Properties

Label 5586.2.a.j.1.1
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
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] = [5586,2,Mod(1,5586)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5586, 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("5586.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.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: \(44.6044345691\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5586.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +2.00000 q^{20} +2.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} +2.00000 q^{30} +10.0000 q^{31} -1.00000 q^{32} +2.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +1.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} +6.00000 q^{41} -4.00000 q^{43} -2.00000 q^{44} +2.00000 q^{45} -6.00000 q^{47} -1.00000 q^{48} +1.00000 q^{50} -4.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{55} +1.00000 q^{57} +6.00000 q^{58} +12.0000 q^{59} -2.00000 q^{60} -10.0000 q^{61} -10.0000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -2.00000 q^{66} -2.00000 q^{67} +4.00000 q^{68} +8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +1.00000 q^{75} -1.00000 q^{76} -2.00000 q^{78} +16.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +8.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} +2.00000 q^{88} -10.0000 q^{89} -2.00000 q^{90} -10.0000 q^{93} +6.00000 q^{94} -2.00000 q^{95} +1.00000 q^{96} +12.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 2.00000 0.365148
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 1.00000 0.162221
\(39\) 2.00000 0.320256
\(40\) −2.00000 −0.316228
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −2.00000 −0.301511
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −4.00000 −0.560112
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 6.00000 0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −2.00000 −0.258199
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −10.0000 −1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −2.00000 −0.246183
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 4.00000 0.431331
\(87\) 6.00000 0.643268
\(88\) 2.00000 0.213201
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 0 0
\(93\) −10.0000 −1.03695
\(94\) 6.00000 0.618853
\(95\) −2.00000 −0.205196
\(96\) 1.00000 0.102062
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) −1.00000 −0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 4.00000 0.396059
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) −2.00000 −0.184900
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) −7.00000 −0.636364
\(122\) 10.0000 0.905357
\(123\) −6.00000 −0.541002
\(124\) 10.0000 0.898027
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 4.00000 0.350823
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) −2.00000 −0.172133
\(136\) −4.00000 −0.342997
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) −8.00000 −0.671345
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) −12.0000 −0.996546
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 1.00000 0.0811107
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 20.0000 1.60644
\(156\) 2.00000 0.160128
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) −16.0000 −1.27289
\(159\) 6.00000 0.475831
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 6.00000 0.468521
\(165\) 4.00000 0.311400
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −8.00000 −0.613572
\(171\) −1.00000 −0.0764719
\(172\) −4.00000 −0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) −12.0000 −0.901975
\(178\) 10.0000 0.749532
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 2.00000 0.149071
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 0 0
\(186\) 10.0000 0.733236
\(187\) −8.00000 −0.585018
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) −12.0000 −0.861550
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 2.00000 0.142134
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 1.00000 0.0707107
\(201\) 2.00000 0.141069
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 12.0000 0.838116
\(206\) −14.0000 −0.975426
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) 26.0000 1.78991 0.894957 0.446153i \(-0.147206\pi\)
0.894957 + 0.446153i \(0.147206\pi\)
\(212\) −6.00000 −0.412082
\(213\) −8.00000 −0.548151
\(214\) 4.00000 0.273434
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −4.00000 −0.270914
\(219\) −6.00000 −0.405442
\(220\) −4.00000 −0.269680
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −2.00000 −0.133038
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 1.00000 0.0662266
