Properties

Label 338.2.a.a.1.1
Level $338$
Weight $2$
Character 338.1
Self dual yes
Analytic conductor $2.699$
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] = [338,2,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, 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("338.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(2.69894358832\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 338.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +3.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} -3.00000 q^{12} +1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -6.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} +3.00000 q^{21} -2.00000 q^{22} -4.00000 q^{23} +3.00000 q^{24} -4.00000 q^{25} -9.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} +3.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} +3.00000 q^{34} -1.00000 q^{35} +6.00000 q^{36} -3.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} -3.00000 q^{42} -5.00000 q^{43} +2.00000 q^{44} +6.00000 q^{45} +4.00000 q^{46} -13.0000 q^{47} -3.00000 q^{48} -6.00000 q^{49} +4.00000 q^{50} +9.00000 q^{51} +12.0000 q^{53} +9.00000 q^{54} +2.00000 q^{55} +1.00000 q^{56} +18.0000 q^{57} -2.00000 q^{58} +10.0000 q^{59} -3.00000 q^{60} -8.00000 q^{61} +4.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} +6.00000 q^{66} +2.00000 q^{67} -3.00000 q^{68} +12.0000 q^{69} +1.00000 q^{70} +5.00000 q^{71} -6.00000 q^{72} +10.0000 q^{73} +3.00000 q^{74} +12.0000 q^{75} -6.00000 q^{76} -2.00000 q^{77} -4.00000 q^{79} +1.00000 q^{80} +9.00000 q^{81} +3.00000 q^{84} -3.00000 q^{85} +5.00000 q^{86} -6.00000 q^{87} -2.00000 q^{88} -6.00000 q^{89} -6.00000 q^{90} -4.00000 q^{92} +12.0000 q^{93} +13.0000 q^{94} -6.00000 q^{95} +3.00000 q^{96} -14.0000 q^{97} +6.00000 q^{98} +12.0000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 3.00000 1.22474
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.00000 −0.353553
\(9\) 6.00000 2.00000
\(10\) −1.00000 −0.316228
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −3.00000 −0.866025
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −6.00000 −1.41421
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.00000 0.654654
\(22\) −2.00000 −0.426401
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 3.00000 0.612372
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) −1.00000 −0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 3.00000 0.547723
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −6.00000 −1.04447
\(34\) 3.00000 0.514496
\(35\) −1.00000 −0.169031
\(36\) 6.00000 1.00000
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −3.00000 −0.462910
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 2.00000 0.301511
\(45\) 6.00000 0.894427
\(46\) 4.00000 0.589768
\(47\) −13.0000 −1.89624 −0.948122 0.317905i \(-0.897021\pi\)
−0.948122 + 0.317905i \(0.897021\pi\)
\(48\) −3.00000 −0.433013
\(49\) −6.00000 −0.857143
\(50\) 4.00000 0.565685
\(51\) 9.00000 1.26025
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 9.00000 1.22474
\(55\) 2.00000 0.269680
\(56\) 1.00000 0.133631
\(57\) 18.0000 2.38416
\(58\) −2.00000 −0.262613
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) −3.00000 −0.387298
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 4.00000 0.508001
\(63\) −6.00000 −0.755929
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −3.00000 −0.363803
\(69\) 12.0000 1.44463
\(70\) 1.00000 0.119523
\(71\) 5.00000 0.593391 0.296695 0.954972i \(-0.404115\pi\)
0.296695 + 0.954972i \(0.404115\pi\)
\(72\) −6.00000 −0.707107
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 3.00000 0.348743
\(75\) 12.0000 1.38564
\(76\) −6.00000 −0.688247
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 1.00000 0.111803
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 3.00000 0.327327
\(85\) −3.00000 −0.325396
\(86\) 5.00000 0.539164
\(87\) −6.00000 −0.643268
\(88\) −2.00000 −0.213201
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −6.00000 −0.632456
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 12.0000 1.24434
\(94\) 13.0000 1.34085
\(95\) −6.00000 −0.615587
\(96\) 3.00000 0.306186
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 6.00000 0.606092
\(99\) 12.0000 1.20605
\(100\) −4.00000 −0.400000
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) −9.00000 −0.891133
