Properties

Label 9702.2.a.db.1.1
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.64575\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.64575 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.64575 q^{5} +1.00000 q^{8} -2.64575 q^{10} -1.00000 q^{11} -4.00000 q^{13} +1.00000 q^{16} +3.00000 q^{17} +5.29150 q^{19} -2.64575 q^{20} -1.00000 q^{22} +2.64575 q^{23} +2.00000 q^{25} -4.00000 q^{26} -2.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} +1.29150 q^{37} +5.29150 q^{38} -2.64575 q^{40} -9.00000 q^{41} +9.29150 q^{43} -1.00000 q^{44} +2.64575 q^{46} +3.93725 q^{47} +2.00000 q^{50} -4.00000 q^{52} -4.00000 q^{53} +2.64575 q^{55} -2.00000 q^{58} -6.58301 q^{59} +11.9373 q^{61} -4.00000 q^{62} +1.00000 q^{64} +10.5830 q^{65} +7.58301 q^{67} +3.00000 q^{68} +2.70850 q^{71} -15.2915 q^{73} +1.29150 q^{74} +5.29150 q^{76} +10.6458 q^{79} -2.64575 q^{80} -9.00000 q^{82} -15.5830 q^{83} -7.93725 q^{85} +9.29150 q^{86} -1.00000 q^{88} -2.70850 q^{89} +2.64575 q^{92} +3.93725 q^{94} -14.0000 q^{95} -11.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} - 8 q^{13} + 2 q^{16} + 6 q^{17} - 2 q^{22} + 4 q^{25} - 8 q^{26} - 4 q^{29} - 8 q^{31} + 2 q^{32} + 6 q^{34} - 8 q^{37} - 18 q^{41} + 8 q^{43} - 2 q^{44} - 8 q^{47} + 4 q^{50} - 8 q^{52} - 8 q^{53} - 4 q^{58} + 8 q^{59} + 8 q^{61} - 8 q^{62} + 2 q^{64} - 6 q^{67} + 6 q^{68} + 16 q^{71} - 20 q^{73} - 8 q^{74} + 16 q^{79} - 18 q^{82} - 10 q^{83} + 8 q^{86} - 2 q^{88} - 16 q^{89} - 8 q^{94} - 28 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −2.64575 −1.18322 −0.591608 0.806226i \(-0.701507\pi\)
−0.591608 + 0.806226i \(0.701507\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.64575 −0.836660
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 5.29150 1.21395 0.606977 0.794719i \(-0.292382\pi\)
0.606977 + 0.794719i \(0.292382\pi\)
\(20\) −2.64575 −0.591608
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 2.64575 0.551677 0.275839 0.961204i \(-0.411044\pi\)
0.275839 + 0.961204i \(0.411044\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 0 0
\(37\) 1.29150 0.212322 0.106161 0.994349i \(-0.466144\pi\)
0.106161 + 0.994349i \(0.466144\pi\)
\(38\) 5.29150 0.858395
\(39\) 0 0
\(40\) −2.64575 −0.418330
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) 9.29150 1.41694 0.708470 0.705740i \(-0.249386\pi\)
0.708470 + 0.705740i \(0.249386\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 2.64575 0.390095
\(47\) 3.93725 0.574308 0.287154 0.957885i \(-0.407291\pi\)
0.287154 + 0.957885i \(0.407291\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 2.00000 0.282843
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) 2.64575 0.356753
\(56\) 0 0
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −6.58301 −0.857034 −0.428517 0.903534i \(-0.640964\pi\)
−0.428517 + 0.903534i \(0.640964\pi\)
\(60\) 0 0
\(61\) 11.9373 1.52841 0.764204 0.644974i \(-0.223132\pi\)
0.764204 + 0.644974i \(0.223132\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 10.5830 1.31266
\(66\) 0 0
\(67\) 7.58301 0.926412 0.463206 0.886251i \(-0.346699\pi\)
0.463206 + 0.886251i \(0.346699\pi\)
\(68\) 3.00000 0.363803
\(69\) 0 0
\(70\) 0 0
\(71\) 2.70850 0.321440 0.160720 0.987000i \(-0.448618\pi\)
0.160720 + 0.987000i \(0.448618\pi\)
\(72\) 0 0
\(73\) −15.2915 −1.78974 −0.894868 0.446332i \(-0.852730\pi\)
−0.894868 + 0.446332i \(0.852730\pi\)
\(74\) 1.29150 0.150134
\(75\) 0 0
\(76\) 5.29150 0.606977
\(77\) 0 0
\(78\) 0 0
\(79\) 10.6458 1.19774 0.598870 0.800846i \(-0.295616\pi\)
0.598870 + 0.800846i \(0.295616\pi\)
\(80\) −2.64575 −0.295804
\(81\) 0 0
\(82\) −9.00000 −0.993884
\(83\) −15.5830 −1.71046 −0.855229 0.518251i \(-0.826583\pi\)
−0.855229 + 0.518251i \(0.826583\pi\)
\(84\) 0 0
\(85\) −7.93725 −0.860916
\(86\) 9.29150 1.00193
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −2.70850 −0.287100 −0.143550 0.989643i \(-0.545852\pi\)
−0.143550 + 0.989643i \(0.545852\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.64575 0.275839
\(93\) 0 0
\(94\) 3.93725 0.406097
\(95\) −14.0000 −1.43637
\(96\) 0 0
\(97\) −11.5830 −1.17608 −0.588038 0.808833i \(-0.700099\pi\)
