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Properties

Label 162.2.a.a.1.1
Level $162$
Weight $2$
Character 162.1
Self dual yes
Analytic conductor $1.294$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,2,Mod(1,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 162.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: \(1.29357651274\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 162.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} +3.00000 q^{10} -1.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -4.00000 q^{19} -3.00000 q^{20} +4.00000 q^{25} +1.00000 q^{26} -4.00000 q^{28} +9.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} +3.00000 q^{34} +12.0000 q^{35} -1.00000 q^{37} +4.00000 q^{38} +3.00000 q^{40} +6.00000 q^{41} +8.00000 q^{43} -12.0000 q^{47} +9.00000 q^{49} -4.00000 q^{50} -1.00000 q^{52} -6.00000 q^{53} +4.00000 q^{56} -9.00000 q^{58} -1.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} -4.00000 q^{67} -3.00000 q^{68} -12.0000 q^{70} -12.0000 q^{71} +11.0000 q^{73} +1.00000 q^{74} -4.00000 q^{76} -16.0000 q^{79} -3.00000 q^{80} -6.00000 q^{82} -12.0000 q^{83} +9.00000 q^{85} -8.00000 q^{86} -3.00000 q^{89} +4.00000 q^{91} +12.0000 q^{94} +12.0000 q^{95} +2.00000 q^{97} -9.00000 q^{98} +O(q^{100})\)

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
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.00000 0.948683
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 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
\(35\) 12.0000 2.02837
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) −9.00000 −1.18176
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 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
\(70\) −12.0000 −1.43427
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 − 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) −3.00000 −0.335410
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 − 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −9.00000 −0.909137
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 − 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) −15.0000 −1.41108 −0.705541 − 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) 0 0
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 1.00000 0.0905357
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −3.00000 −0.263117
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 9.00000 0.768922 0.384461 − 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 12.0000 1.01419
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) 0 0
\(145\) −27.0000 −2.24223
\(146\) −11.0000 −0.910366
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) 9.00000 0.737309 0.368654 − 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 − 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 16.0000 1.27289
\(159\) 0 0
\(160\) 3.00000 0.237171
\(161\) 0 0
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 − 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −9.00000 −0.690268
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −3.00000 −0.228086 −0.114043 − 0.993476i \(-0.536380\pi\)
−0.114043 + 0.993476i \(0.536380\pi\)
\(174\) 0 0
\(175\) −16.0000 −1.20949
\(176\) 0 0
\(177\) 0 0
\(178\) 3.00000 0.224860
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 0 0
\(185\) 3.00000 0.220564
\(186\) 0 0
\(187\) 0 0
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) −12.0000 −0.870572
\(191\) 12.0000 0.868290 0.434145 − 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −13.0000 −0.935760 −0.467880 − 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −3.00000 −0.213741 −0.106871 − 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) −36.0000 −2.52670
\(204\) 0 0
\(205\) −18.0000 −1.25717
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −24.0000 −1.63679
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) −11.0000 −0.745014
\(219\) 0 0
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) 15.0000 0.997785
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 23.0000 1.51988 0.759941 − 0.649992i \(-0.225228\pi\)
0.759941 + 0.649992i \(0.225228\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −9.00000 −0.590879
\(233\) 21.0000 1.37576 0.687878 − 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) 0 0
\(235\) 36.0000 2.34838
\(236\) 0 0
\(237\) 0 0
\(238\) −12.0000 −0.777844
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −13.0000 −0.837404 −0.418702 − 0.908124i \(-0.637515\pi\)
−0.418702 + 0.908124i \(0.637515\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) −1.00000 −0.0640184
\(245\) −27.0000 −1.72497