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) −12.0000 −1.16554
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −9.00000 −0.866025
\(109\) −19.0000 −1.81987 −0.909935 0.414751i \(-0.863869\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) −2.00000 −0.190693
\(111\) 9.00000 0.854242
\(112\) −1.00000 −0.0944911
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −18.0000 −1.68585
\(115\) −4.00000 −0.373002
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) 3.00000 0.275010
\(120\) 3.00000 0.273861
\(121\) −7.00000 −0.636364
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −9.00000 −0.804984
\(126\) 6.00000 0.534522
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 15.0000 1.32068
\(130\) 0 0
\(131\) −1.00000 −0.0873704 −0.0436852 0.999045i \(-0.513910\pi\)
−0.0436852 + 0.999045i \(0.513910\pi\)
\(132\) −6.00000 −0.522233
\(133\) 6.00000 0.520266
\(134\) −2.00000 −0.172774
\(135\) −9.00000 −0.774597
\(136\) 3.00000 0.257248
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) −12.0000 −1.02151
\(139\) 7.00000 0.593732 0.296866 0.954919i \(-0.404058\pi\)
0.296866 + 0.954919i \(0.404058\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 39.0000 3.28439
\(142\) −5.00000 −0.419591
\(143\) 0 0
\(144\) 6.00000 0.500000
\(145\) 2.00000 0.166091
\(146\) −10.0000 −0.827606
\(147\) 18.0000 1.48461
\(148\) −3.00000 −0.246598
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −12.0000 −0.979796
\(151\) 9.00000 0.732410 0.366205 0.930534i \(-0.380657\pi\)
0.366205 + 0.930534i \(0.380657\pi\)
\(152\) 6.00000 0.486664
\(153\) −18.0000 −1.45521
\(154\) 2.00000 0.161165
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 4.00000 0.318223
\(159\) −36.0000 −2.85499
\(160\) −1.00000 −0.0790569
\(161\) 4.00000 0.315244
\(162\) −9.00000 −0.707107
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −3.00000 −0.231455
\(169\) 0 0
\(170\) 3.00000 0.230089
\(171\) −36.0000 −2.75299
\(172\) −5.00000 −0.381246
\(173\) 20.0000 1.52057 0.760286 0.649589i \(-0.225059\pi\)
0.760286 + 0.649589i \(0.225059\pi\)
\(174\) 6.00000 0.454859
\(175\) 4.00000 0.302372
\(176\) 2.00000 0.150756
\(177\) −30.0000 −2.25494
\(178\) 6.00000 0.449719
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 6.00000 0.447214
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 24.0000 1.77413
\(184\) 4.00000 0.294884
\(185\) −3.00000 −0.220564
\(186\) −12.0000 −0.879883
\(187\) −6.00000 −0.438763
\(188\) −13.0000 −0.948122
\(189\) 9.00000 0.654654
\(190\) 6.00000 0.435286
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) −3.00000 −0.216506
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −9.00000 −0.641223 −0.320612 0.947211i \(-0.603888\pi\)
−0.320612 + 0.947211i \(0.603888\pi\)
\(198\) −12.0000 −0.852803
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 4.00000 0.282843
\(201\) −6.00000 −0.423207
\(202\) −4.00000 −0.281439
\(203\) −2.00000 −0.140372
\(204\) 9.00000 0.630126
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) −24.0000 −1.66812
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) −3.00000 −0.207020
\(211\) 23.0000 1.58339 0.791693 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) 12.0000 0.824163
\(213\) −15.0000 −1.02778
\(214\) 4.00000 0.273434
\(215\) −5.00000 −0.340997
\(216\) 9.00000 0.612372
\(217\) 4.00000 0.271538
\(218\) 19.0000 1.28684
\(219\) −30.0000 −2.02721
\(220\) 2.00000 0.134840
\(221\) 0 0
\(222\) −9.00000 −0.604040
\(223\) 21.0000 1.40626 0.703132 0.711059i \(-0.251784\pi\)
0.703132 + 0.711059i \(0.251784\pi\)
\(224\) 1.00000 0.0668153
\(225\) −24.0000 −1.60000
\(226\) −2.00000 −0.133038
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 18.0000 1.19208
\(229\) 15.0000 0.991228 0.495614 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 4.00000 0.263752
\(231\) 6.00000 0.394771
\(232\) −2.00000 −0.131306
\(233\) −11.0000 −0.720634 −0.360317 0.932830i \(-0.617331\pi\)
−0.360317 + 0.932830i \(0.617331\pi\)
\(234\) 0 0
\(235\) −13.0000 −0.848026
\(236\) 10.0000 0.650945
\(237\) 12.0000 0.779484
\(238\) −3.00000 −0.194461
\(239\) −9.00000 −0.582162 −0.291081 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) −3.00000 −0.193649
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 9.00000 0.569210
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −6.00000 −0.377964
