Properties

Label 560.2.q.a.401.1
Level $560$
Weight $2$
Character 560.401
Analytic conductor $4.472$
Analytic rank $1$
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: \(1\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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.a.81.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 - 2.59808i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(-3.00000 + 5.19615i) q^{9} +O(q^{10})\) \(q+(-1.50000 - 2.59808i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(-3.00000 + 5.19615i) q^{9} +(-1.00000 - 1.73205i) q^{11} -6.00000 q^{13} -3.00000 q^{15} +(-1.00000 - 1.73205i) q^{17} +(7.50000 - 2.59808i) q^{21} +(-4.50000 + 7.79423i) q^{23} +(-0.500000 - 0.866025i) q^{25} +9.00000 q^{27} +3.00000 q^{29} +(1.00000 + 1.73205i) q^{31} +(-3.00000 + 5.19615i) q^{33} +(2.00000 + 1.73205i) q^{35} +(-4.00000 + 6.92820i) q^{37} +(9.00000 + 15.5885i) q^{39} +5.00000 q^{41} -1.00000 q^{43} +(3.00000 + 5.19615i) q^{45} +(4.00000 - 6.92820i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(-3.00000 + 5.19615i) q^{51} +(-2.00000 - 3.46410i) q^{53} -2.00000 q^{55} +(-4.00000 - 6.92820i) q^{59} +(-3.50000 + 6.06218i) q^{61} +(-12.0000 - 10.3923i) q^{63} +(-3.00000 + 5.19615i) q^{65} +(-1.50000 - 2.59808i) q^{67} +27.0000 q^{69} -8.00000 q^{71} +(-7.00000 - 12.1244i) q^{73} +(-1.50000 + 2.59808i) q^{75} +(5.00000 - 1.73205i) q^{77} +(2.00000 - 3.46410i) q^{79} +(-4.50000 - 7.79423i) q^{81} +1.00000 q^{83} -2.00000 q^{85} +(-4.50000 - 7.79423i) q^{87} +(-6.50000 + 11.2583i) q^{89} +(3.00000 - 15.5885i) q^{91} +(3.00000 - 5.19615i) q^{93} -10.0000 q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + q^{5} - q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + q^{5} - q^{7} - 6 q^{9} - 2 q^{11} - 12 q^{13} - 6 q^{15} - 2 q^{17} + 15 q^{21} - 9 q^{23} - q^{25} + 18 q^{27} + 6 q^{29} + 2 q^{31} - 6 q^{33} + 4 q^{35} - 8 q^{37} + 18 q^{39} + 10 q^{41} - 2 q^{43} + 6 q^{45} + 8 q^{47} - 13 q^{49} - 6 q^{51} - 4 q^{53} - 4 q^{55} - 8 q^{59} - 7 q^{61} - 24 q^{63} - 6 q^{65} - 3 q^{67} + 54 q^{69} - 16 q^{71} - 14 q^{73} - 3 q^{75} + 10 q^{77} + 4 q^{79} - 9 q^{81} + 2 q^{83} - 4 q^{85} - 9 q^{87} - 13 q^{89} + 6 q^{91} + 6 q^{93} - 20 q^{97} + 24 q^{99}+O(q^{100}) \) 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
\(3\) −1.50000 2.59808i −0.866025 1.50000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(-0.5\pi\)
\(4\) 0 0
\(5\) 0.500000 0.866025i 0.223607 0.387298i
\(6\) 0 0
\(7\) −0.500000 + 2.59808i −0.188982 + 0.981981i
\(8\) 0 0
\(9\) −3.00000 + 5.19615i −1.00000 + 1.73205i
\(10\) 0 0
\(11\) −1.00000 1.73205i −0.301511 0.522233i 0.674967 0.737848i \(-0.264158\pi\)
−0.976478 + 0.215615i \(0.930824\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) −1.00000 1.73205i −0.242536 0.420084i 0.718900 0.695113i \(-0.244646\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) 0 0
\(19\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 0 0
\(21\) 7.50000 2.59808i 1.63663 0.566947i
\(22\) 0 0
\(23\) −4.50000 + 7.79423i −0.938315 + 1.62521i −0.169701 + 0.985496i \(0.554280\pi\)
−0.768613 + 0.639713i \(0.779053\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 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
