Properties

Label 560.2.x.a.127.1
Level $560$
Weight $2$
Character 560.127
Analytic conductor $4.472$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,2,Mod(127,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.x (of order \(4\), degree \(2\), 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: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.426337261060096.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 127.1
Root \(1.19252 - 0.760198i\) of defining polynomial
Character \(\chi\) \(=\) 560.127
Dual form 560.2.x.a.463.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.95272 - 1.95272i) q^{3} +(0.432320 + 2.19388i) q^{5} +(-0.707107 + 0.707107i) q^{7} +4.62620i q^{9} +O(q^{10})\) \(q+(-1.95272 - 1.95272i) q^{3} +(0.432320 + 2.19388i) q^{5} +(-0.707107 + 0.707107i) q^{7} +4.62620i q^{9} -1.68840i q^{11} +(4.19388 - 4.19388i) q^{13} +(3.43982 - 5.12822i) q^{15} +(-1.32924 - 1.32924i) q^{17} +6.35101 q^{19} +2.76156 q^{21} +(-5.12822 - 5.12822i) q^{23} +(-4.62620 + 1.89692i) q^{25} +(3.17550 - 3.17550i) q^{27} -5.35548i q^{29} +0.528636i q^{31} +(-3.29696 + 3.29696i) q^{33} +(-1.85700 - 1.24561i) q^{35} +(-5.76156 - 5.76156i) q^{37} -16.3789 q^{39} +0.658473 q^{41} +(2.91118 + 2.91118i) q^{43} +(-10.1493 + 2.00000i) q^{45} +(8.83236 - 8.83236i) q^{47} -1.00000i q^{49} +5.19124i q^{51} +(7.38776 - 7.38776i) q^{53} +(3.70414 - 0.729929i) q^{55} +(-12.4017 - 12.4017i) q^{57} -2.97421 q^{59} +8.77551 q^{61} +(-3.27122 - 3.27122i) q^{63} +(11.0140 + 7.38776i) q^{65} +(4.37104 - 4.37104i) q^{67} +20.0279i q^{69} +7.81086i q^{71} +(-0.896916 + 0.896916i) q^{73} +(12.7378 + 5.32951i) q^{75} +(1.19388 + 1.19388i) q^{77} -13.0021 q^{79} +1.47689 q^{81} +(-5.65685 - 5.65685i) q^{83} +(2.34153 - 3.49084i) q^{85} +(-10.4577 + 10.4577i) q^{87} -6.27072i q^{89} +5.93104i q^{91} +(1.03228 - 1.03228i) q^{93} +(2.74567 + 13.9333i) q^{95} +(10.3109 + 10.3109i) q^{97} +7.81086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 20 q^{13} + 4 q^{17} + 8 q^{21} - 20 q^{25} - 24 q^{33} - 44 q^{37} - 32 q^{41} - 36 q^{45} + 28 q^{53} + 8 q^{57} - 16 q^{61} + 36 q^{65} + 4 q^{73} - 16 q^{77} + 68 q^{81} + 68 q^{85} + 8 q^{93} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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
\(3\) −1.95272 1.95272i −1.12740 1.12740i −0.990598 0.136803i \(-0.956317\pi\)
−0.136803 0.990598i \(-0.543683\pi\)
\(4\) 0 0
\(5\) 0.432320 + 2.19388i 0.193340 + 0.981132i
\(6\) 0 0
\(7\) −0.707107 + 0.707107i −0.267261 + 0.267261i
\(8\) 0 0
\(9\) 4.62620i 1.54207i
\(10\) 0 0
\(11\) 1.68840i 0.509071i −0.967063 0.254536i \(-0.918077\pi\)
0.967063 0.254536i \(-0.0819226\pi\)
\(12\) 0 0
\(13\) 4.19388 4.19388i 1.16317 1.16317i 0.179395 0.983777i \(-0.442586\pi\)
0.983777 0.179395i \(-0.0574141\pi\)
\(14\) 0 0
\(15\) 3.43982 5.12822i 0.888158 1.32410i
\(16\) 0 0
\(17\) −1.32924 1.32924i −0.322387 0.322387i 0.527295 0.849682i \(-0.323206\pi\)
−0.849682 + 0.527295i \(0.823206\pi\)
\(18\) 0 0
\(19\) 6.35101 1.45702 0.728510 0.685035i \(-0.240213\pi\)
0.728510 + 0.685035i \(0.240213\pi\)
\(20\) 0 0
\(21\) 2.76156 0.602621
\(22\) 0 0
\(23\) −5.12822 5.12822i −1.06931 1.06931i −0.997412 0.0718953i \(-0.977095\pi\)
−0.0718953 0.997412i \(-0.522905\pi\)
\(24\) 0 0
\(25\) −4.62620 + 1.89692i −0.925240 + 0.379383i
\(26\) 0 0
\(27\) 3.17550 3.17550i 0.611126 0.611126i
\(28\) 0 0
\(29\) 5.35548i 0.994488i −0.867611 0.497244i \(-0.834345\pi\)
0.867611 0.497244i \(-0.165655\pi\)
\(30\) 0 0
\(31\) 0.528636i 0.0949458i 0.998873 + 0.0474729i \(0.0151168\pi\)
−0.998873 + 0.0474729i \(0.984883\pi\)
\(32\) 0 0
\(33\) −3.29696 + 3.29696i −0.573927 + 0.573927i
\(34\) 0 0
\(35\) −1.85700 1.24561i −0.313891 0.210546i
\(36\) 0 0
\(37\) −5.76156 5.76156i −0.947194 0.947194i 0.0514799 0.998674i \(-0.483606\pi\)
−0.998674 + 0.0514799i \(0.983606\pi\)
\(38\) 0 0
\(39\) −16.3789 −2.62272
\(40\) 0 0
\(41\) 0.658473 0.102836 0.0514181 0.998677i \(-0.483626\pi\)
0.0514181 + 0.998677i \(0.483626\pi\)
\(42\) 0 0
\(43\) 2.91118 + 2.91118i 0.443952 + 0.443952i 0.893338 0.449386i \(-0.148357\pi\)
−0.449386 + 0.893338i \(0.648357\pi\)
\(44\) 0 0
\(45\) −10.1493 + 2.00000i −1.51297 + 0.298142i
\(46\) 0 0
\(47\) 8.83236 8.83236i 1.28833 1.28833i 0.352532 0.935800i \(-0.385321\pi\)
0.935800 0.352532i \(-0.114679\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) 5.19124i 0.726919i
\(52\) 0 0
\(53\) 7.38776 7.38776i 1.01479 1.01479i 0.0148972 0.999889i \(-0.495258\pi\)
0.999889 0.0148972i \(-0.00474210\pi\)
\(54\) 0 0
\(55\) 3.70414 0.729929i 0.499466 0.0984236i
\(56\) 0 0
\(57\) −12.4017 12.4017i −1.64265 1.64265i
\(58\) 0 0
\(59\) −2.97421 −0.387209 −0.193605 0.981080i \(-0.562018\pi\)
−0.193605 + 0.981080i \(0.562018\pi\)
\(60\) 0 0
\(61\) 8.77551 1.12359 0.561794 0.827277i \(-0.310111\pi\)
0.561794 + 0.827277i \(0.310111\pi\)
\(62\) 0 0
\(63\) −3.27122 3.27122i −0.412134 0.412134i
\(64\) 0 0
\(65\) 11.0140 + 7.38776i 1.36611 + 0.916338i
\(66\) 0 0
\(67\) 4.37104 4.37104i 0.534008 0.534008i −0.387755 0.921763i \(-0.626749\pi\)
