Properties

Label 560.2.q.f.401.1
Level $560$
Weight $2$
Character 560.401
Analytic conductor $4.472$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,2,Mod(81,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.q (of order \(3\), 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: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 401.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 560.401
Dual form 560.2.q.f.81.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(1.00000 - 1.73205i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(1.00000 - 1.73205i) q^{9} +(3.00000 + 5.19615i) q^{11} +2.00000 q^{13} +1.00000 q^{15} +(3.00000 + 5.19615i) q^{17} +(4.00000 - 6.92820i) q^{19} +(-0.500000 - 2.59808i) q^{21} +(1.50000 - 2.59808i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.00000 q^{27} +3.00000 q^{29} +(1.00000 + 1.73205i) q^{31} +(-3.00000 + 5.19615i) q^{33} +(-2.00000 + 1.73205i) q^{35} +(-4.00000 + 6.92820i) q^{37} +(1.00000 + 1.73205i) q^{39} -3.00000 q^{41} -5.00000 q^{43} +(-1.00000 - 1.73205i) q^{45} +(5.50000 + 4.33013i) q^{49} +(-3.00000 + 5.19615i) q^{51} +(-6.00000 - 10.3923i) q^{53} +6.00000 q^{55} +8.00000 q^{57} +(0.500000 - 0.866025i) q^{61} +(-4.00000 + 3.46410i) q^{63} +(1.00000 - 1.73205i) q^{65} +(-3.50000 - 6.06218i) q^{67} +3.00000 q^{69} +(5.00000 + 8.66025i) q^{73} +(0.500000 - 0.866025i) q^{75} +(-3.00000 - 15.5885i) q^{77} +(-2.00000 + 3.46410i) q^{79} +(-0.500000 - 0.866025i) q^{81} -3.00000 q^{83} +6.00000 q^{85} +(1.50000 + 2.59808i) q^{87} +(1.50000 - 2.59808i) q^{89} +(-5.00000 - 1.73205i) q^{91} +(-1.00000 + 1.73205i) q^{93} +(-4.00000 - 6.92820i) q^{95} -10.0000 q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{5} - 5 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + q^{5} - 5 q^{7} + 2 q^{9} + 6 q^{11} + 4 q^{13} + 2 q^{15} + 6 q^{17} + 8 q^{19} - q^{21} + 3 q^{23} - q^{25} + 10 q^{27} + 6 q^{29} + 2 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} + 2 q^{39} - 6 q^{41} - 10 q^{43} - 2 q^{45} + 11 q^{49} - 6 q^{51} - 12 q^{53} + 12 q^{55} + 16 q^{57} + q^{61} - 8 q^{63} + 2 q^{65} - 7 q^{67} + 6 q^{69} + 10 q^{73} + q^{75} - 6 q^{77} - 4 q^{79} - q^{81} - 6 q^{83} + 12 q^{85} + 3 q^{87} + 3 q^{89} - 10 q^{91} - 2 q^{93} - 8 q^{95} - 20 q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i 0.973494 0.228714i \(-0.0734519\pi\)
−0.684819 + 0.728714i \(0.740119\pi\)
\(4\) 0 0
\(5\) 0.500000 0.866025i 0.223607 0.387298i
\(6\) 0 0
\(7\) −2.50000 0.866025i −0.944911 0.327327i
\(8\) 0 0
\(9\) 1.00000 1.73205i 0.333333 0.577350i
\(10\) 0 0
\(11\) 3.00000 + 5.19615i 0.904534 + 1.56670i 0.821541 + 0.570149i \(0.193114\pi\)
0.0829925 + 0.996550i \(0.473552\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 3.00000 + 5.19615i 0.727607 + 1.26025i 0.957892 + 0.287129i \(0.0927008\pi\)
−0.230285 + 0.973123i \(0.573966\pi\)
\(18\) 0 0
\(19\) 4.00000 6.92820i 0.917663 1.58944i 0.114708 0.993399i \(-0.463407\pi\)
0.802955 0.596040i \(-0.203260\pi\)
\(20\) 0 0
\(21\) −0.500000 2.59808i −0.109109 0.566947i
\(22\) 0 0
\(23\) 1.50000 2.59808i 0.312772 0.541736i −0.666190 0.745782i \(-0.732076\pi\)
0.978961 + 0.204046i \(0.0654092\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 1.00000 + 1.73205i 0.179605 + 0.311086i 0.941745 0.336327i \(-0.109185\pi\)
−0.762140 + 0.647412i \(0.775851\pi\)
\(32\) 0 0
\(33\) −3.00000 + 5.19615i −0.522233 + 0.904534i
\(34\) 0 0
\(35\) −2.00000 + 1.73205i −0.338062 + 0.292770i
\(36\) 0 0
\(37\) −4.00000 + 6.92820i −0.657596 + 1.13899i 0.323640 + 0.946180i \(0.395093\pi\)
−0.981236 + 0.192809i \(0.938240\pi\)
\(38\) 0 0
\(39\) 1.00000 + 1.73205i 0.160128 + 0.277350i
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) −1.00000 1.73205i −0.149071 0.258199i
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 5.50000 + 4.33013i 0.785714 + 0.618590i
\(50\) 0 0
\(51\) −3.00000 + 5.19615i −0.420084 + 0.727607i
\(52\) 0 0
\(53\) −6.00000 10.3923i −0.824163 1.42749i −0.902557 0.430570i \(-0.858312\pi\)
0.0783936 0.996922i \(-0.475021\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 0.500000 0.866025i 0.0640184 0.110883i −0.832240 0.554416i \(-0.812942\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) −4.00000 + 3.46410i −0.503953 + 0.436436i
\(64\) 0 0
\(65\) 1.00000 1.73205i 0.124035 0.214834i
\(66\) 0 0
\(67\) −3.50000 6.06218i −0.427593 0.740613i 0.569066 0.822292i \(-0.307305\pi\)
−0.996659 + 0.0816792i \(0.973972\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 5.00000 + 8.66025i 0.585206 + 1.01361i 0.994850 + 0.101361i \(0.0323196\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0.500000 0.866025i 0.0577350 0.100000i
