Properties

Label 208.2.f.a.129.2
Level $208$
Weight $2$
Character 208.129
Analytic conductor $1.661$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [208,2,Mod(129,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 208.f (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(1.66088836204\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 208.129
Dual form 208.2.f.a.129.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.00000i q^{5} +3.00000i q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.00000i q^{5} +3.00000i q^{7} -2.00000 q^{9} +(2.00000 - 3.00000i) q^{13} +3.00000i q^{15} +3.00000 q^{17} -6.00000i q^{19} +3.00000i q^{21} +6.00000 q^{23} -4.00000 q^{25} -5.00000 q^{27} -9.00000 q^{35} -3.00000i q^{37} +(2.00000 - 3.00000i) q^{39} +1.00000 q^{43} -6.00000i q^{45} +3.00000i q^{47} -2.00000 q^{49} +3.00000 q^{51} -6.00000 q^{53} -6.00000i q^{57} -6.00000i q^{59} -8.00000 q^{61} -6.00000i q^{63} +(9.00000 + 6.00000i) q^{65} -12.0000i q^{67} +6.00000 q^{69} +15.0000i q^{71} -6.00000i q^{73} -4.00000 q^{75} -10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{83} +9.00000i q^{85} +6.00000i q^{89} +(9.00000 + 6.00000i) q^{91} +18.0000 q^{95} +12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 4 q^{9} + 4 q^{13} + 6 q^{17} + 12 q^{23} - 8 q^{25} - 10 q^{27} - 18 q^{35} + 4 q^{39} + 2 q^{43} - 4 q^{49} + 6 q^{51} - 12 q^{53} - 16 q^{61} + 18 q^{65} + 12 q^{69} - 8 q^{75} - 20 q^{79} + 2 q^{81} + 18 q^{91} + 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 3.00000i 1.34164i 0.741620 + 0.670820i \(0.234058\pi\)
−0.741620 + 0.670820i \(0.765942\pi\)
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 2.00000 3.00000i 0.554700 0.832050i
\(14\) 0 0
\(15\) 3.00000i 0.774597i
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) 0 0
\(21\) 3.00000i 0.654654i
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.00000 −1.52128
\(36\) 0 0
\(37\) 3.00000i 0.493197i −0.969118 0.246598i \(-0.920687\pi\)
0.969118 0.246598i \(-0.0793129\pi\)
\(38\) 0 0
\(39\) 2.00000 3.00000i 0.320256 0.480384i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 6.00000i 0.894427i
\(46\) 0 0
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000i 0.794719i
\(58\) 0 0
\(59\) 6.00000i 0.781133i −0.920575 0.390567i \(-0.872279\pi\)
0.920575 0.390567i \(-0.127721\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 6.00000i 0.755929i
\(64\) 0 0
\(65\) 9.00000 + 6.00000i 1.11631 + 0.744208i
\(66\) 0 0
\(67\) 12.0000i 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 15.0000i 1.78017i 0.455792 + 0.890086i \(0.349356\pi\)
−0.455792 + 0.890086i \(0.650644\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 9.00000i 0.976187i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 0 0
\(91\) 9.00000 + 6.00000i 0.943456 + 0.628971i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 18.0000 1.84676
\(96\) 0 0
\(97\) 12.0000i 1.21842i 0.793011 + 0.609208i \(0.208512\pi\)
−0.793011 + 0.609208i \(0.791488\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −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
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 9.00000i 0.862044i −0.902342 0.431022i \(-0.858153\pi\)
0.902342 0.431022i \(-0.141847\pi\)
\(110\) 0 0
\(111\) 3.00000i 0.284747i
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 18.0000i 1.67851i