0.921763 + 0.387755i \(0.126749\pi\)
\(68\) 0 0
\(69\) 20.0279i 2.41108i
\(70\) 0 0
\(71\) 7.81086i 0.926979i 0.886102 + 0.463489i \(0.153403\pi\)
−0.886102 + 0.463489i \(0.846597\pi\)
\(72\) 0 0
\(73\) −0.896916 + 0.896916i −0.104976 + 0.104976i −0.757644 0.652668i \(-0.773650\pi\)
0.652668 + 0.757644i \(0.273650\pi\)
\(74\) 0 0
\(75\) 12.7378 + 5.32951i 1.47083 + 0.615399i
\(76\) 0 0
\(77\) 1.19388 + 1.19388i 0.136055 + 0.136055i
\(78\) 0 0
\(79\) −13.0021 −1.46285 −0.731426 0.681921i \(-0.761145\pi\)
−0.731426 + 0.681921i \(0.761145\pi\)
\(80\) 0 0
\(81\) 1.47689 0.164098
\(82\) 0 0
\(83\) −5.65685 5.65685i −0.620920 0.620920i 0.324846 0.945767i \(-0.394687\pi\)
−0.945767 + 0.324846i \(0.894687\pi\)
\(84\) 0 0
\(85\) 2.34153 3.49084i 0.253974 0.378635i
\(86\) 0 0
\(87\) −10.4577 + 10.4577i −1.12119 + 1.12119i
\(88\) 0 0
\(89\) 6.27072i 0.664695i −0.943157 0.332347i \(-0.892159\pi\)
0.943157 0.332347i \(-0.107841\pi\)
\(90\) 0 0
\(91\) 5.93104i 0.621742i
\(92\) 0 0
\(93\) 1.03228 1.03228i 0.107042 0.107042i
\(94\) 0 0
\(95\) 2.74567 + 13.9333i 0.281700 + 1.42953i
\(96\) 0 0
\(97\) 10.3109 + 10.3109i 1.04691 + 1.04691i 0.998844 + 0.0480708i \(0.0153073\pi\)
0.0480708 + 0.998844i \(0.484693\pi\)
\(98\) 0 0
\(99\) 7.81086 0.785021
\(100\) 0 0
\(101\) −0.387755 −0.0385831 −0.0192915 0.999814i \(-0.506141\pi\)
−0.0192915 + 0.999814i \(0.506141\pi\)
\(102\) 0 0
\(103\) 5.32951 + 5.32951i 0.525132 + 0.525132i 0.919117 0.393985i \(-0.128904\pi\)
−0.393985 + 0.919117i \(0.628904\pi\)
\(104\) 0 0
\(105\) 1.19388 + 6.05852i 0.116511 + 0.591251i
\(106\) 0 0
\(107\) −2.91118 + 2.91118i −0.281435 + 0.281435i −0.833681 0.552246i \(-0.813771\pi\)
0.552246 + 0.833681i \(0.313771\pi\)
\(108\) 0 0
\(109\) 3.14931i 0.301649i −0.988561 0.150825i \(-0.951807\pi\)
0.988561 0.150825i \(-0.0481929\pi\)
\(110\) 0 0
\(111\) 22.5014i 2.13574i
\(112\) 0 0
\(113\) −7.55539 + 7.55539i −0.710751 + 0.710751i −0.966692 0.255941i \(-0.917615\pi\)
0.255941 + 0.966692i \(0.417615\pi\)
\(114\) 0 0
\(115\) 9.03365 13.4677i 0.842392 1.25587i
\(116\) 0 0
\(117\) 19.4017 + 19.4017i 1.79369 + 1.79369i
\(118\) 0 0
\(119\) 1.87982 0.172323
\(120\) 0 0
\(121\) 8.14931 0.740847
\(122\) 0 0
\(123\) −1.28581 1.28581i −0.115938 0.115938i
\(124\) 0 0
\(125\) −6.16160 9.32924i −0.551110 0.834432i
\(126\) 0 0
\(127\) 3.43982 3.43982i 0.305235 0.305235i −0.537823 0.843058i \(-0.680753\pi\)
0.843058 + 0.537823i \(0.180753\pi\)
\(128\) 0 0
\(129\) 11.3694i 1.00102i
\(130\) 0 0
\(131\) 6.87964i 0.601077i 0.953770 + 0.300539i \(0.0971664\pi\)
−0.953770 + 0.300539i \(0.902834\pi\)
\(132\) 0 0
\(133\) −4.49084 + 4.49084i −0.389405 + 0.389405i
\(134\) 0 0
\(135\) 8.33950 + 5.59383i 0.717750 + 0.481440i
\(136\) 0 0
\(137\) −6.62620 6.62620i −0.566114 0.566114i 0.364923 0.931038i \(-0.381095\pi\)
−0.931038 + 0.364923i \(0.881095\pi\)
\(138\) 0 0
\(139\) 18.9985 1.61143 0.805717 0.592301i \(-0.201780\pi\)
0.805717 + 0.592301i \(0.201780\pi\)
\(140\) 0 0
\(141\) −34.4942 −2.90493
\(142\) 0 0
\(143\) −7.08093 7.08093i −0.592137 0.592137i
\(144\) 0 0
\(145\) 11.7493 2.31528i 0.975724 0.192274i
\(146\) 0 0
\(147\) −1.95272 + 1.95272i −0.161057 + 0.161057i
\(148\) 0 0
\(149\) 3.52311i 0.288625i 0.989532 + 0.144312i \(0.0460970\pi\)
−0.989532 + 0.144312i \(0.953903\pi\)
\(150\) 0 0
\(151\) 3.20275i 0.260636i 0.991472 + 0.130318i \(0.0415998\pi\)
−0.991472 + 0.130318i \(0.958400\pi\)
\(152\) 0 0
\(153\) 6.14931 6.14931i 0.497142 0.497142i
\(154\) 0 0
\(155\) −1.15976 + 0.228540i −0.0931543 + 0.0183568i
\(156\) 0 0
\(157\) −8.08476 8.08476i −0.645234 0.645234i 0.306603 0.951837i \(-0.400808\pi\)
−0.951837 + 0.306603i \(0.900808\pi\)
\(158\) 0 0
\(159\) −28.8524 −2.28814
\(160\) 0 0
\(161\) 7.25240 0.571569
\(162\) 0 0
\(163\) 8.50501 + 8.50501i 0.666164 + 0.666164i 0.956826 0.290662i \(-0.0938754\pi\)
−0.290662 + 0.956826i \(0.593875\pi\)
\(164\) 0 0
\(165\) −8.65847 5.80779i −0.674061 0.452136i
\(166\) 0 0
\(167\) −15.5859 + 15.5859i −1.20608 + 1.20608i −0.233790 + 0.972287i \(0.575113\pi\)
−0.972287 + 0.233790i \(0.924887\pi\)
\(168\) 0 0
\(169\) 22.1772i 1.70594i
\(170\) 0 0
\(171\) 29.3810i 2.24682i
\(172\) 0 0
\(173\) −5.80612 + 5.80612i −0.441431 + 0.441431i −0.892493 0.451062i \(-0.851045\pi\)
0.451062 + 0.892493i \(0.351045\pi\)
\(174\) 0 0
\(175\) 1.92989 4.61254i 0.145886 0.348675i
\(176\) 0 0
\(177\) 5.80779 + 5.80779i 0.436540 + 0.436540i
\(178\) 0 0
\(179\) −20.9870 −1.56864 −0.784322 0.620354i \(-0.786989\pi\)
−0.784322 + 0.620354i \(0.786989\pi\)
\(180\) 0 0
\(181\) −5.13536 −0.381708 −0.190854 0.981618i \(-0.561126\pi\)
−0.190854 + 0.981618i \(0.561126\pi\)
\(182\) 0 0
\(183\) −17.1361 17.1361i −1.26674 1.26674i
\(184\) 0 0
\(185\) 10.1493 15.1310i 0.746192 1.11245i
\(186\) 0 0
\(187\) −2.24428 + 2.24428i −0.164118 + 0.164118i
\(188\) 0 0
\(189\) 4.49084i 0.326660i
\(190\) 0 0
\(191\) 11.0136i 0.796917i −0.917186 0.398459i \(-0.869545\pi\)
0.917186 0.398459i \(-0.130455\pi\)
\(192\) 0 0
\(193\) −13.1955 + 13.1955i −0.949836 + 0.949836i −0.998801 0.0489647i \(-0.984408\pi\)
