Properties

Label 416.2.e.a.337.2
Level $416$
Weight $2$
Character 416.337
Analytic conductor $3.322$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(337,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.e (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: \(3.32177672409\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 416.337
Dual form 416.2.e.a.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -1.00000 q^{5} -3.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -1.00000 q^{5} -3.00000i q^{7} +2.00000 q^{9} +2.00000 q^{11} +(3.00000 + 2.00000i) q^{13} -1.00000i q^{15} +3.00000 q^{17} +3.00000 q^{21} +6.00000 q^{23} -4.00000 q^{25} +5.00000i q^{27} +6.00000i q^{29} +2.00000i q^{33} +3.00000i q^{35} -3.00000 q^{37} +(-2.00000 + 3.00000i) q^{39} -10.0000i q^{41} -9.00000i q^{43} -2.00000 q^{45} +7.00000i q^{47} -2.00000 q^{49} +3.00000i q^{51} -6.00000i q^{53} -2.00000 q^{55} +10.0000 q^{59} -10.0000i q^{61} -6.00000i q^{63} +(-3.00000 - 2.00000i) q^{65} -12.0000 q^{67} +6.00000i q^{69} +5.00000i q^{71} +6.00000i q^{73} -4.00000i q^{75} -6.00000i q^{77} +1.00000 q^{81} -16.0000 q^{83} -3.00000 q^{85} -6.00000 q^{87} +4.00000i q^{89} +(6.00000 - 9.00000i) q^{91} +18.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 4 q^{9} + 4 q^{11} + 6 q^{13} + 6 q^{17} + 6 q^{21} + 12 q^{23} - 8 q^{25} - 6 q^{37} - 4 q^{39} - 4 q^{45} - 4 q^{49} - 4 q^{55} + 20 q^{59} - 6 q^{65} - 24 q^{67} + 2 q^{81} - 32 q^{83} - 6 q^{85} - 12 q^{87} + 12 q^{91} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 3.00000i 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 3.00000 + 2.00000i 0.832050 + 0.554700i
\(14\) 0 0
\(15\) 1.00000i 0.258199i
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) 3.00000i 0.507093i
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 0 0
\(39\) −2.00000 + 3.00000i −0.320256 + 0.480384i
\(40\) 0 0
\(41\) 10.0000i 1.56174i −0.624695 0.780869i \(-0.714777\pi\)
0.624695 0.780869i \(-0.285223\pi\)
\(42\) 0 0
\(43\) 9.00000i 1.37249i −0.727372 0.686244i \(-0.759258\pi\)
0.727372 0.686244i \(-0.240742\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 7.00000i 1.02105i 0.859861 + 0.510527i \(0.170550\pi\)
−0.859861 + 0.510527i \(0.829450\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 3.00000i 0.420084i
\(52\) 0 0
\(53\) 6.00000i 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 10.0000i 1.28037i −0.768221 0.640184i \(-0.778858\pi\)
0.768221 0.640184i \(-0.221142\pi\)
\(62\) 0 0
\(63\) 6.00000i 0.755929i
\(64\) 0 0
\(65\) −3.00000 2.00000i −0.372104 0.248069i
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 6.00000i 0.722315i
\(70\) 0 0
\(71\) 5.00000i 0.593391i 0.954972 + 0.296695i \(0.0958846\pi\)
−0.954972 + 0.296695i \(0.904115\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 4.00000i 0.461880i
\(76\) 0 0
\(77\) 6.00000i 0.683763i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 4.00000i 0.423999i 0.977270 + 0.212000i \(0.0679975\pi\)
−0.977270 + 0.212000i \(0.932002\pi\)
\(90\) 0 0
\(91\) 6.00000 9.00000i 0.628971 0.943456i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.0000i 1.82762i 0.406138 + 0.913812i \(0.366875\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) 12.0000i 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) −15.0000 −1.43674 −0.718370 0.695662i \(-0.755111\pi\)
−0.718370 + 0.695662i \(0.755111\pi\)
\(110\) 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
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) 6.00000 + 4.00000i 0.554700 + 0.369800i