\(253\) −8.00000 −0.502956
\(254\) −16.0000 −1.00393
\(255\) 9.00000 0.563602
\(256\) 1.00000 0.0625000
\(257\) −15.0000 −0.935674 −0.467837 0.883815i \(-0.654967\pi\)
−0.467837 + 0.883815i \(0.654967\pi\)
\(258\) −15.0000 −0.933859
\(259\) 3.00000 0.186411
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 1.00000 0.0617802
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 6.00000 0.369274
\(265\) 12.0000 0.737154
\(266\) −6.00000 −0.367884
\(267\) 18.0000 1.10158
\(268\) 2.00000 0.122169
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 9.00000 0.547723
\(271\) −13.0000 −0.789694 −0.394847 0.918747i \(-0.629202\pi\)
−0.394847 + 0.918747i \(0.629202\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) −8.00000 −0.482418
\(276\) 12.0000 0.722315
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) −7.00000 −0.419832
\(279\) −24.0000 −1.43684
\(280\) 1.00000 0.0597614
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) −39.0000 −2.32242
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 5.00000 0.296695
\(285\) 18.0000 1.06623
\(286\) 0 0
\(287\) 0 0
\(288\) −6.00000 −0.353553
\(289\) −8.00000 −0.470588
\(290\) −2.00000 −0.117444
\(291\) 42.0000 2.46208
\(292\) 10.0000 0.585206
\(293\) −7.00000 −0.408944 −0.204472 0.978872i \(-0.565548\pi\)
−0.204472 + 0.978872i \(0.565548\pi\)
\(294\) −18.0000 −1.04978
\(295\) 10.0000 0.582223
\(296\) 3.00000 0.174371
\(297\) −18.0000 −1.04447
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 12.0000 0.692820
\(301\) 5.00000 0.288195
\(302\) −9.00000 −0.517892
\(303\) −12.0000 −0.689382
\(304\) −6.00000 −0.344124
\(305\) −8.00000 −0.458079
\(306\) 18.0000 1.02899
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) −2.00000 −0.113961
\(309\) 24.0000 1.36531
\(310\) 4.00000 0.227185
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 10.0000 0.564333
\(315\) −6.00000 −0.338062
\(316\) −4.00000 −0.225018
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 36.0000 2.01878
\(319\) 4.00000 0.223957
\(320\) 1.00000 0.0559017
\(321\) 12.0000 0.669775
\(322\) −4.00000 −0.222911
\(323\) 18.0000 1.00155
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 57.0000 3.15211
\(328\) 0 0
\(329\) 13.0000 0.716713
\(330\) 6.00000 0.330289
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) −18.0000 −0.986394
\(334\) 0 0
\(335\) 2.00000 0.109272
\(336\) 3.00000 0.163663
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) −3.00000 −0.162698
\(341\) −8.00000 −0.433224
\(342\) 36.0000 1.94666
\(343\) 13.0000 0.701934
\(344\) 5.00000 0.269582
\(345\) 12.0000 0.646058
\(346\) −20.0000 −1.07521
\(347\) −9.00000 −0.483145 −0.241573 0.970383i \(-0.577663\pi\)
−0.241573 + 0.970383i \(0.577663\pi\)
\(348\) −6.00000 −0.321634
\(349\) −7.00000 −0.374701 −0.187351 0.982293i \(-0.559990\pi\)
−0.187351 + 0.982293i \(0.559990\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) −4.00000 −0.212899 −0.106449 0.994318i \(-0.533948\pi\)
−0.106449 + 0.994318i \(0.533948\pi\)
\(354\) 30.0000 1.59448
\(355\) 5.00000 0.265372
\(356\) −6.00000 −0.317999
\(357\) −9.00000 −0.476331
\(358\) 9.00000 0.475665
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −6.00000 −0.316228
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 21.0000 1.10221
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) −24.0000 −1.25450
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 3.00000 0.155963
\(371\) −12.0000 −0.623009
\(372\) 12.0000 0.622171
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 6.00000 0.310253
\(375\) 27.0000 1.39427
\(376\) 13.0000 0.670424
\(377\) 0 0
\(378\) −9.00000 −0.462910
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) −6.00000 −0.307794
\(381\) −48.0000 −2.45911
\(382\) −10.0000 −0.511645
\(383\) −27.0000 −1.37964 −0.689818 0.723983i \(-0.742309\pi\)
−0.689818 + 0.723983i \(0.742309\pi\)
\(384\) 3.00000 0.153093
\(385\) −2.00000 −0.101929
\(386\) −16.0000 −0.814379
\(387\) −30.0000 −1.52499
\(388\) −14.0000 −0.710742
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 6.00000 0.303046
\(393\) 3.00000 0.151330
\(394\) 9.00000 0.453413
\(395\) −4.00000 −0.201262
\(396\) 12.0000 0.603023
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 10.0000 0.501255
\(399\) −18.0000 −0.901127
\(400\) −4.00000 −0.200000
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 6.00000 0.299253