0.0489647 + 0.998801i \(0.484408\pi\)
\(194\) 0 0
\(195\) −7.08093 35.9333i −0.507076 2.57324i
\(196\) 0 0
\(197\) −0.640151 0.640151i −0.0456089 0.0456089i 0.683935 0.729543i \(-0.260267\pi\)
−0.729543 + 0.683935i \(0.760267\pi\)
\(198\) 0 0
\(199\) 25.3495 1.79698 0.898490 0.438994i \(-0.144665\pi\)
0.898490 + 0.438994i \(0.144665\pi\)
\(200\) 0 0
\(201\) −17.0708 −1.20408
\(202\) 0 0
\(203\) 3.78690 + 3.78690i 0.265788 + 0.265788i
\(204\) 0 0
\(205\) 0.284672 + 1.44461i 0.0198823 + 0.100896i
\(206\) 0 0
\(207\) 23.7242 23.7242i 1.64894 1.64894i
\(208\) 0 0
\(209\) 10.7230i 0.741727i
\(210\) 0 0
\(211\) 19.8817i 1.36872i 0.729146 + 0.684358i \(0.239917\pi\)
−0.729146 + 0.684358i \(0.760083\pi\)
\(212\) 0 0
\(213\) 15.2524 15.2524i 1.04508 1.04508i
\(214\) 0 0
\(215\) −5.12822 + 7.64535i −0.349742 + 0.521408i
\(216\) 0 0
\(217\) −0.373802 0.373802i −0.0253753 0.0253753i
\(218\) 0 0
\(219\) 3.50285 0.236700
\(220\) 0 0
\(221\) −11.1493 −0.749984
\(222\) 0 0
\(223\) −2.88394 2.88394i −0.193123 0.193123i 0.603921 0.797044i \(-0.293604\pi\)
−0.797044 + 0.603921i \(0.793604\pi\)
\(224\) 0 0
\(225\) −8.77551 21.4017i −0.585034 1.42678i
\(226\) 0 0
\(227\) 10.7493 10.7493i 0.713456 0.713456i −0.253801 0.967257i \(-0.581681\pi\)
0.967257 + 0.253801i \(0.0816808\pi\)
\(228\) 0 0
\(229\) 12.8925i 0.851964i 0.904732 + 0.425982i \(0.140071\pi\)
−0.904732 + 0.425982i \(0.859929\pi\)
\(230\) 0 0
\(231\) 4.66261i 0.306777i
\(232\) 0 0
\(233\) −7.67243 + 7.67243i −0.502637 + 0.502637i −0.912257 0.409619i \(-0.865662\pi\)
0.409619 + 0.912257i \(0.365662\pi\)
\(234\) 0 0
\(235\) 23.1955 + 15.5587i 1.51311 + 1.01494i
\(236\) 0 0
\(237\) 25.3894 + 25.3894i 1.64922 + 1.64922i
\(238\) 0 0
\(239\) −9.62531 −0.622610 −0.311305 0.950310i \(-0.600766\pi\)
−0.311305 + 0.950310i \(0.600766\pi\)
\(240\) 0 0
\(241\) −8.56934 −0.552000 −0.276000 0.961158i \(-0.589009\pi\)
−0.276000 + 0.961158i \(0.589009\pi\)
\(242\) 0 0
\(243\) −12.4104 12.4104i −0.796130 0.796130i
\(244\) 0 0
\(245\) 2.19388 0.432320i 0.140162 0.0276199i
\(246\) 0 0
\(247\) 26.6353 26.6353i 1.69477 1.69477i
\(248\) 0 0
\(249\) 22.0925i 1.40005i
\(250\) 0 0
\(251\) 5.82237i 0.367505i −0.982973 0.183752i \(-0.941176\pi\)
0.982973 0.183752i \(-0.0588244\pi\)
\(252\) 0 0
\(253\) −8.65847 + 8.65847i −0.544354 + 0.544354i
\(254\) 0 0
\(255\) −11.3890 + 2.24428i −0.713204 + 0.140542i
\(256\) 0 0
\(257\) 20.1493 + 20.1493i 1.25688 + 1.25688i 0.952575 + 0.304305i \(0.0984242\pi\)
0.304305 + 0.952575i \(0.401576\pi\)
\(258\) 0 0
\(259\) 8.14807 0.506297
\(260\) 0 0
\(261\) 24.7755 1.53357
\(262\) 0 0
\(263\) 22.0358 + 22.0358i 1.35878 + 1.35878i 0.875420 + 0.483363i \(0.160585\pi\)
0.483363 + 0.875420i \(0.339415\pi\)
\(264\) 0 0
\(265\) 19.4017 + 13.0140i 1.19184 + 0.799441i
\(266\) 0 0
\(267\) −12.2449 + 12.2449i −0.749378 + 0.749378i
\(268\) 0 0
\(269\) 6.00000i 0.365826i 0.983129 + 0.182913i \(0.0585527\pi\)
−0.983129 + 0.182913i \(0.941447\pi\)
\(270\) 0 0
\(271\) 32.6318i 1.98224i −0.132978 0.991119i \(-0.542454\pi\)
0.132978 0.991119i \(-0.457546\pi\)
\(272\) 0 0
\(273\) 11.5816 11.5816i 0.700952 0.700952i
\(274\) 0 0
\(275\) 3.20275 + 7.81086i 0.193133 + 0.471013i
\(276\) 0 0
\(277\) −6.03228 6.03228i −0.362444 0.362444i 0.502268 0.864712i \(-0.332499\pi\)
−0.864712 + 0.502268i \(0.832499\pi\)
\(278\) 0 0
\(279\) −2.44557 −0.146413
\(280\) 0 0
\(281\) 5.56165 0.331780 0.165890 0.986144i \(-0.446950\pi\)
0.165890 + 0.986144i \(0.446950\pi\)
\(282\) 0 0
\(283\) −17.1719 17.1719i −1.02076 1.02076i −0.999780 0.0209810i \(-0.993321\pi\)
−0.0209810 0.999780i \(-0.506679\pi\)
\(284\) 0 0
\(285\) 21.8463 32.5693i 1.29406 1.92924i
\(286\) 0 0
\(287\) −0.465611 + 0.465611i −0.0274842 + 0.0274842i
\(288\) 0 0
\(289\) 13.4663i 0.792133i
\(290\) 0 0
\(291\) 40.2686i 2.36059i
\(292\) 0 0
\(293\) −18.4280 + 18.4280i −1.07657 + 1.07657i −0.0797582 + 0.996814i \(0.525415\pi\)
−0.996814 + 0.0797582i \(0.974585\pi\)
\(294\) 0 0
\(295\) −1.28581 6.52505i −0.0748628 0.379903i
\(296\) 0 0
\(297\) −5.36151 5.36151i −0.311106 0.311106i
\(298\) 0 0
\(299\) −43.0142 −2.48758
\(300\) 0 0
\(301\) −4.11704 −0.237302
\(302\) 0 0
\(303\) 0.757176 + 0.757176i 0.0434986 + 0.0434986i
\(304\) 0 0
\(305\) 3.79383 + 19.2524i 0.217234 + 1.10239i
\(306\) 0 0
\(307\) −10.8209 + 10.8209i −0.617579 + 0.617579i −0.944910 0.327331i \(-0.893851\pi\)
0.327331 + 0.944910i \(0.393851\pi\)
\(308\) 0 0
\(309\) 20.8140i 1.18407i
\(310\) 0 0
\(311\) 10.9111i 0.618713i −0.950946 0.309356i \(-0.899886\pi\)
0.950946 0.309356i \(-0.100114\pi\)
\(312\) 0 0
\(313\) −14.3109 + 14.3109i −0.808901 + 0.808901i −0.984468 0.175567i \(-0.943824\pi\)
0.175567 + 0.984468i \(0.443824\pi\)
\(314\) 0 0
\(315\) 5.76243 8.59086i 0.324676 0.484040i
\(316\) 0 0
\(317\) 5.50479 + 5.50479i 0.309180 + 0.309180i 0.844591 0.535411i \(-0.179843\pi\)
−0.535411 + 0.844591i \(0.679843\pi\)
\(318\) 0 0
\(319\) −9.04218 −0.506265
\(320\) 0 0
\(321\) 11.3694 0.634580
\(322\) 0 0
\(323\) −8.44199 8.44199i −0.469725 0.469725i
\(324\) 0 0
\(325\) −11.4463 + 27.3571i −0.634925 + 1.51750i