\(116\) 0 0
\(117\) −4.00000 + 6.00000i −0.369800 + 0.554700i
\(118\) 0 0
\(119\) 9.00000i 0.825029i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000i 0.268328i
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 0 0
\(133\) 18.0000 1.56080
\(134\) 0 0
\(135\) 15.0000i 1.29099i
\(136\) 0 0
\(137\) 18.0000i 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 3.00000i 0.252646i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) 0 0
\(151\) 15.0000i 1.22068i −0.792139 0.610341i \(-0.791032\pi\)
0.792139 0.610341i \(-0.208968\pi\)
\(152\) 0 0
\(153\) −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 \(0.841053\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 18.0000i 1.41860i
\(162\) 0 0
\(163\) 6.00000i 0.469956i 0.972001 + 0.234978i \(0.0755019\pi\)
−0.972001 + 0.234978i \(0.924498\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000i 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 0 0
\(169\) −5.00000 12.0000i −0.384615 0.923077i
\(170\) 0 0
\(171\) 12.0000i 0.917663i
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 12.0000i 0.907115i
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) 9.00000 0.661693
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 15.0000i 1.09109i
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 6.00000i 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) 0 0
\(195\) 9.00000 + 6.00000i 0.644503 + 0.429669i
\(196\) 0 0
\(197\) 3.00000i 0.213741i −0.994273 0.106871i \(-0.965917\pi\)
0.994273 0.106871i \(-0.0340831\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 12.0000i 0.846415i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −12.0000 −0.834058
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) 0 0
\(213\) 15.0000i 1.02778i
\(214\) 0 0
\(215\) 3.00000i 0.204598i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 6.00000i 0.405442i
\(220\) 0 0
\(221\) 6.00000 9.00000i 0.403604 0.605406i
\(222\) 0 0
\(223\) 9.00000i 0.602685i −0.953516 0.301342i \(-0.902565\pi\)
0.953516 0.301342i \(-0.0974347\pi\)
\(224\) 0 0
\(225\) 8.00000 0.533333
\(226\) 0 0
\(227\) 12.0000i 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) 9.00000i 0.594737i −0.954763 0.297368i \(-0.903891\pi\)
0.954763 0.297368i \(-0.0961089\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −21.0000 −1.37576 −0.687878 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 0 0
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) 9.00000i 0.582162i 0.956698 + 0.291081i \(0.0940149\pi\)
−0.956698 + 0.291081i \(0.905985\pi\)
\(240\) 0 0
\(241\) 30.0000i 1.93247i 0.257663 + 0.966235i \(0.417048\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 6.00000i 0.383326i
\(246\) 0 0
\(247\) −18.0000 12.0000i −1.14531 0.763542i
\(248\) 0 0
\(249\) 6.00000i 0.380235i
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 9.00000i 0.563602i
\(256\) 0 0
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 0 0
\(259\) 9.00000 0.559233
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 18.0000i 1.10573i
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 15.0000i 0.911185i 0.890188 + 0.455593i \(0.150573\pi\)
−0.890188 + 0.455593i \(0.849427\pi\)
\(272\) 0 0
\(273\) 9.00000 + 6.00000i 0.544705 + 0.363137i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 30.0000i 1.78965i 0.446417 + 0.894825i \(0.352700\pi\)
