↑

Properties

Label 9702.2.a.bx.1.1
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9702,2,Mod(1,9702)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9702, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9702.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(77.4708600410\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1386)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9702.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{8} +2.00000 q^{10} -1.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} +3.00000 q^{17} -7.00000 q^{19} +2.00000 q^{20} -1.00000 q^{22} -7.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} -5.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} +3.00000 q^{37} -7.00000 q^{38} +2.00000 q^{40} -6.00000 q^{41} +11.0000 q^{43} -1.00000 q^{44} -7.00000 q^{46} +7.00000 q^{47} -1.00000 q^{50} -2.00000 q^{52} -4.00000 q^{53} -2.00000 q^{55} -5.00000 q^{58} -11.0000 q^{59} -10.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} +3.00000 q^{68} -5.00000 q^{71} +8.00000 q^{73} +3.00000 q^{74} -7.00000 q^{76} -8.00000 q^{79} +2.00000 q^{80} -6.00000 q^{82} -14.0000 q^{83} +6.00000 q^{85} +11.0000 q^{86} -1.00000 q^{88} -2.00000 q^{89} -7.00000 q^{92} +7.00000 q^{94} -14.0000 q^{95} -15.0000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 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
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −7.00000 −1.45960 −0.729800 − 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 − 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −7.00000 −1.13555
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 11.0000 1.67748 0.838742 − 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −7.00000 −1.03209
\(47\) 7.00000 1.02105 0.510527 − 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) −5.00000 −0.656532
\(59\) −11.0000 −1.43208 −0.716039 − 0.698060i \(-0.754047\pi\)
−0.716039 + 0.698060i \(0.754047\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 3.00000 0.363803
\(69\) 0 0
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 − 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) −7.00000 −0.802955
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −14.0000 −1.53670 −0.768350 − 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 11.0000 1.18616
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −7.00000 −0.729800
\(93\) 0 0
\(94\) 7.00000 0.721995
\(95\) −14.0000 −1.43637
\(96\) 0 0
\(97\) −15.0000 −1.52302 −0.761510 − 0.648154i \(-0.775541\pi\)
−0.761510 + 0.648154i \(0.775541\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −3.00000 −0.298511 −0.149256 − 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) 10.0000 0.985329 0.492665 − 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) 18.0000 1.74013 0.870063 − 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 − 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −14.0000 −1.30551
\(116\) −5.00000 −0.464238
\(117\) 0 0
\(118\) −11.0000 −1.01263
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −19.0000 −1.68598 −0.842989 − 0.537931i \(-0.819206\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) 8.00000 0.698963 0.349482 − 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −17.0000 −1.44192 −0.720961 − 0.692976i \(-0.756299\pi\)
−0.720961 + 0.692976i \(0.756299\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −5.00000 −0.419591
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −10.0000 −0.830455
\(146\) 8.00000 0.662085
\(147\) 0 0
\(148\) 3.00000 0.246598
\(149\) 17.0000 1.39269 0.696347 − 0.717705i \(-0.254807\pi\)
0.696347 + 0.717705i \(0.254807\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 − 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) −7.00000 −0.567775
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −1.00000 −0.0798087 −0.0399043 − 0.999204i \(-0.512705\pi\)