\(76\) 0 0
\(77\) −3.00000 15.5885i −0.341882 1.77647i
\(78\) 0 0
\(79\) −2.00000 + 3.46410i −0.225018 + 0.389742i −0.956325 0.292306i \(-0.905577\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 1.50000 + 2.59808i 0.160817 + 0.278543i
\(88\) 0 0
\(89\) 1.50000 2.59808i 0.159000 0.275396i −0.775509 0.631337i \(-0.782506\pi\)
0.934508 + 0.355942i \(0.115840\pi\)
\(90\) 0 0
\(91\) −5.00000 1.73205i −0.524142 0.181568i
\(92\) 0 0
\(93\) −1.00000 + 1.73205i −0.103695 + 0.179605i
\(94\) 0 0
\(95\) −4.00000 6.92820i −0.410391 0.710819i
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 12.0000 1.20605
\(100\) 0 0
\(101\) 1.50000 + 2.59808i 0.149256 + 0.258518i 0.930953 0.365140i \(-0.118979\pi\)
−0.781697 + 0.623658i \(0.785646\pi\)
\(102\) 0 0
\(103\) −3.50000 + 6.06218i −0.344865 + 0.597324i −0.985329 0.170664i \(-0.945409\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) 0 0
\(105\) −2.50000 0.866025i −0.243975 0.0845154i
\(106\) 0 0
\(107\) −1.50000 + 2.59808i −0.145010 + 0.251166i −0.929377 0.369132i \(-0.879655\pi\)
0.784366 + 0.620298i \(0.212988\pi\)
\(108\) 0 0
\(109\) −8.50000 14.7224i −0.814152 1.41015i −0.909935 0.414751i \(-0.863869\pi\)
0.0957826 0.995402i \(-0.469465\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) −1.50000 2.59808i −0.139876 0.242272i
\(116\) 0 0
\(117\) 2.00000 3.46410i 0.184900 0.320256i
\(118\) 0 0
\(119\) −3.00000 15.5885i −0.275010 1.42899i
\(120\) 0 0
\(121\) −12.5000 + 21.6506i −1.13636 + 1.96824i
\(122\) 0 0
\(123\) −1.50000 2.59808i −0.135250 0.234261i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −2.50000 4.33013i −0.220113 0.381246i
\(130\) 0 0
\(131\) 6.00000 10.3923i 0.524222 0.907980i −0.475380 0.879781i \(-0.657689\pi\)
0.999602 0.0281993i \(-0.00897729\pi\)
\(132\) 0 0
\(133\) −16.0000 + 13.8564i −1.38738 + 1.20150i
\(134\) 0 0
\(135\) 2.50000 4.33013i 0.215166 0.372678i
\(136\) 0 0
\(137\) 6.00000 + 10.3923i 0.512615 + 0.887875i 0.999893 + 0.0146279i \(0.00465636\pi\)
−0.487278 + 0.873247i \(0.662010\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.00000 + 10.3923i 0.501745 + 0.869048i
\(144\) 0 0
\(145\) 1.50000 2.59808i 0.124568 0.215758i
\(146\) 0 0
\(147\) −1.00000 + 6.92820i −0.0824786 + 0.571429i
\(148\) 0 0
\(149\) 7.50000 12.9904i 0.614424 1.06421i −0.376061 0.926595i \(-0.622722\pi\)
0.990485 0.137619i \(-0.0439449\pi\)
\(150\) 0 0
\(151\) −5.00000 8.66025i −0.406894 0.704761i 0.587646 0.809118i \(-0.300055\pi\)
−0.994540 + 0.104357i \(0.966722\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) −7.00000 12.1244i −0.558661 0.967629i −0.997609 0.0691164i \(-0.977982\pi\)
0.438948 0.898513i \(-0.355351\pi\)
\(158\) 0 0
\(159\) 6.00000 10.3923i 0.475831 0.824163i
\(160\) 0 0
\(161\) −6.00000 + 5.19615i −0.472866 + 0.409514i
\(162\) 0 0
\(163\) −8.00000 + 13.8564i −0.626608 + 1.08532i 0.361619 + 0.932326i \(0.382224\pi\)
−0.988227 + 0.152992i \(0.951109\pi\)
\(164\) 0 0
\(165\) 3.00000 + 5.19615i 0.233550 + 0.404520i
\(166\) 0 0
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −8.00000 13.8564i −0.611775 1.05963i
\(172\) 0 0
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 0.500000 + 2.59808i 0.0377964 + 0.196396i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3.00000 5.19615i −0.224231 0.388379i 0.731858 0.681457i \(-0.238654\pi\)
−0.956088 + 0.293079i \(0.905320\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 0 0
\(183\) 1.00000 0.0739221
\(184\) 0 0
\(185\) 4.00000 + 6.92820i 0.294086 + 0.509372i
\(186\) 0 0
\(187\) −18.0000 + 31.1769i −1.31629 + 2.27988i
\(188\) 0 0
\(189\) −12.5000 4.33013i −0.909241 0.314970i
\(190\) 0 0
\(191\) −9.00000 + 15.5885i −0.651217 + 1.12794i 0.331611 + 0.943416i \(0.392408\pi\)
−0.982828 + 0.184525i \(0.940925\pi\)
\(192\) 0 0
\(193\) −1.00000 1.73205i −0.0719816 0.124676i 0.827788 0.561041i \(-0.189599\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 10.0000 + 17.3205i 0.708881 + 1.22782i 0.965272 + 0.261245i \(0.0841331\pi\)
−0.256391 + 0.966573i \(0.582534\pi\)
\(200\) 0 0
\(201\) 3.50000 6.06218i 0.246871 0.427593i
\(202\) 0 0
\(203\) −7.50000 2.59808i −0.526397 0.182349i
\(204\) 0 0
\(205\) −1.50000 + 2.59808i −0.104765 + 0.181458i
\(206\) 0 0
\(207\) −3.00000 5.19615i −0.208514 0.361158i
\(208\) 0 0
\(209\) 48.0000 3.32023
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.50000 + 4.33013i −0.170499 + 0.295312i
\(216\) 0 0
\(217\) −1.00000 5.19615i −0.0678844 0.352738i
\(218\) 0 0
\(219\) −5.00000 + 8.66025i −0.337869 + 0.585206i
\(220\) 0 0
\(221\) 6.00000 + 10.3923i 0.403604 + 0.699062i