\(403\) 0 0
\(404\) 4.00000 0.199007
\(405\) 9.00000 0.447214
\(406\) 2.00000 0.0992583
\(407\) −6.00000 −0.297409
\(408\) −9.00000 −0.445566
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 36.0000 1.77575
\(412\) −8.00000 −0.394132
\(413\) −10.0000 −0.492068
\(414\) 24.0000 1.17954
\(415\) 0 0
\(416\) 0 0
\(417\) −21.0000 −1.02837
\(418\) 12.0000 0.586939
\(419\) 21.0000 1.02592 0.512959 0.858413i \(-0.328549\pi\)
0.512959 + 0.858413i \(0.328549\pi\)
\(420\) 3.00000 0.146385
\(421\) 5.00000 0.243685 0.121843 0.992549i \(-0.461120\pi\)
0.121843 + 0.992549i \(0.461120\pi\)
\(422\) −23.0000 −1.11962
\(423\) −78.0000 −3.79249
\(424\) −12.0000 −0.582772
\(425\) 12.0000 0.582086
\(426\) 15.0000 0.726752
\(427\) 8.00000 0.387147
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) 5.00000 0.241121
\(431\) −33.0000 −1.58955 −0.794777 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) −9.00000 −0.433013
\(433\) 7.00000 0.336399 0.168199 0.985753i \(-0.446205\pi\)
0.168199 + 0.985753i \(0.446205\pi\)
\(434\) −4.00000 −0.192006
\(435\) −6.00000 −0.287678
\(436\) −19.0000 −0.909935
\(437\) 24.0000 1.14808
\(438\) 30.0000 1.43346
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) −2.00000 −0.0953463
\(441\) −36.0000 −1.71429
\(442\) 0 0
\(443\) −39.0000 −1.85295 −0.926473 0.376361i \(-0.877175\pi\)
−0.926473 + 0.376361i \(0.877175\pi\)
\(444\) 9.00000 0.427121
\(445\) −6.00000 −0.284427
\(446\) −21.0000 −0.994379
\(447\) −54.0000 −2.55411
\(448\) −1.00000 −0.0472456
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 24.0000 1.13137
\(451\) 0 0
\(452\) 2.00000 0.0940721
\(453\) −27.0000 −1.26857
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) −18.0000 −0.842927
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −15.0000 −0.700904
\(459\) 27.0000 1.26025
\(460\) −4.00000 −0.186501
\(461\) 21.0000 0.978068 0.489034 0.872265i \(-0.337349\pi\)
0.489034 + 0.872265i \(0.337349\pi\)
\(462\) −6.00000 −0.279145
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 2.00000 0.0928477
\(465\) 12.0000 0.556487
\(466\) 11.0000 0.509565
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) −2.00000 −0.0923514
\(470\) 13.0000 0.599645
\(471\) 30.0000 1.38233
\(472\) −10.0000 −0.460287
\(473\) −10.0000 −0.459800
\(474\) −12.0000 −0.551178
\(475\) 24.0000 1.10120
\(476\) 3.00000 0.137505
\(477\) 72.0000 3.29665
\(478\) 9.00000 0.411650
\(479\) 3.00000 0.137073 0.0685367 0.997649i \(-0.478167\pi\)
0.0685367 + 0.997649i \(0.478167\pi\)
\(480\) 3.00000 0.136931
\(481\) 0 0
\(482\) 18.0000 0.819878
\(483\) −12.0000 −0.546019
\(484\) −7.00000 −0.318182
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 8.00000 0.362143
\(489\) −12.0000 −0.542659
\(490\) 6.00000 0.271052
\(491\) −5.00000 −0.225647 −0.112823 0.993615i \(-0.535989\pi\)
−0.112823 + 0.993615i \(0.535989\pi\)
\(492\) 0 0
\(493\) −6.00000 −0.270226
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) −4.00000 −0.179605
\(497\) −5.00000 −0.224281
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −9.00000 −0.402492
\(501\) 0 0
\(502\) 0 0
\(503\) −14.0000 −0.624229 −0.312115 0.950044i \(-0.601037\pi\)
−0.312115 + 0.950044i \(0.601037\pi\)
\(504\) 6.00000 0.267261
\(505\) 4.00000 0.177998
\(506\) 8.00000 0.355643
\(507\) 0 0
\(508\) 16.0000 0.709885
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) −9.00000 −0.398527
\(511\) −10.0000 −0.442374
\(512\) −1.00000 −0.0441942
\(513\) 54.0000 2.38416
\(514\) 15.0000 0.661622
\(515\) −8.00000 −0.352522
\(516\) 15.0000 0.660338
\(517\) −26.0000 −1.14348
\(518\) −3.00000 −0.131812
\(519\) −60.0000 −2.63371
\(520\) 0 0
\(521\) 39.0000 1.70862 0.854311 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) −12.0000 −0.525226
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) −1.00000 −0.0436852
\(525\) −12.0000 −0.523723
\(526\) −12.0000 −0.523225
\(527\) 12.0000 0.522728
\(528\) −6.00000 −0.261116
\(529\) −7.00000 −0.304348
\(530\) −12.0000 −0.521247
\(531\) 60.0000 2.60378
\(532\) 6.00000 0.260133
\(533\) 0 0
\(534\) −18.0000 −0.778936
\(535\) −4.00000 −0.172935
\(536\) −2.00000 −0.0863868
\(537\) 27.0000 1.16514
\(538\) 24.0000 1.03471
\(539\) −12.0000 −0.516877
\(540\) −9.00000 −0.387298
\(541\) −17.0000 −0.730887 −0.365444 0.930834i \(-0.619083\pi\)
−0.365444 + 0.930834i \(0.619083\pi\)