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) −3.00000 −0.189737
\(251\) −24.0000 −1.51487 −0.757433 − 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −15.0000 −0.935674 −0.467837 − 0.883815i \(-0.654967\pi\)
−0.467837 + 0.883815i \(0.654967\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 3.00000 0.186052
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 18.0000 1.10573
\(266\) −16.0000 −0.981023
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 21.0000 1.28039 0.640196 − 0.768211i \(-0.278853\pi\)
0.640196 + 0.768211i \(0.278853\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) −9.00000 −0.543710
\(275\) 0 0
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) −12.0000 −0.717137
\(281\) −27.0000 −1.61068 −0.805342 − 0.592810i \(-0.798019\pi\)
−0.805342 + 0.592810i \(0.798019\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 27.0000 1.58549
\(291\) 0 0
\(292\) 11.0000 0.643726
\(293\) 9.00000 0.525786 0.262893 − 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −9.00000 −0.521356
\(299\) 0 0
\(300\) 0 0
\(301\) −32.0000 −1.84445
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 3.00000 0.171780
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −12.0000 −0.681554
\(311\) 24.0000 1.36092 0.680458 − 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 23.0000 1.30004 0.650018 − 0.759918i \(-0.274761\pi\)
0.650018 + 0.759918i \(0.274761\pi\)
\(314\) 13.0000 0.733632
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 21.0000 1.17948 0.589739 − 0.807594i \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.00000 −0.167705
\(321\) 0 0
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) −8.00000 −0.443079
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 48.0000 2.64633
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 − 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) 9.00000 0.488094
\(341\) 0 0
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 3.00000 0.161281
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 − 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 16.0000 0.855236
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 36.0000 1.91068
\(356\) −3.00000 −0.159000
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −33.0000 −1.72730
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 − 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −3.00000 −0.155963
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 − 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 − 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 12.0000 0.615587
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) −12.0000 −0.613171 −0.306586 − 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 13.0000 0.661683
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) 3.00000 0.151138
\(395\) 48.0000 2.41514
\(396\) 0 0
\(397\) −25.0000 −1.25471 −0.627357 − 0.778732i \(-0.715863\pi\)
−0.627357 + 0.778732i \(0.715863\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −3.00000 −0.149813 −0.0749064 − 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 36.0000 1.78665
\(407\) 0 0
\(408\) 0 0
\(409\) −25.0000 −1.23617 −0.618085 − 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 18.0000 0.888957
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 36.0000 1.76717
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 − 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 − 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 24.0000 1.15738
\(431\) −12.0000 −0.578020 −0.289010 − 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 11.0000 0.528626 0.264313 − 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 11.0000 0.526804
\(437\) 0 0
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 − 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.00000 −0.142695
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 9.00000 0.426641
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) −4.00000 −0.188982
\(449\) −18.0000 −0.849473 −0.424736 − 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −15.0000 −0.705541
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) −1.00000 −0.0467780 −0.0233890 − 0.999726i \(-0.507446\pi\)
−0.0233890 + 0.999726i \(0.507446\pi\)
\(458\) −23.0000 −1.07472
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 − 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) −21.0000 −0.972806
\(467\) −24.0000 −1.11059 −0.555294 − 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) −36.0000 −1.66056
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −16.0000 −0.734130
\(476\) 12.0000 0.550019