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −2.00000 −0.131024 −0.0655122 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 2.00000 0.130744
\(235\) −12.0000 −0.782794
\(236\) 12.0000 0.781133
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) −2.00000 −0.129099
\(241\) 24.0000 1.54598 0.772988 0.634421i \(-0.218761\pi\)
0.772988 + 0.634421i \(0.218761\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 2.00000 0.127257
\(248\) −10.0000 −0.635001
\(249\) −12.0000 −0.760469
\(250\) 12.0000 0.758947
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) −8.00000 −0.500979
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) −4.00000 −0.248069
\(261\) −6.00000 −0.371391
\(262\) 8.00000 0.494242
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) −2.00000 −0.123091
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) −2.00000 −0.122169
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 2.00000 0.121716
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 4.00000 0.239904
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) −6.00000 −0.357295
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 8.00000 0.474713
\(285\) 2.00000 0.118470
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 12.0000 0.704664
\(291\) −12.0000 −0.703452
\(292\) 6.00000 0.351123
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) −10.0000 −0.574485
\(304\) −1.00000 −0.0573539
\(305\) −20.0000 −1.14520
\(306\) −4.00000 −0.228665
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −14.0000 −0.796432
\(310\) −20.0000 −1.13592
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) −2.00000 −0.113228
\(313\) 30.0000 1.69570 0.847850 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −6.00000 −0.336463
\(319\) 12.0000 0.671871
\(320\) 2.00000 0.111803
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 8.00000 0.443079
\(327\) −4.00000 −0.221201
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) −6.00000 −0.329790 −0.164895 0.986311i \(-0.552728\pi\)
−0.164895 + 0.986311i \(0.552728\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 9.00000 0.489535
\(339\) −2.00000 −0.108625
\(340\) 8.00000 0.433861
\(341\) −20.0000 −1.08306
\(342\) 1.00000 0.0540738
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 10.0000 0.536828 0.268414 0.963304i \(-0.413500\pi\)
0.268414 + 0.963304i \(0.413500\pi\)
\(348\) 6.00000 0.321634
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 2.00000 0.106600
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 12.0000 0.637793
\(355\) 16.0000 0.849192
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −24.0000 −1.26844
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) −2.00000 −0.105409
\(361\) 1.00000 0.0526316
\(362\) 2.00000 0.105118
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) −10.0000 −0.522708
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) −10.0000 −0.518476
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 8.00000 0.413670
\(375\) 12.0000 0.619677
\(376\) 6.00000 0.309426
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) −2.00000 −0.102598
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) −4.00000 −0.203331
\(388\) 12.0000 0.609208
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) −4.00000 −0.202548
\(391\) 0 0
\(392\) 0 0
\(393\) 8.00000 0.403547
\(394\) 8.00000 0.403034
\(395\) 32.0000 1.61009
\(396\) −2.00000 −0.100504
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) −2.00000 −0.0997509
\(403\) −20.0000 −0.996271
\(404\) 10.0000 0.497519
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 4.00000 0.198030
\(409\) 28.0000 1.38451 0.692255 0.721653i \(-0.256617\pi\)
0.692255 + 0.721653i \(0.256617\pi\)
\(410\) −12.0000 −0.592638
\(411\) −2.00000 −0.0986527
\(412\) 14.0000 0.689730
\(413\) 0 0
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 2.00000 0.0980581
\(417\) 4.00000 0.195881
\(418\) −2.00000 −0.0978232
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 36.0000 1.75453 0.877266 0.480004i \(-0.159365\pi\)
0.877266 + 0.480004i \(0.159365\pi\)
\(422\) −26.0000 −1.26566
\(423\) −6.00000 −0.291730
\(424\) 6.00000 0.291386
\(425\) −4.00000 −0.194029
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) −4.00000 −0.193122
\(430\) 8.00000 0.385794
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 20.0000 0.961139 0.480569 0.876957i \(-0.340430\pi\)
0.480569 + 0.876957i \(0.340430\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) 4.00000 0.191565
\(437\) 0 0
\(438\) 6.00000 0.286691
\(439\) 34.0000 1.62273 0.811366 0.584539i \(-0.198725\pi\)
0.811366 + 0.584539i \(0.198725\pi\)
\(440\) 4.00000 0.190693
\(441\) 0 0
