Properties

Label 637.2.r.b.116.2
Level $637$
Weight $2$
Character 637.116
Analytic conductor $5.086$
Analytic rank $0$
Dimension $4$
CM discriminant -91
Inner twists $8$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(116,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.116");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.r (of order \(6\), degree \(2\), 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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 13x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 116.2
Root \(-3.12250 + 1.80278i\) of defining polynomial
Character \(\chi\) \(=\) 637.116
Dual form 637.2.r.b.324.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.73205i) q^{4} +(3.12250 - 1.80278i) q^{5} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\) \(q+(-1.00000 + 1.73205i) q^{4} +(3.12250 - 1.80278i) q^{5} +(1.50000 + 2.59808i) q^{9} +3.60555i q^{13} +(-2.00000 - 3.46410i) q^{16} +(3.12250 - 1.80278i) q^{19} +7.21110i q^{20} +(-0.500000 - 0.866025i) q^{23} +(4.00000 - 6.92820i) q^{25} -5.00000 q^{29} +(9.36750 + 5.40833i) q^{31} -6.00000 q^{36} +7.21110i q^{41} +9.00000 q^{43} +(9.36750 + 5.40833i) q^{45} +(3.12250 - 1.80278i) q^{47} +(-6.24500 - 3.60555i) q^{52} +(-5.50000 + 9.52628i) q^{53} +(-12.4900 - 7.21110i) q^{59} +8.00000 q^{64} +(6.50000 + 11.2583i) q^{65} +(9.36750 + 5.40833i) q^{73} +7.21110i q^{76} +(-7.50000 - 12.9904i) q^{79} +(-12.4900 - 7.21110i) q^{80} +(-4.50000 + 7.79423i) q^{81} -18.0278i q^{83} +(3.12250 - 1.80278i) q^{89} +2.00000 q^{92} +(6.50000 - 11.2583i) q^{95} -18.0278i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 6 q^{9} - 8 q^{16} - 2 q^{23} + 16 q^{25} - 20 q^{29} - 24 q^{36} + 36 q^{43} - 22 q^{53} + 32 q^{64} + 26 q^{65} - 30 q^{79} - 18 q^{81} + 8 q^{92} + 26 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) −1.00000 + 1.73205i −0.500000 + 0.866025i
\(5\) 3.12250 1.80278i 1.39642 0.806226i 0.402408 0.915460i \(-0.368173\pi\)
0.994016 + 0.109235i \(0.0348400\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 3.60555i 1.00000i
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 3.46410i −0.500000 0.866025i
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 3.12250 1.80278i 0.716350 0.413585i −0.0970575 0.995279i \(-0.530943\pi\)
0.813408 + 0.581694i \(0.197610\pi\)
\(20\) 7.21110i 1.61245i
\(21\) 0 0
\(22\) 0 0
\(23\) −0.500000 0.866025i −0.104257 0.180579i 0.809177 0.587565i \(-0.199913\pi\)
−0.913434 + 0.406986i \(0.866580\pi\)
\(24\) 0 0
\(25\) 4.00000 6.92820i 0.800000 1.38564i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 9.36750 + 5.40833i 1.68245 + 0.971364i 0.960024 + 0.279918i \(0.0903074\pi\)
0.722428 + 0.691446i \(0.243026\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −6.00000 −1.00000
\(37\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.21110i 1.12619i 0.826394 + 0.563093i \(0.190389\pi\)
−0.826394 + 0.563093i \(0.809611\pi\)
\(42\) 0 0
\(43\) 9.00000 1.37249 0.686244 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) 0 0
\(45\) 9.36750 + 5.40833i 1.39642 + 0.806226i
\(46\) 0 0
\(47\) 3.12250 1.80278i 0.455463 0.262962i −0.254671 0.967028i \(-0.581967\pi\)
0.710135 + 0.704066i \(0.248634\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) −6.24500 3.60555i −0.866025 0.500000i
\(53\) −5.50000 + 9.52628i −0.755483 + 1.30854i 0.189651 + 0.981852i \(0.439264\pi\)
−0.945134 + 0.326683i \(0.894069\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.4900 7.21110i −1.62606 0.938806i −0.985253 0.171103i \(-0.945267\pi\)