\(39\) 9.00000 + 15.5885i 1.44115 + 2.49615i
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 3.00000 + 5.19615i 0.447214 + 0.774597i
\(46\) 0 0
\(47\) 4.00000 6.92820i 0.583460 1.01058i −0.411606 0.911362i \(-0.635032\pi\)
0.995066 0.0992202i \(-0.0316348\pi\)
\(48\) 0 0
\(49\) −6.50000 2.59808i −0.928571 0.371154i
\(50\) 0 0
\(51\) −3.00000 + 5.19615i −0.420084 + 0.727607i
\(52\) 0 0
\(53\) −2.00000 3.46410i −0.274721 0.475831i 0.695344 0.718677i \(-0.255252\pi\)
−0.970065 + 0.242846i \(0.921919\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 6.92820i −0.520756 0.901975i −0.999709 0.0241347i \(-0.992317\pi\)
0.478953 0.877841i \(-0.341016\pi\)
\(60\) 0 0
\(61\) −3.50000 + 6.06218i −0.448129 + 0.776182i −0.998264 0.0588933i \(-0.981243\pi\)
0.550135 + 0.835076i \(0.314576\pi\)
\(62\) 0 0
\(63\) −12.0000 10.3923i −1.51186 1.30931i
\(64\) 0 0
\(65\) −3.00000 + 5.19615i −0.372104 + 0.644503i
\(66\) 0 0
\(67\) −1.50000 2.59808i −0.183254 0.317406i 0.759733 0.650236i \(-0.225330\pi\)
−0.942987 + 0.332830i \(0.891996\pi\)
\(68\) 0 0
\(69\) 27.0000 3.25042
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −7.00000 12.1244i −0.819288 1.41905i −0.906208 0.422833i \(-0.861036\pi\)
0.0869195 0.996215i \(-0.472298\pi\)
\(74\) 0 0
\(75\) −1.50000 + 2.59808i −0.173205 + 0.300000i
\(76\) 0 0
\(77\) 5.00000 1.73205i 0.569803 0.197386i
\(78\) 0 0
\(79\) 2.00000 3.46410i 0.225018 0.389742i −0.731307 0.682048i \(-0.761089\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) 1.00000 0.109764 0.0548821 0.998493i \(-0.482522\pi\)
0.0548821 + 0.998493i \(0.482522\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) −4.50000 7.79423i −0.482451 0.835629i
\(88\) 0 0
\(89\) −6.50000 + 11.2583i −0.688999 + 1.19338i 0.283164 + 0.959072i \(0.408616\pi\)
−0.972162 + 0.234309i \(0.924717\pi\)
\(90\) 0 0
\(91\) 3.00000 15.5885i 0.314485 1.63411i
\(92\) 0 0
\(93\) 3.00000 5.19615i 0.311086 0.538816i
\(94\) 0 0
\(95\) 0 0
\(96\) 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
\(103\) 6.50000 11.2583i 0.640464 1.10932i −0.344865 0.938652i \(-0.612075\pi\)
0.985329 0.170664i \(-0.0545913\pi\)
\(104\) 0 0
\(105\) 1.50000 7.79423i 0.146385 0.760639i
\(106\) 0 0
\(107\) −7.50000 + 12.9904i −0.725052 + 1.25583i 0.233900 + 0.972261i \(0.424851\pi\)
−0.958952 + 0.283567i \(0.908482\pi\)
\(108\) 0 0
\(109\) −4.50000 7.79423i −0.431022 0.746552i 0.565940 0.824447i \(-0.308513\pi\)
−0.996962 + 0.0778949i \(0.975180\pi\)
\(110\) 0 0
\(111\) 24.0000 2.27798
\(112\) 0 0
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) 4.50000 + 7.79423i 0.419627 + 0.726816i
\(116\) 0 0
\(117\) 18.0000 31.1769i 1.66410 2.88231i
\(118\) 0 0
\(119\) 5.00000 1.73205i 0.458349 0.158777i
\(120\) 0 0
\(121\) 3.50000 6.06218i 0.318182 0.551107i
\(122\) 0 0
\(123\) −7.50000 12.9904i −0.676252 1.17130i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 1.50000 + 2.59808i 0.132068 + 0.228748i
\(130\) 0 0
\(131\) −2.00000 + 3.46410i −0.174741 + 0.302660i −0.940072 0.340977i \(-0.889242\pi\)
0.765331 + 0.643637i \(0.222575\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 4.50000 7.79423i 0.387298 0.670820i
\(136\) 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