−0.446417 + 0.894825i \(0.647300\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 18.0000 1.06623
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 12.0000i 0.703452i
\(292\) 0 0
\(293\) 9.00000i 0.525786i 0.964825 + 0.262893i \(0.0846766\pi\)
−0.964825 + 0.262893i \(0.915323\pi\)
\(294\) 0 0
\(295\) 18.0000 1.04800
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000 18.0000i 0.693978 1.04097i
\(300\) 0 0
\(301\) 3.00000i 0.172917i
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) 0 0
\(305\) 24.0000i 1.37424i
\(306\) 0 0
\(307\) 18.0000i 1.02731i 0.857996 + 0.513657i \(0.171710\pi\)
−0.857996 + 0.513657i \(0.828290\pi\)
\(308\) 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 \(0.319568\pi\)
\(314\) 0 0
\(315\) 18.0000 1.01419
\(316\) 0 0
\(317\) 18.0000i 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 18.0000i 1.00155i
\(324\) 0 0
\(325\) −8.00000 + 12.0000i −0.443760 + 0.665640i
\(326\) 0 0
\(327\) 9.00000i 0.497701i
\(328\) 0 0
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) 30.0000i 1.64895i 0.565899 + 0.824475i \(0.308529\pi\)
−0.565899 + 0.824475i \(0.691471\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 36.0000 1.96689
\(336\) 0 0
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 18.0000i 0.969087i
\(346\) 0 0
\(347\) −33.0000 −1.77153 −0.885766 0.464131i \(-0.846367\pi\)
−0.885766 + 0.464131i \(0.846367\pi\)
\(348\) 0 0
\(349\) 21.0000i 1.12410i 0.827102 + 0.562052i \(0.189988\pi\)
−0.827102 + 0.562052i \(0.810012\pi\)
\(350\) 0 0
\(351\) −10.0000 + 15.0000i −0.533761 + 0.800641i
\(352\) 0 0
\(353\) 6.00000i 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) −45.0000 −2.38835
\(356\) 0 0
\(357\) 9.00000i 0.476331i
\(358\) 0 0
\(359\) 24.0000i 1.26667i 0.773877 + 0.633336i \(0.218315\pi\)
−0.773877 + 0.633336i \(0.781685\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 18.0000 0.942163
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 18.0000i 0.934513i
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) 3.00000i 0.154919i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 6.00000i 0.308199i −0.988055 0.154100i \(-0.950752\pi\)
0.988055 0.154100i \(-0.0492477\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 0 0
\(383\) 9.00000i 0.459879i −0.973205 0.229939i \(-0.926147\pi\)
0.973205 0.229939i \(-0.0738528\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.00000 −0.101666
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) 3.00000 0.151330
\(394\) 0 0
\(395\) 30.0000i 1.50946i
\(396\) 0 0
\(397\) 18.0000i 0.903394i −0.892171 0.451697i \(-0.850819\pi\)
0.892171 0.451697i \(-0.149181\pi\)
\(398\) 0 0
\(399\) 18.0000 0.901127
\(400\) 0 0
\(401\) 30.0000i 1.49813i −0.662497 0.749064i \(-0.730503\pi\)
0.662497 0.749064i \(-0.269497\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.00000i 0.149071i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.00000i 0.296681i 0.988936 + 0.148340i \(0.0473931\pi\)
−0.988936 + 0.148340i \(0.952607\pi\)
\(410\) 0 0
\(411\) 18.0000i 0.887875i
\(412\) 0 0
\(413\) 18.0000 0.885722
\(414\) 0 0
\(415\) −18.0000 −0.883585
\(416\) 0 0
\(417\) −5.00000 −0.244851
\(418\) 0 0
\(419\) −15.0000 −0.732798 −0.366399 0.930458i \(-0.619409\pi\)
−0.366399 + 0.930458i \(0.619409\pi\)
\(420\) 0 0
\(421\) 15.0000i 0.731055i 0.930800 + 0.365528i \(0.119111\pi\)
