Properties

Label 9702.2.a.ck.1.1
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.23607 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.23607 q^{5} -1.00000 q^{8} +3.23607 q^{10} +1.00000 q^{11} +1.00000 q^{16} -0.763932 q^{17} -5.70820 q^{19} -3.23607 q^{20} -1.00000 q^{22} -6.47214 q^{23} +5.47214 q^{25} +4.47214 q^{29} +7.23607 q^{31} -1.00000 q^{32} +0.763932 q^{34} +6.94427 q^{37} +5.70820 q^{38} +3.23607 q^{40} -0.763932 q^{41} -2.47214 q^{43} +1.00000 q^{44} +6.47214 q^{46} -9.70820 q^{47} -5.47214 q^{50} +6.00000 q^{53} -3.23607 q^{55} -4.47214 q^{58} -10.4721 q^{59} -7.23607 q^{62} +1.00000 q^{64} -11.4164 q^{67} -0.763932 q^{68} -6.47214 q^{71} +2.29180 q^{73} -6.94427 q^{74} -5.70820 q^{76} -15.4164 q^{79} -3.23607 q^{80} +0.763932 q^{82} -3.23607 q^{83} +2.47214 q^{85} +2.47214 q^{86} -1.00000 q^{88} +2.47214 q^{89} -6.47214 q^{92} +9.70820 q^{94} +18.4721 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} + 2 q^{11} + 2 q^{16} - 6 q^{17} + 2 q^{19} - 2 q^{20} - 2 q^{22} - 4 q^{23} + 2 q^{25} + 10 q^{31} - 2 q^{32} + 6 q^{34} - 4 q^{37} - 2 q^{38} + 2 q^{40} - 6 q^{41} + 4 q^{43} + 2 q^{44} + 4 q^{46} - 6 q^{47} - 2 q^{50} + 12 q^{53} - 2 q^{55} - 12 q^{59} - 10 q^{62} + 2 q^{64} + 4 q^{67} - 6 q^{68} - 4 q^{71} + 18 q^{73} + 4 q^{74} + 2 q^{76} - 4 q^{79} - 2 q^{80} + 6 q^{82} - 2 q^{83} - 4 q^{85} - 4 q^{86} - 2 q^{88} - 4 q^{89} - 4 q^{92} + 6 q^{94} + 28 q^{95}+O(q^{100}) \) 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
\(5\) −3.23607 −1.44721 −0.723607 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.23607 1.02333
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.763932 −0.185281 −0.0926404 0.995700i \(-0.529531\pi\)
−0.0926404 + 0.995700i \(0.529531\pi\)
\(18\) 0 0
\(19\) −5.70820 −1.30955 −0.654776 0.755823i \(-0.727237\pi\)
−0.654776 + 0.755823i \(0.727237\pi\)
\(20\) −3.23607 −0.723607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −6.47214 −1.34953 −0.674767 0.738031i \(-0.735756\pi\)
−0.674767 + 0.738031i \(0.735756\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) 7.23607 1.29964 0.649818 0.760090i \(-0.274845\pi\)
0.649818 + 0.760090i \(0.274845\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0.763932 0.131013
\(35\) 0 0
\(36\) 0 0
\(37\) 6.94427 1.14163 0.570816 0.821078i \(-0.306627\pi\)
0.570816 + 0.821078i \(0.306627\pi\)
\(38\) 5.70820 0.925993
\(39\) 0 0
\(40\) 3.23607 0.511667
\(41\) −0.763932 −0.119306 −0.0596531 0.998219i \(-0.518999\pi\)
−0.0596531 + 0.998219i \(0.518999\pi\)
\(42\) 0 0
\(43\) −2.47214 −0.376997 −0.188499 0.982073i \(-0.560362\pi\)
−0.188499 + 0.982073i \(0.560362\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 6.47214 0.954264
\(47\) −9.70820 −1.41609 −0.708044 0.706169i \(-0.750422\pi\)
−0.708044 + 0.706169i \(0.750422\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −5.47214 −0.773877
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −3.23607 −0.436351
\(56\) 0 0
\(57\) 0 0
\(58\) −4.47214 −0.587220
\(59\) −10.4721 −1.36336 −0.681678 0.731652i \(-0.738749\pi\)
−0.681678 + 0.731652i \(0.738749\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −7.23607 −0.918982
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −11.4164 −1.39474 −0.697368 0.716713i \(-0.745646\pi\)
−0.697368 + 0.716713i \(0.745646\pi\)
\(68\) −0.763932 −0.0926404
\(69\) 0 0
\(70\) 0 0
\(71\) −6.47214 −0.768101 −0.384051 0.923312i \(-0.625471\pi\)
−0.384051 + 0.923312i \(0.625471\pi\)
\(72\) 0 0
\(73\) 2.29180 0.268234 0.134117 0.990965i \(-0.457180\pi\)
0.134117 + 0.990965i \(0.457180\pi\)
\(74\) −6.94427 −0.807255
\(75\) 0 0
\(76\) −5.70820 −0.654776
\(77\) 0 0
\(78\) 0 0
\(79\) −15.4164 −1.73448 −0.867241 0.497889i \(-0.834109\pi\)
−0.867241 + 0.497889i \(0.834109\pi\)
\(80\) −3.23607 −0.361803
\(81\) 0 0
\(82\) 0.763932 0.0843622
\(83\) −3.23607 −0.355205 −0.177602 0.984102i \(-0.556834\pi\)
−0.177602 + 0.984102i \(0.556834\pi\)
\(84\) 0 0
\(85\) 2.47214 0.268141
\(86\) 2.47214 0.266577