−0.588038 + 0.808833i \(0.700099\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 2.00000 0.200000
\(101\) −7.29150 −0.725532 −0.362766 0.931880i \(-0.618168\pi\)
−0.362766 + 0.931880i \(0.618168\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 0 0
\(109\) −3.93725 −0.377121 −0.188560 0.982062i \(-0.560382\pi\)
−0.188560 + 0.982062i \(0.560382\pi\)
\(110\) 2.64575 0.252262
\(111\) 0 0
\(112\) 0 0
\(113\) −12.5830 −1.18371 −0.591855 0.806045i \(-0.701604\pi\)
−0.591855 + 0.806045i \(0.701604\pi\)
\(114\) 0 0
\(115\) −7.00000 −0.652753
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −6.58301 −0.606015
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 11.9373 1.08075
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 7.93725 0.709930
\(126\) 0 0
\(127\) 2.64575 0.234772 0.117386 0.993086i \(-0.462548\pi\)
0.117386 + 0.993086i \(0.462548\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 10.5830 0.928191
\(131\) 18.5830 1.62360 0.811802 0.583932i \(-0.198487\pi\)
0.811802 + 0.583932i \(0.198487\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 7.58301 0.655072
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 13.2915 1.13557 0.567785 0.823177i \(-0.307801\pi\)
0.567785 + 0.823177i \(0.307801\pi\)
\(138\) 0 0
\(139\) −12.5830 −1.06728 −0.533638 0.845713i \(-0.679176\pi\)
−0.533638 + 0.845713i \(0.679176\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.70850 0.227292
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 5.29150 0.439435
\(146\) −15.2915 −1.26553
\(147\) 0 0
\(148\) 1.29150 0.106161
\(149\) −19.8745 −1.62818 −0.814092 0.580737i \(-0.802765\pi\)
−0.814092 + 0.580737i \(0.802765\pi\)
\(150\) 0 0
\(151\) 22.6458 1.84289 0.921443 0.388515i \(-0.127012\pi\)
0.921443 + 0.388515i \(0.127012\pi\)
\(152\) 5.29150 0.429198
\(153\) 0 0
\(154\) 0 0
\(155\) 10.5830 0.850047
\(156\) 0 0
\(157\) −0.708497 −0.0565442 −0.0282721 0.999600i \(-0.509000\pi\)
−0.0282721 + 0.999600i \(0.509000\pi\)
\(158\) 10.6458 0.846931
\(159\) 0 0
\(160\) −2.64575 −0.209165
\(161\) 0 0
\(162\) 0 0
\(163\) −5.58301 −0.437295 −0.218647 0.975804i \(-0.570164\pi\)
−0.218647 + 0.975804i \(0.570164\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) −15.5830 −1.20948
\(167\) −8.70850 −0.673884 −0.336942 0.941525i \(-0.609393\pi\)
−0.336942 + 0.941525i \(0.609393\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −7.93725 −0.608760
\(171\) 0 0
\(172\) 9.29150 0.708470
\(173\) −4.00000 −0.304114 −0.152057 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −2.70850 −0.203010
\(179\) −4.70850 −0.351930 −0.175965 0.984396i \(-0.556304\pi\)
−0.175965 + 0.984396i \(0.556304\pi\)
\(180\) 0 0
\(181\) −5.29150 −0.393314 −0.196657 0.980472i \(-0.563009\pi\)
−0.196657 + 0.980472i \(0.563009\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2.64575 0.195047
\(185\) −3.41699 −0.251222
\(186\) 0 0
\(187\) −3.00000 −0.219382
\(188\) 3.93725 0.287154
\(189\) 0 0
\(190\) −14.0000 −1.01567
\(191\) −6.70850 −0.485410 −0.242705 0.970100i \(-0.578035\pi\)
−0.242705 + 0.970100i \(0.578035\pi\)
\(192\) 0 0
\(193\) −15.8745 −1.14267 −0.571336 0.820716i \(-0.693575\pi\)
−0.571336 + 0.820716i \(0.693575\pi\)
\(194\) −11.5830 −0.831611
\(195\) 0 0
\(196\) 0 0
\(197\) −21.8745 −1.55849 −0.779247 0.626717i \(-0.784398\pi\)
−0.779247 + 0.626717i \(0.784398\pi\)
\(198\) 0 0
\(199\) −19.1660 −1.35864 −0.679321 0.733841i \(-0.737726\pi\)
−0.679321 + 0.733841i \(0.737726\pi\)
\(200\) 2.00000 0.141421
\(201\) 0 0
\(202\) −7.29150 −0.513028
\(203\) 0 0
\(204\) 0 0
\(205\) 23.8118 1.66309
\(206\) −10.0000 −0.696733
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −5.29150 −0.366021
\(210\) 0 0
\(211\) 21.2915 1.46577 0.732884 0.680354i \(-0.238174\pi\)
0.732884 + 0.680354i \(0.238174\pi\)
\(212\) −4.00000 −0.274721
\(213\) 0 0
\(214\) −9.00000 −0.615227
\(215\) −24.5830 −1.67655
\(216\) 0 0
\(217\) 0 0
\(218\) −3.93725 −0.266664
\(219\) 0 0
\(220\) 2.64575 0.178377
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 28.4575 1.90566 0.952828 0.303511i \(-0.0981588\pi\)
0.952828 + 0.303511i \(0.0981588\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −12.5830 −0.837009