−0.0399043 + 0.999204i \(0.512705\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 0 0
\(163\) 6.00000 0.469956 0.234978 − 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −14.0000 −1.08661
\(167\) 10.0000 0.773823 0.386912 − 0.922117i \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) 11.0000 0.838742
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −2.00000 −0.149906
\(179\) −9.00000 −0.672692 −0.336346 − 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −7.00000 −0.516047
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) −3.00000 −0.219382
\(188\) 7.00000 0.510527
\(189\) 0 0
\(190\) −14.0000 −1.01567
\(191\) −8.00000 −0.578860 −0.289430 − 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 24.0000 1.72756 0.863779 − 0.503871i \(-0.168091\pi\)
0.863779 + 0.503871i \(0.168091\pi\)
\(194\) −15.0000 −1.07694
\(195\) 0 0
\(196\) 0 0
\(197\) 15.0000 1.06871 0.534353 − 0.845262i \(-0.320555\pi\)
0.534353 + 0.845262i \(0.320555\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 − 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −3.00000 −0.211079
\(203\) 0 0
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 10.0000 0.696733
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 7.00000 0.484200
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −4.00000 −0.274721
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) 22.0000 1.50039
\(216\) 0 0
\(217\) 0 0
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) −2.00000 −0.134840
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 − 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) −10.0000 −0.663723 −0.331862 − 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 − 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −14.0000 −0.923133
\(231\) 0 0
\(232\) −5.00000 −0.328266
\(233\) −9.00000 −0.589610 −0.294805 − 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) 0 0
\(235\) 14.0000 0.913259
\(236\) −11.0000 −0.716039
\(237\) 0 0
\(238\) 0 0
\(239\) −10.0000 −0.646846 −0.323423 − 0.946254i \(-0.604834\pi\)
−0.323423 + 0.946254i \(0.604834\pi\)
\(240\) 0 0
\(241\) 4.00000 0.257663 0.128831 − 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 14.0000 0.890799
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) −5.00000 −0.315597 −0.157799 − 0.987471i \(-0.550440\pi\)
−0.157799 + 0.987471i \(0.550440\pi\)
\(252\) 0 0
\(253\) 7.00000 0.440086
\(254\) −19.0000 −1.19217
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 − 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) 8.00000 0.494242
\(263\) 30.0000 1.84988 0.924940 − 0.380114i \(-0.124115\pi\)
0.924940 + 0.380114i \(0.124115\pi\)
\(264\) 0 0
\(265\) −8.00000 −0.491436
\(266\) 0 0
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 20.0000 1.21942 0.609711 − 0.792624i \(-0.291286\pi\)
0.609711 + 0.792624i \(0.291286\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 − 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 24.0000 1.44202 0.721010 − 0.692925i \(-0.243678\pi\)
0.721010 + 0.692925i \(0.243678\pi\)
\(278\) −17.0000 −1.01959
\(279\) 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
\(283\) −24.0000 −1.42665 −0.713326 − 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) −5.00000 −0.296695
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) −10.0000 −0.587220
\(291\) 0 0
\(292\) 8.00000 0.468165
\(293\) 3.00000 0.175262 0.0876309 − 0.996153i \(-0.472070\pi\)
0.0876309 + 0.996153i \(0.472070\pi\)
\(294\) 0 0
\(295\) −22.0000 −1.28089
\(296\) 3.00000 0.174371
\(297\) 0 0
\(298\) 17.0000 0.984784
\(299\) 14.0000 0.809641
\(300\) 0 0
\(301\) 0 0
\(302\) −5.00000 −0.287718
\(303\) 0 0
\(304\) −7.00000 −0.401478
\(305\) −20.0000 −1.14520
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) −27.0000 −1.53103 −0.765515 − 0.643418i \(-0.777516\pi\)
−0.765515 + 0.643418i \(0.777516\pi\)
\(312\) 0 0