\(542\) 13.0000 0.558398
\(543\) 0 0
\(544\) 3.00000 0.128624
\(545\) −19.0000 −0.813871
\(546\) 0 0
\(547\) 37.0000 1.58201 0.791003 0.611812i \(-0.209559\pi\)
0.791003 + 0.611812i \(0.209559\pi\)
\(548\) −12.0000 −0.512615
\(549\) −48.0000 −2.04859
\(550\) 8.00000 0.341121
\(551\) −12.0000 −0.511217
\(552\) −12.0000 −0.510754
\(553\) 4.00000 0.170097
\(554\) −12.0000 −0.509831
\(555\) 9.00000 0.382029
\(556\) 7.00000 0.296866
\(557\) −33.0000 −1.39825 −0.699127 0.714997i \(-0.746428\pi\)
−0.699127 + 0.714997i \(0.746428\pi\)
\(558\) 24.0000 1.01600
\(559\) 0 0
\(560\) −1.00000 −0.0422577
\(561\) 18.0000 0.759961
\(562\) −26.0000 −1.09674
\(563\) 11.0000 0.463595 0.231797 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) 39.0000 1.64220
\(565\) 2.00000 0.0841406
\(566\) −4.00000 −0.168133
\(567\) −9.00000 −0.377964
\(568\) −5.00000 −0.209795
\(569\) 31.0000 1.29959 0.649794 0.760111i \(-0.274855\pi\)
0.649794 + 0.760111i \(0.274855\pi\)
\(570\) −18.0000 −0.753937
\(571\) 33.0000 1.38101 0.690504 0.723329i \(-0.257389\pi\)
0.690504 + 0.723329i \(0.257389\pi\)
\(572\) 0 0
\(573\) −30.0000 −1.25327
\(574\) 0 0
\(575\) 16.0000 0.667246
\(576\) 6.00000 0.250000
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 8.00000 0.332756
\(579\) −48.0000 −1.99481
\(580\) 2.00000 0.0830455
\(581\) 0 0
\(582\) −42.0000 −1.74096
\(583\) 24.0000 0.993978
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 7.00000 0.289167
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 18.0000 0.742307
\(589\) 24.0000 0.988903
\(590\) −10.0000 −0.411693
\(591\) 27.0000 1.11063
\(592\) −3.00000 −0.123299
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 18.0000 0.738549
\(595\) 3.00000 0.122988
\(596\) 18.0000 0.737309
\(597\) 30.0000 1.22782
\(598\) 0 0
\(599\) −2.00000 −0.0817178 −0.0408589 0.999165i \(-0.513009\pi\)
−0.0408589 + 0.999165i \(0.513009\pi\)
\(600\) −12.0000 −0.489898
\(601\) −35.0000 −1.42768 −0.713840 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) −5.00000 −0.203785
\(603\) 12.0000 0.488678
\(604\) 9.00000 0.366205
\(605\) −7.00000 −0.284590
\(606\) 12.0000 0.487467
\(607\) 6.00000 0.243532 0.121766 0.992559i \(-0.461144\pi\)
0.121766 + 0.992559i \(0.461144\pi\)
\(608\) 6.00000 0.243332
\(609\) 6.00000 0.243132
\(610\) 8.00000 0.323911
\(611\) 0 0
\(612\) −18.0000 −0.727607
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 14.0000 0.564994
\(615\) 0 0
\(616\) 2.00000 0.0805823
\(617\) −16.0000 −0.644136 −0.322068 0.946717i \(-0.604378\pi\)
−0.322068 + 0.946717i \(0.604378\pi\)
\(618\) −24.0000 −0.965422
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) −4.00000 −0.160644
\(621\) 36.0000 1.44463
\(622\) −18.0000 −0.721734
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 1.00000 0.0399680
\(627\) 36.0000 1.43770
\(628\) −10.0000 −0.399043
\(629\) 9.00000 0.358854
\(630\) 6.00000 0.239046
\(631\) 5.00000 0.199047 0.0995234 0.995035i \(-0.468268\pi\)
0.0995234 + 0.995035i \(0.468268\pi\)
\(632\) 4.00000 0.159111
\(633\) −69.0000 −2.74250
\(634\) −18.0000 −0.714871
\(635\) 16.0000 0.634941
\(636\) −36.0000 −1.42749
\(637\) 0 0
\(638\) −4.00000 −0.158362
\(639\) 30.0000 1.18678
\(640\) −1.00000 −0.0395285
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) −12.0000 −0.473602
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 4.00000 0.157622
\(645\) 15.0000 0.590624
\(646\) −18.0000 −0.708201
\(647\) −38.0000 −1.49393 −0.746967 0.664861i \(-0.768491\pi\)
−0.746967 + 0.664861i \(0.768491\pi\)
\(648\) −9.00000 −0.353553
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) −12.0000 −0.470317
\(652\) 4.00000 0.156652
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) −57.0000 −2.22888
\(655\) −1.00000 −0.0390732
\(656\) 0 0
\(657\) 60.0000 2.34082
\(658\) −13.0000 −0.506793
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) −6.00000 −0.233550
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 0 0
\(665\) 6.00000 0.232670
\(666\) 18.0000 0.697486
\(667\) −8.00000 −0.309761
\(668\) 0 0
\(669\) −63.0000 −2.43572
\(670\) −2.00000 −0.0772667
\(671\) −16.0000 −0.617673
\(672\) −3.00000 −0.115728
\(673\) 37.0000 1.42625 0.713123 0.701039i \(-0.247280\pi\)
0.713123 + 0.701039i \(0.247280\pi\)
\(674\) −23.0000 −0.885927
\(675\) 36.0000 1.38564
\(676\) 0 0
\(677\) −36.0000 −1.38359 −0.691796 0.722093i \(-0.743180\pi\)