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) −6.00000 10.3923i −0.398234 0.689761i 0.595274 0.803523i \(-0.297043\pi\)
−0.993508 + 0.113761i \(0.963710\pi\)
\(228\) 0 0
\(229\) −1.00000 + 1.73205i −0.0660819 + 0.114457i −0.897173 0.441679i \(-0.854383\pi\)
0.831092 + 0.556136i \(0.187717\pi\)
\(230\) 0 0
\(231\) 12.0000 10.3923i 0.789542 0.683763i
\(232\) 0 0
\(233\) −3.00000 + 5.19615i −0.196537 + 0.340411i −0.947403 0.320043i \(-0.896303\pi\)
0.750867 + 0.660454i \(0.229636\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −13.0000 22.5167i −0.837404 1.45043i −0.892058 0.451920i \(-0.850739\pi\)
0.0546547 0.998505i \(-0.482594\pi\)
\(242\) 0 0
\(243\) 8.00000 13.8564i 0.513200 0.888889i
\(244\) 0 0
\(245\) 6.50000 2.59808i 0.415270 0.165985i
\(246\) 0 0
\(247\) 8.00000 13.8564i 0.509028 0.881662i
\(248\) 0 0
\(249\) −1.50000 2.59808i −0.0950586 0.164646i
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 3.00000 + 5.19615i 0.187867 + 0.325396i
\(256\) 0 0
\(257\) 12.0000 20.7846i 0.748539 1.29651i −0.199983 0.979799i \(-0.564089\pi\)
0.948523 0.316709i \(-0.102578\pi\)
\(258\) 0 0
\(259\) 16.0000 13.8564i 0.994192 0.860995i
\(260\) 0 0
\(261\) 3.00000 5.19615i 0.185695 0.321634i
\(262\) 0 0
\(263\) −1.50000 2.59808i −0.0924940 0.160204i 0.816066 0.577959i \(-0.196151\pi\)
−0.908560 + 0.417755i \(0.862817\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 3.00000 0.183597
\(268\) 0 0
\(269\) 7.50000 + 12.9904i 0.457283 + 0.792038i 0.998816 0.0486418i \(-0.0154893\pi\)
−0.541533 + 0.840679i \(0.682156\pi\)
\(270\) 0 0
\(271\) −8.00000 + 13.8564i −0.485965 + 0.841717i −0.999870 0.0161307i \(-0.994865\pi\)
0.513905 + 0.857847i \(0.328199\pi\)
\(272\) 0 0
\(273\) −1.00000 5.19615i −0.0605228 0.314485i
\(274\) 0 0
\(275\) 3.00000 5.19615i 0.180907 0.313340i
\(276\) 0 0
\(277\) 11.0000 + 19.0526i 0.660926 + 1.14476i 0.980373 + 0.197153i \(0.0631696\pi\)
−0.319447 + 0.947604i \(0.603497\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 10.0000 + 17.3205i 0.594438 + 1.02960i 0.993626 + 0.112728i \(0.0359589\pi\)
−0.399188 + 0.916869i \(0.630708\pi\)
\(284\) 0 0
\(285\) 4.00000 6.92820i 0.236940 0.410391i
\(286\) 0 0
\(287\) 7.50000 + 2.59808i 0.442711 + 0.153360i
\(288\) 0 0
\(289\) −9.50000 + 16.4545i −0.558824 + 0.967911i
\(290\) 0 0
\(291\) −5.00000 8.66025i −0.293105 0.507673i
\(292\) 0 0
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 15.0000 + 25.9808i 0.870388 + 1.50756i
\(298\) 0 0
\(299\) 3.00000 5.19615i 0.173494 0.300501i
\(300\) 0 0
\(301\) 12.5000 + 4.33013i 0.720488 + 0.249584i
\(302\) 0 0
\(303\) −1.50000 + 2.59808i −0.0861727 + 0.149256i
\(304\) 0 0
\(305\) −0.500000 0.866025i −0.0286299 0.0495885i
\(306\) 0 0
\(307\) 19.0000 1.08439 0.542194 0.840254i \(-0.317594\pi\)
0.542194 + 0.840254i \(0.317594\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 2.00000 3.46410i 0.113047 0.195803i −0.803951 0.594696i \(-0.797272\pi\)
0.916997 + 0.398894i \(0.130606\pi\)
\(314\) 0 0
\(315\) 1.00000 + 5.19615i 0.0563436 + 0.292770i
\(316\) 0 0
\(317\) −9.00000 + 15.5885i −0.505490 + 0.875535i 0.494489 + 0.869184i \(0.335355\pi\)
−0.999980 + 0.00635137i \(0.997978\pi\)
\(318\) 0 0
\(319\) 9.00000 + 15.5885i 0.503903 + 0.872786i
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 48.0000 2.67079
\(324\) 0 0
\(325\) −1.00000 1.73205i −0.0554700 0.0960769i
\(326\) 0 0
\(327\) 8.50000 14.7224i 0.470051 0.814152i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −11.0000 + 19.0526i −0.604615 + 1.04722i 0.387498 + 0.921871i \(0.373340\pi\)
−0.992112 + 0.125353i \(0.959994\pi\)
\(332\) 0 0
\(333\) 8.00000 + 13.8564i 0.438397 + 0.759326i
\(334\) 0 0
\(335\) −7.00000 −0.382451
\(336\) 0 0
\(337\) 20.0000 1.08947 0.544735 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 0 0
\(339\) −6.00000 10.3923i −0.325875 0.564433i
\(340\) 0 0
\(341\) −6.00000 + 10.3923i −0.324918 + 0.562775i
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) 1.50000 2.59808i 0.0807573 0.139876i
\(346\) 0 0
\(347\) −13.5000 23.3827i −0.724718 1.25525i −0.959090 0.283101i \(-0.908637\pi\)
0.234372 0.972147i \(-0.424697\pi\)
\(348\) 0 0
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) 10.0000 0.533761
\(352\) 0 0
\(353\) 6.00000 + 10.3923i 0.319348 + 0.553127i 0.980352 0.197256i \(-0.0632029\pi\)
−0.661004 + 0.750382i \(0.729870\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 12.0000 10.3923i 0.635107 0.550019i
\(358\) 0 0
\(359\) 3.00000 5.19615i 0.158334 0.274242i −0.775934 0.630814i \(-0.782721\pi\)
0.934268 + 0.356572i \(0.116054\pi\)
\(360\) 0 0
\(361\) −22.5000 38.9711i −1.18421 2.05111i
\(362\) 0 0