\(313\) −15.0000 −0.847850 −0.423925 − 0.905697i \(-0.639348\pi\)
−0.423925 + 0.905697i \(0.639348\pi\)
\(314\) −1.00000 −0.0564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 30.0000 1.68497 0.842484 − 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 5.00000 0.279946
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) 0 0
\(323\) −21.0000 −1.16847
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 6.00000 0.332309
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −14.0000 −0.769510 −0.384755 − 0.923019i \(-0.625714\pi\)
−0.384755 + 0.923019i \(0.625714\pi\)
\(332\) −14.0000 −0.768350
\(333\) 0 0
\(334\) 10.0000 0.547176
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 − 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) 0 0
\(344\) 11.0000 0.593080
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) 24.0000 1.28839 0.644194 − 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 − 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −10.0000 −0.532246 −0.266123 − 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) −10.0000 −0.530745
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) −9.00000 −0.475665
\(359\) −18.0000 −0.950004 −0.475002 − 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) 0 0
\(365\) 16.0000 0.837478
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −7.00000 −0.364900
\(369\) 0 0
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −3.00000 −0.155126
\(375\) 0 0
\(376\) 7.00000 0.360997
\(377\) 10.0000 0.515026
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −14.0000 −0.718185
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) −31.0000 −1.58403 −0.792013 − 0.610504i \(-0.790967\pi\)
−0.792013 + 0.610504i \(0.790967\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 24.0000 1.22157
\(387\) 0 0
\(388\) −15.0000 −0.761510
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −21.0000 −1.06202
\(392\) 0 0
\(393\) 0 0
\(394\) 15.0000 0.755689
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) 3.00000 0.150566 0.0752828 − 0.997162i \(-0.476014\pi\)
0.0752828 + 0.997162i \(0.476014\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 32.0000 1.59800 0.799002 − 0.601329i \(-0.205362\pi\)
0.799002 + 0.601329i \(0.205362\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) −3.00000 −0.149256
\(405\) 0 0
\(406\) 0 0
\(407\) −3.00000 −0.148704
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 − 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) −12.0000 −0.592638
\(411\) 0 0
\(412\) 10.0000 0.492665
\(413\) 0 0
\(414\) 0 0
\(415\) −28.0000 −1.37447
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 7.00000 0.342381
\(419\) −3.00000 −0.146560 −0.0732798 − 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) −17.0000 −0.828529 −0.414265 − 0.910156i \(-0.635961\pi\)
−0.414265 + 0.910156i \(0.635961\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) −4.00000 −0.194257
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) 0 0
\(428\) 18.0000 0.870063
\(429\) 0 0
\(430\) 22.0000 1.06093
\(431\) 40.0000 1.92673 0.963366 − 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) 0 0
\(433\) −5.00000 −0.240285 −0.120142 − 0.992757i \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) 49.0000 2.34399
\(438\) 0 0
\(439\) −5.00000 −0.238637 −0.119318 − 0.992856i \(-0.538071\pi\)
−0.119318 + 0.992856i \(0.538071\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) −6.00000 −0.285391
\(443\) 41.0000 1.94797 0.973984 − 0.226615i \(-0.0727659\pi\)
0.973984 + 0.226615i \(0.0727659\pi\)
\(444\) 0 0
\(445\) −4.00000 −0.189618
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) 0 0
\(449\) −36.0000 −1.69895 −0.849473 − 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 2.00000 0.0940721
\(453\) 0 0
\(454\) −10.0000 −0.469323
\(455\) 0 0
\(456\) 0 0
\(457\) 40.0000 1.87112 0.935561 − 0.353166i \(-0.114895\pi\)
0.935561 + 0.353166i \(0.114895\pi\)