−0.930800 + 0.365528i \(0.880889\pi\)
\(422\) 0 0
\(423\) 6.00000i 0.291730i
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) 24.0000i 1.16144i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.0000i 0.722525i −0.932464 0.361262i \(-0.882346\pi\)
0.932464 0.361262i \(-0.117654\pi\)
\(432\) 0 0
\(433\) −11.0000 −0.528626 −0.264313 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 36.0000i 1.72211i
\(438\) 0 0
\(439\) 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 \(-0.333748\pi\)
0.498870 + 0.866677i \(0.333748\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) 6.00000i 0.283790i
\(448\) 0 0
\(449\) 24.0000i 1.13263i −0.824189 0.566315i \(-0.808369\pi\)
0.824189 0.566315i \(-0.191631\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 15.0000i 0.704761i
\(454\) 0 0
\(455\) −18.0000 + 27.0000i −0.843853 + 1.26578i
\(456\) 0 0
\(457\) 18.0000i 0.842004i −0.907060 0.421002i \(-0.861678\pi\)
0.907060 0.421002i \(-0.138322\pi\)
\(458\) 0 0
\(459\) −15.0000 −0.700140
\(460\) 0 0
\(461\) 15.0000i 0.698620i 0.937007 + 0.349310i \(0.113584\pi\)
−0.937007 + 0.349310i \(0.886416\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i −0.830051 0.557687i \(-0.811689\pi\)
0.830051 0.557687i \(-0.188311\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 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
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 24.0000i 1.10120i
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) 39.0000i 1.78196i 0.454047 + 0.890978i \(0.349980\pi\)
−0.454047 + 0.890978i \(0.650020\pi\)
\(480\) 0 0
\(481\) −9.00000 6.00000i −0.410365 0.273576i
\(482\) 0 0
\(483\) 18.0000i 0.819028i
\(484\) 0 0
\(485\) −36.0000 −1.63468
\(486\) 0 0
\(487\) 12.0000i 0.543772i −0.962329 0.271886i \(-0.912353\pi\)
0.962329 0.271886i \(-0.0876473\pi\)
\(488\) 0 0
\(489\) 6.00000i 0.271329i
\(490\) 0 0
\(491\) −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
\(498\) 0 0
\(499\) 36.0000i 1.61158i −0.592200 0.805791i \(-0.701741\pi\)
0.592200 0.805791i \(-0.298259\pi\)
\(500\) 0 0
\(501\) 12.0000i 0.536120i
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 36.0000i 1.60198i
\(506\) 0 0
\(507\) −5.00000 12.0000i −0.222058 0.532939i
\(508\) 0 0
\(509\) 6.00000i 0.265945i 0.991120 + 0.132973i \(0.0424523\pi\)
−0.991120 + 0.132973i \(0.957548\pi\)
\(510\) 0 0
\(511\) 18.0000 0.796273
\(512\) 0 0
\(513\) 30.0000i 1.32453i
\(514\) 0 0
\(515\) 42.0000i 1.85074i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −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 \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 12.0000i 0.523723i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 12.0000i 0.520756i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 36.0000i 1.55642i
\(536\) 0 0
\(537\) −15.0000 −0.647298
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 15.0000i 0.644900i 0.946586 + 0.322450i \(0.104506\pi\)
−0.946586 + 0.322450i \(0.895494\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) 0 0
\(545\) 27.0000 1.15655
\(546\) 0 0
\(547\) 37.0000 1.58201 0.791003 0.611812i \(-0.209559\pi\)
0.791003 + 0.611812i \(0.209559\pi\)
\(548\) 0 0
\(549\) 16.0000 0.682863
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 30.0000i 1.27573i
\(554\) 0 0
\(555\) 9.00000 0.382029
\(556\) 0 0
