Properties

Label 338.2.a.d.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -3.00000 q^{10} -1.00000 q^{12} -3.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} -6.00000 q^{19} -3.00000 q^{20} +3.00000 q^{21} +6.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +5.00000 q^{27} -3.00000 q^{28} +3.00000 q^{30} +1.00000 q^{32} -3.00000 q^{34} +9.00000 q^{35} -2.00000 q^{36} -3.00000 q^{37} -6.00000 q^{38} -3.00000 q^{40} +3.00000 q^{42} +1.00000 q^{43} +6.00000 q^{45} +6.00000 q^{46} -3.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +4.00000 q^{50} +3.00000 q^{51} -6.00000 q^{53} +5.00000 q^{54} -3.00000 q^{56} +6.00000 q^{57} +6.00000 q^{59} +3.00000 q^{60} -8.00000 q^{61} +6.00000 q^{63} +1.00000 q^{64} -12.0000 q^{67} -3.00000 q^{68} -6.00000 q^{69} +9.00000 q^{70} +15.0000 q^{71} -2.00000 q^{72} -6.00000 q^{73} -3.00000 q^{74} -4.00000 q^{75} -6.00000 q^{76} +10.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +6.00000 q^{83} +3.00000 q^{84} +9.00000 q^{85} +1.00000 q^{86} +6.00000 q^{89} +6.00000 q^{90} +6.00000 q^{92} -3.00000 q^{94} +18.0000 q^{95} -1.00000 q^{96} -12.0000 q^{97} +2.00000 q^{98} +O(q^{100})\)

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 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) −1.00000 −0.408248
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) −3.00000 −0.948683
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −3.00000 −0.801784
\(15\) 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 \(0.618522\pi\)
\(18\) −2.00000 −0.471405
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −3.00000 −0.670820
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) −3.00000 −0.566947
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 3.00000 0.547723
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 9.00000 1.52128
\(36\) −2.00000 −0.333333
\(37\) −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
\(40\) −3.00000 −0.474342
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 3.00000 0.462910
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 6.00000 0.884652
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) 4.00000 0.565685
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 3.00000 0.387298
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −3.00000 −0.363803
\(69\) −6.00000 −0.722315
\(70\) 9.00000 1.07571
\(71\) 15.0000 1.78017 0.890086 0.455792i \(-0.150644\pi\)
0.890086 + 0.455792i \(0.150644\pi\)
\(72\) −2.00000 −0.235702
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −3.00000 −0.348743
\(75\) −4.00000 −0.461880
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 3.00000 0.327327
\(85\) 9.00000 0.976187
\(86\) 1.00000 0.107833
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 6.00000 0.632456
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) −3.00000 −0.309426
\(95\) 18.0000 1.84676
\(96\) −1.00000 −0.102062
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 2.00000 0.202031
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 3.00000 0.297044
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) −9.00000 −0.878310
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 5.00000 0.481125
\(109\) 9.00000 0.862044 0.431022 0.902342i \(-0.358153\pi\)
0.431022 + 0.902342i \(0.358153\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) −3.00000 −0.283473
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 6.00000 0.561951
\(115\) −18.0000 −1.67851
\(116\) 0 0
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 9.00000 0.825029
\(120\) 3.00000 0.273861
\(121\) −11.0000 −1.00000
\(122\) −8.00000 −0.724286
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 6.00000 0.534522