\(458\) 10.0000 0.467269
\(459\) 0 0
\(460\) −14.0000 −0.652753
\(461\) −13.0000 −0.605470 −0.302735 − 0.953075i \(-0.597900\pi\)
−0.302735 + 0.953075i \(0.597900\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −5.00000 −0.232119
\(465\) 0 0
\(466\) −9.00000 −0.416917
\(467\) −13.0000 −0.601568 −0.300784 − 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 14.0000 0.645772
\(471\) 0 0
\(472\) −11.0000 −0.506316
\(473\) −11.0000 −0.505781
\(474\) 0 0
\(475\) 7.00000 0.321182
\(476\) 0 0
\(477\) 0 0
\(478\) −10.0000 −0.457389
\(479\) −18.0000 −0.822441 −0.411220 − 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 4.00000 0.182195
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −30.0000 −1.36223
\(486\) 0 0
\(487\) 18.0000 0.815658 0.407829 − 0.913058i \(-0.366286\pi\)
0.407829 + 0.913058i \(0.366286\pi\)
\(488\) −10.0000 −0.452679
\(489\) 0 0
\(490\) 0 0
\(491\) −26.0000 −1.17336 −0.586682 − 0.809818i \(-0.699566\pi\)
−0.586682 + 0.809818i \(0.699566\pi\)
\(492\) 0 0
\(493\) −15.0000 −0.675566
\(494\) 14.0000 0.629890
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) 0 0
\(499\) −40.0000 −1.79065 −0.895323 − 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) −5.00000 −0.223161
\(503\) 30.0000 1.33763 0.668817 − 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 7.00000 0.311188
\(507\) 0 0
\(508\) −19.0000 −0.842989
\(509\) −4.00000 −0.177297 −0.0886484 − 0.996063i \(-0.528255\pi\)
−0.0886484 + 0.996063i \(0.528255\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) 20.0000 0.881305
\(516\) 0 0
\(517\) −7.00000 −0.307860
\(518\) 0 0
\(519\) 0 0
\(520\) −4.00000 −0.175412
\(521\) 18.0000 0.788594 0.394297 − 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) 30.0000 1.30806
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) −8.00000 −0.347498
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 36.0000 1.55642
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 20.0000 0.862261
\(539\) 0 0
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 − 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) −8.00000 −0.343629
\(543\) 0 0
\(544\) 3.00000 0.128624
\(545\) 8.00000 0.342682
\(546\) 0 0
\(547\) −11.0000 −0.470326 −0.235163 − 0.971956i \(-0.575562\pi\)
−0.235163 + 0.971956i \(0.575562\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 35.0000 1.49105
\(552\) 0 0
\(553\) 0 0
\(554\) 24.0000 1.01966
\(555\) 0 0
\(556\) −17.0000 −0.720961
\(557\) 9.00000 0.381342 0.190671 − 0.981654i \(-0.438934\pi\)
0.190671 + 0.981654i \(0.438934\pi\)
\(558\) 0 0
\(559\) −22.0000 −0.930501
\(560\) 0 0
\(561\) 0 0
\(562\) 3.00000 0.126547
\(563\) 42.0000 1.77009 0.885044 − 0.465506i \(-0.154128\pi\)
0.885044 + 0.465506i \(0.154128\pi\)
\(564\) 0 0
\(565\) 4.00000 0.168281
\(566\) −24.0000 −1.00880
\(567\) 0 0
\(568\) −5.00000 −0.209795
\(569\) 13.0000 0.544988 0.272494 − 0.962157i \(-0.412151\pi\)
0.272494 + 0.962157i \(0.412151\pi\)
\(570\) 0 0
\(571\) −43.0000 −1.79949 −0.899747 − 0.436412i \(-0.856249\pi\)
−0.899747 + 0.436412i \(0.856249\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 0 0
\(575\) 7.00000 0.291920
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 − 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) −10.0000 −0.415227
\(581\) 0 0
\(582\) 0 0
\(583\) 4.00000 0.165663
\(584\) 8.00000 0.331042
\(585\) 0 0
\(586\) 3.00000 0.123929
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 14.0000 0.576860
\(590\) −22.0000 −0.905726
\(591\) 0 0
\(592\) 3.00000 0.123299
\(593\) 21.0000 0.862367 0.431183 − 0.902264i \(-0.358096\pi\)
0.431183 + 0.902264i \(0.358096\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 17.0000 0.696347
\(597\) 0 0
\(598\) 14.0000 0.572503
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −5.00000 −0.203447
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −36.0000 −1.46119 −0.730597 − 0.682808i \(-0.760758\pi\)