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) 2.50000 + 4.33013i 0.130499 + 0.226031i 0.923869 0.382709i \(-0.125009\pi\)
−0.793370 + 0.608740i \(0.791675\pi\)
\(368\) 0 0
\(369\) −3.00000 + 5.19615i −0.156174 + 0.270501i
\(370\) 0 0
\(371\) 6.00000 + 31.1769i 0.311504 + 1.61862i
\(372\) 0 0
\(373\) 8.00000 13.8564i 0.414224 0.717458i −0.581122 0.813816i \(-0.697386\pi\)
0.995347 + 0.0963587i \(0.0307196\pi\)
\(374\) 0 0
\(375\) −0.500000 0.866025i −0.0258199 0.0447214i
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) −4.00000 6.92820i −0.204926 0.354943i
\(382\) 0 0
\(383\) 1.50000 2.59808i 0.0766464 0.132755i −0.825155 0.564907i \(-0.808912\pi\)
0.901801 + 0.432151i \(0.142245\pi\)
\(384\) 0 0
\(385\) −15.0000 5.19615i −0.764471 0.264820i
\(386\) 0 0
\(387\) −5.00000 + 8.66025i −0.254164 + 0.440225i
\(388\) 0 0
\(389\) −3.00000 5.19615i −0.152106 0.263455i 0.779895 0.625910i \(-0.215272\pi\)
−0.932002 + 0.362454i \(0.881939\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 2.00000 + 3.46410i 0.100631 + 0.174298i
\(396\) 0 0
\(397\) −13.0000 + 22.5167i −0.652451 + 1.13008i 0.330075 + 0.943955i \(0.392926\pi\)
−0.982526 + 0.186124i \(0.940407\pi\)
\(398\) 0 0
\(399\) −20.0000 6.92820i −1.00125 0.346844i
\(400\) 0 0
\(401\) −1.50000 + 2.59808i −0.0749064 + 0.129742i −0.901046 0.433724i \(-0.857199\pi\)
0.826139 + 0.563466i \(0.190532\pi\)
\(402\) 0 0
\(403\) 2.00000 + 3.46410i 0.0996271 + 0.172559i
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −48.0000 −2.37927
\(408\) 0 0
\(409\) −5.50000 9.52628i −0.271957 0.471044i 0.697406 0.716677i \(-0.254338\pi\)
−0.969363 + 0.245633i \(0.921004\pi\)
\(410\) 0 0
\(411\) −6.00000 + 10.3923i −0.295958 + 0.512615i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1.50000 + 2.59808i −0.0736321 + 0.127535i
\(416\) 0 0
\(417\) −1.00000 1.73205i −0.0489702 0.0848189i
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) 23.0000 1.12095 0.560476 0.828171i \(-0.310618\pi\)
0.560476 + 0.828171i \(0.310618\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.00000 5.19615i 0.145521 0.252050i
\(426\) 0 0
\(427\) −2.00000 + 1.73205i −0.0967868 + 0.0838198i
\(428\) 0 0
\(429\) −6.00000 + 10.3923i −0.289683 + 0.501745i
\(430\) 0 0
\(431\) 15.0000 + 25.9808i 0.722525 + 1.25145i 0.959985 + 0.280052i \(0.0903517\pi\)
−0.237460 + 0.971397i \(0.576315\pi\)
\(432\) 0 0
\(433\) −28.0000 −1.34559 −0.672797 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) 0 0
\(435\) 3.00000 0.143839
\(436\) 0 0
\(437\) −12.0000 20.7846i −0.574038 0.994263i
\(438\) 0 0
\(439\) −14.0000 + 24.2487i −0.668184 + 1.15733i 0.310228 + 0.950662i \(0.399595\pi\)
−0.978412 + 0.206666i \(0.933739\pi\)
\(440\) 0 0
\(441\) 13.0000 5.19615i 0.619048 0.247436i
\(442\) 0 0
\(443\) 4.50000 7.79423i 0.213801 0.370315i −0.739100 0.673596i \(-0.764749\pi\)
0.952901 + 0.303281i \(0.0980821\pi\)
\(444\) 0 0
\(445\) −1.50000 2.59808i −0.0711068 0.123161i
\(446\) 0 0
\(447\) 15.0000 0.709476
\(448\) 0 0
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) −9.00000 15.5885i −0.423793 0.734032i
\(452\) 0 0
\(453\) 5.00000 8.66025i 0.234920 0.406894i
\(454\) 0 0
\(455\) −4.00000 + 3.46410i −0.187523 + 0.162400i
\(456\) 0 0
\(457\) 2.00000 3.46410i 0.0935561 0.162044i −0.815449 0.578829i \(-0.803510\pi\)
0.909005 + 0.416785i \(0.136843\pi\)
\(458\) 0 0
\(459\) 15.0000 + 25.9808i 0.700140 + 1.21268i
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) −29.0000 −1.34774 −0.673872 0.738848i \(-0.735370\pi\)
−0.673872 + 0.738848i \(0.735370\pi\)
\(464\) 0 0
\(465\) 1.00000 + 1.73205i 0.0463739 + 0.0803219i
\(466\) 0 0
\(467\) 16.5000 28.5788i 0.763529 1.32247i −0.177492 0.984122i \(-0.556798\pi\)
0.941021 0.338349i \(-0.109868\pi\)
\(468\) 0 0
\(469\) 3.50000 + 18.1865i 0.161615 + 0.839776i
\(470\) 0 0
\(471\) 7.00000 12.1244i 0.322543 0.558661i
\(472\) 0 0
\(473\) −15.0000 25.9808i −0.689701 1.19460i
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) 0 0
\(477\) −24.0000 −1.09888
\(478\) 0 0
\(479\) −15.0000 25.9808i −0.685367 1.18709i −0.973321 0.229447i \(-0.926308\pi\)
0.287954 0.957644i \(-0.407025\pi\)
\(480\) 0 0
\(481\) −8.00000 + 13.8564i −0.364769 + 0.631798i
\(482\) 0 0
\(483\) −7.50000 2.59808i −0.341262 0.118217i
\(484\) 0 0
\(485\) −5.00000 + 8.66025i −0.227038 + 0.393242i
\(486\) 0 0
\(487\) 4.00000 + 6.92820i 0.181257 + 0.313947i 0.942309 0.334744i \(-0.108650\pi\)
−0.761052 + 0.648691i \(0.775317\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 0 0
\(493\) 9.00000 + 15.5885i 0.405340 + 0.702069i
\(494\) 0 0
\(495\) 6.00000 10.3923i 0.269680 0.467099i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.00000 1.73205i 0.0447661 0.0775372i −0.842774 0.538267i \(-0.819079\pi\)