\(326\) 0 0
\(327\) −6.14971 + 6.14971i −0.340080 + 0.340080i
\(328\) 0 0
\(329\) 12.4908i 0.688642i
\(330\) 0 0
\(331\) 2.91972i 0.160482i −0.996775 0.0802410i \(-0.974431\pi\)
0.996775 0.0802410i \(-0.0255690\pi\)
\(332\) 0 0
\(333\) 26.6541 26.6541i 1.46064 1.46064i
\(334\) 0 0
\(335\) 11.4792 + 7.69984i 0.627177 + 0.420687i
\(336\) 0 0
\(337\) −1.01395 1.01395i −0.0552336 0.0552336i 0.678950 0.734184i \(-0.262435\pi\)
−0.734184 + 0.678950i \(0.762435\pi\)
\(338\) 0 0
\(339\) 29.5071 1.60260
\(340\) 0 0
\(341\) 0.892548 0.0483342
\(342\) 0 0
\(343\) 0.707107 + 0.707107i 0.0381802 + 0.0381802i
\(344\) 0 0
\(345\) −43.9388 + 8.65847i −2.36558 + 0.466157i
\(346\) 0 0
\(347\) 13.9964 13.9964i 0.751364 0.751364i −0.223370 0.974734i \(-0.571706\pi\)
0.974734 + 0.223370i \(0.0717058\pi\)
\(348\) 0 0
\(349\) 12.3232i 0.659646i −0.944043 0.329823i \(-0.893011\pi\)
0.944043 0.329823i \(-0.106989\pi\)
\(350\) 0 0
\(351\) 26.6353i 1.42169i
\(352\) 0 0
\(353\) −13.0585 + 13.0585i −0.695035 + 0.695035i −0.963335 0.268300i \(-0.913538\pi\)
0.268300 + 0.963335i \(0.413538\pi\)
\(354\) 0 0
\(355\) −17.1361 + 3.37680i −0.909489 + 0.179222i
\(356\) 0 0
\(357\) −3.67076 3.67076i −0.194277 0.194277i
\(358\) 0 0
\(359\) −6.42256 −0.338970 −0.169485 0.985533i \(-0.554210\pi\)
−0.169485 + 0.985533i \(0.554210\pi\)
\(360\) 0 0
\(361\) 21.3353 1.12291
\(362\) 0 0
\(363\) −15.9133 15.9133i −0.835231 0.835231i
\(364\) 0 0
\(365\) −2.35548 1.57997i −0.123291 0.0826994i
\(366\) 0 0
\(367\) 0.895444 0.895444i 0.0467418 0.0467418i −0.683350 0.730091i \(-0.739477\pi\)
0.730091 + 0.683350i \(0.239477\pi\)
\(368\) 0 0
\(369\) 3.04623i 0.158580i
\(370\) 0 0
\(371\) 10.4479i 0.542426i
\(372\) 0 0
\(373\) 20.6541 20.6541i 1.06943 1.06943i 0.0720263 0.997403i \(-0.477053\pi\)
0.997403 0.0720263i \(-0.0229465\pi\)
\(374\) 0 0
\(375\) −6.18549 + 30.2492i −0.319417 + 1.56206i
\(376\) 0 0
\(377\) −22.4602 22.4602i −1.15676 1.15676i
\(378\) 0 0
\(379\) 23.5586 1.21013 0.605063 0.796178i \(-0.293148\pi\)
0.605063 + 0.796178i \(0.293148\pi\)
\(380\) 0 0
\(381\) −13.4340 −0.688244
\(382\) 0 0
\(383\) 6.58808 + 6.58808i 0.336635 + 0.336635i 0.855099 0.518464i \(-0.173496\pi\)
−0.518464 + 0.855099i \(0.673496\pi\)
\(384\) 0 0
\(385\) −2.10308 + 3.13536i −0.107183 + 0.159793i
\(386\) 0 0
\(387\) −13.4677 + 13.4677i −0.684603 + 0.684603i
\(388\) 0 0
\(389\) 26.2234i 1.32958i −0.747030 0.664791i \(-0.768521\pi\)
0.747030 0.664791i \(-0.231479\pi\)
\(390\) 0 0
\(391\) 13.6332i 0.689462i
\(392\) 0 0
\(393\) 13.4340 13.4340i 0.677655 0.677655i
\(394\) 0 0
\(395\) −5.62108 28.5250i −0.282827 1.43525i
\(396\) 0 0
\(397\) −7.35714 7.35714i −0.369244 0.369244i 0.497957 0.867202i \(-0.334084\pi\)
−0.867202 + 0.497957i \(0.834084\pi\)
\(398\) 0 0
\(399\) 17.5387 0.878031
\(400\) 0 0
\(401\) −8.33716 −0.416338 −0.208169 0.978093i \(-0.566750\pi\)
−0.208169 + 0.978093i \(0.566750\pi\)
\(402\) 0 0
\(403\) 2.21703 + 2.21703i 0.110438 + 0.110438i
\(404\) 0 0
\(405\) 0.638488 + 3.24011i 0.0317267 + 0.161002i
\(406\) 0 0
\(407\) −9.72780 + 9.72780i −0.482189 + 0.482189i
\(408\) 0 0
\(409\) 15.8463i 0.783550i 0.920061 + 0.391775i \(0.128139\pi\)
−0.920061 + 0.391775i \(0.871861\pi\)
\(410\) 0 0
\(411\) 25.8782i 1.27648i
\(412\) 0 0
\(413\) 2.10308 2.10308i 0.103486 0.103486i
\(414\) 0 0
\(415\) 9.96487 14.8560i 0.489156 0.729253i
\(416\) 0 0
\(417\) −37.0987 37.0987i −1.81673 1.81673i
\(418\) 0 0
\(419\) 13.7048 0.669523 0.334761 0.942303i \(-0.391344\pi\)
0.334761 + 0.942303i \(0.391344\pi\)
\(420\) 0 0
\(421\) −13.7678 −0.671002 −0.335501 0.942040i \(-0.608906\pi\)
−0.335501 + 0.942040i \(0.608906\pi\)
\(422\) 0 0
\(423\) 40.8602 + 40.8602i 1.98669 + 1.98669i
\(424\) 0 0
\(425\) 8.67076 + 3.62786i 0.420594 + 0.175977i
\(426\) 0 0
\(427\) −6.20522 + 6.20522i −0.300292 + 0.300292i
\(428\) 0 0
\(429\) 27.6541i 1.33515i
\(430\) 0 0
\(431\) 22.8015i 1.09831i −0.835721 0.549154i \(-0.814950\pi\)
0.835721 0.549154i \(-0.185050\pi\)
\(432\) 0 0
\(433\) −8.98605 + 8.98605i −0.431842 + 0.431842i −0.889255 0.457413i \(-0.848776\pi\)
0.457413 + 0.889255i \(0.348776\pi\)
\(434\) 0 0
\(435\) −27.4641 18.4219i −1.31680 0.883262i
\(436\) 0 0
\(437\) −32.5693 32.5693i −1.55800 1.55800i
\(438\) 0 0
\(439\) 17.1361 0.817860 0.408930 0.912566i \(-0.365902\pi\)
0.408930 + 0.912566i \(0.365902\pi\)
\(440\) 0 0
\(441\) 4.62620 0.220295
\(442\) 0 0
\(443\) 10.2650 + 10.2650i 0.487703 + 0.487703i 0.907581 0.419877i \(-0.137927\pi\)
−0.419877 + 0.907581i \(0.637927\pi\)
\(444\) 0 0
\(445\) 13.7572 2.71096i 0.652153 0.128512i
\(446\) 0 0
\(447\) 6.87964 6.87964i 0.325396 0.325396i
\(448\) 0 0
\(449\) 31.1772i 1.47134i 0.677338 + 0.735672i \(0.263133\pi\)
−0.677338 + 0.735672i \(0.736867\pi\)
\(450\) 0 0
\(451\) 1.11177i 0.0523510i
\(452\) 0 0
\(453\) 6.25406 6.25406i 0.293841 0.293841i
\(454\) 0 0
\(455\) −13.0120 + 2.56411i −0.610011 + 0.120207i
\(456\) 0 0
\(457\) −3.72302 3.72302i −0.174156 0.174156i 0.614647 0.788802i \(-0.289299\pi\)
−0.788802 + 0.614647i \(0.789299\pi\)
\(458\) 0 0
\(459\) −8.44199 −0.394038