−0.730597 + 0.682808i \(0.760758\pi\)
\(608\) −7.00000 −0.283887
\(609\) 0 0
\(610\) −20.0000 −0.809776
\(611\) −14.0000 −0.566379
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) −20.0000 −0.805170 −0.402585 − 0.915383i \(-0.631888\pi\)
−0.402585 + 0.915383i \(0.631888\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 − 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) −27.0000 −1.08260
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −15.0000 −0.599521
\(627\) 0 0
\(628\) −1.00000 −0.0399043
\(629\) 9.00000 0.358854
\(630\) 0 0
\(631\) −10.0000 −0.398094 −0.199047 − 0.979990i \(-0.563785\pi\)
−0.199047 + 0.979990i \(0.563785\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) 30.0000 1.19145
\(635\) −38.0000 −1.50798
\(636\) 0 0
\(637\) 0 0
\(638\) 5.00000 0.197952
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) −12.0000 −0.473972 −0.236986 − 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) −40.0000 −1.57745 −0.788723 − 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −21.0000 −0.826234
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 11.0000 0.431788
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 6.00000 0.234978
\(653\) −10.0000 −0.391330 −0.195665 − 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 − 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 11.0000 0.427850 0.213925 − 0.976850i \(-0.431375\pi\)
0.213925 + 0.976850i \(0.431375\pi\)
\(662\) −14.0000 −0.544125
\(663\) 0 0
\(664\) −14.0000 −0.543305
\(665\) 0 0
\(666\) 0 0
\(667\) 35.0000 1.35521
\(668\) 10.0000 0.386912
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 − 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 21.0000 0.807096 0.403548 − 0.914959i \(-0.367777\pi\)
0.403548 + 0.914959i \(0.367777\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) 2.00000 0.0765840
\(683\) 39.0000 1.49229 0.746147 − 0.665782i \(-0.231902\pi\)
0.746147 + 0.665782i \(0.231902\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 11.0000 0.419371
\(689\) 8.00000 0.304776
\(690\) 0 0
\(691\) −18.0000 −0.684752 −0.342376 − 0.939563i \(-0.611232\pi\)
−0.342376 + 0.939563i \(0.611232\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) −34.0000 −1.28969
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) −8.00000 −0.302804
\(699\) 0 0
\(700\) 0 0
\(701\) −27.0000 −1.01978 −0.509888 − 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(702\) 0 0
\(703\) −21.0000 −0.792030
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −10.0000 −0.376355
\(707\) 0 0
\(708\) 0 0
\(709\) −31.0000 −1.16423 −0.582115 − 0.813107i \(-0.697775\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) −10.0000 −0.375293
\(711\) 0 0
\(712\) −2.00000 −0.0749532
\(713\) 14.0000 0.524304
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) −9.00000 −0.336346
\(717\) 0 0
\(718\) −18.0000 −0.671754
\(719\) −9.00000 −0.335643 −0.167822 − 0.985817i \(-0.553673\pi\)
−0.167822 + 0.985817i \(0.553673\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 30.0000 1.11648
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) 5.00000 0.185695
\(726\) 0 0
\(727\) 14.0000 0.519231 0.259616 − 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 16.0000 0.592187
\(731\) 33.0000 1.22055
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 − 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −7.00000 −0.258023
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) 0 0
\(743\) −4.00000 −0.146746 −0.0733729 − 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) 0 0
\(745\) 34.0000 1.24566
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) −3.00000 −0.109691
\(749\) 0 0
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 − 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 7.00000 0.255264
\(753\) 0 0
\(754\) 10.0000 0.364179
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) 15.0000 0.545184 0.272592 − 0.962130i \(-0.412119\pi\)