−0.640806 0.767703i \(-0.721400\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 6.50000 + 11.2583i 0.806226 + 1.39642i
\(66\) 0 0
\(67\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 9.36750 + 5.40833i 1.09638 + 0.632997i 0.935269 0.353939i \(-0.115158\pi\)
0.161114 + 0.986936i \(0.448491\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 7.21110i 0.827170i
\(77\) 0 0
\(78\) 0 0
\(79\) −7.50000 12.9904i −0.843816 1.46153i −0.886646 0.462450i \(-0.846971\pi\)
0.0428296 0.999082i \(-0.486363\pi\)
\(80\) −12.4900 7.21110i −1.39642 0.806226i
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 18.0278i 1.97880i −0.145204 0.989402i \(-0.546384\pi\)
0.145204 0.989402i \(-0.453616\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.12250 1.80278i 0.330984 0.191094i −0.325294 0.945613i \(-0.605463\pi\)
0.656278 + 0.754519i \(0.272130\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) 0 0
\(95\) 6.50000 11.2583i 0.666886 1.15508i
\(96\) 0 0
\(97\) 18.0278i 1.83044i −0.402953 0.915221i \(-0.632016\pi\)
0.402953 0.915221i \(-0.367984\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 8.00000 + 13.8564i 0.800000 + 1.38564i
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 6.92820i −0.386695 0.669775i 0.605308 0.795991i \(-0.293050\pi\)
−0.992003 + 0.126217i \(0.959717\pi\)
\(108\) 0 0
\(109\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −19.0000 −1.78737 −0.893685 0.448695i \(-0.851889\pi\)
−0.893685 + 0.448695i \(0.851889\pi\)
\(114\) 0 0
\(115\) −3.12250 1.80278i −0.291175 0.168110i
\(116\) 5.00000 8.66025i 0.464238 0.804084i
\(117\) −9.36750 + 5.40833i −0.866025 + 0.500000i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.50000 9.52628i −0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) −18.7350 + 10.8167i −1.68245 + 0.971364i
\(125\) 10.8167i 0.967471i
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 6.00000 10.3923i 0.500000 0.866025i
\(145\) −15.6125 + 9.01388i −1.29655 + 0.748562i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 39.0000 3.13256
\(156\) 0 0
\(157\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(164\) −12.4900 7.21110i −0.975305 0.563093i
\(165\) 0 0
\(166\) 0 0
\(167\) 18.0278i 1.39503i −0.716570 0.697515i \(-0.754289\pi\)
0.716570 0.697515i \(-0.245711\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 9.36750 + 5.40833i 0.716350 + 0.413585i
\(172\) −9.00000 + 15.5885i −0.686244 + 1.18861i
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.5000 + 21.6506i −0.934294 + 1.61824i −0.158406 + 0.987374i \(0.550635\pi\)
−0.775888 + 0.630870i \(0.782698\pi\)
\(180\) −18.7350 + 10.8167i −1.39642 + 0.806226i
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 7.21110i 0.525924i
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0000 + 17.3205i 0.723575 + 1.25327i 0.959558 + 0.281511i \(0.0908356\pi\)
−0.235983 + 0.971757i \(0.575831\pi\)
\(192\) 0 0
\(193\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 13.0000 + 22.5167i 0.907959 + 1.57263i
\(206\) 0 0
\(207\) 1.50000 2.59808i 0.104257 0.180579i
\(208\) 12.4900 7.21110i 0.866025 0.500000i
\(209\) 0 0
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) −11.0000 19.0526i −0.755483 1.30854i
\(213\) 0 0
\(214\) 0 0
\(215\) 28.1025 16.2250i 1.91657 1.10653i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 18.0278i 1.20723i −0.797277 0.603614i \(-0.793727\pi\)
0.797277 0.603614i \(-0.206273\pi\)
\(224\) 0 0
\(225\) 24.0000 1.60000
\(226\) 0 0