\(442\) 8.00000 0.380521
\(443\) −10.0000 −0.475114 −0.237557 0.971374i \(-0.576347\pi\)
−0.237557 + 0.971374i \(0.576347\pi\)
\(444\) 0 0
\(445\) −20.0000 −0.948091
\(446\) −6.00000 −0.284108
\(447\) 0 0
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 1.00000 0.0471405
\(451\) −12.0000 −0.565058
\(452\) 2.00000 0.0940721
\(453\) −8.00000 −0.375873
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −10.0000 −0.467269
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) −6.00000 −0.278543
\(465\) −20.0000 −0.927478
\(466\) 2.00000 0.0926482
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 12.0000 0.553519
\(471\) 6.00000 0.276465
\(472\) −12.0000 −0.552345
\(473\) 8.00000 0.367840
\(474\) 16.0000 0.734904
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −12.0000 −0.548867
\(479\) 26.0000 1.18797 0.593985 0.804476i \(-0.297554\pi\)
0.593985 + 0.804476i \(0.297554\pi\)
\(480\) 2.00000 0.0912871
\(481\) 0 0
\(482\) −24.0000 −1.09317
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 24.0000 1.08978
\(486\) 1.00000 0.0453609
\(487\) 4.00000 0.181257 0.0906287 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 10.0000 0.452679
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) −6.00000 −0.270501
\(493\) −24.0000 −1.08091
\(494\) −2.00000 −0.0899843
\(495\) −4.00000 −0.179787
\(496\) 10.0000 0.449013
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) 24.0000 1.07117
\(503\) −26.0000 −1.15928 −0.579641 0.814872i \(-0.696807\pi\)
−0.579641 + 0.814872i \(0.696807\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 8.00000 0.354943
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 8.00000 0.354246
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) 6.00000 0.264649
\(515\) 28.0000 1.23383
\(516\) 4.00000 0.176090
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 4.00000 0.175412
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 6.00000 0.262613
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 40.0000 1.74243
\(528\) 2.00000 0.0870388
\(529\) −23.0000 −1.00000
\(530\) 12.0000 0.521247
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) −10.0000 −0.432742
\(535\) −8.00000 −0.345870
\(536\) 2.00000 0.0863868
\(537\) −24.0000 −1.03568
\(538\) 18.0000 0.776035
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) −8.00000 −0.343629
\(543\) 2.00000 0.0858282
\(544\) −4.00000 −0.171499
\(545\) 8.00000 0.342682
\(546\) 0 0
\(547\) 30.0000 1.28271 0.641354 0.767245i \(-0.278373\pi\)
0.641354 + 0.767245i \(0.278373\pi\)
\(548\) 2.00000 0.0854358
\(549\) −10.0000 −0.426790
\(550\) −2.00000 −0.0852803
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 0 0
\(554\) −6.00000 −0.254916
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) −10.0000 −0.423334
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 30.0000 1.26547
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 6.00000 0.252646
\(565\) 4.00000 0.168281
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) −2.00000 −0.0837708
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) 1.00000 0.0415945
\(579\) −22.0000 −0.914289
\(580\) −12.0000 −0.498273
\(581\) 0 0
\(582\) 12.0000 0.497416
\(583\) 12.0000 0.496989
\(584\) −6.00000 −0.248282
\(585\) −4.00000 −0.165380
\(586\) −30.0000 −1.23929
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −10.0000 −0.412043
\(590\) −24.0000 −0.988064
\(591\) 8.00000 0.329076
\(592\) 0 0
\(593\) −4.00000 −0.164260 −0.0821302 0.996622i \(-0.526172\pi\)
−0.0821302 + 0.996622i \(0.526172\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −16.0000 −0.652654 −0.326327 0.945257i \(-0.605811\pi\)
−0.326327 + 0.945257i \(0.605811\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 8.00000 0.325515
\(605\) −14.0000 −0.569181
\(606\) 10.0000 0.406222
\(607\) 26.0000 1.05531 0.527654 0.849460i \(-0.323072\pi\)
0.527654 + 0.849460i \(0.323072\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 20.0000 0.809776
\(611\) 12.0000 0.485468
\(612\) 4.00000 0.161690
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 20.0000 0.807134
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) −26.0000 −1.04672 −0.523360 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 14.0000 0.563163
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 20.0000 0.803219
\(621\) 0 0
\(622\) 10.0000 0.400963
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) −30.0000 −1.19904
\(627\) −2.00000 −0.0798723
\(628\) −6.00000 −0.239426
\(629\) 0 0
\(630\) 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
\(633\) −26.0000 −1.03341
\(634\) 18.0000 0.714871
\(635\) 16.0000 0.634941
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) −12.0000 −0.475085