0.887540 + 0.460730i \(0.152412\pi\)
\(500\) 0 0
\(501\) 10.5000 + 18.1865i 0.469105 + 0.812514i
\(502\) 0 0
\(503\) 9.00000 0.401290 0.200645 0.979664i \(-0.435696\pi\)
0.200645 + 0.979664i \(0.435696\pi\)
\(504\) 0 0
\(505\) 3.00000 0.133498
\(506\) 0 0
\(507\) −4.50000 7.79423i −0.199852 0.346154i
\(508\) 0 0
\(509\) −19.5000 + 33.7750i −0.864322 + 1.49705i 0.00339621 + 0.999994i \(0.498919\pi\)
−0.867719 + 0.497056i \(0.834414\pi\)
\(510\) 0 0
\(511\) −5.00000 25.9808i −0.221187 1.14932i
\(512\) 0 0
\(513\) 20.0000 34.6410i 0.883022 1.52944i
\(514\) 0 0
\(515\) 3.50000 + 6.06218i 0.154228 + 0.267131i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −15.0000 25.9808i −0.657162 1.13824i −0.981347 0.192244i \(-0.938423\pi\)
0.324185 0.945994i \(-0.394910\pi\)
\(522\) 0 0
\(523\) 22.0000 38.1051i 0.961993 1.66622i 0.244507 0.969648i \(-0.421374\pi\)
0.717486 0.696573i \(-0.245293\pi\)
\(524\) 0 0
\(525\) −2.00000 + 1.73205i −0.0872872 + 0.0755929i
\(526\) 0 0
\(527\) −6.00000 + 10.3923i −0.261364 + 0.452696i
\(528\) 0 0
\(529\) 7.00000 + 12.1244i 0.304348 + 0.527146i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 1.50000 + 2.59808i 0.0648507 + 0.112325i
\(536\) 0 0
\(537\) 3.00000 5.19615i 0.129460 0.224231i
\(538\) 0 0
\(539\) −6.00000 + 41.5692i −0.258438 + 1.79051i
\(540\) 0 0
\(541\) −8.50000 + 14.7224i −0.365444 + 0.632967i −0.988847 0.148933i \(-0.952416\pi\)
0.623404 + 0.781900i \(0.285749\pi\)
\(542\) 0 0
\(543\) 8.50000 + 14.7224i 0.364770 + 0.631800i
\(544\) 0 0
\(545\) −17.0000 −0.728200
\(546\) 0 0
\(547\) 19.0000 0.812381 0.406191 0.913788i \(-0.366857\pi\)
0.406191 + 0.913788i \(0.366857\pi\)
\(548\) 0 0
\(549\) −1.00000 1.73205i −0.0426790 0.0739221i
\(550\) 0 0
\(551\) 12.0000 20.7846i 0.511217 0.885454i
\(552\) 0 0
\(553\) 8.00000 6.92820i 0.340195 0.294617i
\(554\) 0 0
\(555\) −4.00000 + 6.92820i −0.169791 + 0.294086i
\(556\) 0 0
\(557\) −15.0000 25.9808i −0.635570 1.10084i −0.986394 0.164399i \(-0.947432\pi\)
0.350824 0.936442i \(-0.385902\pi\)
\(558\) 0 0
\(559\) −10.0000 −0.422955
\(560\) 0 0
\(561\) −36.0000 −1.51992
\(562\) 0 0
\(563\) −4.50000 7.79423i −0.189652 0.328488i 0.755482 0.655169i \(-0.227403\pi\)
−0.945134 + 0.326682i \(0.894069\pi\)
\(564\) 0 0
\(565\) −6.00000 + 10.3923i −0.252422 + 0.437208i
\(566\) 0 0
\(567\) 0.500000 + 2.59808i 0.0209980 + 0.109109i
\(568\) 0 0
\(569\) 9.00000 15.5885i 0.377300 0.653502i −0.613369 0.789797i \(-0.710186\pi\)
0.990668 + 0.136295i \(0.0435194\pi\)
\(570\) 0 0
\(571\) −8.00000 13.8564i −0.334790 0.579873i 0.648655 0.761083i \(-0.275332\pi\)
−0.983444 + 0.181210i \(0.941999\pi\)
\(572\) 0 0
\(573\) −18.0000 −0.751961
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) 0 0
\(577\) 5.00000 + 8.66025i 0.208153 + 0.360531i 0.951133 0.308783i \(-0.0999216\pi\)
−0.742980 + 0.669314i \(0.766588\pi\)
\(578\) 0 0
\(579\) 1.00000 1.73205i 0.0415586 0.0719816i
\(580\) 0 0
\(581\) 7.50000 + 2.59808i 0.311152 + 0.107786i
\(582\) 0 0
\(583\) 36.0000 62.3538i 1.49097 2.58243i
\(584\) 0 0
\(585\) −2.00000 3.46410i −0.0826898 0.143223i
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) −9.00000 15.5885i −0.370211 0.641223i
\(592\) 0 0
\(593\) 6.00000 10.3923i 0.246390 0.426761i −0.716131 0.697966i \(-0.754089\pi\)
0.962522 + 0.271205i \(0.0874221\pi\)
\(594\) 0 0
\(595\) −15.0000 5.19615i −0.614940 0.213021i
\(596\) 0 0
\(597\) −10.0000 + 17.3205i −0.409273 + 0.708881i
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) −14.0000 −0.570124
\(604\) 0 0
\(605\) 12.5000 + 21.6506i 0.508197 + 0.880223i
\(606\) 0 0
\(607\) −21.5000 + 37.2391i −0.872658 + 1.51149i −0.0134214 + 0.999910i \(0.504272\pi\)
−0.859237 + 0.511578i \(0.829061\pi\)
\(608\) 0 0
\(609\) −1.50000 7.79423i −0.0607831 0.315838i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.00000 1.73205i −0.0403896 0.0699569i 0.845124 0.534570i \(-0.179527\pi\)
−0.885514 + 0.464614i \(0.846193\pi\)
\(614\) 0 0
\(615\) −3.00000 −0.120972
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) 7.00000 + 12.1244i 0.281354 + 0.487319i 0.971718 0.236143i \(-0.0758832\pi\)
−0.690365 + 0.723462i \(0.742550\pi\)
\(620\) 0 0
\(621\) 7.50000 12.9904i 0.300965 0.521286i
\(622\) 0 0
\(623\) −6.00000 + 5.19615i −0.240385 + 0.208179i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 24.0000 + 41.5692i 0.958468 + 1.66011i
\(628\) 0 0
\(629\) −48.0000 −1.91389
\(630\) 0 0
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) 0 0
\(633\) 2.00000 + 3.46410i 0.0794929 + 0.137686i
\(634\) 0 0