\(227\) −12.4900 7.21110i −0.828990 0.478618i 0.0245166 0.999699i \(-0.492195\pi\)
−0.853507 + 0.521082i \(0.825529\pi\)
\(228\) 0 0
\(229\) −18.7350 + 10.8167i −1.23804 + 0.714785i −0.968694 0.248258i \(-0.920142\pi\)
−0.269349 + 0.963043i \(0.586809\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.5000 25.1147i −0.949927 1.64532i −0.745573 0.666424i \(-0.767824\pi\)
−0.204354 0.978897i \(-0.565509\pi\)
\(234\) 0 0
\(235\) 6.50000 11.2583i 0.424013 0.734412i
\(236\) 24.9800 14.4222i 1.62606 0.938806i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 9.36750 + 5.40833i 0.603414 + 0.348381i 0.770383 0.637581i \(-0.220065\pi\)
−0.166970 + 0.985962i \(0.553398\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.50000 + 11.2583i 0.413585 + 0.716350i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 + 13.8564i −0.500000 + 0.866025i
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −26.0000 −1.61245
\(261\) −7.50000 12.9904i −0.464238 0.804084i
\(262\) 0 0
\(263\) 15.5000 26.8468i 0.955771 1.65544i 0.223177 0.974778i \(-0.428357\pi\)
0.732594 0.680666i \(-0.238309\pi\)
\(264\) 0 0
\(265\) 39.6611i 2.43636i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 24.9800 14.4222i 1.51743 0.876087i 0.517636 0.855601i \(-0.326812\pi\)
0.999790 0.0204858i \(-0.00652129\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.50000 14.7224i 0.510716 0.884585i −0.489207 0.872167i \(-0.662714\pi\)
0.999923 0.0124177i \(-0.00395278\pi\)
\(278\) 0 0
\(279\) 32.4500i 1.94273i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 + 14.7224i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) −18.7350 + 10.8167i −1.09638 + 0.632997i
\(293\) 18.0278i 1.05319i −0.850115 0.526596i \(-0.823468\pi\)
0.850115 0.526596i \(-0.176532\pi\)
\(294\) 0 0
\(295\) −52.0000 −3.02756
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.12250 1.80278i 0.180579 0.104257i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −12.4900 7.21110i −0.716350 0.413585i
\(305\) 0 0
\(306\) 0 0
\(307\) 32.4500i 1.85202i 0.377503 + 0.926009i \(0.376783\pi\)
−0.377503 + 0.926009i \(0.623217\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 30.0000 1.68763
\(317\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 24.9800 14.4222i 1.39642 0.806226i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −9.00000 15.5885i −0.500000 0.866025i
\(325\) 24.9800 + 14.4222i 1.38564 + 0.800000i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) 31.2250 + 18.0278i 1.71369 + 0.989402i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −16.0000 + 27.7128i −0.858925 + 1.48770i 0.0140303 + 0.999902i \(0.495534\pi\)
−0.872955 + 0.487800i \(0.837799\pi\)
\(348\) 0 0
\(349\) 32.4500i 1.73701i 0.495683 + 0.868503i \(0.334918\pi\)
−0.495683 + 0.868503i \(0.665082\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 31.2250 + 18.0278i 1.66194 + 0.959521i 0.971788 + 0.235854i \(0.0757889\pi\)
0.690150 + 0.723666i \(0.257544\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 7.21110i 0.382188i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) −3.00000 + 5.19615i −0.157895 + 0.273482i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 39.0000 2.04135
\(366\) 0 0
\(367\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(368\) −2.00000 + 3.46410i −0.104257 + 0.180579i
\(369\) −18.7350 + 10.8167i −0.975305 + 0.563093i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 3.00000 + 5.19615i 0.155334 + 0.269047i 0.933181 0.359408i \(-0.117021\pi\)
−0.777847 + 0.628454i \(0.783688\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.0278i 0.928477i