\(227\) −2.41699 −0.160422 −0.0802108 0.996778i \(-0.525559\pi\)
−0.0802108 + 0.996778i \(0.525559\pi\)
\(228\) 0 0
\(229\) 6.70850 0.443310 0.221655 0.975125i \(-0.428854\pi\)
0.221655 + 0.975125i \(0.428854\pi\)
\(230\) −7.00000 −0.461566
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −20.1660 −1.32112 −0.660560 0.750774i \(-0.729681\pi\)
−0.660560 + 0.750774i \(0.729681\pi\)
\(234\) 0 0
\(235\) −10.4170 −0.679530
\(236\) −6.58301 −0.428517
\(237\) 0 0
\(238\) 0 0
\(239\) −2.70850 −0.175198 −0.0875991 0.996156i \(-0.527919\pi\)
−0.0875991 + 0.996156i \(0.527919\pi\)
\(240\) 0 0
\(241\) −27.1660 −1.74992 −0.874958 0.484198i \(-0.839111\pi\)
−0.874958 + 0.484198i \(0.839111\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 11.9373 0.764204
\(245\) 0 0
\(246\) 0 0
\(247\) −21.1660 −1.34676
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) 7.93725 0.501996
\(251\) −23.2915 −1.47015 −0.735073 0.677988i \(-0.762852\pi\)
−0.735073 + 0.677988i \(0.762852\pi\)
\(252\) 0 0
\(253\) −2.64575 −0.166337
\(254\) 2.64575 0.166009
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.29150 −0.579588 −0.289794 0.957089i \(-0.593587\pi\)
−0.289794 + 0.957089i \(0.593587\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 10.5830 0.656330
\(261\) 0 0
\(262\) 18.5830 1.14806
\(263\) −21.8745 −1.34884 −0.674420 0.738348i \(-0.735606\pi\)
−0.674420 + 0.738348i \(0.735606\pi\)
\(264\) 0 0
\(265\) 10.5830 0.650109
\(266\) 0 0
\(267\) 0 0
\(268\) 7.58301 0.463206
\(269\) 10.6458 0.649083 0.324541 0.945871i \(-0.394790\pi\)
0.324541 + 0.945871i \(0.394790\pi\)
\(270\) 0 0
\(271\) 9.29150 0.564419 0.282209 0.959353i \(-0.408933\pi\)
0.282209 + 0.959353i \(0.408933\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) 13.2915 0.802969
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) −17.1660 −1.03141 −0.515703 0.856768i \(-0.672469\pi\)
−0.515703 + 0.856768i \(0.672469\pi\)
\(278\) −12.5830 −0.754679
\(279\) 0 0
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) 30.4575 1.81051 0.905256 0.424867i \(-0.139679\pi\)
0.905256 + 0.424867i \(0.139679\pi\)
\(284\) 2.70850 0.160720
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 5.29150 0.310728
\(291\) 0 0
\(292\) −15.2915 −0.894868
\(293\) 14.5830 0.851948 0.425974 0.904735i \(-0.359931\pi\)
0.425974 + 0.904735i \(0.359931\pi\)
\(294\) 0 0
\(295\) 17.4170 1.01406
\(296\) 1.29150 0.0750671
\(297\) 0 0
\(298\) −19.8745 −1.15130
\(299\) −10.5830 −0.612031
\(300\) 0 0
\(301\) 0 0
\(302\) 22.6458 1.30312
\(303\) 0 0
\(304\) 5.29150 0.303488
\(305\) −31.5830 −1.80844
\(306\) 0 0
\(307\) −11.4170 −0.651602 −0.325801 0.945438i \(-0.605634\pi\)
−0.325801 + 0.945438i \(0.605634\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 10.5830 0.601074
\(311\) −22.5203 −1.27701 −0.638503 0.769619i \(-0.720446\pi\)
−0.638503 + 0.769619i \(0.720446\pi\)
\(312\) 0 0
\(313\) −20.5830 −1.16342 −0.581710 0.813396i \(-0.697616\pi\)
−0.581710 + 0.813396i \(0.697616\pi\)
\(314\) −0.708497 −0.0399828
\(315\) 0 0
\(316\) 10.6458 0.598870
\(317\) −9.22876 −0.518339 −0.259169 0.965832i \(-0.583449\pi\)
−0.259169 + 0.965832i \(0.583449\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) −2.64575 −0.147902
\(321\) 0 0
\(322\) 0 0
\(323\) 15.8745 0.883281
\(324\) 0 0
\(325\) −8.00000 −0.443760
\(326\) −5.58301 −0.309214
\(327\) 0 0
\(328\) −9.00000 −0.496942
\(329\) 0 0
\(330\) 0 0
\(331\) 13.0000 0.714545 0.357272 0.934000i \(-0.383707\pi\)
0.357272 + 0.934000i \(0.383707\pi\)
\(332\) −15.5830 −0.855229
\(333\) 0 0
\(334\) −8.70850 −0.476508
\(335\) −20.0627 −1.09614
\(336\) 0 0
\(337\) 6.70850 0.365435 0.182718 0.983165i \(-0.441511\pi\)
0.182718 + 0.983165i \(0.441511\pi\)
\(338\) 3.00000 0.163178
\(339\) 0 0
\(340\) −7.93725 −0.430458
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) 9.29150 0.500964
\(345\) 0 0
\(346\) −4.00000 −0.215041
\(347\) −29.5830 −1.58810 −0.794049 0.607853i \(-0.792031\pi\)
−0.794049 + 0.607853i \(0.792031\pi\)
\(348\) 0 0
\(349\) −13.2288 −0.708119 −0.354060 0.935223i \(-0.615199\pi\)
−0.354060 + 0.935223i \(0.615199\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 23.2915 1.23968 0.619841 0.784728i \(-0.287197\pi\)