\(635\) −4.00000 + 6.92820i −0.158735 + 0.274937i
\(636\) 0 0
\(637\) 11.0000 + 8.66025i 0.435836 + 0.343132i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 19.5000 + 33.7750i 0.770204 + 1.33403i 0.937451 + 0.348117i \(0.113179\pi\)
−0.167247 + 0.985915i \(0.553488\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) −5.00000 −0.196875
\(646\) 0 0
\(647\) 10.5000 + 18.1865i 0.412798 + 0.714986i 0.995194 0.0979182i \(-0.0312184\pi\)
−0.582397 + 0.812905i \(0.697885\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 4.00000 3.46410i 0.156772 0.135769i
\(652\) 0 0
\(653\) 21.0000 36.3731i 0.821794 1.42339i −0.0825519 0.996587i \(-0.526307\pi\)
0.904345 0.426801i \(-0.140360\pi\)
\(654\) 0 0
\(655\) −6.00000 10.3923i −0.234439 0.406061i
\(656\) 0 0
\(657\) 20.0000 0.780274
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 12.5000 + 21.6506i 0.486194 + 0.842112i 0.999874 0.0158695i \(-0.00505163\pi\)
−0.513680 + 0.857982i \(0.671718\pi\)
\(662\) 0 0
\(663\) −6.00000 + 10.3923i −0.233021 + 0.403604i
\(664\) 0 0
\(665\) 4.00000 + 20.7846i 0.155113 + 0.805993i
\(666\) 0 0
\(667\) 4.50000 7.79423i 0.174241 0.301794i
\(668\) 0 0
\(669\) −4.00000 6.92820i −0.154649 0.267860i
\(670\) 0 0
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) 32.0000 1.23351 0.616755 0.787155i \(-0.288447\pi\)
0.616755 + 0.787155i \(0.288447\pi\)
\(674\) 0 0
\(675\) −2.50000 4.33013i −0.0962250 0.166667i
\(676\) 0 0
\(677\) 9.00000 15.5885i 0.345898 0.599113i −0.639618 0.768693i \(-0.720908\pi\)
0.985517 + 0.169580i \(0.0542410\pi\)
\(678\) 0 0
\(679\) 25.0000 + 8.66025i 0.959412 + 0.332350i
\(680\) 0 0
\(681\) 6.00000 10.3923i 0.229920 0.398234i
\(682\) 0 0
\(683\) −7.50000 12.9904i −0.286980 0.497063i 0.686108 0.727500i \(-0.259318\pi\)
−0.973087 + 0.230437i \(0.925985\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) −2.00000 −0.0763048
\(688\) 0 0
\(689\) −12.0000 20.7846i −0.457164 0.791831i
\(690\) 0 0
\(691\) 7.00000 12.1244i 0.266293 0.461232i −0.701609 0.712562i \(-0.747535\pi\)
0.967901 + 0.251330i \(0.0808679\pi\)
\(692\) 0 0
\(693\) −30.0000 10.3923i −1.13961 0.394771i
\(694\) 0 0
\(695\) −1.00000 + 1.73205i −0.0379322 + 0.0657004i
\(696\) 0 0
\(697\) −9.00000 15.5885i −0.340899 0.590455i
\(698\) 0 0
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 27.0000 1.01978 0.509888 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) 0 0
\(703\) 32.0000 + 55.4256i 1.20690 + 2.09042i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.50000 7.79423i −0.0564133 0.293132i
\(708\) 0 0
\(709\) 0.500000 0.866025i 0.0187779 0.0325243i −0.856484 0.516174i \(-0.827356\pi\)
0.875262 + 0.483650i \(0.160689\pi\)
\(710\) 0 0
\(711\) 4.00000 + 6.92820i 0.150012 + 0.259828i
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 0 0
\(717\) −3.00000 5.19615i −0.112037 0.194054i
\(718\) 0 0
\(719\) −15.0000 + 25.9808i −0.559406 + 0.968919i 0.438141 + 0.898906i \(0.355637\pi\)
−0.997546 + 0.0700124i \(0.977696\pi\)
\(720\) 0 0
\(721\) 14.0000 12.1244i 0.521387 0.451535i
\(722\) 0 0
\(723\) 13.0000 22.5167i 0.483475 0.837404i
\(724\) 0 0
\(725\) −1.50000 2.59808i −0.0557086 0.0964901i
\(726\) 0 0
\(727\) −23.0000 −0.853023 −0.426511 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −15.0000 25.9808i −0.554795 0.960933i
\(732\) 0 0
\(733\) −1.00000 + 1.73205i −0.0369358 + 0.0639748i −0.883902 0.467671i \(-0.845093\pi\)
0.846967 + 0.531646i \(0.178426\pi\)
\(734\) 0 0
\(735\) 5.50000 + 4.33013i 0.202871 + 0.159719i
\(736\) 0 0
\(737\) 21.0000 36.3731i 0.773545 1.33982i
\(738\) 0 0
\(739\) −23.0000 39.8372i −0.846069 1.46543i −0.884690 0.466180i \(-0.845630\pi\)
0.0386212 0.999254i \(-0.487703\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) 0 0
\(743\) 39.0000 1.43077 0.715386 0.698730i \(-0.246251\pi\)
0.715386 + 0.698730i \(0.246251\pi\)
\(744\) 0 0
\(745\) −7.50000 12.9904i −0.274779 0.475931i
\(746\) 0 0
\(747\) −3.00000 + 5.19615i −0.109764 + 0.190117i
\(748\) 0 0
\(749\) 6.00000 5.19615i 0.219235 0.189863i
\(750\) 0 0
\(751\) 10.0000 17.3205i 0.364905 0.632034i −0.623856 0.781540i \(-0.714435\pi\)
0.988761 + 0.149505i \(0.0477681\pi\)
\(752\) 0 0
\(753\) 9.00000 + 15.5885i 0.327978 + 0.568075i
\(754\) 0 0
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 9.00000 + 15.5885i 0.326679 + 0.565825i
\(760\) 0 0
\(761\) −15.0000 + 25.9808i −0.543750 + 0.941802i 0.454935 + 0.890525i \(0.349663\pi\)
−0.998684 + 0.0512772i \(0.983671\pi\)
\(762\) 0 0
\(763\) 8.50000 + 44.1673i 0.307721 + 1.59896i
\(764\) 0 0
\(765\) 6.00000 10.3923i 0.216930 0.375735i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) 0 0