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 13.0000 + 22.5167i 0.666886 + 1.15508i
\(381\) 0 0
\(382\) 0 0
\(383\) 24.9800 14.4222i 1.27642 0.736940i 0.300230 0.953867i \(-0.402937\pi\)
0.976188 + 0.216927i \(0.0696032\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 13.5000 + 23.3827i 0.686244 + 1.18861i
\(388\) 31.2250 + 18.0278i 1.58521 + 0.915221i
\(389\) 5.00000 8.66025i 0.253510 0.439092i −0.710980 0.703213i \(-0.751748\pi\)
0.964490 + 0.264120i \(0.0850816\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −46.8375 27.0416i −2.35665 1.36061i
\(396\) 0 0
\(397\) 3.12250 1.80278i 0.156714 0.0904787i −0.419592 0.907713i \(-0.637827\pi\)
0.576306 + 0.817234i \(0.304494\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −32.0000 −1.60000
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) −19.5000 + 33.7750i −0.971364 + 1.68245i
\(404\) 0 0
\(405\) 32.4500i 1.61245i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −34.3475 19.8305i −1.69837 0.980557i −0.947305 0.320333i \(-0.896205\pi\)
−0.751069 0.660224i \(-0.770461\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −32.5000 56.2917i −1.59536 2.76325i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 9.36750 + 5.40833i 0.455463 + 0.262962i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.12250 1.80278i −0.149369 0.0862385i
\(438\) 0 0
\(439\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.5000 + 35.5070i 0.973984 + 1.68699i 0.683247 + 0.730188i \(0.260567\pi\)
0.290738 + 0.956803i \(0.406099\pi\)
\(444\) 0 0
\(445\) 6.50000 11.2583i 0.308130 0.533696i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 19.0000 32.9090i 0.893685 1.54791i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 6.24500 3.60555i 0.291175 0.168110i
\(461\) 7.21110i 0.335855i 0.985799 + 0.167927i \(0.0537074\pi\)
−0.985799 + 0.167927i \(0.946293\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 10.0000 + 17.3205i 0.464238 + 0.804084i
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) 21.6333i 1.00000i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 28.8444i 1.32347i
\(476\) 0 0
\(477\) −33.0000 −1.51097
\(478\) 0 0
\(479\) −34.3475 19.8305i −1.56938 0.906080i −0.996241 0.0866194i \(-0.972394\pi\)
−0.573135 0.819461i \(-0.694273\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000 1.00000
\(485\) −32.5000 56.2917i −1.47575 2.55607i
\(486\) 0 0
\(487\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −40.0000 −1.80517 −0.902587 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 43.2666i 1.94273i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(500\) 18.7350 + 10.8167i 0.837854 + 0.483735i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 12.0000 20.7846i 0.532414 0.922168i
\(509\) 3.12250 1.80278i 0.138402 0.0799066i −0.429200 0.903209i \(-0.641204\pi\)
0.567602 + 0.823303i \(0.307871\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(522\) 0 0
\(523\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.0000 19.0526i 0.478261 0.828372i
\(530\) 0 0
\(531\) 43.2666i 1.87761i
\(532\) 0 0
\(533\) −26.0000 −1.12619
\(534\) 0 0
\(535\) −24.9800 14.4222i −1.07998 0.623526i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 37.0000 1.58201 0.791003 0.611812i \(-0.209559\pi\)
0.791003 + 0.611812i \(0.209559\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −15.6125 + 9.01388i −0.665115 + 0.384004i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) 32.4500i 1.37249i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) −59.3275 + 34.2527i −2.49593 + 1.44102i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.500000 0.866025i −0.0209611 0.0363057i 0.855355 0.518043i \(-0.173339\pi\)