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) −12.0000 −0.548294 −0.274147 − 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 1.00000 0.0455961
\(482\) 13.0000 0.592134
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) −4.00000 −0.181257 −0.0906287 − 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 1.00000 0.0452679
\(489\) 0 0
\(490\) 27.0000 1.21974
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −27.0000 −1.21602
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 48.0000 2.15309
\(498\) 0 0
\(499\) −40.0000 −1.79065 −0.895323 − 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 3.00000 0.134164
\(501\) 0 0
\(502\) 24.0000 1.07117
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) −30.0000 −1.32973 −0.664863 − 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) −44.0000 −1.94645
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 15.0000 0.661622
\(515\) 12.0000 0.528783
\(516\) 0 0
\(517\) 0 0
\(518\) −4.00000 −0.175750
\(519\) 0 0
\(520\) −3.00000 −0.131559
\(521\) 6.00000 0.262865 0.131432 − 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 − 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −18.0000 −0.781870
\(531\) 0 0
\(532\) 16.0000 0.693688
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) −36.0000 −1.55642
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −21.0000 −0.905374
\(539\) 0 0
\(540\) 0 0
\(541\) −1.00000 −0.0429934 −0.0214967 − 0.999769i \(-0.506843\pi\)
−0.0214967 + 0.999769i \(0.506843\pi\)
\(542\) 16.0000 0.687259
\(543\) 0 0
\(544\) 3.00000 0.128624
\(545\) −33.0000 −1.41356
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 − 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 9.00000 0.384461
\(549\) 0 0
\(550\) 0 0
\(551\) −36.0000 −1.53365
\(552\) 0 0
\(553\) 64.0000 2.72156
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) −3.00000 −0.127114 −0.0635570 − 0.997978i \(-0.520244\pi\)
−0.0635570 + 0.997978i \(0.520244\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 12.0000 0.507093
\(561\) 0 0
\(562\) 27.0000 1.13893
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 45.0000 1.89316
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) −15.0000 −0.628833 −0.314416 − 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 − 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 0 0
\(577\) −25.0000 −1.04076 −0.520382 − 0.853934i \(-0.674210\pi\)
−0.520382 + 0.853934i \(0.674210\pi\)
\(578\) 8.00000 0.332756
\(579\) 0 0
\(580\) −27.0000 −1.12111
\(581\) 48.0000 1.99138
\(582\) 0 0
\(583\) 0 0
\(584\) −11.0000 −0.455183
\(585\) 0 0
\(586\) −9.00000 −0.371787
\(587\) −24.0000 −0.990586 −0.495293 − 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) 33.0000 1.35515 0.677574 − 0.735455i \(-0.263031\pi\)
0.677574 + 0.735455i \(0.263031\pi\)
\(594\) 0 0
\(595\) −36.0000 −1.47586
\(596\) 9.00000 0.368654
\(597\) 0 0
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 − 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 − 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 32.0000 1.30422
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) 33.0000 1.34164
\(606\) 0 0
\(607\) 20.0000 0.811775 0.405887 − 0.913923i \(-0.366962\pi\)
0.405887 + 0.913923i \(0.366962\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −3.00000 −0.121466
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 − 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) −3.00000 −0.120775 −0.0603877 − 0.998175i \(-0.519234\pi\)
−0.0603877 + 0.998175i \(0.519234\pi\)
\(618\) 0 0
\(619\) 8.00000 0.321547 0.160774 − 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 12.0000 0.481932
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −23.0000 −0.919265
\(627\) 0 0
\(628\) −13.0000 −0.518756
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 16.0000 0.636446
\(633\) 0 0
\(634\) −21.0000 −0.834017
\(635\) 48.0000 1.90482
\(636\) 0 0
\(637\) −9.00000 −0.356593
\(638\) 0 0
\(639\) 0 0
\(640\) 3.00000 0.118585
\(641\) 45.0000 1.77739 0.888697 − 0.458496i \(-0.151612\pi\)
0.888697 + 0.458496i \(0.151612\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 − 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) 36.0000 1.41531 0.707653 − 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 18.0000 0.704394 0.352197 − 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 36.0000 1.40664
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) −48.0000 −1.87123
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 23.0000 0.894596 0.447298 − 0.894385i \(-0.352386\pi\)
0.447298 + 0.894385i \(0.352386\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) −48.0000 −1.86136