0.619841 + 0.784728i \(0.287197\pi\)
\(354\) 0 0
\(355\) −7.16601 −0.380332
\(356\) −2.70850 −0.143550
\(357\) 0 0
\(358\) −4.70850 −0.248852
\(359\) −20.5830 −1.08633 −0.543165 0.839626i \(-0.682774\pi\)
−0.543165 + 0.839626i \(0.682774\pi\)
\(360\) 0 0
\(361\) 9.00000 0.473684
\(362\) −5.29150 −0.278115
\(363\) 0 0
\(364\) 0 0
\(365\) 40.4575 2.11764
\(366\) 0 0
\(367\) 10.7085 0.558979 0.279490 0.960149i \(-0.409835\pi\)
0.279490 + 0.960149i \(0.409835\pi\)
\(368\) 2.64575 0.137919
\(369\) 0 0
\(370\) −3.41699 −0.177641
\(371\) 0 0
\(372\) 0 0
\(373\) −15.8118 −0.818702 −0.409351 0.912377i \(-0.634245\pi\)
−0.409351 + 0.912377i \(0.634245\pi\)
\(374\) −3.00000 −0.155126
\(375\) 0 0
\(376\) 3.93725 0.203048
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) −27.5830 −1.41684 −0.708422 0.705789i \(-0.750593\pi\)
−0.708422 + 0.705789i \(0.750593\pi\)
\(380\) −14.0000 −0.718185
\(381\) 0 0
\(382\) −6.70850 −0.343237
\(383\) −18.7085 −0.955960 −0.477980 0.878371i \(-0.658631\pi\)
−0.477980 + 0.878371i \(0.658631\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −15.8745 −0.807991
\(387\) 0 0
\(388\) −11.5830 −0.588038
\(389\) −31.9373 −1.61928 −0.809642 0.586925i \(-0.800338\pi\)
−0.809642 + 0.586925i \(0.800338\pi\)
\(390\) 0 0
\(391\) 7.93725 0.401404
\(392\) 0 0
\(393\) 0 0
\(394\) −21.8745 −1.10202
\(395\) −28.1660 −1.41719
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) −19.1660 −0.960705
\(399\) 0 0
\(400\) 2.00000 0.100000
\(401\) 35.8745 1.79149 0.895744 0.444571i \(-0.146644\pi\)
0.895744 + 0.444571i \(0.146644\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) −7.29150 −0.362766
\(405\) 0 0
\(406\) 0 0
\(407\) −1.29150 −0.0640174
\(408\) 0 0
\(409\) 35.8745 1.77388 0.886940 0.461884i \(-0.152827\pi\)
0.886940 + 0.461884i \(0.152827\pi\)
\(410\) 23.8118 1.17598
\(411\) 0 0
\(412\) −10.0000 −0.492665
\(413\) 0 0
\(414\) 0 0
\(415\) 41.2288 2.02384
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) −5.29150 −0.258816
\(419\) 9.87451 0.482401 0.241201 0.970475i \(-0.422459\pi\)
0.241201 + 0.970475i \(0.422459\pi\)
\(420\) 0 0
\(421\) 6.12549 0.298538 0.149269 0.988797i \(-0.452308\pi\)
0.149269 + 0.988797i \(0.452308\pi\)
\(422\) 21.2915 1.03645
\(423\) 0 0
\(424\) −4.00000 −0.194257
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) −9.00000 −0.435031
\(429\) 0 0
\(430\) −24.5830 −1.18550
\(431\) 17.2915 0.832902 0.416451 0.909158i \(-0.363274\pi\)
0.416451 + 0.909158i \(0.363274\pi\)
\(432\) 0 0
\(433\) 1.58301 0.0760744 0.0380372 0.999276i \(-0.487889\pi\)
0.0380372 + 0.999276i \(0.487889\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.93725 −0.188560
\(437\) 14.0000 0.669711
\(438\) 0 0
\(439\) −9.35425 −0.446454 −0.223227 0.974766i \(-0.571659\pi\)
−0.223227 + 0.974766i \(0.571659\pi\)
\(440\) 2.64575 0.126131
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) 31.1660 1.48074 0.740371 0.672199i \(-0.234650\pi\)
0.740371 + 0.672199i \(0.234650\pi\)
\(444\) 0 0
\(445\) 7.16601 0.339701
\(446\) 28.4575 1.34750
\(447\) 0 0
\(448\) 0 0
\(449\) −30.4575 −1.43738 −0.718689 0.695331i \(-0.755258\pi\)
−0.718689 + 0.695331i \(0.755258\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) −12.5830 −0.591855
\(453\) 0 0
\(454\) −2.41699 −0.113435
\(455\) 0 0
\(456\) 0 0
\(457\) −40.5830 −1.89839 −0.949196 0.314684i \(-0.898101\pi\)
−0.949196 + 0.314684i \(0.898101\pi\)
\(458\) 6.70850 0.313467
\(459\) 0 0
\(460\) −7.00000 −0.326377
\(461\) 24.5830 1.14494 0.572472 0.819924i \(-0.305984\pi\)
0.572472 + 0.819924i \(0.305984\pi\)
\(462\) 0 0
\(463\) 39.0405 1.81437 0.907183 0.420735i \(-0.138228\pi\)
0.907183 + 0.420735i \(0.138228\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −20.1660 −0.934172
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −10.4170 −0.480500
\(471\) 0 0
\(472\) −6.58301 −0.303007
\(473\) −9.29150 −0.427224
\(474\) 0 0
\(475\) 10.5830 0.485582
\(476\) 0 0
\(477\) 0 0
\(478\) −2.70850 −0.123884
\(479\) 33.8745 1.54777 0.773883 0.633329i \(-0.218312\pi\)
0.773883 + 0.633329i \(0.218312\pi\)
\(480\) 0 0
\(481\) −5.16601 −0.235550
\(482\) −27.1660 −1.23738