−0.876316 + 0.481737i \(0.840006\pi\)
\(570\) 0 0
\(571\) 1.50000 2.59808i 0.0627730 0.108726i −0.832931 0.553377i \(-0.813339\pi\)
0.895704 + 0.444651i \(0.146672\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 12.0000 + 20.7846i 0.500000 + 0.866025i
\(577\) 31.2250 + 18.0278i 1.29991 + 0.750505i 0.980389 0.197073i \(-0.0631435\pi\)
0.319524 + 0.947578i \(0.396477\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 36.0555i 1.49712i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −19.5000 + 33.7750i −0.806226 + 1.39642i
\(586\) 0 0
\(587\) 18.0278i 0.744085i −0.928216 0.372043i \(-0.878658\pi\)
0.928216 0.372043i \(-0.121342\pi\)
\(588\) 0 0
\(589\) 39.0000 1.60697
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −40.5925 + 23.4361i −1.66693 + 0.962405i −0.697657 + 0.716432i \(0.745774\pi\)
−0.969277 + 0.245973i \(0.920893\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.50000 + 9.52628i −0.224724 + 0.389233i −0.956237 0.292595i \(-0.905481\pi\)
0.731513 + 0.681828i \(0.238815\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −34.3475 19.8305i −1.39642 0.806226i
\(606\) 0 0
\(607\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.50000 + 11.2583i 0.262962 + 0.455463i
\(612\) 0 0
\(613\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −12.4900 7.21110i −0.502015 0.289839i 0.227530 0.973771i \(-0.426935\pi\)
−0.729545 + 0.683932i \(0.760268\pi\)
\(620\) −39.0000 + 67.5500i −1.56628 + 2.71287i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.500000 + 0.866025i 0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −37.4700 + 21.6333i −1.48695 + 0.858492i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8.50000 14.7224i 0.335730 0.581501i −0.647895 0.761730i \(-0.724350\pi\)
0.983625 + 0.180229i \(0.0576838\pi\)
\(642\) 0 0
\(643\) 43.2666i 1.70627i −0.521691 0.853134i \(-0.674699\pi\)
0.521691 0.853134i \(-0.325301\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17.0000 + 29.4449i 0.665261 + 1.15227i 0.979214 + 0.202828i \(0.0650132\pi\)
−0.313953 + 0.949439i \(0.601653\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 24.9800 14.4222i 0.975305 0.563093i
\(657\) 32.4500i 1.26599i
\(658\) 0 0
\(659\) −19.0000 −0.740135 −0.370067 0.929005i \(-0.620665\pi\)
−0.370067 + 0.929005i \(0.620665\pi\)
\(660\) 0 0
\(661\) 9.36750 + 5.40833i 0.364353 + 0.210360i 0.670989 0.741468i \(-0.265870\pi\)
−0.306635 + 0.951827i \(0.599203\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 2.50000 + 4.33013i 0.0968004 + 0.167663i
\(668\) 31.2250 + 18.0278i 1.20813 + 0.697515i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 51.0000 1.96591 0.982953 0.183858i \(-0.0588587\pi\)
0.982953 + 0.183858i \(0.0588587\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 13.0000 22.5167i 0.500000 0.866025i
\(677\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) −18.7350 + 10.8167i −0.716350 + 0.413585i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −18.0000 31.1769i −0.686244 1.18861i
\(689\) −34.3475 19.8305i −1.30854 0.755483i
\(690\) 0 0
\(691\) −40.5925 + 23.4361i −1.54421 + 0.891551i −0.545645 + 0.838016i \(0.683715\pi\)
−0.998566 + 0.0535342i \(0.982951\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 23.0000 0.868698 0.434349 0.900745i \(-0.356978\pi\)
0.434349 + 0.900745i \(0.356978\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) 0 0