\(460\) 0 0
\(461\) 13.3815 0.623238 0.311619 0.950207i \(-0.399129\pi\)
0.311619 + 0.950207i \(0.399129\pi\)
\(462\) 0 0
\(463\) 13.9248 + 13.9248i 0.647140 + 0.647140i 0.952301 0.305161i \(-0.0987102\pi\)
−0.305161 + 0.952301i \(0.598710\pi\)
\(464\) 0 0
\(465\) 2.71096 + 1.81841i 0.125718 + 0.0843268i
\(466\) 0 0
\(467\) 11.6805 11.6805i 0.540510 0.540510i −0.383169 0.923678i \(-0.625167\pi\)
0.923678 + 0.383169i \(0.125167\pi\)
\(468\) 0 0
\(469\) 6.18159i 0.285439i
\(470\) 0 0
\(471\) 31.5745i 1.45488i
\(472\) 0 0
\(473\) 4.91524 4.91524i 0.226003 0.226003i
\(474\) 0 0
\(475\) −29.3810 + 12.0473i −1.34809 + 0.552769i
\(476\) 0 0
\(477\) 34.1772 + 34.1772i 1.56487 + 1.56487i
\(478\) 0 0
\(479\) −23.0845 −1.05476 −0.527379 0.849630i \(-0.676825\pi\)
−0.527379 + 0.849630i \(0.676825\pi\)
\(480\) 0 0
\(481\) −48.3265 −2.20350
\(482\) 0 0
\(483\) −14.1619 14.1619i −0.644387 0.644387i
\(484\) 0 0
\(485\) −18.1633 + 27.0785i −0.824751 + 1.22957i
\(486\) 0 0
\(487\) −4.19700 + 4.19700i −0.190184 + 0.190184i −0.795776 0.605592i \(-0.792937\pi\)
0.605592 + 0.795776i \(0.292937\pi\)
\(488\) 0 0
\(489\) 33.2158i 1.50207i
\(490\) 0 0
\(491\) 16.2529i 0.733481i 0.930323 + 0.366740i \(0.119526\pi\)
−0.930323 + 0.366740i \(0.880474\pi\)
\(492\) 0 0
\(493\) −7.11870 + 7.11870i −0.320610 + 0.320610i
\(494\) 0 0
\(495\) 3.37680 + 17.1361i 0.151776 + 0.770209i
\(496\) 0 0
\(497\) −5.52311 5.52311i −0.247746 0.247746i
\(498\) 0 0
\(499\) 26.3043 1.17754 0.588771 0.808300i \(-0.299612\pi\)
0.588771 + 0.808300i \(0.299612\pi\)
\(500\) 0 0
\(501\) 60.8699 2.71946
\(502\) 0 0
\(503\) 13.2664 + 13.2664i 0.591521 + 0.591521i 0.938042 0.346521i \(-0.112637\pi\)
−0.346521 + 0.938042i \(0.612637\pi\)
\(504\) 0 0
\(505\) −0.167635 0.850688i −0.00745964 0.0378551i
\(506\) 0 0
\(507\) −43.3058 + 43.3058i −1.92328 + 1.92328i
\(508\) 0 0
\(509\) 33.3449i 1.47798i −0.673714 0.738992i \(-0.735302\pi\)
0.673714 0.738992i \(-0.264698\pi\)
\(510\) 0 0
\(511\) 1.26843i 0.0561121i
\(512\) 0 0
\(513\) 20.1676 20.1676i 0.890423 0.890423i
\(514\) 0 0
\(515\) −9.38824 + 13.9964i −0.413695 + 0.616753i
\(516\) 0 0
\(517\) −14.9125 14.9125i −0.655852 0.655852i
\(518\) 0 0
\(519\) 22.6754 0.995340
\(520\) 0 0
\(521\) 44.9850 1.97083 0.985414 0.170172i \(-0.0544322\pi\)
0.985414 + 0.170172i \(0.0544322\pi\)
\(522\) 0 0
\(523\) 12.9391 + 12.9391i 0.565787 + 0.565787i 0.930945 0.365159i \(-0.118985\pi\)
−0.365159 + 0.930945i \(0.618985\pi\)
\(524\) 0 0
\(525\) −12.7755 + 5.23844i −0.557569 + 0.228624i
\(526\) 0 0
\(527\) 0.702682 0.702682i 0.0306093 0.0306093i
\(528\) 0 0
\(529\) 29.5972i 1.28684i
\(530\) 0 0
\(531\) 13.7593i 0.597102i
\(532\) 0 0
\(533\) 2.76156 2.76156i 0.119616 0.119616i
\(534\) 0 0
\(535\) −7.64535 5.12822i −0.330537 0.221712i
\(536\) 0 0
\(537\) 40.9817 + 40.9817i 1.76849 + 1.76849i
\(538\) 0 0
\(539\) −1.68840 −0.0727244
\(540\) 0 0
\(541\) 29.3188 1.26052 0.630258 0.776386i \(-0.282949\pi\)
0.630258 + 0.776386i \(0.282949\pi\)
\(542\) 0 0
\(543\) 10.0279 + 10.0279i 0.430338 + 0.430338i
\(544\) 0 0
\(545\) 6.90921 1.36151i 0.295958 0.0583208i
\(546\) 0 0
\(547\) −21.7357 + 21.7357i −0.929350 + 0.929350i −0.997664 0.0683141i \(-0.978238\pi\)
0.0683141 + 0.997664i \(0.478238\pi\)
\(548\) 0 0
\(549\) 40.5972i 1.73265i
\(550\) 0 0
\(551\) 34.0127i 1.44899i
\(552\) 0 0
\(553\) 9.19388 9.19388i 0.390963 0.390963i
\(554\) 0 0
\(555\) −49.3652 + 9.72780i −2.09544 + 0.412922i
\(556\) 0 0
\(557\) 0.0847616 + 0.0847616i 0.00359146 + 0.00359146i 0.708900 0.705309i \(-0.249192\pi\)
−0.705309 + 0.708900i \(0.749192\pi\)
\(558\) 0 0
\(559\) 24.4183 1.03278
\(560\) 0 0
\(561\) 8.76488 0.370054
\(562\) 0 0
\(563\) 1.22279 + 1.22279i 0.0515343 + 0.0515343i 0.732404 0.680870i \(-0.238398\pi\)
−0.680870 + 0.732404i \(0.738398\pi\)
\(564\) 0 0
\(565\) −19.8420 13.3093i −0.834757 0.559924i
\(566\) 0 0
\(567\) −1.04432 + 1.04432i −0.0438571 + 0.0438571i
\(568\) 0 0
\(569\) 19.1108i 0.801166i 0.916261 + 0.400583i \(0.131192\pi\)
−0.916261 + 0.400583i \(0.868808\pi\)
\(570\) 0 0
\(571\) 28.3237i 1.18531i 0.805456 + 0.592656i \(0.201921\pi\)
−0.805456 + 0.592656i \(0.798079\pi\)
\(572\) 0 0
\(573\) −21.5065 + 21.5065i −0.898445 + 0.898445i
\(574\) 0 0
\(575\) 33.4520 + 13.9964i 1.39504 + 0.583688i
\(576\) 0 0
\(577\) 8.10475 + 8.10475i 0.337405 + 0.337405i 0.855390 0.517985i \(-0.173318\pi\)
−0.517985 + 0.855390i \(0.673318\pi\)
\(578\) 0 0
\(579\) 51.5343 2.14169
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) −12.4735 12.4735i −0.516598 0.516598i
\(584\) 0 0
\(585\) −34.1772 + 50.9527i −1.41305 + 2.10664i
\(586\) 0 0
\(587\) −5.53080 + 5.53080i −0.228281 + 0.228281i −0.811974 0.583693i \(-0.801607\pi\)
0.583693 + 0.811974i \(0.301607\pi\)
\(588\) 0 0
\(589\) 3.35737i 0.138338i
\(590\) 0 0
\(591\) 2.50007i 0.102839i
\(592\) 0 0
\(593\) 4.24636 4.24636i 0.174377 0.174377i −0.614522 0.788899i \(-0.710651\pi\)
0.788899 + 0.614522i \(0.210651\pi\)
\(594\) 0 0
\(595\) 0.812687 + 4.12411i 0.0333169 + 0.169072i
\(596\) 0 0
\(597\) −49.5004 49.5004i −2.02592 2.02592i