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 2.47214 0.262046 0.131023 0.991379i \(-0.458174\pi\)
0.131023 + 0.991379i \(0.458174\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.47214 −0.674767
\(93\) 0 0
\(94\) 9.70820 1.00132
\(95\) 18.4721 1.89520
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.47214 0.547214
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) −15.2361 −1.50125 −0.750627 0.660726i \(-0.770249\pi\)
−0.750627 + 0.660726i \(0.770249\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 16.9443 1.63806 0.819032 0.573747i \(-0.194511\pi\)
0.819032 + 0.573747i \(0.194511\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 3.23607 0.308547
\(111\) 0 0
\(112\) 0 0
\(113\) 13.4164 1.26211 0.631055 0.775738i \(-0.282622\pi\)
0.631055 + 0.775738i \(0.282622\pi\)
\(114\) 0 0
\(115\) 20.9443 1.95306
\(116\) 4.47214 0.415227
\(117\) 0 0
\(118\) 10.4721 0.964038
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 7.23607 0.649818
\(125\) −1.52786 −0.136656
\(126\) 0 0
\(127\) 2.47214 0.219367 0.109683 0.993967i \(-0.465016\pi\)
0.109683 + 0.993967i \(0.465016\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 11.2361 0.981700 0.490850 0.871244i \(-0.336686\pi\)
0.490850 + 0.871244i \(0.336686\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 11.4164 0.986227
\(135\) 0 0
\(136\) 0.763932 0.0655066
\(137\) 7.52786 0.643149 0.321574 0.946884i \(-0.395788\pi\)
0.321574 + 0.946884i \(0.395788\pi\)
\(138\) 0 0
\(139\) −12.1803 −1.03312 −0.516561 0.856250i \(-0.672788\pi\)
−0.516561 + 0.856250i \(0.672788\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.47214 0.543130
\(143\) 0 0
\(144\) 0 0
\(145\) −14.4721 −1.20185
\(146\) −2.29180 −0.189670
\(147\) 0 0
\(148\) 6.94427 0.570816
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 5.70820 0.462996
\(153\) 0 0
\(154\) 0 0
\(155\) −23.4164 −1.88085
\(156\) 0 0
\(157\) 19.2361 1.53521 0.767603 0.640926i \(-0.221449\pi\)
0.767603 + 0.640926i \(0.221449\pi\)
\(158\) 15.4164 1.22646
\(159\) 0 0
\(160\) 3.23607 0.255834
\(161\) 0 0
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −0.763932 −0.0596531
\(165\) 0 0
\(166\) 3.23607 0.251168
\(167\) 9.52786 0.737288 0.368644 0.929571i \(-0.379822\pi\)
0.368644 + 0.929571i \(0.379822\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −2.47214 −0.189604
\(171\) 0 0
\(172\) −2.47214 −0.188499
\(173\) −7.41641 −0.563859 −0.281930 0.959435i \(-0.590974\pi\)
−0.281930 + 0.959435i \(0.590974\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −2.47214 −0.185294
\(179\) 14.4721 1.08170 0.540849 0.841120i \(-0.318103\pi\)
0.540849 + 0.841120i \(0.318103\pi\)
\(180\) 0 0
\(181\) 1.70820 0.126970 0.0634849 0.997983i \(-0.479779\pi\)
0.0634849 + 0.997983i \(0.479779\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 6.47214 0.477132
\(185\) −22.4721 −1.65218
\(186\) 0 0
\(187\) −0.763932 −0.0558642
\(188\) −9.70820 −0.708044
\(189\) 0 0
\(190\) −18.4721 −1.34011
\(191\) 1.52786 0.110552 0.0552762 0.998471i \(-0.482396\pi\)
0.0552762 + 0.998471i \(0.482396\pi\)
\(192\) 0 0
\(193\) 22.9443 1.65156 0.825782 0.563989i \(-0.190734\pi\)
0.825782 + 0.563989i \(0.190734\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −22.3607 −1.59313 −0.796566 0.604551i \(-0.793352\pi\)
−0.796566 + 0.604551i \(0.793352\pi\)
\(198\) 0 0
\(199\) −17.1246 −1.21393 −0.606966 0.794728i \(-0.707614\pi\)
−0.606966 + 0.794728i \(0.707614\pi\)
\(200\) −5.47214 −0.386938
\(201\) 0 0
\(202\) 12.0000 0.844317
\(203\) 0 0
\(204\) 0 0
\(205\) 2.47214 0.172661
\(206\) 15.2361 1.06155
\(207\) 0 0
\(208\) 0 0
\(209\) −5.70820 −0.394845
\(210\) 0 0
\(211\) −23.4164 −1.61205 −0.806026 0.591880i \(-0.798386\pi\)
−0.806026 + 0.591880i \(0.798386\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −16.9443 −1.15829
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 0 0
\(220\) −3.23607 −0.218176
\(221\) 0 0
\(222\) 0 0
\(223\) 2.29180 0.153470 0.0767350 0.997052i \(-0.475550\pi\)