\(557\) 27.0000i 1.14403i 0.820244 + 0.572013i \(0.193837\pi\)
−0.820244 + 0.572013i \(0.806163\pi\)
\(558\) 0 0
\(559\) 2.00000 3.00000i 0.0845910 0.126886i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −39.0000 −1.64365 −0.821827 0.569737i \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) 0 0
\(565\) 18.0000i 0.757266i
\(566\) 0 0
\(567\) 3.00000i 0.125988i
\(568\) 0 0
\(569\) 45.0000 1.88650 0.943249 0.332086i \(-0.107752\pi\)
0.943249 + 0.332086i \(0.107752\pi\)
\(570\) 0 0
\(571\) 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
\(576\) 0 0
\(577\) 42.0000i 1.74848i 0.485491 + 0.874241i \(0.338641\pi\)
−0.485491 + 0.874241i \(0.661359\pi\)
\(578\) 0 0
\(579\) 6.00000i 0.249351i
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −18.0000 12.0000i −0.744208 0.496139i
\(586\) 0 0
\(587\) 18.0000i 0.742940i 0.928445 + 0.371470i \(0.121146\pi\)
−0.928445 + 0.371470i \(0.878854\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 3.00000i 0.123404i
\(592\) 0 0
\(593\) 36.0000i 1.47834i −0.673517 0.739171i \(-0.735217\pi\)
0.673517 0.739171i \(-0.264783\pi\)
\(594\) 0 0
\(595\) −27.0000 −1.10689
\(596\) 0 0
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) 37.0000 1.50926 0.754631 0.656150i \(-0.227816\pi\)
0.754631 + 0.656150i \(0.227816\pi\)
\(602\) 0 0
\(603\) 24.0000i 0.977356i
\(604\) 0 0
\(605\) 33.0000i 1.34164i
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.00000 + 6.00000i 0.364101 + 0.242734i
\(612\) 0 0
\(613\) 6.00000i 0.242338i −0.992632 0.121169i \(-0.961336\pi\)
0.992632 0.121169i \(-0.0386643\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000i 0.483102i 0.970388 + 0.241551i \(0.0776561\pi\)
−0.970388 + 0.241551i \(0.922344\pi\)
\(618\) 0 0
\(619\) 24.0000i 0.964641i 0.875995 + 0.482321i \(0.160206\pi\)
−0.875995 + 0.482321i \(0.839794\pi\)
\(620\) 0 0
\(621\) −30.0000 −1.20386
\(622\) 0 0
\(623\) −18.0000 −0.721155
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9.00000i 0.358854i
\(630\) 0 0
\(631\) 15.0000i 0.597141i −0.954388 0.298570i \(-0.903490\pi\)
0.954388 0.298570i \(-0.0965097\pi\)
\(632\) 0 0
\(633\) 23.0000 0.914168
\(634\) 0 0
\(635\) 6.00000i 0.238103i
\(636\) 0 0
\(637\) −4.00000 + 6.00000i −0.158486 + 0.237729i
\(638\) 0 0
\(639\) 30.0000i 1.18678i
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 36.0000i 1.41970i 0.704352 + 0.709851i \(0.251238\pi\)
−0.704352 + 0.709851i \(0.748762\pi\)
\(644\) 0 0
\(645\) 3.00000i 0.118125i
\(646\) 0 0
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 0 0
\(655\) 9.00000i 0.351659i
\(656\) 0 0
\(657\) 12.0000i 0.468165i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 30.0000i 1.16686i −0.812162 0.583432i \(-0.801709\pi\)
0.812162 0.583432i \(-0.198291\pi\)
\(662\) 0 0
\(663\) 6.00000 9.00000i 0.233021 0.349531i
\(664\) 0 0
\(665\) 54.0000i 2.09403i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 9.00000i 0.347960i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 0 0
\(675\) 20.0000 0.769800
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) −36.0000 −1.38155
\(680\) 0 0
\(681\) 12.0000i 0.459841i
\(682\) 0 0
\(683\) 6.00000i 0.229584i 0.993390 + 0.114792i \(0.0366201\pi\)
−0.993390 + 0.114792i \(0.963380\pi\)
\(684\) 0 0
\(685\) 54.0000 2.06323
\(686\) 0 0
\(687\) 9.00000i 0.343371i
\(688\) 0 0
\(689\) −12.0000 + 18.0000i −0.457164 + 0.685745i