\(118\) 0 0
\(119\) 9.00000i 0.825029i
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) 0 0
\(129\) 9.00000 0.792406
\(130\) 0 0
\(131\) 15.0000i 1.31056i 0.755388 + 0.655278i \(0.227449\pi\)
−0.755388 + 0.655278i \(0.772551\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 5.00000i 0.430331i
\(136\) 0 0
\(137\) 2.00000i 0.170872i −0.996344 0.0854358i \(-0.972772\pi\)
0.996344 0.0854358i \(-0.0272282\pi\)
\(138\) 0 0
\(139\) 9.00000i 0.763370i 0.924292 + 0.381685i \(0.124656\pi\)
−0.924292 + 0.381685i \(0.875344\pi\)
\(140\) 0 0
\(141\) −7.00000 −0.589506
\(142\) 0 0
\(143\) 6.00000 + 4.00000i 0.501745 + 0.334497i
\(144\) 0 0
\(145\) 6.00000i 0.498273i
\(146\) 0 0
\(147\) 2.00000i 0.164957i
\(148\) 0 0
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 15.0000i 1.22068i 0.792139 + 0.610341i \(0.208968\pi\)
−0.792139 + 0.610341i \(0.791032\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000i 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 18.0000i 1.41860i
\(162\) 0 0
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) 0 0
\(165\) 2.00000i 0.155700i
\(166\) 0 0
\(167\) 8.00000i 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 0 0
\(169\) 5.00000 + 12.0000i 0.384615 + 0.923077i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 24.0000i 1.82469i 0.409426 + 0.912343i \(0.365729\pi\)
−0.409426 + 0.912343i \(0.634271\pi\)
\(174\) 0 0
\(175\) 12.0000i 0.907115i
\(176\) 0 0
\(177\) 10.0000i 0.751646i
\(178\) 0 0
\(179\) 9.00000i 0.672692i 0.941739 + 0.336346i \(0.109191\pi\)
−0.941739 + 0.336346i \(0.890809\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 3.00000 0.220564
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 0 0
\(189\) 15.0000 1.09109
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) 24.0000i 1.72756i −0.503871 0.863779i \(-0.668091\pi\)
0.503871 0.863779i \(-0.331909\pi\)
\(194\) 0 0
\(195\) 2.00000 3.00000i 0.143223 0.214834i
\(196\) 0 0
\(197\) −23.0000 −1.63868 −0.819341 0.573306i \(-0.805660\pi\)
−0.819341 + 0.573306i \(0.805660\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 12.0000i 0.846415i
\(202\) 0 0
\(203\) 18.0000 1.26335
\(204\) 0 0
\(205\) 10.0000i 0.698430i
\(206\) 0 0
\(207\) 12.0000 0.834058
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 5.00000i 0.344214i −0.985078 0.172107i \(-0.944942\pi\)
0.985078 0.172107i \(-0.0550575\pi\)
\(212\) 0 0
\(213\) −5.00000 −0.342594
\(214\) 0 0
\(215\) 9.00000i 0.613795i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 9.00000 + 6.00000i 0.605406 + 0.403604i
\(222\) 0 0
\(223\) 21.0000i 1.40626i −0.711059 0.703132i \(-0.751784\pi\)
0.711059 0.703132i \(-0.248216\pi\)
\(224\) 0 0
\(225\) −8.00000 −0.533333
\(226\) 0 0
\(227\) −2.00000 −0.132745 −0.0663723 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 0 0
\(229\) 15.0000 0.991228 0.495614 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) 9.00000 0.589610 0.294805 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 0 0
\(235\) 7.00000i 0.456630i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.0000i 1.22901i −0.788914 0.614504i \(-0.789356\pi\)
0.788914 0.614504i \(-0.210644\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 16.0000i 1.02640i
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 16.0000i 1.01396i
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 3.00000i 0.187867i
\(256\) 0 0
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 0 0
\(259\) 9.00000i 0.559233i