0.272592 + 0.962130i \(0.412119\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) −14.0000 −0.507833
\(761\) 30.0000 1.08750 0.543750 − 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −31.0000 −1.12008
\(767\) 22.0000 0.794374
\(768\) 0 0
\(769\) −28.0000 −1.00971 −0.504853 − 0.863205i \(-0.668453\pi\)
−0.504853 + 0.863205i \(0.668453\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 24.0000 0.863779
\(773\) −4.00000 −0.143870 −0.0719350 − 0.997409i \(-0.522917\pi\)
−0.0719350 + 0.997409i \(0.522917\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) −15.0000 −0.538469
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 42.0000 1.50481
\(780\) 0 0
\(781\) 5.00000 0.178914
\(782\) −21.0000 −0.750958
\(783\) 0 0
\(784\) 0 0
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 37.0000 1.31891 0.659454 − 0.751745i \(-0.270788\pi\)
0.659454 + 0.751745i \(0.270788\pi\)
\(788\) 15.0000 0.534353
\(789\) 0 0
\(790\) −16.0000 −0.569254
\(791\) 0 0
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) 3.00000 0.106466
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) 50.0000 1.77109 0.885545 − 0.464553i \(-0.153785\pi\)
0.885545 + 0.464553i \(0.153785\pi\)
\(798\) 0 0
\(799\) 21.0000 0.742927
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 32.0000 1.12996
\(803\) −8.00000 −0.282314
\(804\) 0 0
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 0 0
\(808\) −3.00000 −0.105540
\(809\) −14.0000 −0.492214 −0.246107 − 0.969243i \(-0.579151\pi\)
−0.246107 + 0.969243i \(0.579151\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 − 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −3.00000 −0.105150
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) −77.0000 −2.69389
\(818\) −32.0000 −1.11885
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) 50.0000 1.74501 0.872506 − 0.488603i \(-0.162493\pi\)
0.872506 + 0.488603i \(0.162493\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 − 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 10.0000 0.348367
\(825\) 0 0
\(826\) 0 0
\(827\) −10.0000 −0.347734 −0.173867 − 0.984769i \(-0.555626\pi\)
−0.173867 + 0.984769i \(0.555626\pi\)
\(828\) 0 0
\(829\) 17.0000 0.590434 0.295217 − 0.955430i \(-0.404608\pi\)
0.295217 + 0.955430i \(0.404608\pi\)
\(830\) −28.0000 −0.971894
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 0 0
\(835\) 20.0000 0.692129
\(836\) 7.00000 0.242100
\(837\) 0 0
\(838\) −3.00000 −0.103633
\(839\) −24.0000 −0.828572 −0.414286 − 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −17.0000 −0.585859
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) −18.0000 −0.619219
\(846\) 0 0
\(847\) 0 0
\(848\) −4.00000 −0.137361
\(849\) 0 0
\(850\) −3.00000 −0.102899
\(851\) −21.0000 −0.719871
\(852\) 0 0
\(853\) −46.0000 −1.57501 −0.787505 − 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 18.0000 0.615227
\(857\) −21.0000 −0.717346 −0.358673 − 0.933463i \(-0.616771\pi\)
−0.358673 + 0.933463i \(0.616771\pi\)
\(858\) 0 0
\(859\) 22.0000 0.750630 0.375315 − 0.926897i \(-0.377534\pi\)
0.375315 + 0.926897i \(0.377534\pi\)
\(860\) 22.0000 0.750194
\(861\) 0 0
\(862\) 40.0000 1.36241
\(863\) −28.0000 −0.953131 −0.476566 − 0.879139i \(-0.658119\pi\)
−0.476566 + 0.879139i \(0.658119\pi\)
\(864\) 0 0
\(865\) −28.0000 −0.952029
\(866\) −5.00000 −0.169907
\(867\) 0 0
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 4.00000 0.135457
\(873\) 0 0
\(874\) 49.0000 1.65745
\(875\) 0 0
\(876\) 0 0
\(877\) 26.0000 0.877958 0.438979 − 0.898497i \(-0.355340\pi\)
0.438979 + 0.898497i \(0.355340\pi\)
\(878\) −5.00000 −0.168742
\(879\) 0 0
\(880\) −2.00000 −0.0674200
\(881\) −36.0000 −1.21287 −0.606435 − 0.795133i \(-0.707401\pi\)
−0.606435 + 0.795133i \(0.707401\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 − 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) 41.0000 1.37742