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 30.6458 1.39155
\(486\) 0 0
\(487\) 21.8745 0.991229 0.495614 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(488\) 11.9373 0.540374
\(489\) 0 0
\(490\) 0 0
\(491\) 1.00000 0.0451294 0.0225647 0.999745i \(-0.492817\pi\)
0.0225647 + 0.999745i \(0.492817\pi\)
\(492\) 0 0
\(493\) −6.00000 −0.270226
\(494\) −21.1660 −0.952304
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −18.5830 −0.831890 −0.415945 0.909390i \(-0.636549\pi\)
−0.415945 + 0.909390i \(0.636549\pi\)
\(500\) 7.93725 0.354965
\(501\) 0 0
\(502\) −23.2915 −1.03955
\(503\) 1.29150 0.0575853 0.0287926 0.999585i \(-0.490834\pi\)
0.0287926 + 0.999585i \(0.490834\pi\)
\(504\) 0 0
\(505\) 19.2915 0.858461
\(506\) −2.64575 −0.117618
\(507\) 0 0
\(508\) 2.64575 0.117386
\(509\) −25.1660 −1.11546 −0.557732 0.830021i \(-0.688328\pi\)
−0.557732 + 0.830021i \(0.688328\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −9.29150 −0.409831
\(515\) 26.4575 1.16586
\(516\) 0 0
\(517\) −3.93725 −0.173160
\(518\) 0 0
\(519\) 0 0
\(520\) 10.5830 0.464095
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) 2.70850 0.118434 0.0592172 0.998245i \(-0.481140\pi\)
0.0592172 + 0.998245i \(0.481140\pi\)
\(524\) 18.5830 0.811802
\(525\) 0 0
\(526\) −21.8745 −0.953774
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −16.0000 −0.695652
\(530\) 10.5830 0.459696
\(531\) 0 0
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) 0 0
\(535\) 23.8118 1.02947
\(536\) 7.58301 0.327536
\(537\) 0 0
\(538\) 10.6458 0.458971
\(539\) 0 0
\(540\) 0 0
\(541\) 3.81176 0.163880 0.0819402 0.996637i \(-0.473888\pi\)
0.0819402 + 0.996637i \(0.473888\pi\)
\(542\) 9.29150 0.399104
\(543\) 0 0
\(544\) 3.00000 0.128624
\(545\) 10.4170 0.446215
\(546\) 0 0
\(547\) 14.7085 0.628890 0.314445 0.949276i \(-0.398182\pi\)
0.314445 + 0.949276i \(0.398182\pi\)
\(548\) 13.2915 0.567785
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) −10.5830 −0.450851
\(552\) 0 0
\(553\) 0 0
\(554\) −17.1660 −0.729314
\(555\) 0 0
\(556\) −12.5830 −0.533638
\(557\) 19.2915 0.817407 0.408704 0.912667i \(-0.365981\pi\)
0.408704 + 0.912667i \(0.365981\pi\)
\(558\) 0 0
\(559\) −37.1660 −1.57195
\(560\) 0 0
\(561\) 0 0
\(562\) 3.00000 0.126547
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 33.2915 1.40058
\(566\) 30.4575 1.28022
\(567\) 0 0
\(568\) 2.70850 0.113646
\(569\) 15.1660 0.635792 0.317896 0.948126i \(-0.397024\pi\)
0.317896 + 0.948126i \(0.397024\pi\)
\(570\) 0 0
\(571\) −29.7490 −1.24496 −0.622479 0.782637i \(-0.713874\pi\)
−0.622479 + 0.782637i \(0.713874\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 5.29150 0.220671
\(576\) 0 0
\(577\) 22.7490 0.947054 0.473527 0.880779i \(-0.342981\pi\)
0.473527 + 0.880779i \(0.342981\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) 5.29150 0.219718
\(581\) 0 0
\(582\) 0 0
\(583\) 4.00000 0.165663
\(584\) −15.2915 −0.632767
\(585\) 0 0
\(586\) 14.5830 0.602418
\(587\) −9.87451 −0.407565 −0.203782 0.979016i \(-0.565323\pi\)
−0.203782 + 0.979016i \(0.565323\pi\)
\(588\) 0 0
\(589\) −21.1660 −0.872130
\(590\) 17.4170 0.717046
\(591\) 0 0
\(592\) 1.29150 0.0530804
\(593\) 25.7490 1.05739 0.528693 0.848813i \(-0.322682\pi\)
0.528693 + 0.848813i \(0.322682\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −19.8745 −0.814092
\(597\) 0 0
\(598\) −10.5830 −0.432771
\(599\) −29.3542 −1.19938 −0.599691 0.800232i \(-0.704710\pi\)
−0.599691 + 0.800232i \(0.704710\pi\)
\(600\) 0 0
\(601\) −38.5830 −1.57383 −0.786917 0.617059i \(-0.788324\pi\)
−0.786917 + 0.617059i \(0.788324\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 22.6458 0.921443
\(605\) −2.64575 −0.107565
\(606\) 0 0
\(607\) −13.1033 −0.531845 −0.265923 0.963994i \(-0.585677\pi\)
−0.265923 + 0.963994i \(0.585677\pi\)
\(608\) 5.29150 0.214599
\(609\) 0 0
\(610\) −31.5830 −1.27876
\(611\) −15.7490 −0.637137
\(612\) 0 0
\(613\) 7.81176 0.315514 0.157757 0.987478i \(-0.449574\pi\)
0.157757 + 0.987478i \(0.449574\pi\)
\(614\) −11.4170 −0.460752
\(615\) 0 0
\(616\) 0 0
\(617\) 14.7085 0.592142 0.296071 0.955166i \(-0.404324\pi\)
0.296071 + 0.955166i \(0.404324\pi\)
\(618\) 0 0