\(711\) 22.5000 38.9711i 0.843816 1.46153i
\(712\) 0 0
\(713\) 10.8167i 0.405087i
\(714\) 0 0
\(715\) 0 0
\(716\) −25.0000 43.3013i −0.934294 1.61824i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 43.2666i 1.61245i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −20.0000 + 34.6410i −0.742781 + 1.28654i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 46.8375 27.0416i 1.72998 0.998806i 0.840560 0.541718i \(-0.182226\pi\)
0.889422 0.457087i \(-0.151107\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 46.8375 27.0416i 1.71369 0.989402i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 13.5000 + 23.3827i 0.492622 + 0.853246i 0.999964 0.00849853i \(-0.00270520\pi\)
−0.507342 + 0.861745i \(0.669372\pi\)
\(752\) −12.4900 7.21110i −0.455463 0.262962i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −47.0000 −1.70824 −0.854122 0.520073i \(-0.825905\pi\)
−0.854122 + 0.520073i \(0.825905\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −40.5925 + 23.4361i −1.47148 + 0.849557i −0.999486 0.0320465i \(-0.989798\pi\)
−0.471990 + 0.881604i \(0.656464\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −40.0000 −1.44715
\(765\) 0 0
\(766\) 0 0
\(767\) 26.0000 45.0333i 0.938806 1.62606i
\(768\) 0 0
\(769\) 32.4500i 1.17018i 0.810970 + 0.585088i \(0.198940\pi\)
−0.810970 + 0.585088i \(0.801060\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 31.2250 + 18.0278i 1.12308 + 0.648413i 0.942187 0.335089i \(-0.108766\pi\)
0.180898 + 0.983502i \(0.442100\pi\)
\(774\) 0 0
\(775\) 74.9400 43.2666i 2.69192 1.55418i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 13.0000 + 22.5167i 0.465773 + 0.806743i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −34.3475 19.8305i −1.22436 0.706882i −0.258512 0.966008i \(-0.583232\pi\)
−0.965844 + 0.259126i \(0.916566\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 9.36750 + 5.40833i 0.330984 + 0.191094i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 15.5000 26.8468i 0.544951 0.943883i −0.453659 0.891175i \(-0.649882\pi\)
0.998610 0.0527074i \(-0.0167851\pi\)
\(810\) 0 0
\(811\) 43.2666i 1.51930i −0.650334 0.759648i \(-0.725371\pi\)
0.650334 0.759648i \(-0.274629\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 28.1025 16.2250i 0.983182 0.567640i
\(818\) 0 0
\(819\) 0 0
\(820\) −52.0000 −1.81592
\(821\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(822\) 0 0
\(823\) −2.00000 + 3.46410i −0.0697156 + 0.120751i −0.898776 0.438408i \(-0.855543\pi\)
0.829060 + 0.559159i \(0.188876\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 3.00000 + 5.19615i 0.104257 + 0.180579i
\(829\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 28.8444i 1.00000i
\(833\) 0 0
\(834\) 0 0
\(835\) −32.5000 56.2917i −1.12471 1.94805i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 57.6888i 1.99164i 0.0913415 + 0.995820i \(0.470885\pi\)
−0.0913415 + 0.995820i \(0.529115\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 0 0
\(844\) 5.00000 8.66025i 0.172107 0.298098i
\(845\) −40.5925 + 23.4361i −1.39642 + 0.806226i
\(846\) 0 0
\(847\) 0 0
\(848\) 44.0000 1.51097
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 18.0278i 0.617259i −0.951182 0.308629i \(-0.900130\pi\)
0.951182 0.308629i \(-0.0998703\pi\)
\(854\) 0 0
\(855\) 39.0000 1.33377
\(856\) 0 0
\(857\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(858\) 0 0
\(859\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(860\) 64.8999i 2.21307i