\(666\) 0 0
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) −12.0000 −0.463600
\(671\) 0 0
\(672\) 0 0
\(673\) 11.0000 0.424019 0.212009 − 0.977268i \(-0.431999\pi\)
0.212009 + 0.977268i \(0.431999\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) −9.00000 −0.345134
\(681\) 0 0
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) −27.0000 −1.03162
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −3.00000 −0.114043
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −60.0000 −2.27593
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) −14.0000 −0.529908
\(699\) 0 0
\(700\) −16.0000 −0.604743
\(701\) −51.0000 −1.92624 −0.963122 − 0.269066i \(-0.913285\pi\)
−0.963122 + 0.269066i \(0.913285\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 24.0000 0.902613
\(708\) 0 0
\(709\) 47.0000 1.76512 0.882561 − 0.470198i \(-0.155817\pi\)
0.882561 + 0.470198i \(0.155817\pi\)
\(710\) −36.0000 −1.35106
\(711\) 0 0
\(712\) 3.00000 0.112430
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 36.0000 1.33701
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 − 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) −4.00000 −0.148250
\(729\) 0 0
\(730\) 33.0000 1.22138
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 − 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 3.00000 0.110282
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) 12.0000 0.440237 0.220119 − 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) −27.0000 −0.989203
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) 0 0
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 − 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 9.00000 0.327761
\(755\) −24.0000 −0.873449
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 − 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 28.0000 1.01701
\(759\) 0 0
\(760\) −12.0000 −0.435286
\(761\) −15.0000 −0.543750 −0.271875 − 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) 0 0
\(763\) −44.0000 −1.59291
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) 0 0
\(768\) 0 0
\(769\) −37.0000 −1.33425 −0.667127 − 0.744944i \(-0.732476\pi\)
−0.667127 + 0.744944i \(0.732476\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −13.0000 −0.467880
\(773\) −27.0000 −0.971123 −0.485561 − 0.874203i \(-0.661385\pi\)
−0.485561 + 0.874203i \(0.661385\pi\)
\(774\) 0 0
\(775\) −16.0000 −0.574737
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 39.0000 1.39197
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 − 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −3.00000 −0.106871
\(789\) 0 0
\(790\) −48.0000 −1.70776
\(791\) 60.0000 2.13335
\(792\) 0 0
\(793\) 1.00000 0.0355110
\(794\) 25.0000 0.887217
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −51.0000 −1.80651 −0.903256 − 0.429101i \(-0.858830\pi\)
−0.903256 + 0.429101i \(0.858830\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) 3.00000 0.105934
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) −39.0000 −1.37117 −0.685583 − 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 − 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) −36.0000 −1.26335
\(813\) 0 0
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 25.0000 0.874105
\(819\) 0 0
\(820\) −18.0000 −0.628587
\(821\) −15.0000 −0.523504 −0.261752 − 0.965135i \(-0.584300\pi\)
−0.261752 + 0.965135i \(0.584300\pi\)
\(822\) 0 0
\(823\) 44.0000 1.53374 0.766872 − 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) −48.0000 −1.66912 −0.834562 − 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 − 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) −36.0000 −1.24958
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) −27.0000 −0.935495
\(834\) 0 0
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) 0 0
\(838\) 24.0000 0.829066
\(839\) 24.0000 0.828572 0.414286 − 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 13.0000 0.448010
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 36.0000 1.23844
\(846\) 0 0
\(847\) 44.0000 1.51186
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 12.0000 0.411597
\(851\) 0 0
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 − 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −39.0000 −1.33221 −0.666107 − 0.745856i \(-0.732041\pi\)
−0.666107 + 0.745856i \(0.732041\pi\)
\(858\) 0 0
\(859\) −52.0000 −1.77422 −0.887109 − 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) −24.0000 −0.818393
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 9.00000 0.306009
\(866\) −11.0000 −0.373795
\(867\) 0 0
\(868\) 16.0000 0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) −11.0000 −0.372507