\(619\) −41.0000 −1.64793 −0.823965 0.566641i \(-0.808243\pi\)
−0.823965 + 0.566641i \(0.808243\pi\)
\(620\) 10.5830 0.425024
\(621\) 0 0
\(622\) −22.5203 −0.902980
\(623\) 0 0
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) −20.5830 −0.822662
\(627\) 0 0
\(628\) −0.708497 −0.0282721
\(629\) 3.87451 0.154487
\(630\) 0 0
\(631\) 41.8745 1.66700 0.833499 0.552521i \(-0.186334\pi\)
0.833499 + 0.552521i \(0.186334\pi\)
\(632\) 10.6458 0.423465
\(633\) 0 0
\(634\) −9.22876 −0.366521
\(635\) −7.00000 −0.277787
\(636\) 0 0
\(637\) 0 0
\(638\) 2.00000 0.0791808
\(639\) 0 0
\(640\) −2.64575 −0.104583
\(641\) 26.5830 1.04997 0.524983 0.851113i \(-0.324072\pi\)
0.524983 + 0.851113i \(0.324072\pi\)
\(642\) 0 0
\(643\) −1.16601 −0.0459830 −0.0229915 0.999736i \(-0.507319\pi\)
−0.0229915 + 0.999736i \(0.507319\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 15.8745 0.624574
\(647\) 26.3948 1.03769 0.518843 0.854870i \(-0.326363\pi\)
0.518843 + 0.854870i \(0.326363\pi\)
\(648\) 0 0
\(649\) 6.58301 0.258406
\(650\) −8.00000 −0.313786
\(651\) 0 0
\(652\) −5.58301 −0.218647
\(653\) 45.1033 1.76503 0.882514 0.470287i \(-0.155850\pi\)
0.882514 + 0.470287i \(0.155850\pi\)
\(654\) 0 0
\(655\) −49.1660 −1.92107
\(656\) −9.00000 −0.351391
\(657\) 0 0
\(658\) 0 0
\(659\) 24.1660 0.941374 0.470687 0.882300i \(-0.344006\pi\)
0.470687 + 0.882300i \(0.344006\pi\)
\(660\) 0 0
\(661\) 20.7085 0.805467 0.402734 0.915317i \(-0.368060\pi\)
0.402734 + 0.915317i \(0.368060\pi\)
\(662\) 13.0000 0.505259
\(663\) 0 0
\(664\) −15.5830 −0.604738
\(665\) 0 0
\(666\) 0 0
\(667\) −5.29150 −0.204888
\(668\) −8.70850 −0.336942
\(669\) 0 0
\(670\) −20.0627 −0.775092
\(671\) −11.9373 −0.460833
\(672\) 0 0
\(673\) −19.4170 −0.748470 −0.374235 0.927334i \(-0.622095\pi\)
−0.374235 + 0.927334i \(0.622095\pi\)
\(674\) 6.70850 0.258402
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 8.12549 0.312288 0.156144 0.987734i \(-0.450094\pi\)
0.156144 + 0.987734i \(0.450094\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −7.93725 −0.304380
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) 17.1660 0.656839 0.328420 0.944532i \(-0.393484\pi\)
0.328420 + 0.944532i \(0.393484\pi\)
\(684\) 0 0
\(685\) −35.1660 −1.34362
\(686\) 0 0
\(687\) 0 0
\(688\) 9.29150 0.354235
\(689\) 16.0000 0.609551
\(690\) 0 0
\(691\) 14.1660 0.538900 0.269450 0.963014i \(-0.413158\pi\)
0.269450 + 0.963014i \(0.413158\pi\)
\(692\) −4.00000 −0.152057
\(693\) 0 0
\(694\) −29.5830 −1.12296
\(695\) 33.2915 1.26282
\(696\) 0 0
\(697\) −27.0000 −1.02270
\(698\) −13.2288 −0.500716
\(699\) 0 0
\(700\) 0 0
\(701\) −42.4575 −1.60360 −0.801799 0.597594i \(-0.796124\pi\)
−0.801799 + 0.597594i \(0.796124\pi\)
\(702\) 0 0
\(703\) 6.83399 0.257749
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 23.2915 0.876587
\(707\) 0 0
\(708\) 0 0
\(709\) 15.7490 0.591467 0.295733 0.955271i \(-0.404436\pi\)
0.295733 + 0.955271i \(0.404436\pi\)
\(710\) −7.16601 −0.268936
\(711\) 0 0
\(712\) −2.70850 −0.101505
\(713\) −10.5830 −0.396337
\(714\) 0 0
\(715\) −10.5830 −0.395782
\(716\) −4.70850 −0.175965
\(717\) 0 0
\(718\) −20.5830 −0.768151
\(719\) 28.0627 1.04656 0.523282 0.852160i \(-0.324707\pi\)
0.523282 + 0.852160i \(0.324707\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 9.00000 0.334945
\(723\) 0 0
\(724\) −5.29150 −0.196657
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) −2.00000 −0.0741759 −0.0370879 0.999312i \(-0.511808\pi\)
−0.0370879 + 0.999312i \(0.511808\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 40.4575 1.49740
\(731\) 27.8745 1.03098
\(732\) 0 0
\(733\) 11.9373 0.440913 0.220456 0.975397i \(-0.429245\pi\)
0.220456 + 0.975397i \(0.429245\pi\)
\(734\) 10.7085 0.395258
\(735\) 0 0
\(736\) 2.64575 0.0975237
\(737\) −7.58301 −0.279324
\(738\) 0 0
\(739\) −24.5830 −0.904300 −0.452150 0.891942i \(-0.649343\pi\)
−0.452150 + 0.891942i \(0.649343\pi\)
\(740\) −3.41699 −0.125611
\(741\) 0 0
\(742\) 0 0
\(743\) 18.7085 0.686348 0.343174 0.939272i \(-0.388498\pi\)
0.343174 + 0.939272i \(0.388498\pi\)
\(744\) 0 0
\(745\) 52.5830 1.92649
\(746\) −15.8118 −0.578910
\(747\) 0 0
\(748\) −3.00000 −0.109691
