184i \(0.627224\pi\)
\(942\) 0 0
\(943\) −15.8375 −0.515739
\(944\) −1.48881 −0.0484567
\(945\) 0 0
\(946\) 4.32666 0.140672
\(947\) 21.8678 0.710610 0.355305 0.934751i \(-0.384377\pi\)
0.355305 + 0.934751i \(0.384377\pi\)
\(948\) 0 0
\(949\) 9.64923 0.313227
\(950\) 0.623909 0.0202423
\(951\) 0 0
\(952\) 0 0
\(953\) −4.42292 −0.143273 −0.0716363 0.997431i \(-0.522822\pi\)
−0.0716363 + 0.997431i \(0.522822\pi\)
\(954\) 0 0
\(955\) −50.2806 −1.62704
\(956\) −24.2047 −0.782835
\(957\) 0 0
\(958\) −39.4155 −1.27346
\(959\) 0 0
\(960\) 0 0
\(961\) −26.9733 −0.870106
\(962\) −0.480818 −0.0155022
\(963\) 0 0
\(964\) −15.3083 −0.493048
\(965\) 32.3453 1.04123
\(966\) 0 0
\(967\) 17.1894 0.552773 0.276386 0.961047i \(-0.410863\pi\)
0.276386 + 0.961047i \(0.410863\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 2.91038 0.0934467
\(971\) 25.4614 0.817094 0.408547 0.912737i \(-0.366036\pi\)
0.408547 + 0.912737i \(0.366036\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −23.1551 −0.741937
\(975\) 0 0
\(976\) −3.19932 −0.102408
\(977\) −16.9143 −0.541136 −0.270568 0.962701i \(-0.587211\pi\)
−0.270568 + 0.962701i \(0.587211\pi\)
\(978\) 0 0
\(979\) 6.62558 0.211754
\(980\) 0 0
\(981\) 0 0
\(982\) −4.47530 −0.142812
\(983\) −20.3763 −0.649902 −0.324951 0.945731i \(-0.605348\pi\)
−0.324951 + 0.945731i \(0.605348\pi\)
\(984\) 0 0
\(985\) −9.40370 −0.299627
\(986\) −25.0768 −0.798608
\(987\) 0 0
\(988\) 1.69038 0.0537782
\(989\) 11.5514 0.367312
\(990\) 0 0
\(991\) 35.1482 1.11652 0.558260 0.829666i \(-0.311469\pi\)
0.558260 + 0.829666i \(0.311469\pi\)
\(992\) 2.00666 0.0637116
\(993\) 0 0
\(994\) 0 0
\(995\) 16.9962 0.538817
\(996\) 0 0
\(997\) 43.8660 1.38925 0.694625 0.719372i \(-0.255571\pi\)
0.694625 + 0.719372i \(0.255571\pi\)
\(998\) 16.7727 0.530931
\(999\) 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 9702.2.a.dz.1.2 4
3.2 odd 2 3234.2.a.bm.1.3 yes 4
7.6 odd 2 9702.2.a.ea.1.3 4
21.20 even 2 3234.2.a.bl.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bl.1.2 4 21.20 even 2
3234.2.a.bm.1.3 yes 4 3.2 odd 2
9702.2.a.dz.1.2 4 1.1 even 1 trivial
9702.2.a.ea.1.3 4 7.6 odd 2