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 15.0000i 0.568982i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −21.0000 −0.794293
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) −18.0000 −0.678883
\(704\) 0 0
\(705\) −9.00000 −0.338960
\(706\) 0 0
\(707\) 36.0000i 1.35392i
\(708\) 0 0
\(709\) 6.00000i 0.225335i 0.993633 + 0.112667i \(0.0359394\pi\)
−0.993633 + 0.112667i \(0.964061\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 9.00000i 0.336111i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 42.0000i 1.56416i
\(722\) 0 0
\(723\) 30.0000i 1.11571i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −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
\(730\) 0 0
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) 9.00000i 0.332423i 0.986090 + 0.166211i \(0.0531534\pi\)
−0.986090 + 0.166211i \(0.946847\pi\)
\(734\) 0 0
\(735\) 6.00000i 0.221313i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 36.0000i 1.32428i −0.749380 0.662141i \(-0.769648\pi\)
0.749380 0.662141i \(-0.230352\pi\)
\(740\) 0 0
\(741\) −18.0000 12.0000i −0.661247 0.440831i
\(742\) 0 0
\(743\) 39.0000i 1.43077i −0.698730 0.715386i \(-0.746251\pi\)
0.698730 0.715386i \(-0.253749\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) 36.0000i 1.31541i
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) 45.0000 1.63772
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000i 1.08750i 0.839248 + 0.543750i \(0.182996\pi\)
−0.839248 + 0.543750i \(0.817004\pi\)
\(762\) 0 0
\(763\) 27.0000 0.977466
\(764\) 0 0
\(765\) 18.0000i 0.650791i
\(766\) 0 0
\(767\) −18.0000 12.0000i −0.649942 0.433295i
\(768\) 0 0
\(769\) 24.0000i 0.865462i −0.901523 0.432731i \(-0.857550\pi\)
0.901523 0.432731i \(-0.142450\pi\)
\(770\) 0 0
\(771\) 3.00000 0.108042
\(772\) 0 0
\(773\) 21.0000i 0.755318i −0.925945 0.377659i \(-0.876729\pi\)
0.925945 0.377659i \(-0.123271\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 9.00000 0.322873
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 66.0000i 2.35564i
\(786\) 0 0
\(787\) 12.0000i 0.427754i −0.976861 0.213877i \(-0.931391\pi\)
0.976861 0.213877i \(-0.0686091\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) 18.0000i 0.640006i
\(792\) 0 0
\(793\) −16.0000 + 24.0000i −0.568177 + 0.852265i
\(794\) 0 0
\(795\) 18.0000i 0.638394i
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 9.00000i 0.318397i
\(800\) 0 0
\(801\) 12.0000i 0.423999i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −54.0000 −1.90325
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 15.0000i 0.526073i
\(814\) 0 0
\(815\) −18.0000 −0.630512
\(816\) 0 0
\(817\) 6.00000i 0.209913i
\(818\) 0 0
\(819\) −18.0000 12.0000i −0.628971 0.419314i
\(820\) 0 0
\(821\) 45.0000i 1.57051i −0.619172 0.785255i \(-0.712532\pi\)
0.619172 0.785255i \(-0.287468\pi\)
\(822\) 0 0
\(823\) −14.0000 −0.488009 −0.244005 0.969774i \(-0.578461\pi\)
−0.244005 + 0.969774i \(0.578461\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 18.0000i 0.625921i 0.949766 + 0.312961i \(0.101321\pi\)
−0.949766 + 0.312961i \(0.898679\pi\)
\(828\) 0 0
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 36.0000 1.24583
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 24.0000i 0.828572i 0.910147 + 0.414286i \(0.135969\pi\)
−0.910147 + 0.414286i \(0.864031\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 30.0000i 1.03325i