\(749\) 0 0
\(750\) 0 0
\(751\) −29.2915 −1.06886 −0.534431 0.845212i \(-0.679474\pi\)
−0.534431 + 0.845212i \(0.679474\pi\)
\(752\) 3.93725 0.143577
\(753\) 0 0
\(754\) 8.00000 0.291343
\(755\) −59.9150 −2.18053
\(756\) 0 0
\(757\) −32.5830 −1.18425 −0.592125 0.805846i \(-0.701711\pi\)
−0.592125 + 0.805846i \(0.701711\pi\)
\(758\) −27.5830 −1.00186
\(759\) 0 0
\(760\) −14.0000 −0.507833
\(761\) −47.5830 −1.72488 −0.862441 0.506157i \(-0.831066\pi\)
−0.862441 + 0.506157i \(0.831066\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −6.70850 −0.242705
\(765\) 0 0
\(766\) −18.7085 −0.675965
\(767\) 26.3320 0.950794
\(768\) 0 0
\(769\) −30.7085 −1.10738 −0.553688 0.832724i \(-0.686780\pi\)
−0.553688 + 0.832724i \(0.686780\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −15.8745 −0.571336
\(773\) 8.06275 0.289997 0.144998 0.989432i \(-0.453682\pi\)
0.144998 + 0.989432i \(0.453682\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) −11.5830 −0.415806
\(777\) 0 0
\(778\) −31.9373 −1.14501
\(779\) −47.6235 −1.70629
\(780\) 0 0
\(781\) −2.70850 −0.0969177
\(782\) 7.93725 0.283836
\(783\) 0 0
\(784\) 0 0
\(785\) 1.87451 0.0669041
\(786\) 0 0
\(787\) 9.29150 0.331206 0.165603 0.986192i \(-0.447043\pi\)
0.165603 + 0.986192i \(0.447043\pi\)
\(788\) −21.8745 −0.779247
\(789\) 0 0
\(790\) −28.1660 −1.00210
\(791\) 0 0
\(792\) 0 0
\(793\) −47.7490 −1.69562
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) −19.1660 −0.679321
\(797\) −45.1033 −1.59764 −0.798820 0.601570i \(-0.794542\pi\)
−0.798820 + 0.601570i \(0.794542\pi\)
\(798\) 0 0
\(799\) 11.8118 0.417870
\(800\) 2.00000 0.0707107
\(801\) 0 0
\(802\) 35.8745 1.26677
\(803\) 15.2915 0.539625
\(804\) 0 0
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) −7.29150 −0.256514
\(809\) −45.3320 −1.59379 −0.796894 0.604119i \(-0.793525\pi\)
−0.796894 + 0.604119i \(0.793525\pi\)
\(810\) 0 0
\(811\) 16.5830 0.582308 0.291154 0.956676i \(-0.405961\pi\)
0.291154 + 0.956676i \(0.405961\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −1.29150 −0.0452671
\(815\) 14.7712 0.517414
\(816\) 0 0
\(817\) 49.1660 1.72010
\(818\) 35.8745 1.25432
\(819\) 0 0
\(820\) 23.8118 0.831543
\(821\) 30.3320 1.05859 0.529297 0.848436i \(-0.322456\pi\)
0.529297 + 0.848436i \(0.322456\pi\)
\(822\) 0 0
\(823\) −12.1255 −0.422668 −0.211334 0.977414i \(-0.567781\pi\)
−0.211334 + 0.977414i \(0.567781\pi\)
\(824\) −10.0000 −0.348367
\(825\) 0 0
\(826\) 0 0
\(827\) −35.3320 −1.22861 −0.614307 0.789067i \(-0.710565\pi\)
−0.614307 + 0.789067i \(0.710565\pi\)
\(828\) 0 0
\(829\) 19.8745 0.690270 0.345135 0.938553i \(-0.387833\pi\)
0.345135 + 0.938553i \(0.387833\pi\)
\(830\) 41.2288 1.43107
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) 0 0
\(834\) 0 0
\(835\) 23.0405 0.797350
\(836\) −5.29150 −0.183010
\(837\) 0 0
\(838\) 9.87451 0.341109
\(839\) 37.1033 1.28095 0.640473 0.767980i \(-0.278738\pi\)
0.640473 + 0.767980i \(0.278738\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 6.12549 0.211098
\(843\) 0 0
\(844\) 21.2915 0.732884
\(845\) −7.93725 −0.273050
\(846\) 0 0
\(847\) 0 0
\(848\) −4.00000 −0.137361
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 3.41699 0.117133
\(852\) 0 0
\(853\) 14.5203 0.497164 0.248582 0.968611i \(-0.420035\pi\)
0.248582 + 0.968611i \(0.420035\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) −9.83399 −0.335923 −0.167961 0.985794i \(-0.553718\pi\)
−0.167961 + 0.985794i \(0.553718\pi\)
\(858\) 0 0
\(859\) 48.7490 1.66329 0.831647 0.555304i \(-0.187398\pi\)
0.831647 + 0.555304i \(0.187398\pi\)
\(860\) −24.5830 −0.838274
\(861\) 0 0
\(862\) 17.2915 0.588951
\(863\) 33.1033 1.12685 0.563424 0.826168i \(-0.309484\pi\)
0.563424 + 0.826168i \(0.309484\pi\)
\(864\) 0 0
\(865\) 10.5830 0.359833
\(866\) 1.58301 0.0537927
\(867\) 0 0
\(868\) 0 0
\(869\) −10.6458 −0.361132
\(870\) 0 0
\(871\) −30.3320 −1.02776
\(872\) −3.93725 −0.133332
\(873\) 0 0
\(874\) 14.0000 0.473557
\(875\) 0 0
\(876\) 0 0
\(877\) 10.7712 0.363719 0.181860 0.983325i \(-0.441788\pi\)
0.181860 + 0.983325i \(0.441788\pi\)
\(878\) −9.35425 −0.315691
\(879\) 0 0
\(880\) 2.64575 0.0891883
\(881\) −23.1660 −0.780483 −0.390241 0.920713i \(-0.627608\pi\)