\(260\) 0 0
\(261\) 12.0000i 0.742781i
\(262\) 0 0
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 6.00000i 0.368577i
\(266\) 0 0
\(267\) −4.00000 −0.244796
\(268\) 0 0
\(269\) 6.00000i 0.365826i 0.983129 + 0.182913i \(0.0585527\pi\)
−0.983129 + 0.182913i \(0.941447\pi\)
\(270\) 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
\(275\) −8.00000 −0.482418
\(276\) 0 0
\(277\) 18.0000i 1.08152i −0.841178 0.540758i \(-0.818138\pi\)
0.841178 0.540758i \(-0.181862\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 20.0000i 1.19310i 0.802576 + 0.596550i \(0.203462\pi\)
−0.802576 + 0.596550i \(0.796538\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −30.0000 −1.77084
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −18.0000 −1.05518
\(292\) 0 0
\(293\) 1.00000 0.0584206 0.0292103 0.999573i \(-0.490701\pi\)
0.0292103 + 0.999573i \(0.490701\pi\)
\(294\) 0 0
\(295\) −10.0000 −0.582223
\(296\) 0 0
\(297\) 10.0000i 0.580259i
\(298\) 0 0
\(299\) 18.0000 + 12.0000i 1.04097 + 0.693978i
\(300\) 0 0
\(301\) −27.0000 −1.55625
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 10.0000i 0.572598i
\(306\) 0 0
\(307\) 18.0000 1.02731 0.513657 0.857996i \(-0.328290\pi\)
0.513657 + 0.857996i \(0.328290\pi\)
\(308\) 0 0
\(309\) 4.00000i 0.227552i
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 9.00000 0.508710 0.254355 0.967111i \(-0.418137\pi\)
0.254355 + 0.967111i \(0.418137\pi\)
\(314\) 0 0
\(315\) 6.00000i 0.338062i
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) 12.0000i 0.671871i
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −12.0000 8.00000i −0.665640 0.443760i
\(326\) 0 0
\(327\) 15.0000i 0.829502i
\(328\) 0 0
\(329\) 21.0000 1.15777
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) −6.00000 −0.328798
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −27.0000 −1.47078 −0.735392 0.677642i \(-0.763002\pi\)
−0.735392 + 0.677642i \(0.763002\pi\)
\(338\) 0 0
\(339\) 6.00000i 0.325875i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 6.00000i 0.323029i
\(346\) 0 0
\(347\) 27.0000i 1.44944i −0.689046 0.724718i \(-0.741970\pi\)
0.689046 0.724718i \(-0.258030\pi\)
\(348\) 0 0
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) 0 0
\(351\) −10.0000 + 15.0000i −0.533761 + 0.800641i
\(352\) 0 0
\(353\) 16.0000i 0.851594i 0.904819 + 0.425797i \(0.140006\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 0 0
\(355\) 5.00000i 0.265372i
\(356\) 0 0
\(357\) 9.00000 0.476331
\(358\) 0 0
\(359\) 4.00000i 0.211112i −0.994413 0.105556i \(-0.966338\pi\)
0.994413 0.105556i \(-0.0336622\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) 6.00000i 0.314054i
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) 20.0000i 1.04116i
\(370\) 0 0
\(371\) −18.0000 −0.934513
\(372\) 0 0
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 0 0
\(375\) 9.00000i 0.464758i
\(376\) 0 0
\(377\) −12.0000 + 18.0000i −0.618031 + 0.927047i
\(378\) 0 0
\(379\) −30.0000 −1.54100 −0.770498 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) 0 0
\(381\) 18.0000i 0.922168i
\(382\) 0 0
\(383\) 1.00000i 0.0510976i −0.999674 0.0255488i \(-0.991867\pi\)
0.999674 0.0255488i \(-0.00813332\pi\)
\(384\) 0 0
\(385\) 6.00000i 0.305788i
\(386\) 0 0
\(387\) 18.0000i 0.914991i
\(388\) 0 0
\(389\) 6.00000i 0.304212i 0.988364 + 0.152106i \(0.0486055\pi\)
−0.988364 + 0.152106i \(0.951394\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) −15.0000 −0.756650