\(844\) 0 0
\(845\) 36.0000 15.0000i 1.23844 0.516016i
\(846\) 0 0
\(847\) 33.0000i 1.13389i
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 18.0000i 0.617032i
\(852\) 0 0
\(853\) 39.0000i 1.33533i 0.744460 + 0.667667i \(0.232707\pi\)
−0.744460 + 0.667667i \(0.767293\pi\)
\(854\) 0 0
\(855\) −36.0000 −1.23117
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 39.0000i 1.32758i −0.747921 0.663788i \(-0.768948\pi\)
0.747921 0.663788i \(-0.231052\pi\)
\(864\) 0 0
\(865\) 18.0000i 0.612018i
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −36.0000 24.0000i −1.21981 0.813209i
\(872\) 0 0
\(873\) 24.0000i 0.812277i
\(874\) 0 0
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) 3.00000i 0.101303i −0.998716 0.0506514i \(-0.983870\pi\)
0.998716 0.0506514i \(-0.0161297\pi\)
\(878\) 0 0
\(879\) 9.00000i 0.303562i
\(880\) 0 0
\(881\) −33.0000 −1.11180 −0.555899 0.831250i \(-0.687626\pi\)
−0.555899 + 0.831250i \(0.687626\pi\)
\(882\) 0 0
\(883\) 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
\(886\) 0 0
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) 6.00000i 0.201234i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 18.0000 0.602347
\(894\) 0 0
\(895\) 45.0000i 1.50418i
\(896\) 0 0
\(897\) 12.0000 18.0000i 0.400668 0.601003i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 0 0
\(903\) 3.00000i 0.0998337i
\(904\) 0 0
\(905\) 6.00000i 0.199447i
\(906\) 0 0
\(907\) 17.0000 0.564476 0.282238 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) 0 0
\(909\) −24.0000 −0.796030
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 24.0000i 0.793416i
\(916\) 0 0
\(917\) 9.00000i 0.297206i
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 18.0000i 0.593120i
\(922\) 0 0
\(923\) 45.0000 + 30.0000i 1.48119 + 0.987462i
\(924\) 0 0
\(925\) 12.0000i 0.394558i
\(926\) 0 0
\(927\) 28.0000 0.919641
\(928\) 0 0
\(929\) 36.0000i 1.18112i 0.806993 + 0.590561i \(0.201093\pi\)
−0.806993 + 0.590561i \(0.798907\pi\)
\(930\) 0 0
\(931\) 12.0000i 0.393284i
\(932\) 0 0
\(933\) 18.0000 0.589294
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 19.0000 0.620042
\(940\) 0 0
\(941\) 45.0000i 1.46696i 0.679712 + 0.733479i \(0.262105\pi\)
−0.679712 + 0.733479i \(0.737895\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 45.0000 1.46385
\(946\) 0 0
\(947\) 48.0000i 1.55979i 0.625910 + 0.779895i \(0.284728\pi\)
−0.625910 + 0.779895i \(0.715272\pi\)
\(948\) 0 0
\(949\) −18.0000 12.0000i −0.584305 0.389536i
\(950\) 0 0
\(951\) 18.0000i 0.583690i
\(952\) 0 0
\(953\) 9.00000 0.291539 0.145769 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(954\) 0 0
\(955\) 36.0000i 1.16493i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 54.0000 1.74375
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) −24.0000 −0.773389
\(964\) 0 0
\(965\) 18.0000 0.579441
\(966\) 0 0
\(967\) 3.00000i 0.0964735i 0.998836 + 0.0482367i \(0.0153602\pi\)
−0.998836 + 0.0482367i \(0.984640\pi\)
\(968\) 0 0
\(969\) 18.0000i 0.578243i
\(970\) 0 0
\(971\) −27.0000 −0.866471 −0.433236 0.901281i \(-0.642628\pi\)
−0.433236 + 0.901281i \(0.642628\pi\)
\(972\) 0 0
\(973\) 15.0000i 0.480878i
\(974\) 0 0
\(975\) −8.00000 + 12.0000i −0.256205 + 0.384308i
\(976\) 0 0
\(977\) 12.0000i 0.383914i 0.981403 + 0.191957i \(0.0614834\pi\)