\(639\) 8.00000 0.316475
\(640\) −2.00000 −0.0790569
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) −4.00000 −0.157867
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 4.00000 0.157378
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −24.0000 −0.942082
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) −12.0000 −0.469596 −0.234798 0.972044i \(-0.575443\pi\)
−0.234798 + 0.972044i \(0.575443\pi\)
\(654\) 4.00000 0.156412
\(655\) −16.0000 −0.625172
\(656\) 6.00000 0.234261
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) −32.0000 −1.24654 −0.623272 0.782006i \(-0.714197\pi\)
−0.623272 + 0.782006i \(0.714197\pi\)
\(660\) 4.00000 0.155700
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 6.00000 0.233197
\(663\) 8.00000 0.310694
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −6.00000 −0.231973
\(670\) 4.00000 0.154533
\(671\) 20.0000 0.772091
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 26.0000 1.00148
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 2.00000 0.0768095
\(679\) 0 0
\(680\) −8.00000 −0.306786
\(681\) 12.0000 0.459841
\(682\) 20.0000 0.765840
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 4.00000 0.152832
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) −10.0000 −0.379595
\(695\) −8.00000 −0.303457
\(696\) −6.00000 −0.227429
\(697\) 24.0000 0.909065
\(698\) −10.0000 −0.378506
\(699\) 2.00000 0.0756469
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 12.0000 0.451946
\(706\) −24.0000 −0.903252
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) −16.0000 −0.600469
\(711\) 16.0000 0.600047
\(712\) 10.0000 0.374766
\(713\) 0 0
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 24.0000 0.896922
\(717\) −12.0000 −0.448148
\(718\) −12.0000 −0.447836
\(719\) −42.0000 −1.56634 −0.783168 0.621810i \(-0.786397\pi\)
−0.783168 + 0.621810i \(0.786397\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) −24.0000 −0.892570
\(724\) −2.00000 −0.0743294
\(725\) 6.00000 0.222834
\(726\) −7.00000 −0.259794
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −12.0000 −0.444140
\(731\) −16.0000 −0.591781
\(732\) 10.0000 0.369611
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) −6.00000 −0.220863
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) 10.0000 0.366618
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) 12.0000 0.439057
\(748\) −8.00000 −0.292509
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) −6.00000 −0.218797
\(753\) 24.0000 0.874609
\(754\) −12.0000 −0.437014
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) −2.00000 −0.0726433
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) 40.0000 1.45000 0.724999 0.688749i \(-0.241840\pi\)
0.724999 + 0.688749i \(0.241840\pi\)
\(762\) 8.00000 0.289809
\(763\) 0 0
\(764\) 0 0
\(765\) 8.00000 0.289241
\(766\) 16.0000 0.578103
\(767\) −24.0000 −0.866590
\(768\) −1.00000 −0.0360844
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 22.0000 0.791797
\(773\) 46.0000 1.65451 0.827253 0.561830i \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) 4.00000 0.143777
\(775\) −10.0000 −0.359211
\(776\) −12.0000 −0.430775
\(777\) 0 0
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 4.00000 0.143223
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) −12.0000 −0.428298
\(786\) −8.00000 −0.285351
\(787\) 24.0000 0.855508 0.427754 0.903895i \(-0.359305\pi\)
0.427754 + 0.903895i \(0.359305\pi\)
\(788\) −8.00000 −0.284988
\(789\) −8.00000 −0.284808
\(790\) −32.0000 −1.13851
\(791\) 0 0
\(792\) 2.00000 0.0710669
\(793\) 20.0000 0.710221
\(794\) −14.0000 −0.496841
\(795\) 12.0000 0.425596
\(796\) 4.00000 0.141776
\(797\) −14.0000 −0.495905 −0.247953 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 1.00000 0.0353553
\(801\) −10.0000 −0.353333
\(802\) 10.0000 0.353112
\(803\) −12.0000 −0.423471
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) 18.0000 0.633630
\(808\) −10.0000 −0.351799
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 0 0
\(815\) −16.0000 −0.560456
\(816\) −4.00000 −0.140028
\(817\) 4.00000 0.139942
\(818\) −28.0000 −0.978997
\(819\) 0 0
\(820\) 12.0000 0.419058
\(821\) −12.0000 −0.418803 −0.209401 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) 2.00000 0.0697580
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) −14.0000 −0.487713
\(825\) −2.00000 −0.0696311
\(826\) 0 0
\(827\) 16.0000 0.556375 0.278187 0.960527i \(-0.410266\pi\)
0.278187 + 0.960527i \(0.410266\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) −24.0000 −0.833052
\(831\) −6.00000 −0.208138
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 2.00000 0.0691714