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) 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
\(147\) 3.00000 + 20.7846i 0.247436 + 1.71429i
\(148\) 0 0
\(149\) −4.50000 + 7.79423i −0.368654 + 0.638528i −0.989355 0.145519i \(-0.953515\pi\)
0.620701 + 0.784047i \(0.286848\pi\)
\(150\) 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
\(157\) 1.00000 + 1.73205i 0.0798087 + 0.138233i 0.903167 0.429289i \(-0.141236\pi\)
−0.823359 + 0.567521i \(0.807902\pi\)
\(158\) 0 0
\(159\) −6.00000 + 10.3923i −0.475831 + 0.824163i
\(160\) 0 0
\(161\) −18.0000 15.5885i −1.41860 1.22854i
\(162\) 0 0
\(163\) 4.00000 6.92820i 0.313304 0.542659i −0.665771 0.746156i \(-0.731897\pi\)
0.979076 + 0.203497i \(0.0652307\pi\)
\(164\) 0 0
\(165\) 3.00000 + 5.19615i 0.233550 + 0.404520i
\(166\) 0 0
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.00000 13.8564i 0.608229 1.05348i −0.383304 0.923622i \(-0.625214\pi\)
0.991532 0.129861i \(-0.0414530\pi\)
\(174\) 0 0
\(175\) 2.50000 0.866025i 0.188982 0.0654654i
\(176\) 0 0
\(177\) −12.0000 + 20.7846i −0.901975 + 1.56227i
\(178\) 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
\(181\) 1.00000 0.0743294 0.0371647 0.999309i \(-0.488167\pi\)
0.0371647 + 0.999309i \(0.488167\pi\)
\(182\) 0 0
\(183\) 21.0000 1.55236
\(184\) 0 0
\(185\) 4.00000 + 6.92820i 0.294086 + 0.509372i
\(186\) 0 0
\(187\) −2.00000 + 3.46410i −0.146254 + 0.253320i
\(188\) 0 0
\(189\) −4.50000 + 23.3827i −0.327327 + 1.70084i
\(190\) 0 0
\(191\) 3.00000 5.19615i 0.217072 0.375980i −0.736839 0.676068i \(-0.763683\pi\)
0.953912 + 0.300088i \(0.0970159\pi\)
\(192\) 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
\(195\) 18.0000 1.28901
\(196\) 0 0
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) 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
\(201\) −4.50000 + 7.79423i −0.317406 + 0.549762i
\(202\) 0 0
\(203\) −1.50000 + 7.79423i −0.105279 + 0.547048i
\(204\) 0 0
\(205\) 2.50000 4.33013i 0.174608 0.302429i
\(206\) 0 0
\(207\) −27.0000 46.7654i −1.87663 3.25042i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 12.0000 + 20.7846i 0.822226 + 1.42414i
\(214\) 0 0
\(215\) −0.500000 + 0.866025i −0.0340997 + 0.0590624i
\(216\) 0 0
\(217\) −5.00000 + 1.73205i −0.339422 + 0.117579i
\(218\) 0 0
\(219\) −21.0000 + 36.3731i −1.41905 + 2.45786i
\(220\) 0 0
\(221\) 6.00000 + 10.3923i 0.403604 + 0.699062i
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 6.00000 0.400000
\(226\) 0 0
\(227\) −2.00000 3.46410i −0.132745 0.229920i 0.791989 0.610535i \(-0.209046\pi\)
−0.924734 + 0.380615i \(0.875712\pi\)
\(228\) 0 0
\(229\) 7.00000 12.1244i 0.462573 0.801200i −0.536515 0.843891i \(-0.680260\pi\)
0.999088 + 0.0426906i \(0.0135930\pi\)
\(230\) 0 0
\(231\) −12.0000 10.3923i −0.789542 0.683763i
\(232\) 0 0
\(233\) 9.00000 15.5885i 0.589610 1.02123i −0.404674 0.914461i \(-0.632615\pi\)
0.994283 0.106773i \(-0.0340517\pi\)
\(234\) 0 0
\(235\) −4.00000 6.92820i −0.260931 0.451946i
\(236\) 0 0
\(237\) −12.0000 −0.779484
\(238\) 0 0
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) −5.00000 8.66025i −0.322078 0.557856i 0.658838 0.752285i \(-0.271048\pi\)