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) −3.00000 −0.262111 −0.131056 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) 0 0
\(133\) 18.0000 1.56080
\(134\) −12.0000 −1.03664
\(135\) −15.0000 −1.29099
\(136\) −3.00000 −0.257248
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) −6.00000 −0.510754
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 9.00000 0.760639
\(141\) 3.00000 0.252646
\(142\) 15.0000 1.25877
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) −2.00000 −0.164957
\(148\) −3.00000 −0.246598
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −4.00000 −0.326599
\(151\) 15.0000 1.22068 0.610341 0.792139i \(-0.291032\pi\)
0.610341 + 0.792139i \(0.291032\pi\)
\(152\) −6.00000 −0.486664
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 10.0000 0.795557
\(159\) 6.00000 0.475831
\(160\) −3.00000 −0.237171
\(161\) −18.0000 −1.41860
\(162\) 1.00000 0.0785674
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 3.00000 0.231455
\(169\) 0 0
\(170\) 9.00000 0.690268
\(171\) 12.0000 0.917663
\(172\) 1.00000 0.0762493
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −12.0000 −0.907115
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 6.00000 0.449719
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 6.00000 0.447214
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 6.00000 0.442326
\(185\) 9.00000 0.661693
\(186\) 0 0
\(187\) 0 0
\(188\) −3.00000 −0.218797
\(189\) −15.0000 −1.09109
\(190\) 18.0000 1.30586
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 3.00000 0.213741 0.106871 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 4.00000 0.282843
\(201\) 12.0000 0.846415
\(202\) −12.0000 −0.844317
\(203\) 0 0
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) −12.0000 −0.834058
\(208\) 0 0
\(209\) 0 0
\(210\) −9.00000 −0.621059
\(211\) −23.0000 −1.58339 −0.791693 0.610920i \(-0.790800\pi\)
−0.791693 + 0.610920i \(0.790800\pi\)
\(212\) −6.00000 −0.412082
\(213\) −15.0000 −1.02778
\(214\) −12.0000 −0.820303
\(215\) −3.00000 −0.204598
\(216\) 5.00000 0.340207
\(217\) 0 0
\(218\) 9.00000 0.609557
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 0 0
\(222\) 3.00000 0.201347
\(223\) −9.00000 −0.602685 −0.301342 0.953516i \(-0.597435\pi\)
−0.301342 + 0.953516i \(0.597435\pi\)
\(224\) −3.00000 −0.200446
\(225\) −8.00000 −0.533333
\(226\) −6.00000 −0.399114
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 6.00000 0.397360
\(229\) −9.00000 −0.594737 −0.297368 0.954763i \(-0.596109\pi\)
−0.297368 + 0.954763i \(0.596109\pi\)
\(230\) −18.0000 −1.18688
\(231\) 0 0
\(232\) 0 0
\(233\) 21.0000 1.37576 0.687878 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 6.00000 0.390567
\(237\) −10.0000 −0.649570
\(238\) 9.00000 0.583383
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 3.00000 0.193649
\(241\) 30.0000 1.93247 0.966235 0.257663i \(-0.0829523\pi\)
0.966235 + 0.257663i \(0.0829523\pi\)
\(242\) −11.0000 −0.707107
\(243\) −16.0000 −1.02640
\(244\) −8.00000 −0.512148
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 3.00000 0.189737
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 6.00000 0.377964
\(253\) 0 0
\(254\) 2.00000 0.125491
\(255\) −9.00000 −0.563602
\(256\) 1.00000 0.0625000
\(257\) −3.00000 −0.187135 −0.0935674 0.995613i \(-0.529827\pi\)
−0.0935674 + 0.995613i \(0.529827\pi\)
\(258\) −1.00000 −0.0622573
\(259\) 9.00000 0.559233
\(260\) 0 0
\(261\) 0 0
\(262\) −3.00000 −0.185341
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 18.0000 1.10573
\(266\) 18.0000 1.10365
\(267\) −6.00000 −0.367194
\(268\) −12.0000 −0.733017
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) −15.0000 −0.912871
\(271\) −15.0000 −0.911185 −0.455593 0.890188i \(-0.650573\pi\)