0.0767350 + 0.997052i \(0.475550\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −13.4164 −0.892446
\(227\) −22.6525 −1.50350 −0.751749 0.659450i \(-0.770789\pi\)
−0.751749 + 0.659450i \(0.770789\pi\)
\(228\) 0 0
\(229\) −6.29180 −0.415774 −0.207887 0.978153i \(-0.566659\pi\)
−0.207887 + 0.978153i \(0.566659\pi\)
\(230\) −20.9443 −1.38102
\(231\) 0 0
\(232\) −4.47214 −0.293610
\(233\) 26.9443 1.76518 0.882589 0.470145i \(-0.155799\pi\)
0.882589 + 0.470145i \(0.155799\pi\)
\(234\) 0 0
\(235\) 31.4164 2.04938
\(236\) −10.4721 −0.681678
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 4.18034 0.269279 0.134640 0.990895i \(-0.457012\pi\)
0.134640 + 0.990895i \(0.457012\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) −7.23607 −0.459491
\(249\) 0 0
\(250\) 1.52786 0.0966306
\(251\) 2.47214 0.156040 0.0780199 0.996952i \(-0.475140\pi\)
0.0780199 + 0.996952i \(0.475140\pi\)
\(252\) 0 0
\(253\) −6.47214 −0.406900
\(254\) −2.47214 −0.155116
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 13.5279 0.843845 0.421922 0.906632i \(-0.361355\pi\)
0.421922 + 0.906632i \(0.361355\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −11.2361 −0.694167
\(263\) 18.4721 1.13904 0.569520 0.821977i \(-0.307129\pi\)
0.569520 + 0.821977i \(0.307129\pi\)
\(264\) 0 0
\(265\) −19.4164 −1.19274
\(266\) 0 0
\(267\) 0 0
\(268\) −11.4164 −0.697368
\(269\) −14.2918 −0.871386 −0.435693 0.900095i \(-0.643497\pi\)
−0.435693 + 0.900095i \(0.643497\pi\)
\(270\) 0 0
\(271\) −6.47214 −0.393154 −0.196577 0.980488i \(-0.562983\pi\)
−0.196577 + 0.980488i \(0.562983\pi\)
\(272\) −0.763932 −0.0463202
\(273\) 0 0
\(274\) −7.52786 −0.454775
\(275\) 5.47214 0.329982
\(276\) 0 0
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 12.1803 0.730528
\(279\) 0 0
\(280\) 0 0
\(281\) 15.8885 0.947831 0.473916 0.880570i \(-0.342840\pi\)
0.473916 + 0.880570i \(0.342840\pi\)
\(282\) 0 0
\(283\) −10.2918 −0.611784 −0.305892 0.952066i \(-0.598955\pi\)
−0.305892 + 0.952066i \(0.598955\pi\)
\(284\) −6.47214 −0.384051
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.4164 −0.965671
\(290\) 14.4721 0.849833
\(291\) 0 0
\(292\) 2.29180 0.134117
\(293\) −31.4164 −1.83537 −0.917683 0.397313i \(-0.869943\pi\)
−0.917683 + 0.397313i \(0.869943\pi\)
\(294\) 0 0
\(295\) 33.8885 1.97307
\(296\) −6.94427 −0.403628
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −5.70820 −0.327388
\(305\) 0 0
\(306\) 0 0
\(307\) 23.5967 1.34674 0.673369 0.739307i \(-0.264847\pi\)
0.673369 + 0.739307i \(0.264847\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 23.4164 1.32996
\(311\) 22.6525 1.28450 0.642252 0.766494i \(-0.278000\pi\)
0.642252 + 0.766494i \(0.278000\pi\)
\(312\) 0 0
\(313\) −14.4721 −0.818013 −0.409007 0.912531i \(-0.634125\pi\)
−0.409007 + 0.912531i \(0.634125\pi\)
\(314\) −19.2361 −1.08555
\(315\) 0 0
\(316\) −15.4164 −0.867241
\(317\) −3.88854 −0.218402 −0.109201 0.994020i \(-0.534829\pi\)
−0.109201 + 0.994020i \(0.534829\pi\)
\(318\) 0 0
\(319\) 4.47214 0.250392
\(320\) −3.23607 −0.180902
\(321\) 0 0
\(322\) 0 0
\(323\) 4.36068 0.242635
\(324\) 0 0
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) 0 0
\(328\) 0.763932 0.0421811
\(329\) 0 0
\(330\) 0 0
\(331\) 19.4164 1.06722 0.533611 0.845730i \(-0.320835\pi\)
0.533611 + 0.845730i \(0.320835\pi\)
\(332\) −3.23607 −0.177602
\(333\) 0 0
\(334\) −9.52786 −0.521342
\(335\) 36.9443 2.01848
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 13.0000 0.707107
\(339\) 0 0
\(340\) 2.47214 0.134070
\(341\) 7.23607 0.391855
\(342\) 0 0
\(343\) 0 0
\(344\) 2.47214 0.133289
\(345\) 0 0
\(346\) 7.41641 0.398709
\(347\) −26.8328 −1.44046 −0.720231 0.693735i \(-0.755964\pi\)
−0.720231 + 0.693735i \(0.755964\pi\)
\(348\) 0 0
\(349\) 27.4164 1.46757 0.733783 0.679384i \(-0.237753\pi\)
0.733783 + 0.679384i \(0.237753\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −20.3607 −1.08369 −0.541845 0.840479i \(-0.682274\pi\)
−0.541845 + 0.840479i \(0.682274\pi\)
\(354\) 0 0