−0.980917 + 0.194429i \(0.937715\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.50000 + 4.33013i −0.351382 + 0.276642i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −1.50000 2.59808i −0.0950586 0.164646i
\(250\) 0 0
\(251\) −30.0000 −1.89358 −0.946792 0.321847i \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 3.00000 + 5.19615i 0.187867 + 0.325396i
\(256\) 0 0
\(257\) −4.00000 + 6.92820i −0.249513 + 0.432169i −0.963391 0.268101i \(-0.913604\pi\)
0.713878 + 0.700270i \(0.246937\pi\)
\(258\) 0 0
\(259\) −16.0000 13.8564i −0.994192 0.860995i
\(260\) 0 0
\(261\) −9.00000 + 15.5885i −0.557086 + 0.964901i
\(262\) 0 0
\(263\) 8.50000 + 14.7224i 0.524132 + 0.907824i 0.999605 + 0.0280936i \(0.00894366\pi\)
−0.475473 + 0.879730i \(0.657723\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 39.0000 2.38676
\(268\) 0 0
\(269\) −4.50000 7.79423i −0.274370 0.475223i 0.695606 0.718423i \(-0.255136\pi\)
−0.969976 + 0.243201i \(0.921803\pi\)
\(270\) 0 0
\(271\) −12.0000 + 20.7846i −0.728948 + 1.26258i 0.228380 + 0.973572i \(0.426657\pi\)
−0.957328 + 0.289003i \(0.906676\pi\)
\(272\) 0 0
\(273\) −45.0000 + 15.5885i −2.72352 + 0.943456i
\(274\) 0 0
\(275\) −1.00000 + 1.73205i −0.0603023 + 0.104447i
\(276\) 0 0
\(277\) −9.00000 15.5885i −0.540758 0.936620i −0.998861 0.0477206i \(-0.984804\pi\)
0.458103 0.888899i \(-0.348529\pi\)
\(278\) 0 0
\(279\) −12.0000 −0.718421
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) −2.00000 3.46410i −0.118888 0.205919i 0.800439 0.599414i \(-0.204600\pi\)
−0.919327 + 0.393494i \(0.871266\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.50000 + 12.9904i −0.147570 + 0.766798i
\(288\) 0 0
\(289\) 6.50000 11.2583i 0.382353 0.662255i
\(290\) 0 0
\(291\) 15.0000 + 25.9808i 0.879316 + 1.52302i
\(292\) 0 0
\(293\) 4.00000 0.233682 0.116841 0.993151i \(-0.462723\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) −9.00000 15.5885i −0.522233 0.904534i
\(298\) 0 0
\(299\) 27.0000 46.7654i 1.56145 2.70451i
\(300\) 0 0
\(301\) 0.500000 2.59808i 0.0288195 0.149751i
\(302\) 0 0
\(303\) 4.50000 7.79423i 0.258518 0.447767i
\(304\) 0 0
\(305\) 3.50000 + 6.06218i 0.200409 + 0.347119i
\(306\) 0 0
\(307\) −1.00000 −0.0570730 −0.0285365 0.999593i \(-0.509085\pi\)
−0.0285365 + 0.999593i \(0.509085\pi\)
\(308\) 0 0
\(309\) −39.0000 −2.21863
\(310\) 0 0
\(311\) 12.0000 + 20.7846i 0.680458 + 1.17859i 0.974841 + 0.222900i \(0.0715523\pi\)
−0.294384 + 0.955687i \(0.595114\pi\)
\(312\) 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
\(315\) −15.0000 + 5.19615i −0.845154 + 0.292770i
\(316\) 0 0
\(317\) −5.00000 + 8.66025i −0.280828 + 0.486408i −0.971589 0.236675i \(-0.923942\pi\)
0.690761 + 0.723083i \(0.257276\pi\)
\(318\) 0 0
\(319\) −3.00000 5.19615i −0.167968 0.290929i
\(320\) 0 0
\(321\) 45.0000 2.51166
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 3.00000 + 5.19615i 0.166410 + 0.288231i
\(326\) 0 0
\(327\) −13.5000 + 23.3827i −0.746552 + 1.29307i
\(328\) 0 0
\(329\) 16.0000 + 13.8564i 0.882109 + 0.763928i
\(330\) 0 0
\(331\) 5.00000 8.66025i 0.274825 0.476011i −0.695266 0.718752i \(-0.744713\pi\)
0.970091 + 0.242742i \(0.0780468\pi\)
\(332\) 0 0
\(333\) −24.0000 41.5692i −1.31519 2.27798i
\(334\) 0 0
\(335\) −3.00000 −0.163908
\(336\) 0 0
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) 0 0