−0.455593 + 0.890188i \(0.650573\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 5.00000 0.299880
\(279\) 0 0
\(280\) 9.00000 0.537853
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 3.00000 0.178647
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 15.0000 0.890086
\(285\) −18.0000 −1.06623
\(286\) 0 0
\(287\) 0 0
\(288\) −2.00000 −0.117851
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) −6.00000 −0.351123
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) −2.00000 −0.116642
\(295\) −18.0000 −1.04800
\(296\) −3.00000 −0.174371
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) −4.00000 −0.230940
\(301\) −3.00000 −0.172917
\(302\) 15.0000 0.863153
\(303\) 12.0000 0.689382
\(304\) −6.00000 −0.344124
\(305\) 24.0000 1.37424
\(306\) 6.00000 0.342997
\(307\) −18.0000 −1.02731 −0.513657 0.857996i \(-0.671710\pi\)
−0.513657 + 0.857996i \(0.671710\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 19.0000 1.07394 0.536972 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) −22.0000 −1.24153
\(315\) −18.0000 −1.01419
\(316\) 10.0000 0.562544
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) −3.00000 −0.167705
\(321\) 12.0000 0.669775
\(322\) −18.0000 −1.00310
\(323\) 18.0000 1.00155
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −6.00000 −0.332309
\(327\) −9.00000 −0.497701
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) 30.0000 1.64895 0.824475 0.565899i \(-0.191471\pi\)
0.824475 + 0.565899i \(0.191471\pi\)
\(332\) 6.00000 0.329293
\(333\) 6.00000 0.328798
\(334\) 12.0000 0.656611
\(335\) 36.0000 1.96689
\(336\) 3.00000 0.163663
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 9.00000 0.488094
\(341\) 0 0
\(342\) 12.0000 0.648886
\(343\) 15.0000 0.809924
\(344\) 1.00000 0.0539164
\(345\) 18.0000 0.969087
\(346\) 6.00000 0.322562
\(347\) 33.0000 1.77153 0.885766 0.464131i \(-0.153633\pi\)
0.885766 + 0.464131i \(0.153633\pi\)
\(348\) 0 0
\(349\) 21.0000 1.12410 0.562052 0.827102i \(-0.310012\pi\)
0.562052 + 0.827102i \(0.310012\pi\)
\(350\) −12.0000 −0.641427
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −6.00000 −0.318896
\(355\) −45.0000 −2.38835
\(356\) 6.00000 0.317999
\(357\) −9.00000 −0.476331
\(358\) −15.0000 −0.792775
\(359\) −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
\(362\) −2.00000 −0.105118
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 18.0000 0.942163
\(366\) 8.00000 0.418167
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) 9.00000 0.467888
\(371\) 18.0000 0.934513
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) −3.00000 −0.154713
\(377\) 0 0
\(378\) −15.0000 −0.771517
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) 18.0000 0.923381
\(381\) −2.00000 −0.102463
\(382\) 12.0000 0.613973
\(383\) −9.00000 −0.459879 −0.229939 0.973205i \(-0.573853\pi\)
−0.229939 + 0.973205i \(0.573853\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) −2.00000 −0.101666
\(388\) −12.0000 −0.609208
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 2.00000 0.101015
\(393\) 3.00000 0.151330
\(394\) 3.00000 0.151138
\(395\) −30.0000 −1.50946
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 20.0000 1.00251
\(399\) −18.0000 −0.901127
\(400\) 4.00000 0.200000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 12.0000 0.598506
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) 0 0
\(408\) 3.00000 0.148522
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) −14.0000 −0.689730
\(413\) −18.0000 −0.885722
\(414\) −12.0000 −0.589768
\(415\) −18.0000 −0.883585
\(416\) 0 0
\(417\) −5.00000 −0.244851
\(418\) 0 0
\(419\) 15.0000 0.732798 0.366399 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) −9.00000 −0.439155
\(421\) −15.0000 −0.731055 −0.365528 0.930800i \(-0.619111\pi\)