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.0000i 0.499376i −0.968326 0.249688i \(-0.919672\pi\)
0.968326 0.249688i \(-0.0803281\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) 6.00000i 0.296681i −0.988936 0.148340i \(-0.952607\pi\)
0.988936 0.148340i \(-0.0473931\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) 30.0000i 1.47620i
\(414\) 0 0
\(415\) 16.0000 0.785409
\(416\) 0 0
\(417\) −9.00000 −0.440732
\(418\) 0 0
\(419\) 9.00000i 0.439679i 0.975536 + 0.219839i \(0.0705533\pi\)
−0.975536 + 0.219839i \(0.929447\pi\)
\(420\) 0 0
\(421\) 3.00000 0.146211 0.0731055 0.997324i \(-0.476709\pi\)
0.0731055 + 0.997324i \(0.476709\pi\)
\(422\) 0 0
\(423\) 14.0000i 0.680703i
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) −30.0000 −1.45180
\(428\) 0 0
\(429\) −4.00000 + 6.00000i −0.193122 + 0.289683i
\(430\) 0 0
\(431\) 5.00000i 0.240842i 0.992723 + 0.120421i \(0.0384244\pi\)
−0.992723 + 0.120421i \(0.961576\pi\)
\(432\) 0 0
\(433\) 9.00000 0.432512 0.216256 0.976337i \(-0.430615\pi\)
0.216256 + 0.976337i \(0.430615\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 0 0
\(437\) 0 0
\(438\) 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
\(443\) 9.00000i 0.427603i −0.976877 0.213801i \(-0.931415\pi\)
0.976877 0.213801i \(-0.0685846\pi\)
\(444\) 0 0
\(445\) 4.00000i 0.189618i
\(446\) 0 0
\(447\) 10.0000i 0.472984i
\(448\) 0 0
\(449\) 14.0000i 0.660701i 0.943858 + 0.330350i \(0.107167\pi\)
−0.943858 + 0.330350i \(0.892833\pi\)
\(450\) 0 0
\(451\) 20.0000i 0.941763i
\(452\) 0 0
\(453\) −15.0000 −0.704761
\(454\) 0 0
\(455\) −6.00000 + 9.00000i −0.281284 + 0.421927i
\(456\) 0 0
\(457\) 12.0000i 0.561336i −0.959805 0.280668i \(-0.909444\pi\)
0.959805 0.280668i \(-0.0905560\pi\)
\(458\) 0 0
\(459\) 15.0000i 0.700140i
\(460\) 0 0
\(461\) −7.00000 −0.326023 −0.163011 0.986624i \(-0.552121\pi\)
−0.163011 + 0.986624i \(0.552121\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i 0.830051 + 0.557687i \(0.188311\pi\)
−0.830051 + 0.557687i \(0.811689\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000i 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) 0 0
\(469\) 36.0000i 1.66233i
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 18.0000i 0.827641i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000i 0.549442i
\(478\) 0 0
\(479\) 29.0000i 1.32504i −0.749043 0.662522i \(-0.769486\pi\)
0.749043 0.662522i \(-0.230514\pi\)
\(480\) 0 0
\(481\) −9.00000 6.00000i −0.410365 0.273576i
\(482\) 0 0
\(483\) 18.0000 0.819028
\(484\) 0 0
\(485\) 18.0000i 0.817338i
\(486\) 0 0
\(487\) 12.0000i 0.543772i 0.962329 + 0.271886i \(0.0876473\pi\)
−0.962329 + 0.271886i \(0.912353\pi\)
\(488\) 0 0
\(489\) 6.00000i 0.271329i
\(490\) 0 0
\(491\) 15.0000i 0.676941i 0.940977 + 0.338470i \(0.109909\pi\)
−0.940977 + 0.338470i \(0.890091\pi\)
\(492\) 0 0
\(493\) 18.0000i 0.810679i
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) 15.0000 0.672842
\(498\) 0 0
\(499\) 30.0000 1.34298 0.671492 0.741012i \(-0.265654\pi\)
0.671492 + 0.741012i \(0.265654\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −12.0000 + 5.00000i −0.532939 + 0.222058i
\(508\) 0 0
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) 18.0000 0.796273
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.00000 0.176261
\(516\) 0 0
\(517\) 14.0000i 0.615719i
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) 36.0000i 1.57417i 0.616844 + 0.787085i \(0.288411\pi\)
−0.616844 + 0.787085i \(0.711589\pi\)
\(524\) 0 0
\(525\) −12.0000 −0.523723