−0.691796 + 0.722093i \(0.743180\pi\)
\(678\) 6.00000 0.230429
\(679\) 14.0000 0.537271
\(680\) 3.00000 0.115045
\(681\) −72.0000 −2.75905
\(682\) 8.00000 0.306336
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) −36.0000 −1.37649
\(685\) −12.0000 −0.458496
\(686\) −13.0000 −0.496342
\(687\) −45.0000 −1.71686
\(688\) −5.00000 −0.190623
\(689\) 0 0
\(690\) −12.0000 −0.456832
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 20.0000 0.760286
\(693\) −12.0000 −0.455842
\(694\) 9.00000 0.341635
\(695\) 7.00000 0.265525
\(696\) 6.00000 0.227429
\(697\) 0 0
\(698\) 7.00000 0.264954
\(699\) 33.0000 1.24817
\(700\) 4.00000 0.151186
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 0 0
\(703\) 18.0000 0.678883
\(704\) 2.00000 0.0753778
\(705\) 39.0000 1.46882
\(706\) 4.00000 0.150542
\(707\) −4.00000 −0.150435
\(708\) −30.0000 −1.12747
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) −5.00000 −0.187647
\(711\) −24.0000 −0.900070
\(712\) 6.00000 0.224860
\(713\) 16.0000 0.599205
\(714\) 9.00000 0.336817
\(715\) 0 0
\(716\) −9.00000 −0.336346
\(717\) 27.0000 1.00833
\(718\) 24.0000 0.895672
\(719\) −22.0000 −0.820462 −0.410231 0.911982i \(-0.634552\pi\)
−0.410231 + 0.911982i \(0.634552\pi\)
\(720\) 6.00000 0.223607
\(721\) 8.00000 0.297936
\(722\) −17.0000 −0.632674
\(723\) 54.0000 2.00828
\(724\) 0 0
\(725\) −8.00000 −0.297113
\(726\) −21.0000 −0.779383
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) −10.0000 −0.370117
\(731\) 15.0000 0.554795
\(732\) 24.0000 0.887066
\(733\) 43.0000 1.58824 0.794121 0.607760i \(-0.207932\pi\)
0.794121 + 0.607760i \(0.207932\pi\)
\(734\) 10.0000 0.369107
\(735\) 18.0000 0.663940
\(736\) 4.00000 0.147442
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) −3.00000 −0.110282
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) 47.0000 1.72426 0.862131 0.506685i \(-0.169129\pi\)
0.862131 + 0.506685i \(0.169129\pi\)
\(744\) −12.0000 −0.439941
\(745\) 18.0000 0.659469
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) −6.00000 −0.219382
\(749\) 4.00000 0.146157
\(750\) −27.0000 −0.985901
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) −13.0000 −0.474061
\(753\) 0 0
\(754\) 0 0
\(755\) 9.00000 0.327544
\(756\) 9.00000 0.327327
\(757\) −12.0000 −0.436147 −0.218074 0.975932i \(-0.569977\pi\)
−0.218074 + 0.975932i \(0.569977\pi\)
\(758\) 16.0000 0.581146
\(759\) 24.0000 0.871145
\(760\) 6.00000 0.217643
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 48.0000 1.73886
\(763\) 19.0000 0.687846
\(764\) 10.0000 0.361787
\(765\) −18.0000 −0.650791
\(766\) 27.0000 0.975550
\(767\) 0 0
\(768\) −3.00000 −0.108253
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 2.00000 0.0720750
\(771\) 45.0000 1.62064
\(772\) 16.0000 0.575853
\(773\) −11.0000 −0.395643 −0.197821 0.980238i \(-0.563387\pi\)
−0.197821 + 0.980238i \(0.563387\pi\)
\(774\) 30.0000 1.07833
\(775\) 16.0000 0.574737
\(776\) 14.0000 0.502571
\(777\) −9.00000 −0.322873
\(778\) 30.0000 1.07555
\(779\) 0 0
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) −12.0000 −0.429119
\(783\) −18.0000 −0.643268
\(784\) −6.00000 −0.214286
\(785\) −10.0000 −0.356915
\(786\) −3.00000 −0.107006
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) −9.00000 −0.320612
\(789\) −36.0000 −1.28163
\(790\) 4.00000 0.142314
\(791\) −2.00000 −0.0711118
\(792\) −12.0000 −0.426401
\(793\) 0 0
\(794\) −22.0000 −0.780751
\(795\) −36.0000 −1.27679
\(796\) −10.0000 −0.354441
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 18.0000 0.637193
\(799\) 39.0000 1.37972
\(800\) 4.00000 0.141421
\(801\) −36.0000 −1.27200
\(802\) 24.0000 0.847469
\(803\) 20.0000 0.705785
\(804\) −6.00000 −0.211604
\(805\) 4.00000 0.140981
\(806\) 0 0
\(807\) 72.0000 2.53452
\(808\) −4.00000 −0.140720
\(809\) −9.00000 −0.316423 −0.158212 0.987405i \(-0.550573\pi\)
−0.158212 + 0.987405i \(0.550573\pi\)
\(810\) −9.00000 −0.316228
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 39.0000 1.36779
\(814\) 6.00000 0.210300
\(815\) 4.00000 0.140114
\(816\) 9.00000 0.315063
\(817\) 30.0000 1.04957
\(818\) 4.00000 0.139857
\(819\) 0 0
\(820\) 0 0
\(821\) 25.0000 0.872506 0.436253 0.899824i \(-0.356305\pi\)
0.436253 + 0.899824i \(0.356305\pi\)
\(822\) −36.0000 −1.25564
\(823\) 54.0000 1.88232 0.941161 0.337959i \(-0.109737\pi\)
0.941161 + 0.337959i \(0.109737\pi\)