\(355\) 20.9443 1.11161
\(356\) 2.47214 0.131023
\(357\) 0 0
\(358\) −14.4721 −0.764876
\(359\) −10.4721 −0.552698 −0.276349 0.961057i \(-0.589125\pi\)
−0.276349 + 0.961057i \(0.589125\pi\)
\(360\) 0 0
\(361\) 13.5836 0.714926
\(362\) −1.70820 −0.0897812
\(363\) 0 0
\(364\) 0 0
\(365\) −7.41641 −0.388193
\(366\) 0 0
\(367\) −21.7082 −1.13316 −0.566580 0.824007i \(-0.691734\pi\)
−0.566580 + 0.824007i \(0.691734\pi\)
\(368\) −6.47214 −0.337383
\(369\) 0 0
\(370\) 22.4721 1.16827
\(371\) 0 0
\(372\) 0 0
\(373\) 25.4164 1.31601 0.658006 0.753013i \(-0.271400\pi\)
0.658006 + 0.753013i \(0.271400\pi\)
\(374\) 0.763932 0.0395020
\(375\) 0 0
\(376\) 9.70820 0.500662
\(377\) 0 0
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 18.4721 0.947601
\(381\) 0 0
\(382\) −1.52786 −0.0781723
\(383\) 30.6525 1.56627 0.783134 0.621853i \(-0.213620\pi\)
0.783134 + 0.621853i \(0.213620\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −22.9443 −1.16783
\(387\) 0 0
\(388\) 0 0
\(389\) −10.9443 −0.554897 −0.277448 0.960741i \(-0.589489\pi\)
−0.277448 + 0.960741i \(0.589489\pi\)
\(390\) 0 0
\(391\) 4.94427 0.250043
\(392\) 0 0
\(393\) 0 0
\(394\) 22.3607 1.12651
\(395\) 49.8885 2.51017
\(396\) 0 0
\(397\) 24.1803 1.21358 0.606788 0.794864i \(-0.292458\pi\)
0.606788 + 0.794864i \(0.292458\pi\)
\(398\) 17.1246 0.858379
\(399\) 0 0
\(400\) 5.47214 0.273607
\(401\) −0.472136 −0.0235773 −0.0117887 0.999931i \(-0.503753\pi\)
−0.0117887 + 0.999931i \(0.503753\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) 6.94427 0.344215
\(408\) 0 0
\(409\) −7.23607 −0.357801 −0.178900 0.983867i \(-0.557254\pi\)
−0.178900 + 0.983867i \(0.557254\pi\)
\(410\) −2.47214 −0.122090
\(411\) 0 0
\(412\) −15.2361 −0.750627
\(413\) 0 0
\(414\) 0 0
\(415\) 10.4721 0.514057
\(416\) 0 0
\(417\) 0 0
\(418\) 5.70820 0.279197
\(419\) 32.9443 1.60943 0.804717 0.593659i \(-0.202317\pi\)
0.804717 + 0.593659i \(0.202317\pi\)
\(420\) 0 0
\(421\) −22.9443 −1.11824 −0.559118 0.829088i \(-0.688860\pi\)
−0.559118 + 0.829088i \(0.688860\pi\)
\(422\) 23.4164 1.13989
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −4.18034 −0.202776
\(426\) 0 0
\(427\) 0 0
\(428\) 16.9443 0.819032
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 13.5279 0.651614 0.325807 0.945436i \(-0.394364\pi\)
0.325807 + 0.945436i \(0.394364\pi\)
\(432\) 0 0
\(433\) 25.8885 1.24412 0.622062 0.782968i \(-0.286295\pi\)
0.622062 + 0.782968i \(0.286295\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 36.9443 1.76728
\(438\) 0 0
\(439\) −4.58359 −0.218763 −0.109381 0.994000i \(-0.534887\pi\)
−0.109381 + 0.994000i \(0.534887\pi\)
\(440\) 3.23607 0.154273
\(441\) 0 0
\(442\) 0 0
\(443\) 19.4164 0.922501 0.461251 0.887270i \(-0.347401\pi\)
0.461251 + 0.887270i \(0.347401\pi\)
\(444\) 0 0
\(445\) −8.00000 −0.379236
\(446\) −2.29180 −0.108520
\(447\) 0 0
\(448\) 0 0
\(449\) −28.4721 −1.34368 −0.671842 0.740695i \(-0.734496\pi\)
−0.671842 + 0.740695i \(0.734496\pi\)
\(450\) 0 0
\(451\) −0.763932 −0.0359722
\(452\) 13.4164 0.631055
\(453\) 0 0
\(454\) 22.6525 1.06313
\(455\) 0 0
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 6.29180 0.293996
\(459\) 0 0
\(460\) 20.9443 0.976532
\(461\) 26.8328 1.24973 0.624864 0.780733i \(-0.285154\pi\)
0.624864 + 0.780733i \(0.285154\pi\)
\(462\) 0 0
\(463\) 33.8885 1.57493 0.787467 0.616357i \(-0.211392\pi\)
0.787467 + 0.616357i \(0.211392\pi\)
\(464\) 4.47214 0.207614
\(465\) 0 0
\(466\) −26.9443 −1.24817
\(467\) 5.88854 0.272489 0.136245 0.990675i \(-0.456497\pi\)
0.136245 + 0.990675i \(0.456497\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −31.4164 −1.44913
\(471\) 0 0
\(472\) 10.4721 0.482019
\(473\) −2.47214 −0.113669
\(474\) 0 0
\(475\) −31.2361 −1.43321
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −35.7771 −1.63470 −0.817348 0.576144i \(-0.804557\pi\)
−0.817348 + 0.576144i \(0.804557\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −4.18034 −0.190409