\(887\) 20.0000 0.671534 0.335767 − 0.941945i \(-0.391004\pi\)
0.335767 + 0.941945i \(0.391004\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −4.00000 −0.134080
\(891\) 0 0
\(892\) 14.0000 0.468755
\(893\) −49.0000 −1.63972
\(894\) 0 0
\(895\) −18.0000 −0.601674
\(896\) 0 0
\(897\) 0 0
\(898\) −36.0000 −1.20134
\(899\) 10.0000 0.333519
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 6.00000 0.199778
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) −24.0000 −0.796907 −0.398453 − 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) −10.0000 −0.331862
\(909\) 0 0
\(910\) 0 0
\(911\) 13.0000 0.430709 0.215355 − 0.976536i \(-0.430909\pi\)
0.215355 + 0.976536i \(0.430909\pi\)
\(912\) 0 0
\(913\) 14.0000 0.463332
\(914\) 40.0000 1.32308
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 0 0
\(919\) 43.0000 1.41844 0.709220 − 0.704988i \(-0.249047\pi\)
0.709220 + 0.704988i \(0.249047\pi\)
\(920\) −14.0000 −0.461566
\(921\) 0 0
\(922\) −13.0000 −0.428132
\(923\) 10.0000 0.329154
\(924\) 0 0
\(925\) −3.00000 −0.0986394
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) −5.00000 −0.164133
\(929\) −4.00000 −0.131236 −0.0656179 − 0.997845i \(-0.520902\pi\)
−0.0656179 + 0.997845i \(0.520902\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −9.00000 −0.294805
\(933\) 0 0
\(934\) −13.0000 −0.425373
\(935\) −6.00000 −0.196221
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 − 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 14.0000 0.456630
\(941\) −15.0000 −0.488986 −0.244493 − 0.969651i \(-0.578622\pi\)
−0.244493 + 0.969651i \(0.578622\pi\)
\(942\) 0 0
\(943\) 42.0000 1.36771
\(944\) −11.0000 −0.358020
\(945\) 0 0
\(946\) −11.0000 −0.357641
\(947\) −27.0000 −0.877382 −0.438691 − 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 0 0
\(949\) −16.0000 −0.519382
\(950\) 7.00000 0.227110
\(951\) 0 0
\(952\) 0 0
\(953\) −10.0000 −0.323932 −0.161966 − 0.986796i \(-0.551783\pi\)
−0.161966 + 0.986796i \(0.551783\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) −10.0000 −0.323423
\(957\) 0 0
\(958\) −18.0000 −0.581554
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −6.00000 −0.193448
\(963\) 0 0
\(964\) 4.00000 0.128831
\(965\) 48.0000 1.54517
\(966\) 0 0
\(967\) −3.00000 −0.0964735 −0.0482367 − 0.998836i \(-0.515360\pi\)
−0.0482367 + 0.998836i \(0.515360\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −30.0000 −0.963242
\(971\) 36.0000 1.15529 0.577647 − 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 18.0000 0.576757
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −44.0000 −1.40768 −0.703842 − 0.710356i \(-0.748534\pi\)
−0.703842 + 0.710356i \(0.748534\pi\)
\(978\) 0 0
\(979\) 2.00000 0.0639203
\(980\) 0 0
\(981\) 0 0
\(982\) −26.0000 −0.829693
\(983\) −49.0000 −1.56286 −0.781429 − 0.623995i \(-0.785509\pi\)
−0.781429 + 0.623995i \(0.785509\pi\)
\(984\) 0 0
\(985\) 30.0000 0.955879
\(986\) −15.0000 −0.477697
\(987\) 0 0
\(988\) 14.0000 0.445399
\(989\) −77.0000 −2.44846
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 − 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) 0 0
\(995\) −20.0000 −0.634043
\(996\) 0 0
\(997\) −30.0000 −0.950110 −0.475055 − 0.879956i \(-0.657572\pi\)
−0.475055 + 0.879956i \(0.657572\pi\)
\(998\) −40.0000 −1.26618
\(999\) 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.bx.1.1 1
3.2 odd 2 9702.2.a.j.1.1 1
7.3 odd 6 1386.2.k.i.793.1 ✓ 2
7.5 odd 6 1386.2.k.i.991.1 yes 2
7.6 odd 2 9702.2.a.bg.1.1 1
21.5 even 6 1386.2.k.k.991.1 yes 2
21.17 even 6 1386.2.k.k.793.1 yes 2
21.20 even 2 9702.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1386.2.k.i.793.1 ✓ 2 7.3 odd 6
1386.2.k.i.991.1 yes 2 7.5 odd 6
1386.2.k.k.793.1 yes 2 21.17 even 6
1386.2.k.k.991.1 yes 2 21.5 even 6
9702.2.a.j.1.1 1 3.2 odd 2
9702.2.a.w.1.1 1 21.20 even 2
9702.2.a.bg.1.1 1 7.6 odd 2
9702.2.a.bx.1.1 1 1.1 even 1 trivial