−0.365528 + 0.930800i \(0.619111\pi\)
\(422\) −23.0000 −1.11962
\(423\) 6.00000 0.291730
\(424\) −6.00000 −0.291386
\(425\) −12.0000 −0.582086
\(426\) −15.0000 −0.726752
\(427\) 24.0000 1.16144
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −3.00000 −0.144673
\(431\) −15.0000 −0.722525 −0.361262 0.932464i \(-0.617654\pi\)
−0.361262 + 0.932464i \(0.617654\pi\)
\(432\) 5.00000 0.240563
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 9.00000 0.431022
\(437\) −36.0000 −1.72211
\(438\) 6.00000 0.286691
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) −21.0000 −0.997740 −0.498870 0.866677i \(-0.666252\pi\)
−0.498870 + 0.866677i \(0.666252\pi\)
\(444\) 3.00000 0.142374
\(445\) −18.0000 −0.853282
\(446\) −9.00000 −0.426162
\(447\) 6.00000 0.283790
\(448\) −3.00000 −0.141737
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) −8.00000 −0.377124
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) −15.0000 −0.704761
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −9.00000 −0.420542
\(459\) −15.0000 −0.700140
\(460\) −18.0000 −0.839254
\(461\) −15.0000 −0.698620 −0.349310 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 21.0000 0.972806
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 36.0000 1.66233
\(470\) 9.00000 0.415139
\(471\) 22.0000 1.01371
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) −10.0000 −0.459315
\(475\) −24.0000 −1.10120
\(476\) 9.00000 0.412514
\(477\) 12.0000 0.549442
\(478\) 9.00000 0.411650
\(479\) −39.0000 −1.78196 −0.890978 0.454047i \(-0.849980\pi\)
−0.890978 + 0.454047i \(0.849980\pi\)
\(480\) 3.00000 0.136931
\(481\) 0 0
\(482\) 30.0000 1.36646
\(483\) 18.0000 0.819028
\(484\) −11.0000 −0.500000
\(485\) 36.0000 1.63468
\(486\) −16.0000 −0.725775
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) −8.00000 −0.362143
\(489\) 6.00000 0.271329
\(490\) −6.00000 −0.271052
\(491\) −27.0000 −1.21849 −0.609246 0.792981i \(-0.708528\pi\)
−0.609246 + 0.792981i \(0.708528\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −45.0000 −2.01853
\(498\) −6.00000 −0.268866
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 3.00000 0.134164
\(501\) −12.0000 −0.536120
\(502\) −12.0000 −0.535586
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 6.00000 0.267261
\(505\) 36.0000 1.60198
\(506\) 0 0
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) −9.00000 −0.398527
\(511\) 18.0000 0.796273
\(512\) 1.00000 0.0441942
\(513\) −30.0000 −1.32453
\(514\) −3.00000 −0.132324
\(515\) 42.0000 1.85074
\(516\) −1.00000 −0.0440225
\(517\) 0 0
\(518\) 9.00000 0.395437
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 27.0000 1.18289 0.591446 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −3.00000 −0.131056
\(525\) 12.0000 0.523723
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 18.0000 0.781870
\(531\) −12.0000 −0.520756
\(532\) 18.0000 0.780399
\(533\) 0 0
\(534\) −6.00000 −0.259645
\(535\) 36.0000 1.55642
\(536\) −12.0000 −0.518321
\(537\) 15.0000 0.647298
\(538\) 0 0
\(539\) 0 0
\(540\) −15.0000 −0.645497
\(541\) 15.0000 0.644900 0.322450 0.946586i \(-0.395494\pi\)
0.322450 + 0.946586i \(0.395494\pi\)
\(542\) −15.0000 −0.644305
\(543\) 2.00000 0.0858282
\(544\) −3.00000 −0.128624
\(545\) −27.0000 −1.15655
\(546\) 0 0
\(547\) −37.0000 −1.58201 −0.791003 0.611812i \(-0.790441\pi\)
−0.791003 + 0.611812i \(0.790441\pi\)
\(548\) −18.0000 −0.768922
\(549\) 16.0000 0.682863
\(550\) 0 0
\(551\) 0 0
\(552\) −6.00000 −0.255377
\(553\) −30.0000 −1.27573
\(554\) −8.00000 −0.339887
\(555\) −9.00000 −0.382029
\(556\) 5.00000 0.212047
\(557\) 27.0000 1.14403 0.572013 0.820244i \(-0.306163\pi\)
0.572013 + 0.820244i \(0.306163\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 9.00000 0.380319