\(773\) 21.0000 + 36.3731i 0.755318 + 1.30825i 0.945216 + 0.326445i \(0.105851\pi\)
−0.189899 + 0.981804i \(0.560816\pi\)
\(774\) 0 0
\(775\) 1.00000 1.73205i 0.0359211 0.0622171i
\(776\) 0 0
\(777\) 20.0000 + 6.92820i 0.717496 + 0.248548i
\(778\) 0 0
\(779\) −12.0000 + 20.7846i −0.429945 + 0.744686i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 15.0000 0.536056
\(784\) 0 0
\(785\) −14.0000 −0.499681
\(786\) 0 0
\(787\) −15.5000 26.8468i −0.552515 0.956985i −0.998092 0.0617409i \(-0.980335\pi\)
0.445577 0.895244i \(-0.352999\pi\)
\(788\) 0 0
\(789\) 1.50000 2.59808i 0.0534014 0.0924940i
\(790\) 0 0
\(791\) 30.0000 + 10.3923i 1.06668 + 0.369508i
\(792\) 0 0
\(793\) 1.00000 1.73205i 0.0355110 0.0615069i
\(794\) 0 0
\(795\) −6.00000 10.3923i −0.212798 0.368577i
\(796\) 0 0
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −3.00000 5.19615i −0.106000 0.183597i
\(802\) 0 0
\(803\) −30.0000 + 51.9615i −1.05868 + 1.83368i
\(804\) 0 0
\(805\) 1.50000 + 7.79423i 0.0528681 + 0.274710i
\(806\) 0 0
\(807\) −7.50000 + 12.9904i −0.264013 + 0.457283i
\(808\) 0 0
\(809\) 25.5000 + 44.1673i 0.896532 + 1.55284i 0.831897 + 0.554930i \(0.187255\pi\)
0.0646355 + 0.997909i \(0.479412\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 8.00000 + 13.8564i 0.280228 + 0.485369i
\(816\) 0 0
\(817\) −20.0000 + 34.6410i −0.699711 + 1.21194i
\(818\) 0 0
\(819\) −8.00000 + 6.92820i −0.279543 + 0.242091i
\(820\) 0 0
\(821\) −9.00000 + 15.5885i −0.314102 + 0.544041i −0.979246 0.202674i \(-0.935037\pi\)
0.665144 + 0.746715i \(0.268370\pi\)
\(822\) 0 0
\(823\) −24.5000 42.4352i −0.854016 1.47920i −0.877555 0.479477i \(-0.840826\pi\)
0.0235383 0.999723i \(-0.492507\pi\)
\(824\) 0 0
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) 45.0000 1.56480 0.782402 0.622774i \(-0.213994\pi\)
0.782402 + 0.622774i \(0.213994\pi\)
\(828\) 0 0
\(829\) 5.00000 + 8.66025i 0.173657 + 0.300783i 0.939696 0.342012i \(-0.111108\pi\)
−0.766039 + 0.642795i \(0.777775\pi\)
\(830\) 0 0
\(831\) −11.0000 + 19.0526i −0.381586 + 0.660926i
\(832\) 0 0
\(833\) −6.00000 + 41.5692i −0.207888 + 1.44029i
\(834\) 0 0
\(835\) 10.5000 18.1865i 0.363367 0.629371i
\(836\) 0 0
\(837\) 5.00000 + 8.66025i 0.172825 + 0.299342i
\(838\) 0 0
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −3.00000 5.19615i −0.103325 0.178965i
\(844\) 0 0
\(845\) −4.50000 + 7.79423i −0.154805 + 0.268130i
\(846\) 0 0
\(847\) 50.0000 43.3013i 1.71802 1.48785i
\(848\) 0 0
\(849\) −10.0000 + 17.3205i −0.343199 + 0.594438i
\(850\) 0 0
\(851\) 12.0000 + 20.7846i 0.411355 + 0.712487i
\(852\) 0 0
\(853\) 32.0000 1.09566 0.547830 0.836590i \(-0.315454\pi\)
0.547830 + 0.836590i \(0.315454\pi\)
\(854\) 0 0
\(855\) −16.0000 −0.547188
\(856\) 0 0
\(857\) −21.0000 36.3731i −0.717346 1.24248i −0.962048 0.272882i \(-0.912023\pi\)
0.244701 0.969599i \(-0.421310\pi\)
\(858\) 0 0
\(859\) −2.00000 + 3.46410i −0.0682391 + 0.118194i −0.898126 0.439738i \(-0.855071\pi\)
0.829887 + 0.557931i \(0.188405\pi\)
\(860\) 0 0
\(861\) 1.50000 + 7.79423i 0.0511199 + 0.265627i
\(862\) 0 0
\(863\) 16.5000 28.5788i 0.561667 0.972835i −0.435685 0.900099i \(-0.643494\pi\)
0.997351 0.0727356i \(-0.0231729\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) −7.00000 12.1244i −0.237186 0.410818i
\(872\) 0 0
\(873\) −10.0000 + 17.3205i −0.338449 + 0.586210i
\(874\) 0 0
\(875\) 2.50000 + 0.866025i 0.0845154 + 0.0292770i
\(876\) 0 0
\(877\) 20.0000 34.6410i 0.675352 1.16974i −0.301014 0.953620i \(-0.597325\pi\)
0.976366 0.216124i \(-0.0693416\pi\)
\(878\) 0 0
\(879\) −6.00000 10.3923i −0.202375 0.350524i
\(880\) 0 0
\(881\) 9.00000 0.303218 0.151609 0.988441i \(-0.451555\pi\)
0.151609 + 0.988441i \(0.451555\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 28.5000 49.3634i 0.956936 1.65746i 0.227063 0.973880i \(-0.427088\pi\)
0.729873 0.683582i \(-0.239579\pi\)
\(888\) 0 0
\(889\) 20.0000 + 6.92820i 0.670778 + 0.232364i
\(890\) 0 0
\(891\) 3.00000 5.19615i 0.100504 0.174078i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −6.00000 −0.200558
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) 0 0
\(899\) 3.00000 + 5.19615i 0.100056 + 0.173301i
\(900\) 0 0
\(901\) 36.0000 62.3538i 1.19933 2.07731i
\(902\) 0 0
\(903\) 2.50000 + 12.9904i 0.0831948 + 0.432293i
\(904\) 0 0
\(905\) 8.50000 14.7224i 0.282550 0.489390i
\(906\) 0 0
\(907\) 11.5000 + 19.9186i 0.381851 + 0.661386i 0.991327 0.131419i \(-0.0419533\pi\)
−0.609476 + 0.792805i \(0.708620\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) −9.00000 15.5885i −0.297857 0.515903i
\(914\) 0 0
\(915\) 0.500000 0.866025i 0.0165295 0.0286299i
\(916\) 0 0
\(917\) −24.0000 + 20.7846i −0.792550 + 0.686368i