−0.981403 + 0.191957i \(0.938517\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 18.0000i 0.574696i
\(982\) 0 0
\(983\) 9.00000i 0.287055i −0.989646 0.143528i \(-0.954155\pi\)
0.989646 0.143528i \(-0.0458446\pi\)
\(984\) 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 \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 0 0
\(993\) 30.0000i 0.952021i
\(994\) 0 0
\(995\) 60.0000i 1.90213i
\(996\) 0 0
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 0 0
\(999\) 15.0000i 0.474579i
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 208.2.f.a.129.2 2
3.2 odd 2 1872.2.c.f.1585.1 2
4.3 odd 2 26.2.b.a.25.1 2
8.3 odd 2 832.2.f.d.129.1 2
8.5 even 2 832.2.f.b.129.1 2
12.11 even 2 234.2.b.b.181.2 2
13.5 odd 4 2704.2.a.k.1.1 1
13.8 odd 4 2704.2.a.j.1.1 1
13.12 even 2 inner 208.2.f.a.129.1 2
20.3 even 4 650.2.c.a.649.1 2
20.7 even 4 650.2.c.d.649.2 2
20.19 odd 2 650.2.d.b.51.2 2
28.3 even 6 1274.2.n.c.961.2 4
28.11 odd 6 1274.2.n.d.961.2 4
28.19 even 6 1274.2.n.c.753.1 4
28.23 odd 6 1274.2.n.d.753.1 4
28.27 even 2 1274.2.d.c.883.1 2
39.38 odd 2 1872.2.c.f.1585.2 2
52.3 odd 6 338.2.e.c.147.2 4
52.7 even 12 338.2.c.b.315.1 2
52.11 even 12 338.2.c.b.191.1 2
52.15 even 12 338.2.c.f.191.1 2
52.19 even 12 338.2.c.f.315.1 2
52.23 odd 6 338.2.e.c.147.1 4
52.31 even 4 338.2.a.b.1.1 1
52.35 odd 6 338.2.e.c.23.1 4
52.43 odd 6 338.2.e.c.23.2 4
52.47 even 4 338.2.a.d.1.1 1
52.51 odd 2 26.2.b.a.25.2 yes 2
104.51 odd 2 832.2.f.d.129.2 2
104.77 even 2 832.2.f.b.129.2 2
156.47 odd 4 3042.2.a.g.1.1 1
156.83 odd 4 3042.2.a.j.1.1 1
156.155 even 2 234.2.b.b.181.1 2
260.99 even 4 8450.2.a.h.1.1 1
260.103 even 4 650.2.c.d.649.1 2
260.207 even 4 650.2.c.a.649.2 2
260.239 even 4 8450.2.a.u.1.1 1
260.259 odd 2 650.2.d.b.51.1 2
364.51 odd 6 1274.2.n.d.753.2 4
364.103 even 6 1274.2.n.c.753.2 4
364.207 odd 6 1274.2.n.d.961.1 4
364.311 even 6 1274.2.n.c.961.1 4
364.363 even 2 1274.2.d.c.883.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.2.b.a.25.1 2 4.3 odd 2
26.2.b.a.25.2 yes 2 52.51 odd 2
208.2.f.a.129.1 2 13.12 even 2 inner
208.2.f.a.129.2 2 1.1 even 1 trivial
234.2.b.b.181.1 2 156.155 even 2
234.2.b.b.181.2 2 12.11 even 2
338.2.a.b.1.1 1 52.31 even 4
338.2.a.d.1.1 1 52.47 even 4
338.2.c.b.191.1 2 52.11 even 12
338.2.c.b.315.1 2 52.7 even 12
338.2.c.f.191.1 2 52.15 even 12
338.2.c.f.315.1 2 52.19 even 12
338.2.e.c.23.1 4 52.35 odd 6
338.2.e.c.23.2 4 52.43 odd 6
338.2.e.c.147.1 4 52.23 odd 6
338.2.e.c.147.2 4 52.3 odd 6
650.2.c.a.649.1 2 20.3 even 4
650.2.c.a.649.2 2 260.207 even 4
650.2.c.d.649.1 2 260.103 even 4
650.2.c.d.649.2 2 20.7 even 4
650.2.d.b.51.1 2 260.259 odd 2
650.2.d.b.51.2 2 20.19 odd 2
832.2.f.b.129.1 2 8.5 even 2
832.2.f.b.129.2 2 104.77 even 2
832.2.f.d.129.1 2 8.3 odd 2
832.2.f.d.129.2 2 104.51 odd 2
1274.2.d.c.883.1 2 28.27 even 2
1274.2.d.c.883.2 2 364.363 even 2
1274.2.n.c.753.1 4 28.19 even 6
1274.2.n.c.753.2 4 364.103 even 6
1274.2.n.c.961.1 4 364.311 even 6
1274.2.n.c.961.2 4 28.3 even 6
1274.2.n.d.753.1 4 28.23 odd 6
1274.2.n.d.753.2 4 364.51 odd 6
1274.2.n.d.961.1 4 364.207 odd 6
1274.2.n.d.961.2 4 28.11 odd 6
1872.2.c.f.1585.1 2 3.2 odd 2
1872.2.c.f.1585.2 2 39.38 odd 2
2704.2.a.j.1.1 1 13.8 odd 4
2704.2.a.k.1.1 1 13.5 odd 4
3042.2.a.g.1.1 1 156.47 odd 4
3042.2.a.j.1.1 1 156.83 odd 4
8450.2.a.h.1.1 1 260.99 even 4
8450.2.a.u.1.1 1 260.239 even 4