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) −39.0000 −1.64365 −0.821827 0.569737i \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) 3.00000 0.126323
\(565\) 18.0000 0.757266
\(566\) −4.00000 −0.168133
\(567\) −3.00000 −0.125988
\(568\) 15.0000 0.629386
\(569\) −45.0000 −1.88650 −0.943249 0.332086i \(-0.892248\pi\)
−0.943249 + 0.332086i \(0.892248\pi\)
\(570\) −18.0000 −0.753937
\(571\) 23.0000 0.962520 0.481260 0.876578i \(-0.340179\pi\)
0.481260 + 0.876578i \(0.340179\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) −2.00000 −0.0833333
\(577\) −42.0000 −1.74848 −0.874241 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) −8.00000 −0.332756
\(579\) 6.00000 0.249351
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) 12.0000 0.497416
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 0 0
\(590\) −18.0000 −0.741048
\(591\) −3.00000 −0.123404
\(592\) −3.00000 −0.123299
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 0 0
\(595\) −27.0000 −1.10689
\(596\) −6.00000 −0.245770
\(597\) −20.0000 −0.818546
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) −4.00000 −0.163299
\(601\) 37.0000 1.50926 0.754631 0.656150i \(-0.227816\pi\)
0.754631 + 0.656150i \(0.227816\pi\)
\(602\) −3.00000 −0.122271
\(603\) 24.0000 0.977356
\(604\) 15.0000 0.610341
\(605\) 33.0000 1.34164
\(606\) 12.0000 0.487467
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) −6.00000 −0.243332
\(609\) 0 0
\(610\) 24.0000 0.971732
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) −18.0000 −0.726421
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 14.0000 0.563163
\(619\) −24.0000 −0.964641 −0.482321 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) 30.0000 1.20386
\(622\) 18.0000 0.721734
\(623\) −18.0000 −0.721155
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 19.0000 0.759393
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) 9.00000 0.358854
\(630\) −18.0000 −0.717137
\(631\) 15.0000 0.597141 0.298570 0.954388i \(-0.403490\pi\)
0.298570 + 0.954388i \(0.403490\pi\)
\(632\) 10.0000 0.397779
\(633\) 23.0000 0.914168
\(634\) 18.0000 0.714871
\(635\) −6.00000 −0.238103
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) −30.0000 −1.18678
\(640\) −3.00000 −0.118585
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 12.0000 0.473602
\(643\) 36.0000 1.41970 0.709851 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) −18.0000 −0.709299
\(645\) 3.00000 0.118125
\(646\) 18.0000 0.708201
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −6.00000 −0.234978
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) −9.00000 −0.351928
\(655\) 9.00000 0.351659
\(656\) 0 0
\(657\) 12.0000 0.468165
\(658\) 9.00000 0.350857
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 30.0000 1.16598
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) −54.0000 −2.09403
\(666\) 6.00000 0.232495
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) 9.00000 0.347960
\(670\) 36.0000 1.39080
\(671\) 0 0
\(672\) 3.00000 0.115728
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) −13.0000 −0.500741
\(675\) 20.0000 0.769800
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 6.00000 0.230429
\(679\) 36.0000 1.38155
\(680\) 9.00000 0.345134
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −6.00000 −0.229584 −0.114792 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(684\) 12.0000 0.458831
\(685\) 54.0000 2.06323
\(686\) 15.0000 0.572703
\(687\) 9.00000 0.343371
\(688\) 1.00000 0.0381246
\(689\) 0 0
\(690\) 18.0000 0.685248
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 33.0000 1.25266
\(695\) −15.0000 −0.568982
\(696\) 0 0
\(697\) 0 0
\(698\) 21.0000 0.794862
\(699\) −21.0000 −0.794293