\(824\) 8.00000 0.278693
\(825\) 24.0000 0.835573
\(826\) 10.0000 0.347945
\(827\) −30.0000 −1.04320 −0.521601 0.853189i \(-0.674665\pi\)
−0.521601 + 0.853189i \(0.674665\pi\)
\(828\) −24.0000 −0.834058
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) −36.0000 −1.24883
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 21.0000 0.727171
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 36.0000 1.24434
\(838\) −21.0000 −0.725433
\(839\) −56.0000 −1.93333 −0.966667 0.256036i \(-0.917584\pi\)
−0.966667 + 0.256036i \(0.917584\pi\)
\(840\) −3.00000 −0.103510
\(841\) −25.0000 −0.862069
\(842\) −5.00000 −0.172311
\(843\) −78.0000 −2.68646
\(844\) 23.0000 0.791693
\(845\) 0 0
\(846\) 78.0000 2.68170
\(847\) 7.00000 0.240523
\(848\) 12.0000 0.412082
\(849\) −12.0000 −0.411839
\(850\) −12.0000 −0.411597
\(851\) 12.0000 0.411355
\(852\) −15.0000 −0.513892
\(853\) −49.0000 −1.67773 −0.838864 0.544341i \(-0.816780\pi\)
−0.838864 + 0.544341i \(0.816780\pi\)
\(854\) −8.00000 −0.273754
\(855\) −36.0000 −1.23117
\(856\) 4.00000 0.136717
\(857\) 46.0000 1.57133 0.785665 0.618652i \(-0.212321\pi\)
0.785665 + 0.618652i \(0.212321\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −5.00000 −0.170499
\(861\) 0 0
\(862\) 33.0000 1.12398
\(863\) 11.0000 0.374444 0.187222 0.982318i \(-0.440052\pi\)
0.187222 + 0.982318i \(0.440052\pi\)
\(864\) 9.00000 0.306186
\(865\) 20.0000 0.680020
\(866\) −7.00000 −0.237870
\(867\) 24.0000 0.815083
\(868\) 4.00000 0.135769
\(869\) −8.00000 −0.271381
\(870\) 6.00000 0.203419
\(871\) 0 0
\(872\) 19.0000 0.643421
\(873\) −84.0000 −2.84297
\(874\) −24.0000 −0.811812
\(875\) 9.00000 0.304256
\(876\) −30.0000 −1.01361
\(877\) 39.0000 1.31694 0.658468 0.752609i \(-0.271205\pi\)
0.658468 + 0.752609i \(0.271205\pi\)
\(878\) 22.0000 0.742464
\(879\) 21.0000 0.708312
\(880\) 2.00000 0.0674200
\(881\) 21.0000 0.707508 0.353754 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) 36.0000 1.21218
\(883\) −47.0000 −1.58168 −0.790838 0.612026i \(-0.790355\pi\)
−0.790838 + 0.612026i \(0.790355\pi\)
\(884\) 0 0
\(885\) −30.0000 −1.00844
\(886\) 39.0000 1.31023
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −9.00000 −0.302020
\(889\) −16.0000 −0.536623
\(890\) 6.00000 0.201120
\(891\) 18.0000 0.603023
\(892\) 21.0000 0.703132
\(893\) 78.0000 2.61017
\(894\) 54.0000 1.80603
\(895\) −9.00000 −0.300837
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −26.0000 −0.867631
\(899\) −8.00000 −0.266815
\(900\) −24.0000 −0.800000
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) −15.0000 −0.499169
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) 27.0000 0.897015
\(907\) −9.00000 −0.298840 −0.149420 0.988774i \(-0.547741\pi\)
−0.149420 + 0.988774i \(0.547741\pi\)
\(908\) 24.0000 0.796468
\(909\) 24.0000 0.796030
\(910\) 0 0
\(911\) −54.0000 −1.78910 −0.894550 0.446968i \(-0.852504\pi\)
−0.894550 + 0.446968i \(0.852504\pi\)
\(912\) 18.0000 0.596040
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 24.0000 0.793416
\(916\) 15.0000 0.495614
\(917\) 1.00000 0.0330229
\(918\) −27.0000 −0.891133
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 4.00000 0.131876
\(921\) 42.0000 1.38395
\(922\) −21.0000 −0.691598
\(923\) 0 0
\(924\) 6.00000 0.197386
\(925\) 12.0000 0.394558
\(926\) 16.0000 0.525793
\(927\) −48.0000 −1.57653
\(928\) −2.00000 −0.0656532
\(929\) 36.0000 1.18112 0.590561 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) −12.0000 −0.393496
\(931\) 36.0000 1.17985
\(932\) −11.0000 −0.360317
\(933\) −54.0000 −1.76788
\(934\) −20.0000 −0.654420
\(935\) −6.00000 −0.196221
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 2.00000 0.0653023
\(939\) 3.00000 0.0979013
\(940\) −13.0000 −0.424013
\(941\) −25.0000 −0.814977 −0.407488 0.913210i \(-0.633595\pi\)
−0.407488 + 0.913210i \(0.633595\pi\)
\(942\) −30.0000 −0.977453
\(943\) 0 0
\(944\) 10.0000 0.325472
\(945\) 9.00000 0.292770
\(946\) 10.0000 0.325128
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 12.0000 0.389742
\(949\) 0 0
\(950\) −24.0000 −0.778663
\(951\) −54.0000 −1.75107
\(952\) −3.00000 −0.0972306
\(953\) 23.0000 0.745043 0.372522 0.928024i \(-0.378493\pi\)
0.372522 + 0.928024i \(0.378493\pi\)
\(954\) −72.0000 −2.33109
\(955\) 10.0000 0.323592
\(956\) −9.00000 −0.291081
\(957\) −12.0000 −0.387905
\(958\) −3.00000 −0.0969256