\(598\) 0 0
\(599\) −35.0293 −1.43126 −0.715630 0.698480i \(-0.753860\pi\)
−0.715630 + 0.698480i \(0.753860\pi\)
\(600\) 0 0
\(601\) 24.5573 1.00171 0.500856 0.865531i \(-0.333019\pi\)
0.500856 + 0.865531i \(0.333019\pi\)
\(602\) 0 0
\(603\) 20.2213 + 20.2213i 0.823475 + 0.823475i
\(604\) 0 0
\(605\) 3.52311 + 17.8786i 0.143235 + 0.726868i
\(606\) 0 0
\(607\) 3.99570 3.99570i 0.162181 0.162181i −0.621351 0.783532i \(-0.713416\pi\)
0.783532 + 0.621351i \(0.213416\pi\)
\(608\) 0 0
\(609\) 14.7895i 0.599299i
\(610\) 0 0
\(611\) 74.0836i 2.99710i
\(612\) 0 0
\(613\) −15.7110 + 15.7110i −0.634560 + 0.634560i −0.949208 0.314649i \(-0.898113\pi\)
0.314649 + 0.949208i \(0.398113\pi\)
\(614\) 0 0
\(615\) 2.26503 3.37680i 0.0913348 0.136166i
\(616\) 0 0
\(617\) 16.4725 + 16.4725i 0.663159 + 0.663159i 0.956123 0.292965i \(-0.0946418\pi\)
−0.292965 + 0.956123i \(0.594642\pi\)
\(618\) 0 0
\(619\) −16.0788 −0.646262 −0.323131 0.946354i \(-0.604735\pi\)
−0.323131 + 0.946354i \(0.604735\pi\)
\(620\) 0 0
\(621\) −32.5693 −1.30696
\(622\) 0 0
\(623\) 4.43407 + 4.43407i 0.177647 + 0.177647i
\(624\) 0 0
\(625\) 17.8034 17.5510i 0.712137 0.702041i
\(626\) 0 0
\(627\) −20.9390 + 20.9390i −0.836224 + 0.836224i
\(628\) 0 0
\(629\) 15.3169i 0.610727i
\(630\) 0 0
\(631\) 3.55084i 0.141357i 0.997499 + 0.0706784i \(0.0225164\pi\)
−0.997499 + 0.0706784i \(0.977484\pi\)
\(632\) 0 0
\(633\) 38.8234 38.8234i 1.54309 1.54309i
\(634\) 0 0
\(635\) 9.03365 + 6.05944i 0.358489 + 0.240462i
\(636\) 0 0
\(637\) −4.19388 4.19388i −0.166167 0.166167i
\(638\) 0 0
\(639\) −36.1346 −1.42946
\(640\) 0 0
\(641\) −43.1020 −1.70243 −0.851214 0.524819i \(-0.824133\pi\)
−0.851214 + 0.524819i \(0.824133\pi\)
\(642\) 0 0
\(643\) 17.7005 + 17.7005i 0.698039 + 0.698039i 0.963987 0.265948i \(-0.0856850\pi\)
−0.265948 + 0.963987i \(0.585685\pi\)
\(644\) 0 0
\(645\) 24.9431 4.91524i 0.982135 0.193537i
\(646\) 0 0
\(647\) 18.8875 18.8875i 0.742544 0.742544i −0.230523 0.973067i \(-0.574044\pi\)
0.973067 + 0.230523i \(0.0740436\pi\)
\(648\) 0 0
\(649\) 5.02165i 0.197117i
\(650\) 0 0
\(651\) 1.45986i 0.0572163i
\(652\) 0 0
\(653\) −4.31695 + 4.31695i −0.168935 + 0.168935i −0.786511 0.617576i \(-0.788115\pi\)
0.617576 + 0.786511i \(0.288115\pi\)
\(654\) 0 0
\(655\) −15.0931 + 2.97421i −0.589736 + 0.116212i
\(656\) 0 0
\(657\) −4.14931 4.14931i −0.161880 0.161880i
\(658\) 0 0
\(659\) −7.05369 −0.274773 −0.137386 0.990518i \(-0.543870\pi\)
−0.137386 + 0.990518i \(0.543870\pi\)
\(660\) 0 0
\(661\) 30.2216 1.17548 0.587741 0.809049i \(-0.300017\pi\)
0.587741 + 0.809049i \(0.300017\pi\)
\(662\) 0 0
\(663\) 21.7714 + 21.7714i 0.845533 + 0.845533i
\(664\) 0 0
\(665\) −11.7938 7.91087i −0.457345 0.306770i
\(666\) 0 0
\(667\) −27.4641 + 27.4641i −1.06341 + 1.06341i
\(668\) 0 0
\(669\) 11.2630i 0.435454i
\(670\) 0 0
\(671\) 14.8166i 0.571987i
\(672\) 0 0
\(673\) 14.9248 14.9248i 0.575310 0.575310i −0.358298 0.933607i \(-0.616643\pi\)
0.933607 + 0.358298i \(0.116643\pi\)
\(674\) 0 0
\(675\) −8.66684 + 20.7142i −0.333587 + 0.797289i
\(676\) 0 0
\(677\) 2.37214 + 2.37214i 0.0911687 + 0.0911687i 0.751220 0.660052i \(-0.229466\pi\)
−0.660052 + 0.751220i \(0.729466\pi\)
\(678\) 0 0
\(679\) −14.5818 −0.559599
\(680\) 0 0
\(681\) −41.9806 −1.60870
\(682\) 0 0
\(683\) −6.58808 6.58808i −0.252086 0.252086i 0.569740 0.821825i \(-0.307044\pi\)
−0.821825 + 0.569740i \(0.807044\pi\)
\(684\) 0 0
\(685\) 11.6724 17.4017i 0.445981 0.664885i
\(686\) 0 0
\(687\) 25.1755 25.1755i 0.960505 0.960505i
\(688\) 0 0
\(689\) 61.9667i 2.36074i
\(690\) 0 0
\(691\) 9.45126i 0.359543i −0.983708 0.179772i \(-0.942464\pi\)
0.983708 0.179772i \(-0.0575358\pi\)
\(692\) 0 0
\(693\) −5.52311 + 5.52311i −0.209806 + 0.209806i
\(694\) 0 0
\(695\) 8.21345 + 41.6804i 0.311554 + 1.58103i
\(696\) 0 0
\(697\) −0.875267 0.875267i −0.0331531 0.0331531i
\(698\) 0 0
\(699\) 29.9641 1.13335
\(700\) 0 0
\(701\) 9.71866 0.367069 0.183534 0.983013i \(-0.441246\pi\)
0.183534 + 0.983013i \(0.441246\pi\)
\(702\) 0 0
\(703\) −36.5917 36.5917i −1.38008 1.38008i
\(704\) 0 0
\(705\) −14.9125 75.6760i −0.561638 2.85012i
\(706\) 0 0
\(707\) 0.274184 0.274184i 0.0103118 0.0103118i
\(708\) 0 0
\(709\) 28.2514i 1.06100i −0.847684 0.530501i \(-0.822004\pi\)
0.847684 0.530501i \(-0.177996\pi\)
\(710\) 0 0
\(711\) 60.1503i 2.25581i
\(712\) 0 0
\(713\) 2.71096 2.71096i 0.101526 0.101526i
\(714\) 0 0
\(715\) 12.4735 18.5959i 0.466481 0.695449i
\(716\) 0 0
\(717\) 18.7955 + 18.7955i 0.701931 + 0.701931i
\(718\) 0 0
\(719\) 38.4541 1.43410 0.717048 0.697023i \(-0.245493\pi\)
0.717048 + 0.697023i \(0.245493\pi\)
\(720\) 0 0
\(721\) −7.53707 −0.280695
\(722\) 0 0
\(723\) 16.7335 + 16.7335i 0.622325 + 0.622325i
\(724\) 0 0
\(725\) 10.1589 + 24.7755i 0.377292 + 0.920139i
\(726\) 0 0
\(727\) 5.53080 5.53080i 0.205126 0.205126i −0.597066 0.802192i \(-0.703667\pi\)
0.802192 + 0.597066i \(0.203667\pi\)
\(728\) 0 0
\(729\) 44.0375i 1.63102i
\(730\) 0 0
\(731\) 7.73931i 0.286249i
\(732\) 0 0
\(733\) −35.8986 + 35.8986i −1.32594 + 1.32594i −0.417070 + 0.908874i \(0.636943\pi\)