\(837\) −10.0000 −0.345651
\(838\) −20.0000 −0.690889
\(839\) 32.0000 1.10476 0.552381 0.833592i \(-0.313719\pi\)
0.552381 + 0.833592i \(0.313719\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −36.0000 −1.24064
\(843\) 30.0000 1.03325
\(844\) 26.0000 0.894957
\(845\) −18.0000 −0.619219
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −12.0000 −0.411839
\(850\) 4.00000 0.137199
\(851\) 0 0
\(852\) −8.00000 −0.274075
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) 4.00000 0.136717
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 4.00000 0.136558
\(859\) 52.0000 1.77422 0.887109 0.461561i \(-0.152710\pi\)
0.887109 + 0.461561i \(0.152710\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) −40.0000 −1.36241
\(863\) −56.0000 −1.90626 −0.953131 0.302558i \(-0.902160\pi\)
−0.953131 + 0.302558i \(0.902160\pi\)
\(864\) 1.00000 0.0340207
\(865\) −28.0000 −0.952029
\(866\) −20.0000 −0.679628
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) −12.0000 −0.406838
\(871\) 4.00000 0.135535
\(872\) −4.00000 −0.135457
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −34.0000 −1.14744
\(879\) −30.0000 −1.01187
\(880\) −4.00000 −0.134840
\(881\) 24.0000 0.808581 0.404290 0.914631i \(-0.367519\pi\)
0.404290 + 0.914631i \(0.367519\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) −8.00000 −0.269069
\(885\) −24.0000 −0.806751
\(886\) 10.0000 0.335957
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 20.0000 0.670402
\(891\) −2.00000 −0.0670025
\(892\) 6.00000 0.200895
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) 48.0000 1.60446
\(896\) 0 0
\(897\) 0 0
\(898\) −14.0000 −0.467186
\(899\) −60.0000 −2.00111
\(900\) −1.00000 −0.0333333
\(901\) −24.0000 −0.799556
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) −4.00000 −0.132964
\(906\) 8.00000 0.265782
\(907\) −22.0000 −0.730498 −0.365249 0.930910i \(-0.619016\pi\)
−0.365249 + 0.930910i \(0.619016\pi\)
\(908\) −12.0000 −0.398234
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −56.0000 −1.85536 −0.927681 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(912\) 1.00000 0.0331133
\(913\) −24.0000 −0.794284
\(914\) 10.0000 0.330771
\(915\) 20.0000 0.661180
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 10.0000 0.329332
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 0 0
\(926\) 32.0000 1.05159
\(927\) 14.0000 0.459820
\(928\) 6.00000 0.196960
\(929\) 28.0000 0.918650 0.459325 0.888268i \(-0.348091\pi\)
0.459325 + 0.888268i \(0.348091\pi\)
\(930\) 20.0000 0.655826
\(931\) 0 0
\(932\) −2.00000 −0.0655122
\(933\) 10.0000 0.327385
\(934\) −12.0000 −0.392652
\(935\) −16.0000 −0.523256
\(936\) 2.00000 0.0653720
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −30.0000 −0.979013
\(940\) −12.0000 −0.391397
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) −6.00000 −0.195491
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) −42.0000 −1.36482 −0.682408 0.730971i \(-0.739067\pi\)
−0.682408 + 0.730971i \(0.739067\pi\)
\(948\) −16.0000 −0.519656
\(949\) −12.0000 −0.389536
\(950\) −1.00000 −0.0324443
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −14.0000 −0.453504 −0.226752 0.973952i \(-0.572811\pi\)
−0.226752 + 0.973952i \(0.572811\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) −12.0000 −0.387905
\(958\) −26.0000 −0.840022
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) 24.0000 0.772988
\(965\) 44.0000 1.41641
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 7.00000 0.224989
\(969\) 4.00000 0.128499
\(970\) −24.0000 −0.770594
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −4.00000 −0.128168
\(975\) −2.00000 −0.0640513
\(976\) −10.0000 −0.320092
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) −8.00000 −0.255812
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) −30.0000 −0.957338
\(983\) −44.0000 −1.40338 −0.701691 0.712481i \(-0.747571\pi\)
−0.701691 + 0.712481i \(0.747571\pi\)
\(984\) 6.00000 0.191273
\(985\) −16.0000 −0.509802
\(986\) 24.0000 0.764316
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) 0 0
\(990\) 4.00000 0.127128
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −10.0000 −0.317500
\(993\) 6.00000 0.190404
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) −12.0000 −0.380235
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) −12.0000 −0.379853
\(999\) 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 5586.2.a.j.1.1 1
7.6 odd 2 798.2.a.b.1.1 1
21.20 even 2 2394.2.a.m.1.1 1
28.27 even 2 6384.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.a.b.1.1 1 7.6 odd 2
2394.2.a.m.1.1 1 21.20 even 2
5586.2.a.j.1.1 1 1.1 even 1 trivial
6384.2.a.g.1.1 1 28.27 even 2