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) 19.0557 0.863497 0.431749 0.901994i \(-0.357897\pi\)
0.431749 + 0.901994i \(0.357897\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −5.88854 −0.265746 −0.132873 0.991133i \(-0.542420\pi\)
−0.132873 + 0.991133i \(0.542420\pi\)
\(492\) 0 0
\(493\) −3.41641 −0.153867
\(494\) 0 0
\(495\) 0 0
\(496\) 7.23607 0.324909
\(497\) 0 0
\(498\) 0 0
\(499\) −19.4164 −0.869198 −0.434599 0.900624i \(-0.643110\pi\)
−0.434599 + 0.900624i \(0.643110\pi\)
\(500\) −1.52786 −0.0683282
\(501\) 0 0
\(502\) −2.47214 −0.110337
\(503\) 21.3050 0.949941 0.474970 0.880002i \(-0.342459\pi\)
0.474970 + 0.880002i \(0.342459\pi\)
\(504\) 0 0
\(505\) 38.8328 1.72804
\(506\) 6.47214 0.287722
\(507\) 0 0
\(508\) 2.47214 0.109683
\(509\) −11.2361 −0.498030 −0.249015 0.968500i \(-0.580107\pi\)
−0.249015 + 0.968500i \(0.580107\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −13.5279 −0.596689
\(515\) 49.3050 2.17264
\(516\) 0 0
\(517\) −9.70820 −0.426966
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 41.3050 1.80960 0.904801 0.425834i \(-0.140019\pi\)
0.904801 + 0.425834i \(0.140019\pi\)
\(522\) 0 0
\(523\) 37.7082 1.64886 0.824432 0.565961i \(-0.191495\pi\)
0.824432 + 0.565961i \(0.191495\pi\)
\(524\) 11.2361 0.490850
\(525\) 0 0
\(526\) −18.4721 −0.805423
\(527\) −5.52786 −0.240798
\(528\) 0 0
\(529\) 18.8885 0.821241
\(530\) 19.4164 0.843395
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −54.8328 −2.37063
\(536\) 11.4164 0.493114
\(537\) 0 0
\(538\) 14.2918 0.616163
\(539\) 0 0
\(540\) 0 0
\(541\) −5.41641 −0.232870 −0.116435 0.993198i \(-0.537147\pi\)
−0.116435 + 0.993198i \(0.537147\pi\)
\(542\) 6.47214 0.278002
\(543\) 0 0
\(544\) 0.763932 0.0327533
\(545\) −19.4164 −0.831708
\(546\) 0 0
\(547\) 29.5279 1.26252 0.631260 0.775571i \(-0.282538\pi\)
0.631260 + 0.775571i \(0.282538\pi\)
\(548\) 7.52786 0.321574
\(549\) 0 0
\(550\) −5.47214 −0.233333
\(551\) −25.5279 −1.08752
\(552\) 0 0
\(553\) 0 0
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) −12.1803 −0.516561
\(557\) 37.4164 1.58538 0.792692 0.609622i \(-0.208679\pi\)
0.792692 + 0.609622i \(0.208679\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −15.8885 −0.670218
\(563\) 11.2361 0.473544 0.236772 0.971565i \(-0.423911\pi\)
0.236772 + 0.971565i \(0.423911\pi\)
\(564\) 0 0
\(565\) −43.4164 −1.82654
\(566\) 10.2918 0.432596
\(567\) 0 0
\(568\) 6.47214 0.271565
\(569\) 17.0557 0.715013 0.357507 0.933911i \(-0.383627\pi\)
0.357507 + 0.933911i \(0.383627\pi\)
\(570\) 0 0
\(571\) −26.4721 −1.10782 −0.553912 0.832575i \(-0.686866\pi\)
−0.553912 + 0.832575i \(0.686866\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −35.4164 −1.47697
\(576\) 0 0
\(577\) 12.5836 0.523862 0.261931 0.965087i \(-0.415641\pi\)
0.261931 + 0.965087i \(0.415641\pi\)
\(578\) 16.4164 0.682833
\(579\) 0 0
\(580\) −14.4721 −0.600923
\(581\) 0 0
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) −2.29180 −0.0948352
\(585\) 0 0
\(586\) 31.4164 1.29780
\(587\) 26.4721 1.09262 0.546311 0.837582i \(-0.316032\pi\)
0.546311 + 0.837582i \(0.316032\pi\)
\(588\) 0 0
\(589\) −41.3050 −1.70194
\(590\) −33.8885 −1.39517
\(591\) 0 0
\(592\) 6.94427 0.285408
\(593\) 23.2361 0.954191 0.477095 0.878851i \(-0.341690\pi\)
0.477095 + 0.878851i \(0.341690\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) 37.3050 1.52424 0.762120 0.647436i \(-0.224159\pi\)
0.762120 + 0.647436i \(0.224159\pi\)
\(600\) 0 0
\(601\) 13.7082 0.559169 0.279585 0.960121i \(-0.409803\pi\)
0.279585 + 0.960121i \(0.409803\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.23607 −0.131565
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 5.70820 0.231498
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 3.52786 0.142489 0.0712445 0.997459i \(-0.477303\pi\)
0.0712445 + 0.997459i \(0.477303\pi\)
\(614\) −23.5967 −0.952287
\(615\) 0 0
\(616\) 0 0
\(617\) 11.8885 0.478615 0.239307 0.970944i \(-0.423080\pi\)
0.239307 + 0.970944i \(0.423080\pi\)
\(618\) 0 0