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 20.0000 0.867926
\(532\) 0 0
\(533\) 20.0000 30.0000i 0.866296 1.29944i
\(534\) 0 0
\(535\) 12.0000i 0.518805i
\(536\) 0 0
\(537\) −9.00000 −0.388379
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 33.0000 1.41878 0.709390 0.704816i \(-0.248970\pi\)
0.709390 + 0.704816i \(0.248970\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 15.0000 0.642529
\(546\) 0 0
\(547\) 27.0000i 1.15444i −0.816590 0.577218i \(-0.804138\pi\)
0.816590 0.577218i \(-0.195862\pi\)
\(548\) 0 0
\(549\) 20.0000i 0.853579i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 3.00000i 0.127343i
\(556\) 0 0
\(557\) −23.0000 −0.974541 −0.487271 0.873251i \(-0.662007\pi\)
−0.487271 + 0.873251i \(0.662007\pi\)
\(558\) 0 0
\(559\) 18.0000 27.0000i 0.761319 1.14198i
\(560\) 0 0
\(561\) 6.00000i 0.253320i
\(562\) 0 0
\(563\) 21.0000i 0.885044i 0.896758 + 0.442522i \(0.145916\pi\)
−0.896758 + 0.442522i \(0.854084\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) 3.00000i 0.125988i
\(568\) 0 0
\(569\) 15.0000 0.628833 0.314416 0.949285i \(-0.398191\pi\)
0.314416 + 0.949285i \(0.398191\pi\)
\(570\) 0 0
\(571\) 5.00000i 0.209243i 0.994512 + 0.104622i \(0.0333632\pi\)
−0.994512 + 0.104622i \(0.966637\pi\)
\(572\) 0 0
\(573\) 18.0000i 0.751961i
\(574\) 0 0
\(575\) −24.0000 −1.00087
\(576\) 0 0
\(577\) 18.0000i 0.749350i 0.927156 + 0.374675i \(0.122246\pi\)
−0.927156 + 0.374675i \(0.877754\pi\)
\(578\) 0 0
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) 48.0000i 1.99138i
\(582\) 0 0
\(583\) 12.0000i 0.496989i
\(584\) 0 0
\(585\) −6.00000 4.00000i −0.248069 0.165380i
\(586\) 0 0
\(587\) −2.00000 −0.0825488 −0.0412744 0.999148i \(-0.513142\pi\)
−0.0412744 + 0.999148i \(0.513142\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 23.0000i 0.946094i
\(592\) 0 0
\(593\) 14.0000i 0.574911i −0.957794 0.287456i \(-0.907191\pi\)
0.957794 0.287456i \(-0.0928094\pi\)
\(594\) 0 0
\(595\) 9.00000i 0.368964i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) 27.0000 1.10135 0.550676 0.834719i \(-0.314370\pi\)
0.550676 + 0.834719i \(0.314370\pi\)
\(602\) 0 0
\(603\) −24.0000 −0.977356
\(604\) 0 0
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) 18.0000i 0.729397i
\(610\) 0 0
\(611\) −14.0000 + 21.0000i −0.566379 + 0.849569i
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 0 0
\(615\) −10.0000 −0.403239
\(616\) 0 0
\(617\) 8.00000i 0.322068i 0.986949 + 0.161034i \(0.0514829\pi\)
−0.986949 + 0.161034i \(0.948517\pi\)
\(618\) 0 0
\(619\) −30.0000 −1.20580 −0.602901 0.797816i \(-0.705989\pi\)
−0.602901 + 0.797816i \(0.705989\pi\)
\(620\) 0 0
\(621\) 30.0000i 1.20386i
\(622\) 0 0
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9.00000 −0.358854
\(630\) 0 0
\(631\) 45.0000i 1.79142i −0.444637 0.895711i \(-0.646667\pi\)
0.444637 0.895711i \(-0.353333\pi\)
\(632\) 0 0
\(633\) 5.00000 0.198732
\(634\) 0 0
\(635\) 18.0000 0.714308
\(636\) 0 0
\(637\) −6.00000 4.00000i −0.237729 0.158486i
\(638\) 0 0
\(639\) 10.0000i 0.395594i
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 24.0000 0.946468 0.473234 0.880937i \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) 0 0
\(645\) −9.00000 −0.354375
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.0000i 0.939193i 0.882881 + 0.469596i \(0.155601\pi\)
−0.882881 + 0.469596i \(0.844399\pi\)
\(654\) 0 0
\(655\) 15.0000i 0.586098i
\(656\) 0 0
\(657\) 12.0000i 0.468165i
\(658\) 0 0
\(659\) 36.0000i 1.40236i −0.712984 0.701180i \(-0.752657\pi\)