−0.390241 + 0.920713i \(0.627608\pi\)
\(882\) 0 0
\(883\) −34.4170 −1.15822 −0.579112 0.815248i \(-0.696601\pi\)
−0.579112 + 0.815248i \(0.696601\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 31.1660 1.04704
\(887\) 31.0405 1.04224 0.521119 0.853484i \(-0.325515\pi\)
0.521119 + 0.853484i \(0.325515\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 7.16601 0.240205
\(891\) 0 0
\(892\) 28.4575 0.952828
\(893\) 20.8340 0.697183
\(894\) 0 0
\(895\) 12.4575 0.416409
\(896\) 0 0
\(897\) 0 0
\(898\) −30.4575 −1.01638
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 9.00000 0.299667
\(903\) 0 0
\(904\) −12.5830 −0.418505
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) −15.8340 −0.525759 −0.262879 0.964829i \(-0.584672\pi\)
−0.262879 + 0.964829i \(0.584672\pi\)
\(908\) −2.41699 −0.0802108
\(909\) 0 0
\(910\) 0 0
\(911\) 18.3948 0.609446 0.304723 0.952441i \(-0.401436\pi\)
0.304723 + 0.952441i \(0.401436\pi\)
\(912\) 0 0
\(913\) 15.5830 0.515722
\(914\) −40.5830 −1.34237
\(915\) 0 0
\(916\) 6.70850 0.221655
\(917\) 0 0
\(918\) 0 0
\(919\) 9.35425 0.308568 0.154284 0.988027i \(-0.450693\pi\)
0.154284 + 0.988027i \(0.450693\pi\)
\(920\) −7.00000 −0.230783
\(921\) 0 0
\(922\) 24.5830 0.809598
\(923\) −10.8340 −0.356605
\(924\) 0 0
\(925\) 2.58301 0.0849287
\(926\) 39.0405 1.28295
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 34.4575 1.13051 0.565257 0.824915i \(-0.308777\pi\)
0.565257 + 0.824915i \(0.308777\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −20.1660 −0.660560
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 7.93725 0.259576
\(936\) 0 0
\(937\) 50.0000 1.63343 0.816714 0.577042i \(-0.195793\pi\)
0.816714 + 0.577042i \(0.195793\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −10.4170 −0.339765
\(941\) −3.41699 −0.111391 −0.0556954 0.998448i \(-0.517738\pi\)
−0.0556954 + 0.998448i \(0.517738\pi\)
\(942\) 0 0
\(943\) −23.8118 −0.775418
\(944\) −6.58301 −0.214259
\(945\) 0 0
\(946\) −9.29150 −0.302093
\(947\) 15.4170 0.500985 0.250493 0.968119i \(-0.419407\pi\)
0.250493 + 0.968119i \(0.419407\pi\)
\(948\) 0 0
\(949\) 61.1660 1.98553
\(950\) 10.5830 0.343358
\(951\) 0 0
\(952\) 0 0
\(953\) −26.7490 −0.866486 −0.433243 0.901277i \(-0.642631\pi\)
−0.433243 + 0.901277i \(0.642631\pi\)
\(954\) 0 0
\(955\) 17.7490 0.574345
\(956\) −2.70850 −0.0875991
\(957\) 0 0
\(958\) 33.8745 1.09444
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −5.16601 −0.166559
\(963\) 0 0
\(964\) −27.1660 −0.874958
\(965\) 42.0000 1.35203
\(966\) 0 0
\(967\) 40.0627 1.28833 0.644166 0.764886i \(-0.277205\pi\)
0.644166 + 0.764886i \(0.277205\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 30.6458 0.983976
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 21.8745 0.700904
\(975\) 0 0
\(976\) 11.9373 0.382102
\(977\) −13.1660 −0.421218 −0.210609 0.977570i \(-0.567545\pi\)
−0.210609 + 0.977570i \(0.567545\pi\)
\(978\) 0 0
\(979\) 2.70850 0.0865640
\(980\) 0 0
\(981\) 0 0
\(982\) 1.00000 0.0319113
\(983\) −42.5203 −1.35619 −0.678093 0.734976i \(-0.737193\pi\)
−0.678093 + 0.734976i \(0.737193\pi\)
\(984\) 0 0
\(985\) 57.8745 1.84404
\(986\) −6.00000 −0.191079
\(987\) 0 0
\(988\) −21.1660 −0.673380
\(989\) 24.5830 0.781694
\(990\) 0 0
\(991\) −43.8745 −1.39372 −0.696860 0.717207i \(-0.745420\pi\)
−0.696860 + 0.717207i \(0.745420\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) 0 0
\(995\) 50.7085 1.60757
\(996\) 0 0
\(997\) 6.83399 0.216435 0.108217 0.994127i \(-0.465486\pi\)
0.108217 + 0.994127i \(0.465486\pi\)
\(998\) −18.5830 −0.588235
\(999\) 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.db.1.1 2
3.2 odd 2 3234.2.a.w.1.2 2
7.2 even 3 1386.2.k.r.991.2 4
7.4 even 3 1386.2.k.r.793.2 4
7.6 odd 2 9702.2.a.dm.1.2 2
21.2 odd 6 462.2.i.e.67.1 4
21.11 odd 6 462.2.i.e.331.1 yes 4
21.20 even 2 3234.2.a.ba.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.e.67.1 4 21.2 odd 6
462.2.i.e.331.1 yes 4 21.11 odd 6
1386.2.k.r.793.2 4 7.4 even 3
1386.2.k.r.991.2 4 7.2 even 3
3234.2.a.w.1.2 2 3.2 odd 2
3234.2.a.ba.1.1 2 21.20 even 2
9702.2.a.db.1.1 2 1.1 even 1 trivial
9702.2.a.dm.1.2 2 7.6 odd 2