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 46.8375 27.0416i 1.58521 0.915221i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 31.2250 + 18.0278i 1.04549 + 0.603614i
\(893\) 6.50000 11.2583i 0.217514 0.376746i
\(894\) 0 0
\(895\) 90.1388i 3.01301i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −46.8375 27.0416i −1.56212 0.901889i
\(900\) −24.0000 + 41.5692i −0.800000 + 1.38564i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −26.5000 + 45.8993i −0.879918 + 1.52406i −0.0284883 + 0.999594i \(0.509069\pi\)
−0.851430 + 0.524469i \(0.824264\pi\)
\(908\) 24.9800 14.4222i 0.828990 0.478618i
\(909\) 0 0
\(910\) 0 0
\(911\) 37.0000 1.22586 0.612932 0.790135i \(-0.289990\pi\)
0.612932 + 0.790135i \(0.289990\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 43.2666i 1.42957i
\(917\) 0 0
\(918\) 0 0
\(919\) 10.0000 + 17.3205i 0.329870 + 0.571351i 0.982486 0.186338i \(-0.0596619\pi\)
−0.652616 + 0.757689i \(0.726329\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.12250 1.80278i 0.102446 0.0591472i −0.447902 0.894083i \(-0.647829\pi\)
0.550348 + 0.834936i \(0.314495\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 58.0000 1.89985
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 13.0000 + 22.5167i 0.424013 + 0.734412i
\(941\) 53.0825 + 30.6472i 1.73044 + 0.999070i 0.886782 + 0.462188i \(0.152935\pi\)
0.843657 + 0.536882i \(0.180398\pi\)
\(942\) 0 0
\(943\) 6.24500 3.60555i 0.203365 0.117413i
\(944\) 57.6888i 1.87761i
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(948\) 0 0
\(949\) −19.5000 + 33.7750i −0.632997 + 1.09638i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −61.0000 −1.97598 −0.987992 0.154506i \(-0.950622\pi\)
−0.987992 + 0.154506i \(0.950622\pi\)
\(954\) 0 0
\(955\) 62.4500 + 36.0555i 2.02083 + 1.16673i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 43.0000 + 74.4782i 1.38710 + 2.40252i
\(962\) 0 0
\(963\) 12.0000 20.7846i 0.386695 0.669775i
\(964\) −18.7350 + 10.8167i −0.603414 + 0.348381i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 53.0825 + 30.6472i 1.69307 + 0.977493i 0.952017 + 0.306045i \(0.0990058\pi\)
0.741051 + 0.671449i \(0.234328\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −26.0000 −0.827170
\(989\) −4.50000 7.79423i −0.143092 0.247842i
\(990\) 0 0
\(991\) −30.0000 + 51.9615i −0.952981 + 1.65061i −0.214060 + 0.976820i \(0.568669\pi\)
−0.738921 + 0.673792i \(0.764664\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 637.2.r.b.116.2 4
7.2 even 3 inner 637.2.r.b.324.1 4
7.3 odd 6 637.2.c.b.246.1 2
7.4 even 3 637.2.c.b.246.2 yes 2
7.5 odd 6 inner 637.2.r.b.324.2 4
7.6 odd 2 inner 637.2.r.b.116.1 4
13.12 even 2 inner 637.2.r.b.116.1 4
91.12 odd 6 inner 637.2.r.b.324.1 4
91.18 odd 12 8281.2.a.u.1.2 2
91.25 even 6 637.2.c.b.246.1 2
91.31 even 12 8281.2.a.u.1.1 2
91.38 odd 6 637.2.c.b.246.2 yes 2
91.51 even 6 inner 637.2.r.b.324.2 4
91.60 odd 12 8281.2.a.u.1.1 2
91.73 even 12 8281.2.a.u.1.2 2
91.90 odd 2 CM 637.2.r.b.116.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.c.b.246.1 2 7.3 odd 6
637.2.c.b.246.1 2 91.25 even 6
637.2.c.b.246.2 yes 2 7.4 even 3
637.2.c.b.246.2 yes 2 91.38 odd 6
637.2.r.b.116.1 4 7.6 odd 2 inner
637.2.r.b.116.1 4 13.12 even 2 inner
637.2.r.b.116.2 4 1.1 even 1 trivial
637.2.r.b.116.2 4 91.90 odd 2 CM
637.2.r.b.324.1 4 7.2 even 3 inner
637.2.r.b.324.1 4 91.12 odd 6 inner
637.2.r.b.324.2 4 7.5 odd 6 inner
637.2.r.b.324.2 4 91.51 even 6 inner
8281.2.a.u.1.1 2 91.31 even 12
8281.2.a.u.1.1 2 91.60 odd 12
8281.2.a.u.1.2 2 91.18 odd 12
8281.2.a.u.1.2 2 91.73 even 12