−0.908874 + 0.417070i \(0.863057\pi\)
\(734\) 0 0
\(735\) −5.12822 3.43982i −0.189157 0.126880i
\(736\) 0 0
\(737\) −7.38006 7.38006i −0.271848 0.271848i
\(738\) 0 0
\(739\) −9.83029 −0.361613 −0.180807 0.983519i \(-0.557871\pi\)
−0.180807 + 0.983519i \(0.557871\pi\)
\(740\) 0 0
\(741\) −104.022 −3.82136
\(742\) 0 0
\(743\) 24.0787 + 24.0787i 0.883363 + 0.883363i 0.993875 0.110511i \(-0.0352489\pi\)
−0.110511 + 0.993875i \(0.535249\pi\)
\(744\) 0 0
\(745\) −7.72928 + 1.52311i −0.283179 + 0.0558026i
\(746\) 0 0
\(747\) 26.1697 26.1697i 0.957500 0.957500i
\(748\) 0 0
\(749\) 4.11704i 0.150433i
\(750\) 0 0
\(751\) 16.0479i 0.585595i 0.956174 + 0.292798i \(0.0945862\pi\)
−0.956174 + 0.292798i \(0.905414\pi\)
\(752\) 0 0
\(753\) −11.3694 + 11.3694i −0.414325 + 0.414325i
\(754\) 0 0
\(755\) −7.02644 + 1.38461i −0.255718 + 0.0503913i
\(756\) 0 0
\(757\) 14.6541 + 14.6541i 0.532612 + 0.532612i 0.921349 0.388736i \(-0.127088\pi\)
−0.388736 + 0.921349i \(0.627088\pi\)
\(758\) 0 0
\(759\) 33.8151 1.22741
\(760\) 0 0
\(761\) −12.2707 −0.444813 −0.222407 0.974954i \(-0.571391\pi\)
−0.222407 + 0.974954i \(0.571391\pi\)
\(762\) 0 0
\(763\) 2.22690 + 2.22690i 0.0806192 + 0.0806192i
\(764\) 0 0
\(765\) 16.1493 + 10.8324i 0.583880 + 0.391645i
\(766\) 0 0
\(767\) −12.4735 + 12.4735i −0.450391 + 0.450391i
\(768\) 0 0
\(769\) 17.2278i 0.621251i −0.950532 0.310625i \(-0.899462\pi\)
0.950532 0.310625i \(-0.100538\pi\)
\(770\) 0 0
\(771\) 78.6918i 2.83401i
\(772\) 0 0
\(773\) −13.4463 + 13.4463i −0.483629 + 0.483629i −0.906288 0.422660i \(-0.861097\pi\)
0.422660 + 0.906288i \(0.361097\pi\)
\(774\) 0 0
\(775\) −1.00278 2.44557i −0.0360208 0.0878476i
\(776\) 0 0
\(777\) −15.9109 15.9109i −0.570799 0.570799i
\(778\) 0 0
\(779\) 4.18197 0.149835
\(780\) 0 0
\(781\) 13.1878 0.471898
\(782\) 0 0
\(783\) −17.0063 17.0063i −0.607757 0.607757i
\(784\) 0 0
\(785\) 14.2418 21.2322i 0.508311 0.757809i
\(786\) 0 0
\(787\) 13.6690 13.6690i 0.487248 0.487248i −0.420189 0.907437i \(-0.638036\pi\)
0.907437 + 0.420189i \(0.138036\pi\)
\(788\) 0 0
\(789\) 86.0591i 3.06379i
\(790\) 0 0
\(791\) 10.6849i 0.379913i
\(792\) 0 0
\(793\) 36.8034 36.8034i 1.30693 1.30693i
\(794\) 0 0
\(795\) −12.4735 63.2986i −0.442388 2.24497i
\(796\) 0 0
\(797\) 5.48292 + 5.48292i 0.194215 + 0.194215i 0.797515 0.603300i \(-0.206148\pi\)
−0.603300 + 0.797515i \(0.706148\pi\)
\(798\) 0 0
\(799\) −23.4806 −0.830683
\(800\) 0 0
\(801\) 29.0096 1.02500
\(802\) 0 0
\(803\) 1.51435 + 1.51435i 0.0534403 + 0.0534403i
\(804\) 0 0
\(805\) 3.13536 + 15.9109i 0.110507 + 0.560784i
\(806\) 0 0
\(807\) 11.7163 11.7163i 0.412433 0.412433i
\(808\) 0 0
\(809\) 17.6907i 0.621974i −0.950414 0.310987i \(-0.899341\pi\)
0.950414 0.310987i \(-0.100659\pi\)
\(810\) 0 0
\(811\) 3.95993i 0.139052i 0.997580 + 0.0695259i \(0.0221486\pi\)
−0.997580 + 0.0695259i \(0.977851\pi\)
\(812\) 0 0
\(813\) −63.7205 + 63.7205i −2.23478 + 2.23478i
\(814\) 0 0
\(815\) −14.9821 + 22.3359i −0.524799 + 0.782391i
\(816\) 0 0
\(817\) 18.4890 + 18.4890i 0.646846 + 0.646846i
\(818\) 0 0
\(819\) −27.4382 −0.958767
\(820\) 0 0
\(821\) −4.42962 −0.154595 −0.0772973 0.997008i \(-0.524629\pi\)
−0.0772973 + 0.997008i \(0.524629\pi\)
\(822\) 0 0
\(823\) −16.0158 16.0158i −0.558275 0.558275i 0.370541 0.928816i \(-0.379172\pi\)
−0.928816 + 0.370541i \(0.879172\pi\)
\(824\) 0 0
\(825\) 8.99834 21.5065i 0.313282 0.748759i
\(826\) 0 0
\(827\) 29.0179 29.0179i 1.00905 1.00905i 0.00909218 0.999959i \(-0.497106\pi\)
0.999959 0.00909218i \(-0.00289417\pi\)
\(828\) 0 0
\(829\) 5.54769i 0.192679i −0.995349 0.0963397i \(-0.969286\pi\)
0.995349 0.0963397i \(-0.0307135\pi\)
\(830\) 0 0
\(831\) 23.5586i 0.817240i
\(832\) 0 0
\(833\) −1.32924 + 1.32924i −0.0460553 + 0.0460553i
\(834\) 0 0
\(835\) −40.9318 27.4555i −1.41650 0.950138i
\(836\) 0 0
\(837\) 1.67868 + 1.67868i 0.0580238 + 0.0580238i
\(838\) 0 0
\(839\) −28.9239 −0.998565 −0.499282 0.866439i \(-0.666403\pi\)
−0.499282 + 0.866439i \(0.666403\pi\)
\(840\) 0 0
\(841\) 0.318836 0.0109943
\(842\) 0 0
\(843\) −10.8603 10.8603i −0.374049 0.374049i
\(844\) 0 0
\(845\) 48.6541 9.58767i 1.67375 0.329826i
\(846\) 0 0
\(847\) −5.76243 + 5.76243i −0.198000 + 0.198000i
\(848\) 0 0
\(849\) 67.0635i 2.30161i
\(850\) 0 0
\(851\) 59.0930i 2.02568i
\(852\) 0 0
\(853\) 23.8140 23.8140i 0.815377 0.815377i −0.170057 0.985434i \(-0.554395\pi\)
0.985434 + 0.170057i \(0.0543952\pi\)
\(854\) 0 0
\(855\) −64.4583 + 12.7020i −2.20443 + 0.434400i
\(856\) 0 0
\(857\) 22.9527 + 22.9527i 0.784050 + 0.784050i 0.980512 0.196461i \(-0.0629450\pi\)
−0.196461 + 0.980512i \(0.562945\pi\)
\(858\) 0 0
\(859\) −11.6447 −0.397313 −0.198657 0.980069i \(-0.563658\pi\)
−0.198657 + 0.980069i \(0.563658\pi\)
\(860\) 0 0
\(861\) 1.81841 0.0619713
\(862\) 0 0
\(863\) −26.7464 26.7464i −0.910457 0.910457i 0.0858513 0.996308i \(-0.472639\pi\)
−0.996308 + 0.0858513i \(0.972639\pi\)
\(864\) 0 0
\(865\) −15.2480 10.2278i −0.518448 0.347756i
\(866\) 0 0
\(867\) −26.2958 + 26.2958i −0.893051 + 0.893051i
\(868\) 0 0
\(869\) 21.9527i 0.744695i
\(870\) 0 0