\(619\) 34.8328 1.40005 0.700025 0.714119i \(-0.253172\pi\)
0.700025 + 0.714119i \(0.253172\pi\)
\(620\) −23.4164 −0.940426
\(621\) 0 0
\(622\) −22.6525 −0.908282
\(623\) 0 0
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) 14.4721 0.578423
\(627\) 0 0
\(628\) 19.2361 0.767603
\(629\) −5.30495 −0.211522
\(630\) 0 0
\(631\) −25.8885 −1.03061 −0.515303 0.857008i \(-0.672321\pi\)
−0.515303 + 0.857008i \(0.672321\pi\)
\(632\) 15.4164 0.613232
\(633\) 0 0
\(634\) 3.88854 0.154434
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 0 0
\(638\) −4.47214 −0.177054
\(639\) 0 0
\(640\) 3.23607 0.127917
\(641\) 0.111456 0.00440225 0.00220113 0.999998i \(-0.499299\pi\)
0.00220113 + 0.999998i \(0.499299\pi\)
\(642\) 0 0
\(643\) −16.5836 −0.653993 −0.326997 0.945026i \(-0.606037\pi\)
−0.326997 + 0.945026i \(0.606037\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −4.36068 −0.171569
\(647\) −31.0132 −1.21925 −0.609626 0.792689i \(-0.708681\pi\)
−0.609626 + 0.792689i \(0.708681\pi\)
\(648\) 0 0
\(649\) −10.4721 −0.411067
\(650\) 0 0
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −2.94427 −0.115218 −0.0576091 0.998339i \(-0.518348\pi\)
−0.0576091 + 0.998339i \(0.518348\pi\)
\(654\) 0 0
\(655\) −36.3607 −1.42073
\(656\) −0.763932 −0.0298265
\(657\) 0 0
\(658\) 0 0
\(659\) 21.8885 0.852657 0.426328 0.904569i \(-0.359807\pi\)
0.426328 + 0.904569i \(0.359807\pi\)
\(660\) 0 0
\(661\) 43.5967 1.69572 0.847858 0.530223i \(-0.177892\pi\)
0.847858 + 0.530223i \(0.177892\pi\)
\(662\) −19.4164 −0.754640
\(663\) 0 0
\(664\) 3.23607 0.125584
\(665\) 0 0
\(666\) 0 0
\(667\) −28.9443 −1.12073
\(668\) 9.52786 0.368644
\(669\) 0 0
\(670\) −36.9443 −1.42728
\(671\) 0 0
\(672\) 0 0
\(673\) 17.0557 0.657450 0.328725 0.944426i \(-0.393381\pi\)
0.328725 + 0.944426i \(0.393381\pi\)
\(674\) 18.0000 0.693334
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) −37.5279 −1.44231 −0.721156 0.692772i \(-0.756389\pi\)
−0.721156 + 0.692772i \(0.756389\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.47214 −0.0948021
\(681\) 0 0
\(682\) −7.23607 −0.277083
\(683\) −40.3607 −1.54436 −0.772179 0.635405i \(-0.780833\pi\)
−0.772179 + 0.635405i \(0.780833\pi\)
\(684\) 0 0
\(685\) −24.3607 −0.930774
\(686\) 0 0
\(687\) 0 0
\(688\) −2.47214 −0.0942493
\(689\) 0 0
\(690\) 0 0
\(691\) 26.4721 1.00705 0.503524 0.863981i \(-0.332037\pi\)
0.503524 + 0.863981i \(0.332037\pi\)
\(692\) −7.41641 −0.281930
\(693\) 0 0
\(694\) 26.8328 1.01856
\(695\) 39.4164 1.49515
\(696\) 0 0
\(697\) 0.583592 0.0221051
\(698\) −27.4164 −1.03773
\(699\) 0 0
\(700\) 0 0
\(701\) 43.8885 1.65765 0.828824 0.559510i \(-0.189011\pi\)
0.828824 + 0.559510i \(0.189011\pi\)
\(702\) 0 0
\(703\) −39.6393 −1.49503
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 20.3607 0.766284
\(707\) 0 0
\(708\) 0 0
\(709\) 22.9443 0.861690 0.430845 0.902426i \(-0.358216\pi\)
0.430845 + 0.902426i \(0.358216\pi\)
\(710\) −20.9443 −0.786025
\(711\) 0 0
\(712\) −2.47214 −0.0926472
\(713\) −46.8328 −1.75390
\(714\) 0 0
\(715\) 0 0
\(716\) 14.4721 0.540849
\(717\) 0 0
\(718\) 10.4721 0.390817
\(719\) 33.7082 1.25710 0.628552 0.777768i \(-0.283648\pi\)
0.628552 + 0.777768i \(0.283648\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −13.5836 −0.505529
\(723\) 0 0
\(724\) 1.70820 0.0634849
\(725\) 24.4721 0.908872
\(726\) 0 0
\(727\) −40.7639 −1.51185 −0.755925 0.654658i \(-0.772813\pi\)
−0.755925 + 0.654658i \(0.772813\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 7.41641 0.274494
\(731\) 1.88854 0.0698503
\(732\) 0 0
\(733\) −30.8328 −1.13884 −0.569418 0.822048i \(-0.692831\pi\)
−0.569418 + 0.822048i \(0.692831\pi\)
\(734\) 21.7082 0.801264
\(735\) 0 0
\(736\) 6.47214 0.238566
\(737\) −11.4164 −0.420529
\(738\) 0 0
\(739\) 16.5836 0.610037 0.305019 0.952346i \(-0.401337\pi\)
0.305019 + 0.952346i \(0.401337\pi\)
\(740\) −22.4721 −0.826092
\(741\) 0 0
\(742\) 0 0
\(743\) −9.88854 −0.362775 −0.181388 0.983412i \(-0.558059\pi\)
−0.181388 + 0.983412i \(0.558059\pi\)
\(744\) 0 0