0.712984 0.701180i \(-0.247343\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 0 0
\(663\) −6.00000 + 9.00000i −0.233021 + 0.349531i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 36.0000i 1.39393i
\(668\) 0 0
\(669\) 21.0000 0.811907
\(670\) 0 0
\(671\) 20.0000i 0.772091i
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 0 0
\(675\) 20.0000i 0.769800i
\(676\) 0 0
\(677\) 12.0000i 0.461197i 0.973049 + 0.230599i \(0.0740685\pi\)
−0.973049 + 0.230599i \(0.925932\pi\)
\(678\) 0 0
\(679\) 54.0000 2.07233
\(680\) 0 0
\(681\) 2.00000i 0.0766402i
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 2.00000i 0.0764161i
\(686\) 0 0
\(687\) 15.0000i 0.572286i
\(688\) 0 0
\(689\) 12.0000 18.0000i 0.457164 0.685745i
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 0 0
\(693\) 12.0000i 0.455842i
\(694\) 0 0
\(695\) 9.00000i 0.341389i
\(696\) 0 0
\(697\) 30.0000i 1.13633i
\(698\) 0 0
\(699\) 9.00000i 0.340411i
\(700\) 0 0
\(701\) 30.0000i 1.13308i −0.824033 0.566542i \(-0.808281\pi\)
0.824033 0.566542i \(-0.191719\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 7.00000 0.263635
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −6.00000 4.00000i −0.224387 0.149592i
\(716\) 0 0
\(717\) 19.0000 0.709568
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 12.0000i 0.446903i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 24.0000i 0.891338i
\(726\) 0 0
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 27.0000i 0.998631i
\(732\) 0 0
\(733\) 51.0000 1.88373 0.941864 0.335994i \(-0.109072\pi\)
0.941864 + 0.335994i \(0.109072\pi\)
\(734\) 0 0
\(735\) 2.00000i 0.0737711i
\(736\) 0 0
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) 30.0000 1.10357 0.551784 0.833987i \(-0.313947\pi\)
0.551784 + 0.833987i \(0.313947\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 19.0000i 0.697042i 0.937301 + 0.348521i \(0.113316\pi\)
−0.937301 + 0.348521i \(0.886684\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) 0 0
\(747\) −32.0000 −1.17082
\(748\) 0 0
\(749\) −36.0000 −1.31541
\(750\) 0 0
\(751\) −22.0000 −0.802791 −0.401396 0.915905i \(-0.631475\pi\)
−0.401396 + 0.915905i \(0.631475\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 15.0000i 0.545906i
\(756\) 0 0
\(757\) 38.0000i 1.38113i −0.723269 0.690567i \(-0.757361\pi\)
0.723269 0.690567i \(-0.242639\pi\)
\(758\) 0 0
\(759\) 12.0000i 0.435572i
\(760\) 0 0
\(761\) 10.0000i 0.362500i 0.983437 + 0.181250i \(0.0580143\pi\)
−0.983437 + 0.181250i \(0.941986\pi\)
\(762\) 0 0
\(763\) 45.0000i 1.62911i
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) 30.0000 + 20.0000i 1.08324 + 0.722158i
\(768\) 0 0
\(769\) 36.0000i 1.29819i −0.760706 0.649097i \(-0.775147\pi\)
0.760706 0.649097i \(-0.224853\pi\)
\(770\) 0 0
\(771\) 3.00000i 0.108042i
\(772\) 0 0
\(773\) −49.0000 −1.76241 −0.881204 0.472737i \(-0.843266\pi\)
−0.881204 + 0.472737i \(0.843266\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −9.00000 −0.322873
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 10.0000i 0.357828i
\(782\) 0 0
\(783\) −30.0000 −1.07211
\(784\) 0 0
\(785\) 18.0000i 0.642448i
\(786\) 0 0
\(787\) 18.0000 0.641631 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) 0 0
\(789\) 6.00000i 0.213606i
\(790\) 0 0
\(791\) 18.0000i 0.640006i
\(792\) 0 0
\(793\) 20.0000 30.0000i 0.710221 1.06533i
\(794\) 0 0
\(795\) −6.00000 −0.212798
\(796\) 0 0
\(797\) 12.0000i 0.425062i 0.977154 + 0.212531i \(0.0681706\pi\)
−0.977154 + 0.212531i \(0.931829\pi\)
\(798\) 0 0