\(871\) 36.6632i 1.24229i
\(872\) 0 0
\(873\) −47.7003 + 47.7003i −1.61441 + 1.61441i
\(874\) 0 0
\(875\) 10.9537 + 2.23986i 0.370302 + 0.0757209i
\(876\) 0 0
\(877\) −15.0646 15.0646i −0.508694 0.508694i 0.405432 0.914125i \(-0.367121\pi\)
−0.914125 + 0.405432i \(0.867121\pi\)
\(878\) 0 0
\(879\) 71.9691 2.42746
\(880\) 0 0
\(881\) −29.6801 −0.999949 −0.499974 0.866040i \(-0.666657\pi\)
−0.499974 + 0.866040i \(0.666657\pi\)
\(882\) 0 0
\(883\) −25.2385 25.2385i −0.849343 0.849343i 0.140708 0.990051i \(-0.455062\pi\)
−0.990051 + 0.140708i \(0.955062\pi\)
\(884\) 0 0
\(885\) −10.2307 + 15.2524i −0.343903 + 0.512704i
\(886\) 0 0
\(887\) 2.28006 2.28006i 0.0765569 0.0765569i −0.667792 0.744348i \(-0.732760\pi\)
0.744348 + 0.667792i \(0.232760\pi\)
\(888\) 0 0
\(889\) 4.86464i 0.163155i
\(890\) 0 0
\(891\) 2.49357i 0.0835378i
\(892\) 0 0
\(893\) 56.0943 56.0943i 1.87713 1.87713i
\(894\) 0 0
\(895\) −9.07312 46.0429i −0.303281 1.53905i
\(896\) 0 0
\(897\) 83.9946 + 83.9946i 2.80450 + 2.80450i
\(898\) 0 0
\(899\) 2.83110 0.0944224
\(900\) 0 0
\(901\) −19.6402 −0.654308
\(902\) 0 0
\(903\) 8.03940 + 8.03940i 0.267535 + 0.267535i
\(904\) 0 0
\(905\) −2.22012 11.2663i −0.0737993 0.374506i
\(906\) 0 0
\(907\) −0.237071 + 0.237071i −0.00787181 + 0.00787181i −0.711032 0.703160i \(-0.751772\pi\)
0.703160 + 0.711032i \(0.251772\pi\)
\(908\) 0 0
\(909\) 1.79383i 0.0594977i
\(910\) 0 0
\(911\) 27.2665i 0.903378i 0.892175 + 0.451689i \(0.149178\pi\)
−0.892175 + 0.451689i \(0.850822\pi\)
\(912\) 0 0
\(913\) −9.55102 + 9.55102i −0.316093 + 0.316093i
\(914\) 0 0
\(915\) 30.1862 45.0027i 0.997924 1.48774i
\(916\) 0 0
\(917\) −4.86464 4.86464i −0.160645 0.160645i
\(918\) 0 0
\(919\) −12.7500 −0.420584 −0.210292 0.977639i \(-0.567441\pi\)
−0.210292 + 0.977639i \(0.567441\pi\)
\(920\) 0 0
\(921\) 42.2601 1.39252
\(922\) 0 0
\(923\) 32.7578 + 32.7578i 1.07824 + 1.07824i
\(924\) 0 0
\(925\) 37.5833 + 15.7249i 1.23573 + 0.517032i
\(926\) 0 0
\(927\) −24.6554 + 24.6554i −0.809789 + 0.809789i
\(928\) 0 0
\(929\) 23.8742i 0.783288i −0.920117 0.391644i \(-0.871906\pi\)
0.920117 0.391644i \(-0.128094\pi\)
\(930\) 0 0
\(931\) 6.35101i 0.208146i
\(932\) 0 0
\(933\) −21.3063 + 21.3063i −0.697538 + 0.697538i
\(934\) 0 0
\(935\) −5.89393 3.95343i −0.192752 0.129291i
\(936\) 0 0
\(937\) −3.71699 3.71699i −0.121429 0.121429i 0.643781 0.765210i \(-0.277365\pi\)
−0.765210 + 0.643781i \(0.777365\pi\)
\(938\) 0 0
\(939\) 55.8903 1.82391
\(940\) 0 0
\(941\) 23.3940 0.762623 0.381312 0.924447i \(-0.375473\pi\)
0.381312 + 0.924447i \(0.375473\pi\)
\(942\) 0 0
\(943\) −3.37680 3.37680i −0.109964 0.109964i
\(944\) 0 0
\(945\) −9.85235 + 1.94148i −0.320497 + 0.0631564i
\(946\) 0 0
\(947\) 13.8223 13.8223i 0.449165 0.449165i −0.445912 0.895077i \(-0.647121\pi\)
0.895077 + 0.445912i \(0.147121\pi\)
\(948\) 0 0
\(949\) 7.52311i 0.244211i
\(950\) 0 0
\(951\) 21.4986i 0.697140i
\(952\) 0 0
\(953\) −4.78177 + 4.78177i −0.154897 + 0.154897i −0.780301 0.625404i \(-0.784934\pi\)
0.625404 + 0.780301i \(0.284934\pi\)
\(954\) 0 0
\(955\) 24.1625 4.76141i 0.781881 0.154076i
\(956\) 0 0
\(957\) 17.6568 + 17.6568i 0.570764 + 0.570764i
\(958\) 0 0
\(959\) 9.37086 0.302601
\(960\) 0 0
\(961\) 30.7205 0.990985
\(962\) 0 0
\(963\) −13.4677 13.4677i −0.433991 0.433991i
\(964\) 0 0
\(965\) −34.6541 23.2447i −1.11556 0.748273i
\(966\) 0 0
\(967\) −5.02573 + 5.02573i −0.161617 + 0.161617i −0.783282 0.621666i \(-0.786456\pi\)
0.621666 + 0.783282i \(0.286456\pi\)
\(968\) 0 0
\(969\) 32.9696i 1.05914i
\(970\) 0 0
\(971\) 25.6806i 0.824128i −0.911155 0.412064i \(-0.864808\pi\)
0.911155 0.412064i \(-0.135192\pi\)
\(972\) 0 0
\(973\) −13.4340 + 13.4340i −0.430674 + 0.430674i
\(974\) 0 0
\(975\) 75.7720 31.0694i 2.42665 0.995017i
\(976\) 0 0
\(977\) 22.5510 + 22.5510i 0.721471 + 0.721471i 0.968905 0.247434i \(-0.0795873\pi\)
−0.247434 + 0.968905i \(0.579587\pi\)
\(978\) 0 0
\(979\) −10.5875 −0.338377
\(980\) 0 0
\(981\) 14.5693 0.465163
\(982\) 0 0
\(983\) 19.4914 + 19.4914i 0.621678 + 0.621678i 0.945960 0.324282i \(-0.105123\pi\)
−0.324282 + 0.945960i \(0.605123\pi\)
\(984\) 0 0
\(985\) 1.12766 1.68116i 0.0359303 0.0535663i
\(986\) 0 0
\(987\) 24.3911 24.3911i 0.776376 0.776376i
\(988\) 0 0
\(989\) 29.8584i 0.949441i
\(990\) 0 0
\(991\) 2.46264i 0.0782282i 0.999235 + 0.0391141i \(0.0124536\pi\)
−0.999235 + 0.0391141i \(0.987546\pi\)
\(992\) 0 0
\(993\) −5.70138 + 5.70138i −0.180928 + 0.180928i
\(994\) 0 0
\(995\) 10.9591 + 55.6138i 0.347427 + 1.76307i
\(996\) 0 0
\(997\) 24.9973 + 24.9973i 0.791672 + 0.791672i 0.981766 0.190094i \(-0.0608792\pi\)
−0.190094 + 0.981766i \(0.560879\pi\)
\(998\) 0 0
\(999\) −36.5917 −1.15771
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 560.2.x.a.127.1 12
4.3 odd 2 inner 560.2.x.a.127.6 yes 12
5.3 odd 4 inner 560.2.x.a.463.6 yes 12
20.3 even 4 inner 560.2.x.a.463.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
560.2.x.a.127.1 12 1.1 even 1 trivial
560.2.x.a.127.6 yes 12 4.3 odd 2 inner
560.2.x.a.463.1 yes 12 20.3 even 4 inner
560.2.x.a.463.6 yes 12 5.3 odd 4 inner