\(959\) 12.0000 0.387500
\(960\) −3.00000 −0.0968246
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −24.0000 −0.773389
\(964\) −18.0000 −0.579741
\(965\) 16.0000 0.515058
\(966\) 12.0000 0.386094
\(967\) −23.0000 −0.739630 −0.369815 0.929105i \(-0.620579\pi\)
−0.369815 + 0.929105i \(0.620579\pi\)
\(968\) 7.00000 0.224989
\(969\) −54.0000 −1.73473
\(970\) 14.0000 0.449513
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) −7.00000 −0.224410
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 12.0000 0.383718
\(979\) −12.0000 −0.383522
\(980\) −6.00000 −0.191663
\(981\) −114.000 −3.63974
\(982\) 5.00000 0.159556
\(983\) 31.0000 0.988746 0.494373 0.869250i \(-0.335398\pi\)
0.494373 + 0.869250i \(0.335398\pi\)
\(984\) 0 0
\(985\) −9.00000 −0.286764
\(986\) 6.00000 0.191079
\(987\) −39.0000 −1.24138
\(988\) 0 0
\(989\) 20.0000 0.635963
\(990\) −12.0000 −0.381385
\(991\) −30.0000 −0.952981 −0.476491 0.879180i \(-0.658091\pi\)
−0.476491 + 0.879180i \(0.658091\pi\)
\(992\) 4.00000 0.127000
\(993\) −12.0000 −0.380808
\(994\) 5.00000 0.158590
\(995\) −10.0000 −0.317021
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −32.0000 −1.01294
\(999\) 27.0000 0.854242
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 338.2.a.a.1.1 1
3.2 odd 2 3042.2.a.l.1.1 1
4.3 odd 2 2704.2.a.n.1.1 1
5.4 even 2 8450.2.a.y.1.1 1
13.2 odd 12 338.2.e.d.147.1 4
13.3 even 3 338.2.c.g.191.1 2
13.4 even 6 338.2.c.c.315.1 2
13.5 odd 4 338.2.b.a.337.2 2
13.6 odd 12 338.2.e.d.23.2 4
13.7 odd 12 338.2.e.d.23.1 4
13.8 odd 4 338.2.b.a.337.1 2
13.9 even 3 338.2.c.g.315.1 2
13.10 even 6 338.2.c.c.191.1 2
13.11 odd 12 338.2.e.d.147.2 4
13.12 even 2 26.2.a.b.1.1 1
39.5 even 4 3042.2.b.f.1351.1 2
39.8 even 4 3042.2.b.f.1351.2 2
39.38 odd 2 234.2.a.b.1.1 1
52.31 even 4 2704.2.f.j.337.1 2
52.47 even 4 2704.2.f.j.337.2 2
52.51 odd 2 208.2.a.d.1.1 1
65.12 odd 4 650.2.b.a.599.2 2
65.38 odd 4 650.2.b.a.599.1 2
65.64 even 2 650.2.a.g.1.1 1
91.12 odd 6 1274.2.f.a.1145.1 2
91.25 even 6 1274.2.f.l.79.1 2
91.38 odd 6 1274.2.f.a.79.1 2
91.51 even 6 1274.2.f.l.1145.1 2
91.90 odd 2 1274.2.a.o.1.1 1
104.51 odd 2 832.2.a.a.1.1 1
104.77 even 2 832.2.a.j.1.1 1
117.25 even 6 2106.2.e.h.1405.1 2
117.38 odd 6 2106.2.e.t.1405.1 2
117.77 odd 6 2106.2.e.t.703.1 2
117.103 even 6 2106.2.e.h.703.1 2
143.142 odd 2 3146.2.a.a.1.1 1
156.155 even 2 1872.2.a.m.1.1 1
195.38 even 4 5850.2.e.v.5149.2 2
195.77 even 4 5850.2.e.v.5149.1 2
195.194 odd 2 5850.2.a.bn.1.1 1
208.51 odd 4 3328.2.b.k.1665.2 2
208.77 even 4 3328.2.b.g.1665.1 2
208.155 odd 4 3328.2.b.k.1665.1 2
208.181 even 4 3328.2.b.g.1665.2 2
221.220 even 2 7514.2.a.i.1.1 1
247.246 odd 2 9386.2.a.f.1.1 1
260.259 odd 2 5200.2.a.c.1.1 1
312.77 odd 2 7488.2.a.w.1.1 1
312.155 even 2 7488.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.a.b.1.1 1 13.12 even 2
208.2.a.d.1.1 1 52.51 odd 2
234.2.a.b.1.1 1 39.38 odd 2
338.2.a.a.1.1 1 1.1 even 1 trivial
338.2.b.a.337.1 2 13.8 odd 4
338.2.b.a.337.2 2 13.5 odd 4
338.2.c.c.191.1 2 13.10 even 6
338.2.c.c.315.1 2 13.4 even 6
338.2.c.g.191.1 2 13.3 even 3
338.2.c.g.315.1 2 13.9 even 3
338.2.e.d.23.1 4 13.7 odd 12
338.2.e.d.23.2 4 13.6 odd 12
338.2.e.d.147.1 4 13.2 odd 12
338.2.e.d.147.2 4 13.11 odd 12
650.2.a.g.1.1 1 65.64 even 2
650.2.b.a.599.1 2 65.38 odd 4
650.2.b.a.599.2 2 65.12 odd 4
832.2.a.a.1.1 1 104.51 odd 2
832.2.a.j.1.1 1 104.77 even 2
1274.2.a.o.1.1 1 91.90 odd 2
1274.2.f.a.79.1 2 91.38 odd 6
1274.2.f.a.1145.1 2 91.12 odd 6
1274.2.f.l.79.1 2 91.25 even 6
1274.2.f.l.1145.1 2 91.51 even 6
1872.2.a.m.1.1 1 156.155 even 2
2106.2.e.h.703.1 2 117.103 even 6
2106.2.e.h.1405.1 2 117.25 even 6
2106.2.e.t.703.1 2 117.77 odd 6
2106.2.e.t.1405.1 2 117.38 odd 6
2704.2.a.n.1.1 1 4.3 odd 2
2704.2.f.j.337.1 2 52.31 even 4
2704.2.f.j.337.2 2 52.47 even 4
3042.2.a.l.1.1 1 3.2 odd 2
3042.2.b.f.1351.1 2 39.5 even 4
3042.2.b.f.1351.2 2 39.8 even 4
3146.2.a.a.1.1 1 143.142 odd 2
3328.2.b.g.1665.1 2 208.77 even 4
3328.2.b.g.1665.2 2 208.181 even 4
3328.2.b.k.1665.1 2 208.155 odd 4
3328.2.b.k.1665.2 2 208.51 odd 4
5200.2.a.c.1.1 1 260.259 odd 2
5850.2.a.bn.1.1 1 195.194 odd 2
5850.2.e.v.5149.1 2 195.77 even 4
5850.2.e.v.5149.2 2 195.38 even 4
7488.2.a.v.1.1 1 312.155 even 2
7488.2.a.w.1.1 1 312.77 odd 2
7514.2.a.i.1.1 1 221.220 even 2
8450.2.a.y.1.1 1 5.4 even 2
9386.2.a.f.1.1 1 247.246 odd 2