\(745\) −19.4164 −0.711362
\(746\) −25.4164 −0.930561
\(747\) 0 0
\(748\) −0.763932 −0.0279321
\(749\) 0 0
\(750\) 0 0
\(751\) 38.8328 1.41703 0.708515 0.705696i \(-0.249366\pi\)
0.708515 + 0.705696i \(0.249366\pi\)
\(752\) −9.70820 −0.354022
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 5.05573 0.183754 0.0918768 0.995770i \(-0.470713\pi\)
0.0918768 + 0.995770i \(0.470713\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) −18.4721 −0.670055
\(761\) −17.1246 −0.620767 −0.310383 0.950611i \(-0.600457\pi\)
−0.310383 + 0.950611i \(0.600457\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 1.52786 0.0552762
\(765\) 0 0
\(766\) −30.6525 −1.10752
\(767\) 0 0
\(768\) 0 0
\(769\) −46.0689 −1.66129 −0.830643 0.556805i \(-0.812027\pi\)
−0.830643 + 0.556805i \(0.812027\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 22.9443 0.825782
\(773\) 43.9574 1.58104 0.790519 0.612437i \(-0.209811\pi\)
0.790519 + 0.612437i \(0.209811\pi\)
\(774\) 0 0
\(775\) 39.5967 1.42236
\(776\) 0 0
\(777\) 0 0
\(778\) 10.9443 0.392371
\(779\) 4.36068 0.156238
\(780\) 0 0
\(781\) −6.47214 −0.231591
\(782\) −4.94427 −0.176807
\(783\) 0 0
\(784\) 0 0
\(785\) −62.2492 −2.22177
\(786\) 0 0
\(787\) −44.5410 −1.58772 −0.793858 0.608103i \(-0.791931\pi\)
−0.793858 + 0.608103i \(0.791931\pi\)
\(788\) −22.3607 −0.796566
\(789\) 0 0
\(790\) −49.8885 −1.77495
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −24.1803 −0.858128
\(795\) 0 0
\(796\) −17.1246 −0.606966
\(797\) 53.1246 1.88177 0.940885 0.338726i \(-0.109996\pi\)
0.940885 + 0.338726i \(0.109996\pi\)
\(798\) 0 0
\(799\) 7.41641 0.262374
\(800\) −5.47214 −0.193469
\(801\) 0 0
\(802\) 0.472136 0.0166717
\(803\) 2.29180 0.0808757
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 12.0000 0.422159
\(809\) 8.83282 0.310545 0.155273 0.987872i \(-0.450374\pi\)
0.155273 + 0.987872i \(0.450374\pi\)
\(810\) 0 0
\(811\) 31.5967 1.10951 0.554756 0.832013i \(-0.312812\pi\)
0.554756 + 0.832013i \(0.312812\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −6.94427 −0.243397
\(815\) 38.8328 1.36025
\(816\) 0 0
\(817\) 14.1115 0.493697
\(818\) 7.23607 0.253003
\(819\) 0 0
\(820\) 2.47214 0.0863307
\(821\) 41.7771 1.45803 0.729015 0.684498i \(-0.239978\pi\)
0.729015 + 0.684498i \(0.239978\pi\)
\(822\) 0 0
\(823\) 22.8328 0.795902 0.397951 0.917407i \(-0.369721\pi\)
0.397951 + 0.917407i \(0.369721\pi\)
\(824\) 15.2361 0.530774
\(825\) 0 0
\(826\) 0 0
\(827\) 52.7214 1.83330 0.916651 0.399689i \(-0.130882\pi\)
0.916651 + 0.399689i \(0.130882\pi\)
\(828\) 0 0
\(829\) 37.4853 1.30192 0.650959 0.759113i \(-0.274367\pi\)
0.650959 + 0.759113i \(0.274367\pi\)
\(830\) −10.4721 −0.363493
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −30.8328 −1.06701
\(836\) −5.70820 −0.197422
\(837\) 0 0
\(838\) −32.9443 −1.13804
\(839\) 20.7639 0.716851 0.358425 0.933558i \(-0.383314\pi\)
0.358425 + 0.933558i \(0.383314\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 22.9443 0.790712
\(843\) 0 0
\(844\) −23.4164 −0.806026
\(845\) 42.0689 1.44721
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 4.18034 0.143384
\(851\) −44.9443 −1.54067
\(852\) 0 0
\(853\) −1.88854 −0.0646625 −0.0323313 0.999477i \(-0.510293\pi\)
−0.0323313 + 0.999477i \(0.510293\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −16.9443 −0.579143
\(857\) 6.87539 0.234859 0.117429 0.993081i \(-0.462535\pi\)
0.117429 + 0.993081i \(0.462535\pi\)
\(858\) 0 0
\(859\) 16.5836 0.565825 0.282912 0.959146i \(-0.408699\pi\)
0.282912 + 0.959146i \(0.408699\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) −13.5279 −0.460761
\(863\) −0.360680 −0.0122777 −0.00613884 0.999981i \(-0.501954\pi\)
−0.00613884 + 0.999981i \(0.501954\pi\)
\(864\) 0 0
\(865\) 24.0000 0.816024
\(866\) −25.8885 −0.879729
\(867\) 0 0
\(868\) 0 0
\(869\) −15.4164 −0.522966
\(870\) 0 0
\(871\) 0 0
\(872\) −6.00000 −0.203186
\(873\) 0 0
\(874\) −36.9443 −1.24966
\(875\) 0 0
\(876\) 0 0
\(877\) 30.0000 1.01303 0.506514 0.862232i \(-0.330934\pi\)