\(700\) −12.0000 −0.453557
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 18.0000 0.678883
\(704\) 0 0
\(705\) −9.00000 −0.338960
\(706\) 6.00000 0.225813
\(707\) 36.0000 1.35392
\(708\) −6.00000 −0.225494
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −45.0000 −1.68882
\(711\) −20.0000 −0.750059
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) −9.00000 −0.336817
\(715\) 0 0
\(716\) −15.0000 −0.560576
\(717\) −9.00000 −0.336111
\(718\) −24.0000 −0.895672
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 6.00000 0.223607
\(721\) 42.0000 1.56416
\(722\) 17.0000 0.632674
\(723\) −30.0000 −1.11571
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 11.0000 0.408248
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 18.0000 0.666210
\(731\) −3.00000 −0.110959
\(732\) 8.00000 0.295689
\(733\) −9.00000 −0.332423 −0.166211 0.986090i \(-0.553153\pi\)
−0.166211 + 0.986090i \(0.553153\pi\)
\(734\) 8.00000 0.295285
\(735\) 6.00000 0.221313
\(736\) 6.00000 0.221163
\(737\) 0 0
\(738\) 0 0
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 9.00000 0.330847
\(741\) 0 0
\(742\) 18.0000 0.660801
\(743\) −39.0000 −1.43077 −0.715386 0.698730i \(-0.753749\pi\)
−0.715386 + 0.698730i \(0.753749\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 4.00000 0.146450
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 36.0000 1.31541
\(750\) −3.00000 −0.109545
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −3.00000 −0.109399
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) −45.0000 −1.63772
\(756\) −15.0000 −0.545545
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −6.00000 −0.217930
\(759\) 0 0
\(760\) 18.0000 0.652929
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) −2.00000 −0.0724524
\(763\) −27.0000 −0.977466
\(764\) 12.0000 0.434145
\(765\) −18.0000 −0.650791
\(766\) −9.00000 −0.325183
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 24.0000 0.865462 0.432731 0.901523i \(-0.357550\pi\)
0.432731 + 0.901523i \(0.357550\pi\)
\(770\) 0 0
\(771\) 3.00000 0.108042
\(772\) −6.00000 −0.215945
\(773\) 21.0000 0.755318 0.377659 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) −12.0000 −0.430775
\(777\) −9.00000 −0.322873
\(778\) 30.0000 1.07555
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −18.0000 −0.643679
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) 66.0000 2.35564
\(786\) 3.00000 0.107006
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 3.00000 0.106871
\(789\) −24.0000 −0.854423
\(790\) −30.0000 −1.06735
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 0 0
\(794\) −18.0000 −0.638796
\(795\) −18.0000 −0.638394
\(796\) 20.0000 0.708881
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) −18.0000 −0.637193
\(799\) 9.00000 0.318397
\(800\) 4.00000 0.141421
\(801\) −12.0000 −0.423999
\(802\) −30.0000 −1.05934
\(803\) 0 0
\(804\) 12.0000 0.423207
\(805\) 54.0000 1.90325
\(806\) 0 0
\(807\) 0 0
\(808\) −12.0000 −0.422159
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) −3.00000 −0.105409
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 15.0000 0.526073
\(814\) 0 0
\(815\) 18.0000 0.630512
\(816\) 3.00000 0.105021
\(817\) −6.00000 −0.209913
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) 0 0
\(821\) 45.0000 1.57051 0.785255 0.619172i \(-0.212532\pi\)
0.785255 + 0.619172i \(0.212532\pi\)
\(822\) 18.0000 0.627822
\(823\) −14.0000 −0.488009 −0.244005 0.969774i \(-0.578461\pi\)
−0.244005 + 0.969774i \(0.578461\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) −18.0000 −0.626300
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) −12.0000 −0.417029
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) −18.0000 −0.624789
\(831\) 8.00000 0.277517