\(873\) 0 0
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) −25.0000 −0.844190 −0.422095 − 0.906552i \(-0.638705\pi\)
−0.422095 + 0.906552i \(0.638705\pi\)
\(878\) 28.0000 0.944954
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 − 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 − 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 3.00000 0.100901
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 24.0000 0.805841 0.402921 − 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) 64.0000 2.14649
\(890\) −9.00000 −0.301681
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) 48.0000 1.60626
\(894\) 0 0
\(895\) −36.0000 −1.20335
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) −36.0000 −1.20067
\(900\) 0 0
\(901\) 18.0000 0.599667
\(902\) 0 0
\(903\) 0 0
\(904\) 15.0000 0.498893
\(905\) 30.0000 0.997234
\(906\) 0 0
\(907\) −16.0000 −0.531271 −0.265636 − 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) 12.0000 0.397796
\(911\) 24.0000 0.795155 0.397578 − 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1.00000 0.0330771
\(915\) 0 0
\(916\) 23.0000 0.759941
\(917\) 48.0000 1.58510
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −18.0000 −0.592798
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −8.00000 −0.262896
\(927\) 0 0
\(928\) −9.00000 −0.295439
\(929\) −3.00000 −0.0984268 −0.0492134 − 0.998788i \(-0.515671\pi\)
−0.0492134 + 0.998788i \(0.515671\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) 21.0000 0.687878
\(933\) 0 0
\(934\) 24.0000 0.785304
\(935\) 0 0
\(936\) 0 0
\(937\) −37.0000 −1.20874 −0.604369 − 0.796705i \(-0.706575\pi\)
−0.604369 + 0.796705i \(0.706575\pi\)
\(938\) −16.0000 −0.522419
\(939\) 0 0
\(940\) 36.0000 1.17419
\(941\) −27.0000 −0.880175 −0.440087 − 0.897955i \(-0.645053\pi\)
−0.440087 + 0.897955i \(0.645053\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −11.0000 −0.357075
\(950\) 16.0000 0.519109
\(951\) 0 0
\(952\) −12.0000 −0.388922
\(953\) 9.00000 0.291539 0.145769 − 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(954\) 0 0
\(955\) −36.0000 −1.16493
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 12.0000 0.387702
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −1.00000 −0.0322413
\(963\) 0 0
\(964\) −13.0000 −0.418702
\(965\) 39.0000 1.25545
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 − 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) 6.00000 0.192648
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) −80.0000 −2.56468
\(974\) 4.00000 0.128168
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) −18.0000 −0.575871 −0.287936 − 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −27.0000 −0.862483
\(981\) 0 0
\(982\) 0 0
\(983\) 48.0000 1.53096 0.765481 − 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 0 0
\(985\) 9.00000 0.286764
\(986\) 27.0000 0.859855
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 − 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) −48.0000 −1.52247
\(995\) 12.0000 0.380426
\(996\) 0 0
\(997\) −37.0000 −1.17180 −0.585901 − 0.810383i \(-0.699259\pi\)
−0.585901 + 0.810383i \(0.699259\pi\)
\(998\) 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 162.2.a.a.1.1 ✓ 1
3.2 odd 2 162.2.a.d.1.1 yes 1
4.3 odd 2 1296.2.a.c.1.1 1
5.2 odd 4 4050.2.c.g.649.1 2
5.3 odd 4 4050.2.c.g.649.2 2
5.4 even 2 4050.2.a.bh.1.1 1
7.6 odd 2 7938.2.a.n.1.1 1
8.3 odd 2 5184.2.a.bd.1.1 1
8.5 even 2 5184.2.a.y.1.1 1
9.2 odd 6 162.2.c.a.109.1 2
9.4 even 3 162.2.c.d.55.1 2
9.5 odd 6 162.2.c.a.55.1 2
9.7 even 3 162.2.c.d.109.1 2
12.11 even 2 1296.2.a.l.1.1 1
15.2 even 4 4050.2.c.n.649.2 2
15.8 even 4 4050.2.c.n.649.1 2
15.14 odd 2 4050.2.a.r.1.1 1
21.20 even 2 7938.2.a.s.1.1 1
24.5 odd 2 5184.2.a.c.1.1 1
24.11 even 2 5184.2.a.h.1.1 1
36.7 odd 6 1296.2.i.n.433.1 2
36.11 even 6 1296.2.i.b.433.1 2
36.23 even 6 1296.2.i.b.865.1 2
36.31 odd 6 1296.2.i.n.865.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
162.2.a.d.1.1 yes 1 3.2 odd 2
162.2.c.a.55.1 2 9.5 odd 6
162.2.c.a.109.1 2 9.2 odd 6
162.2.c.d.55.1 2 9.4 even 3
162.2.c.d.109.1 2 9.7 even 3
1296.2.a.c.1.1 1 4.3 odd 2
1296.2.a.l.1.1 1 12.11 even 2
1296.2.i.b.433.1 2 36.11 even 6
1296.2.i.b.865.1 2 36.23 even 6
1296.2.i.n.433.1 2 36.7 odd 6
1296.2.i.n.865.1 2 36.31 odd 6
4050.2.a.r.1.1 1 15.14 odd 2
4050.2.a.bh.1.1 1 5.4 even 2
4050.2.c.g.649.1 2 5.2 odd 4
4050.2.c.g.649.2 2 5.3 odd 4
4050.2.c.n.649.1 2 15.8 even 4
4050.2.c.n.649.2 2 15.2 even 4
5184.2.a.c.1.1 1 24.5 odd 2
5184.2.a.h.1.1 1 24.11 even 2
5184.2.a.y.1.1 1 8.5 even 2
5184.2.a.bd.1.1 1 8.3 odd 2
7938.2.a.n.1.1 1 7.6 odd 2
7938.2.a.s.1.1 1 21.20 even 2