0.506514 + 0.862232i \(0.330934\pi\)
\(878\) 4.58359 0.154689
\(879\) 0 0
\(880\) −3.23607 −0.109088
\(881\) −26.8328 −0.904021 −0.452010 0.892013i \(-0.649293\pi\)
−0.452010 + 0.892013i \(0.649293\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −19.4164 −0.652307
\(887\) −19.0557 −0.639829 −0.319914 0.947446i \(-0.603654\pi\)
−0.319914 + 0.947446i \(0.603654\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 8.00000 0.268161
\(891\) 0 0
\(892\) 2.29180 0.0767350
\(893\) 55.4164 1.85444
\(894\) 0 0
\(895\) −46.8328 −1.56545
\(896\) 0 0
\(897\) 0 0
\(898\) 28.4721 0.950127
\(899\) 32.3607 1.07929
\(900\) 0 0
\(901\) −4.58359 −0.152702
\(902\) 0.763932 0.0254362
\(903\) 0 0
\(904\) −13.4164 −0.446223
\(905\) −5.52786 −0.183752
\(906\) 0 0
\(907\) 29.3050 0.973055 0.486527 0.873665i \(-0.338263\pi\)
0.486527 + 0.873665i \(0.338263\pi\)
\(908\) −22.6525 −0.751749
\(909\) 0 0
\(910\) 0 0
\(911\) 4.58359 0.151861 0.0759306 0.997113i \(-0.475807\pi\)
0.0759306 + 0.997113i \(0.475807\pi\)
\(912\) 0 0
\(913\) −3.23607 −0.107098
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) −6.29180 −0.207887
\(917\) 0 0
\(918\) 0 0
\(919\) 53.6656 1.77027 0.885133 0.465338i \(-0.154067\pi\)
0.885133 + 0.465338i \(0.154067\pi\)
\(920\) −20.9443 −0.690512
\(921\) 0 0
\(922\) −26.8328 −0.883692
\(923\) 0 0
\(924\) 0 0
\(925\) 38.0000 1.24943
\(926\) −33.8885 −1.11365
\(927\) 0 0
\(928\) −4.47214 −0.146805
\(929\) −36.3607 −1.19296 −0.596478 0.802630i \(-0.703434\pi\)
−0.596478 + 0.802630i \(0.703434\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 26.9443 0.882589
\(933\) 0 0
\(934\) −5.88854 −0.192679
\(935\) 2.47214 0.0808475
\(936\) 0 0
\(937\) −47.5967 −1.55492 −0.777459 0.628934i \(-0.783492\pi\)
−0.777459 + 0.628934i \(0.783492\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 31.4164 1.02469
\(941\) −52.3607 −1.70691 −0.853455 0.521167i \(-0.825497\pi\)
−0.853455 + 0.521167i \(0.825497\pi\)
\(942\) 0 0
\(943\) 4.94427 0.161008
\(944\) −10.4721 −0.340839
\(945\) 0 0
\(946\) 2.47214 0.0803761
\(947\) 2.11146 0.0686131 0.0343066 0.999411i \(-0.489078\pi\)
0.0343066 + 0.999411i \(0.489078\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 31.2361 1.01343
\(951\) 0 0
\(952\) 0 0
\(953\) 13.0557 0.422917 0.211458 0.977387i \(-0.432179\pi\)
0.211458 + 0.977387i \(0.432179\pi\)
\(954\) 0 0
\(955\) −4.94427 −0.159993
\(956\) 0 0
\(957\) 0 0
\(958\) 35.7771 1.15591
\(959\) 0 0
\(960\) 0 0
\(961\) 21.3607 0.689054
\(962\) 0 0
\(963\) 0 0
\(964\) 4.18034 0.134640
\(965\) −74.2492 −2.39017
\(966\) 0 0
\(967\) 1.16718 0.0375341 0.0187671 0.999824i \(-0.494026\pi\)
0.0187671 + 0.999824i \(0.494026\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 0 0
\(971\) −56.9443 −1.82743 −0.913714 0.406357i \(-0.866799\pi\)
−0.913714 + 0.406357i \(0.866799\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −19.0557 −0.610585
\(975\) 0 0
\(976\) 0 0
\(977\) −23.8885 −0.764262 −0.382131 0.924108i \(-0.624810\pi\)
−0.382131 + 0.924108i \(0.624810\pi\)
\(978\) 0 0
\(979\) 2.47214 0.0790098
\(980\) 0 0
\(981\) 0 0
\(982\) 5.88854 0.187911
\(983\) 38.2918 1.22132 0.610659 0.791893i \(-0.290904\pi\)
0.610659 + 0.791893i \(0.290904\pi\)
\(984\) 0 0
\(985\) 72.3607 2.30560
\(986\) 3.41641 0.108801
\(987\) 0 0
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 22.8328 0.725308 0.362654 0.931924i \(-0.381871\pi\)
0.362654 + 0.931924i \(0.381871\pi\)
\(992\) −7.23607 −0.229745
\(993\) 0 0
\(994\) 0 0
\(995\) 55.4164 1.75682
\(996\) 0 0
\(997\) 26.2492 0.831321 0.415661 0.909520i \(-0.363550\pi\)
0.415661 + 0.909520i \(0.363550\pi\)
\(998\) 19.4164 0.614616
\(999\) 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.ck.1.1 2
3.2 odd 2 3234.2.a.be.1.2 yes 2
7.6 odd 2 9702.2.a.cw.1.2 2
21.20 even 2 3234.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bb.1.1 2 21.20 even 2
3234.2.a.be.1.2 yes 2 3.2 odd 2
9702.2.a.ck.1.1 2 1.1 even 1 trivial
9702.2.a.cw.1.2 2 7.6 odd 2