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) −5.00000 −0.173136
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) 0 0
\(838\) 15.0000 0.518166
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) −9.00000 −0.310530
\(841\) −29.0000 −1.00000
\(842\) −15.0000 −0.516934
\(843\) −30.0000 −1.03325
\(844\) −23.0000 −0.791693
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) 33.0000 1.13389
\(848\) −6.00000 −0.206041
\(849\) 4.00000 0.137280
\(850\) −12.0000 −0.411597
\(851\) −18.0000 −0.617032
\(852\) −15.0000 −0.513892
\(853\) 39.0000 1.33533 0.667667 0.744460i \(-0.267293\pi\)
0.667667 + 0.744460i \(0.267293\pi\)
\(854\) 24.0000 0.821263
\(855\) −36.0000 −1.23117
\(856\) −12.0000 −0.410152
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) −3.00000 −0.102299
\(861\) 0 0
\(862\) −15.0000 −0.510902
\(863\) −39.0000 −1.32758 −0.663788 0.747921i \(-0.731052\pi\)
−0.663788 + 0.747921i \(0.731052\pi\)
\(864\) 5.00000 0.170103
\(865\) −18.0000 −0.612018
\(866\) 11.0000 0.373795
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 9.00000 0.304778
\(873\) 24.0000 0.812277
\(874\) −36.0000 −1.21772
\(875\) −9.00000 −0.304256
\(876\) 6.00000 0.202721
\(877\) 3.00000 0.101303 0.0506514 0.998716i \(-0.483870\pi\)
0.0506514 + 0.998716i \(0.483870\pi\)
\(878\) 10.0000 0.337484
\(879\) −9.00000 −0.303562
\(880\) 0 0
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) −4.00000 −0.134687
\(883\) 11.0000 0.370179 0.185090 0.982722i \(-0.440742\pi\)
0.185090 + 0.982722i \(0.440742\pi\)
\(884\) 0 0
\(885\) 18.0000 0.605063
\(886\) −21.0000 −0.705509
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 3.00000 0.100673
\(889\) −6.00000 −0.201234
\(890\) −18.0000 −0.603361
\(891\) 0 0
\(892\) −9.00000 −0.301342
\(893\) 18.0000 0.602347
\(894\) 6.00000 0.200670
\(895\) 45.0000 1.50418
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) −24.0000 −0.800890
\(899\) 0 0
\(900\) −8.00000 −0.266667
\(901\) 18.0000 0.599667
\(902\) 0 0
\(903\) 3.00000 0.0998337
\(904\) −6.00000 −0.199557
\(905\) 6.00000 0.199447
\(906\) −15.0000 −0.498342
\(907\) 17.0000 0.564476 0.282238 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) −12.0000 −0.398234
\(909\) 24.0000 0.796030
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 6.00000 0.198680
\(913\) 0 0
\(914\) 18.0000 0.595387
\(915\) −24.0000 −0.793416
\(916\) −9.00000 −0.297368
\(917\) 9.00000 0.297206
\(918\) −15.0000 −0.495074
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) −18.0000 −0.593442
\(921\) 18.0000 0.593120
\(922\) −15.0000 −0.493999
\(923\) 0 0
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) 24.0000 0.788689
\(927\) 28.0000 0.919641
\(928\) 0 0
\(929\) −36.0000 −1.18112 −0.590561 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 21.0000 0.687878
\(933\) −18.0000 −0.589294
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 36.0000 1.17544
\(939\) −19.0000 −0.620042
\(940\) 9.00000 0.293548
\(941\) −45.0000 −1.46696 −0.733479 0.679712i \(-0.762105\pi\)
−0.733479 + 0.679712i \(0.762105\pi\)
\(942\) 22.0000 0.716799
\(943\) 0 0
\(944\) 6.00000 0.195283
\(945\) 45.0000 1.46385
\(946\) 0 0
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) −10.0000 −0.324785
\(949\) 0 0
\(950\) −24.0000 −0.778663
\(951\) −18.0000 −0.583690
\(952\) 9.00000 0.291692
\(953\) −9.00000 −0.291539 −0.145769 0.989319i \(-0.546566\pi\)
−0.145769 + 0.989319i \(0.546566\pi\)
\(954\) 12.0000 0.388514
\(955\) −36.0000 −1.16493
\(956\) 9.00000 0.291081
\(957\) 0 0
\(958\) −39.0000 −1.26003
\(959\) 54.0000 1.74375
\(960\) 3.00000 0.0968246
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 24.0000 0.773389
\(964\) 30.0000 0.966235
\(965\) 18.0000 0.579441
\(966\) 18.0000 0.579141