\(918\) 0 0
\(919\) 4.00000 6.92820i 0.131948 0.228540i −0.792480 0.609898i \(-0.791210\pi\)
0.924427 + 0.381358i \(0.124544\pi\)
\(920\) 0 0
\(921\) 9.50000 + 16.4545i 0.313036 + 0.542194i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) 7.00000 + 12.1244i 0.229910 + 0.398216i
\(928\) 0 0
\(929\) 22.5000 38.9711i 0.738201 1.27860i −0.215104 0.976591i \(-0.569009\pi\)
0.953305 0.302010i \(-0.0976578\pi\)
\(930\) 0 0
\(931\) 52.0000 20.7846i 1.70423 0.681188i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 18.0000 + 31.1769i 0.588663 + 1.01959i
\(936\) 0 0
\(937\) 44.0000 1.43742 0.718709 0.695311i \(-0.244734\pi\)
0.718709 + 0.695311i \(0.244734\pi\)
\(938\) 0 0
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) −27.0000 46.7654i −0.880175 1.52451i −0.851146 0.524929i \(-0.824092\pi\)
−0.0290288 0.999579i \(-0.509241\pi\)
\(942\) 0 0
\(943\) −4.50000 + 7.79423i −0.146540 + 0.253815i
\(944\) 0 0
\(945\) −10.0000 + 8.66025i −0.325300 + 0.281718i
\(946\) 0 0
\(947\) 4.50000 7.79423i 0.146230 0.253278i −0.783601 0.621264i \(-0.786619\pi\)
0.929831 + 0.367986i \(0.119953\pi\)
\(948\) 0 0
\(949\) 10.0000 + 17.3205i 0.324614 + 0.562247i
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 9.00000 + 15.5885i 0.291233 + 0.504431i
\(956\) 0 0
\(957\) −9.00000 + 15.5885i −0.290929 + 0.503903i
\(958\) 0 0
\(959\) −6.00000 31.1769i −0.193750 1.00676i
\(960\) 0 0
\(961\) 13.5000 23.3827i 0.435484 0.754280i
\(962\) 0 0
\(963\) 3.00000 + 5.19615i 0.0966736 + 0.167444i
\(964\) 0 0
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) 7.00000 0.225105 0.112552 0.993646i \(-0.464097\pi\)
0.112552 + 0.993646i \(0.464097\pi\)
\(968\) 0 0
\(969\) 24.0000 + 41.5692i 0.770991 + 1.33540i
\(970\) 0 0
\(971\) −6.00000 + 10.3923i −0.192549 + 0.333505i −0.946094 0.323891i \(-0.895009\pi\)
0.753545 + 0.657396i \(0.228342\pi\)
\(972\) 0 0
\(973\) 5.00000 + 1.73205i 0.160293 + 0.0555270i
\(974\) 0 0
\(975\) 1.00000 1.73205i 0.0320256 0.0554700i
\(976\) 0 0
\(977\) 27.0000 + 46.7654i 0.863807 + 1.49616i 0.868227 + 0.496167i \(0.165259\pi\)
−0.00442082 + 0.999990i \(0.501407\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) −34.0000 −1.08554
\(982\) 0 0
\(983\) −19.5000 33.7750i −0.621953 1.07725i −0.989122 0.147100i \(-0.953006\pi\)
0.367168 0.930155i \(-0.380327\pi\)
\(984\) 0 0
\(985\) −9.00000 + 15.5885i −0.286764 + 0.496690i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −7.50000 + 12.9904i −0.238486 + 0.413070i
\(990\) 0 0
\(991\) 13.0000 + 22.5167i 0.412959 + 0.715265i 0.995212 0.0977423i \(-0.0311621\pi\)
−0.582253 + 0.813008i \(0.697829\pi\)
\(992\) 0 0
\(993\) −22.0000 −0.698149
\(994\) 0 0
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) −16.0000 27.7128i −0.506725 0.877674i −0.999970 0.00778294i \(-0.997523\pi\)
0.493245 0.869891i \(-0.335811\pi\)
\(998\) 0 0
\(999\) −20.0000 + 34.6410i −0.632772 + 1.09599i
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.q.f.401.1 2
4.3 odd 2 140.2.i.a.121.1 yes 2
7.2 even 3 3920.2.a.k.1.1 1
7.4 even 3 inner 560.2.q.f.81.1 2
7.5 odd 6 3920.2.a.w.1.1 1
12.11 even 2 1260.2.s.c.541.1 2
20.3 even 4 700.2.r.a.149.2 4
20.7 even 4 700.2.r.a.149.1 4
20.19 odd 2 700.2.i.b.401.1 2
28.3 even 6 980.2.i.f.361.1 2
28.11 odd 6 140.2.i.a.81.1 2
28.19 even 6 980.2.a.e.1.1 1
28.23 odd 6 980.2.a.g.1.1 1
28.27 even 2 980.2.i.f.961.1 2
84.11 even 6 1260.2.s.c.361.1 2
84.23 even 6 8820.2.a.p.1.1 1
84.47 odd 6 8820.2.a.a.1.1 1
140.19 even 6 4900.2.a.q.1.1 1
140.23 even 12 4900.2.e.m.2549.2 2
140.39 odd 6 700.2.i.b.501.1 2
140.47 odd 12 4900.2.e.n.2549.2 2
140.67 even 12 700.2.r.a.249.2 4
140.79 odd 6 4900.2.a.i.1.1 1
140.103 odd 12 4900.2.e.n.2549.1 2
140.107 even 12 4900.2.e.m.2549.1 2
140.123 even 12 700.2.r.a.249.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.i.a.81.1 2 28.11 odd 6
140.2.i.a.121.1 yes 2 4.3 odd 2
560.2.q.f.81.1 2 7.4 even 3 inner
560.2.q.f.401.1 2 1.1 even 1 trivial
700.2.i.b.401.1 2 20.19 odd 2
700.2.i.b.501.1 2 140.39 odd 6
700.2.r.a.149.1 4 20.7 even 4
700.2.r.a.149.2 4 20.3 even 4
700.2.r.a.249.1 4 140.123 even 12
700.2.r.a.249.2 4 140.67 even 12
980.2.a.e.1.1 1 28.19 even 6
980.2.a.g.1.1 1 28.23 odd 6
980.2.i.f.361.1 2 28.3 even 6
980.2.i.f.961.1 2 28.27 even 2
1260.2.s.c.361.1 2 84.11 even 6
1260.2.s.c.541.1 2 12.11 even 2
3920.2.a.k.1.1 1 7.2 even 3
3920.2.a.w.1.1 1 7.5 odd 6
4900.2.a.i.1.1 1 140.79 odd 6
4900.2.a.q.1.1 1 140.19 even 6
4900.2.e.m.2549.1 2 140.107 even 12
4900.2.e.m.2549.2 2 140.23 even 12
4900.2.e.n.2549.1 2 140.103 odd 12
4900.2.e.n.2549.2 2 140.47 odd 12
8820.2.a.a.1.1 1 84.47 odd 6
8820.2.a.p.1.1 1 84.23 even 6