\(967\) 3.00000 0.0964735 0.0482367 0.998836i \(-0.484640\pi\)
0.0482367 + 0.998836i \(0.484640\pi\)
\(968\) −11.0000 −0.353553
\(969\) −18.0000 −0.578243
\(970\) 36.0000 1.15589
\(971\) 27.0000 0.866471 0.433236 0.901281i \(-0.357372\pi\)
0.433236 + 0.901281i \(0.357372\pi\)
\(972\) −16.0000 −0.513200
\(973\) −15.0000 −0.480878
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 6.00000 0.191859
\(979\) 0 0
\(980\) −6.00000 −0.191663
\(981\) −18.0000 −0.574696
\(982\) −27.0000 −0.861605
\(983\) 9.00000 0.287055 0.143528 0.989646i \(-0.454155\pi\)
0.143528 + 0.989646i \(0.454155\pi\)
\(984\) 0 0
\(985\) −9.00000 −0.286764
\(986\) 0 0
\(987\) −9.00000 −0.286473
\(988\) 0 0
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) 0 0
\(993\) −30.0000 −0.952021
\(994\) −45.0000 −1.42731
\(995\) −60.0000 −1.90213
\(996\) −6.00000 −0.190117
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) −36.0000 −1.13956
\(999\) −15.0000 −0.474579
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.d.1.1 1
3.2 odd 2 3042.2.a.g.1.1 1
4.3 odd 2 2704.2.a.j.1.1 1
5.4 even 2 8450.2.a.h.1.1 1
13.2 odd 12 338.2.e.c.147.2 4
13.3 even 3 338.2.c.b.191.1 2
13.4 even 6 338.2.c.f.315.1 2
13.5 odd 4 26.2.b.a.25.1 2
13.6 odd 12 338.2.e.c.23.1 4
13.7 odd 12 338.2.e.c.23.2 4
13.8 odd 4 26.2.b.a.25.2 yes 2
13.9 even 3 338.2.c.b.315.1 2
13.10 even 6 338.2.c.f.191.1 2
13.11 odd 12 338.2.e.c.147.1 4
13.12 even 2 338.2.a.b.1.1 1
39.5 even 4 234.2.b.b.181.2 2
39.8 even 4 234.2.b.b.181.1 2
39.38 odd 2 3042.2.a.j.1.1 1
52.31 even 4 208.2.f.a.129.2 2
52.47 even 4 208.2.f.a.129.1 2
52.51 odd 2 2704.2.a.k.1.1 1
65.8 even 4 650.2.c.d.649.1 2
65.18 even 4 650.2.c.a.649.1 2
65.34 odd 4 650.2.d.b.51.1 2
65.44 odd 4 650.2.d.b.51.2 2
65.47 even 4 650.2.c.a.649.2 2
65.57 even 4 650.2.c.d.649.2 2
65.64 even 2 8450.2.a.u.1.1 1
91.5 even 12 1274.2.n.c.753.1 4
91.18 odd 12 1274.2.n.d.961.2 4
91.31 even 12 1274.2.n.c.961.2 4
91.34 even 4 1274.2.d.c.883.2 2
91.44 odd 12 1274.2.n.d.753.1 4
91.47 even 12 1274.2.n.c.753.2 4
91.60 odd 12 1274.2.n.d.961.1 4
91.73 even 12 1274.2.n.c.961.1 4
91.83 even 4 1274.2.d.c.883.1 2
91.86 odd 12 1274.2.n.d.753.2 4
104.5 odd 4 832.2.f.d.129.1 2
104.21 odd 4 832.2.f.d.129.2 2
104.83 even 4 832.2.f.b.129.1 2
104.99 even 4 832.2.f.b.129.2 2
156.47 odd 4 1872.2.c.f.1585.2 2
156.83 odd 4 1872.2.c.f.1585.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.b.a.25.1 2 13.5 odd 4
26.2.b.a.25.2 yes 2 13.8 odd 4
208.2.f.a.129.1 2 52.47 even 4
208.2.f.a.129.2 2 52.31 even 4
234.2.b.b.181.1 2 39.8 even 4
234.2.b.b.181.2 2 39.5 even 4
338.2.a.b.1.1 1 13.12 even 2
338.2.a.d.1.1 1 1.1 even 1 trivial
338.2.c.b.191.1 2 13.3 even 3
338.2.c.b.315.1 2 13.9 even 3
338.2.c.f.191.1 2 13.10 even 6
338.2.c.f.315.1 2 13.4 even 6
338.2.e.c.23.1 4 13.6 odd 12
338.2.e.c.23.2 4 13.7 odd 12
338.2.e.c.147.1 4 13.11 odd 12
338.2.e.c.147.2 4 13.2 odd 12
650.2.c.a.649.1 2 65.18 even 4
650.2.c.a.649.2 2 65.47 even 4
650.2.c.d.649.1 2 65.8 even 4
650.2.c.d.649.2 2 65.57 even 4
650.2.d.b.51.1 2 65.34 odd 4
650.2.d.b.51.2 2 65.44 odd 4
832.2.f.b.129.1 2 104.83 even 4
832.2.f.b.129.2 2 104.99 even 4
832.2.f.d.129.1 2 104.5 odd 4
832.2.f.d.129.2 2 104.21 odd 4
1274.2.d.c.883.1 2 91.83 even 4
1274.2.d.c.883.2 2 91.34 even 4
1274.2.n.c.753.1 4 91.5 even 12
1274.2.n.c.753.2 4 91.47 even 12
1274.2.n.c.961.1 4 91.73 even 12
1274.2.n.c.961.2 4 91.31 even 12
1274.2.n.d.753.1 4 91.44 odd 12
1274.2.n.d.753.2 4 91.86 odd 12
1274.2.n.d.961.1 4 91.60 odd 12
1274.2.n.d.961.2 4 91.18 odd 12
1872.2.c.f.1585.1 2 156.83 odd 4
1872.2.c.f.1585.2 2 156.47 odd 4
2704.2.a.j.1.1 1 4.3 odd 2
2704.2.a.k.1.1 1 52.51 odd 2
3042.2.a.g.1.1 1 3.2 odd 2
3042.2.a.j.1.1 1 39.38 odd 2
8450.2.a.h.1.1 1 5.4 even 2
8450.2.a.u.1.1 1 65.64 even 2