\(339\) 6.00000 + 10.3923i 0.325875 + 0.564433i
\(340\) 0 0
\(341\) 2.00000 3.46410i 0.108306 0.187592i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 13.5000 23.3827i 0.726816 1.25888i
\(346\) 0 0
\(347\) 12.5000 + 21.6506i 0.671035 + 1.16227i 0.977611 + 0.210421i \(0.0674834\pi\)
−0.306576 + 0.951846i \(0.599183\pi\)
\(348\) 0 0
\(349\) −17.0000 −0.909989 −0.454995 0.890494i \(-0.650359\pi\)
−0.454995 + 0.890494i \(0.650359\pi\)
\(350\) 0 0
\(351\) −54.0000 −2.88231
\(352\) 0 0
\(353\) 18.0000 + 31.1769i 0.958043 + 1.65938i 0.727245 + 0.686378i \(0.240800\pi\)
0.230799 + 0.973002i \(0.425866\pi\)
\(354\) 0 0
\(355\) −4.00000 + 6.92820i −0.212298 + 0.367711i
\(356\) 0 0
\(357\) −12.0000 10.3923i −0.635107 0.550019i
\(358\) 0 0
\(359\) −5.00000 + 8.66025i −0.263890 + 0.457071i −0.967272 0.253741i \(-0.918339\pi\)
0.703382 + 0.710812i \(0.251672\pi\)
\(360\) 0 0
\(361\) 9.50000 + 16.4545i 0.500000 + 0.866025i
\(362\) 0 0
\(363\) −21.0000 −1.10221
\(364\) 0 0
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) 0.500000 + 0.866025i 0.0260998 + 0.0452062i 0.878780 0.477227i \(-0.158358\pi\)
−0.852680 + 0.522433i \(0.825025\pi\)
\(368\) 0 0
\(369\) −15.0000 + 25.9808i −0.780869 + 1.35250i
\(370\) 0 0
\(371\) 10.0000 3.46410i 0.519174 0.179847i
\(372\) 0 0
\(373\) −16.0000 + 27.7128i −0.828449 + 1.43492i 0.0708063 + 0.997490i \(0.477443\pi\)
−0.899255 + 0.437425i \(0.855891\pi\)
\(374\) 0 0
\(375\) 1.50000 + 2.59808i 0.0774597 + 0.134164i
\(376\) 0 0
\(377\) −18.0000 −0.927047
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 24.0000 + 41.5692i 1.22956 + 2.12966i
\(382\) 0 0
\(383\) −4.50000 + 7.79423i −0.229939 + 0.398266i −0.957790 0.287469i \(-0.907186\pi\)
0.727851 + 0.685736i \(0.240519\pi\)
\(384\) 0 0
\(385\) 1.00000 5.19615i 0.0509647 0.264820i
\(386\) 0 0
\(387\) 3.00000 5.19615i 0.152499 0.264135i
\(388\) 0 0
\(389\) 5.00000 + 8.66025i 0.253510 + 0.439092i 0.964490 0.264120i \(-0.0850816\pi\)
−0.710980 + 0.703213i \(0.751748\pi\)
\(390\) 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
\(397\) −5.00000 + 8.66025i −0.250943 + 0.434646i −0.963786 0.266678i \(-0.914074\pi\)
0.712843 + 0.701324i \(0.247407\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.5000 25.1147i 0.724095 1.25417i −0.235250 0.971935i \(-0.575591\pi\)
0.959345 0.282235i \(-0.0910758\pi\)
\(402\) 0 0
\(403\) −6.00000 10.3923i −0.298881 0.517678i
\(404\) 0 0
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) 16.0000 0.793091
\(408\) 0 0
\(409\) −1.50000 2.59808i −0.0741702 0.128467i 0.826555 0.562856i \(-0.190297\pi\)
−0.900725 + 0.434389i \(0.856964\pi\)
\(410\) 0 0
\(411\) 18.0000 31.1769i 0.887875 1.53784i
\(412\) 0 0
\(413\) 20.0000 6.92820i 0.984136 0.340915i
\(414\) 0 0
\(415\) 0.500000 0.866025i 0.0245440 0.0425115i
\(416\) 0 0
\(417\) 15.0000 + 25.9808i 0.734553 + 1.27228i
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) 24.0000 + 41.5692i 1.16692 + 2.02116i
\(424\) 0 0
\(425\) −1.00000 + 1.73205i −0.0485071 + 0.0840168i
\(426\) 0 0
\(427\) −14.0000 12.1244i −0.677507 0.586739i
\(428\) 0 0
\(429\) 18.0000 31.1769i 0.869048 1.50524i
\(430\) 0 0
\(431\) 3.00000 + 5.19615i 0.144505 + 0.250290i 0.929188 0.369607i \(-0.120508\pi\)
−0.784683 + 0.619897i \(0.787174\pi\)
\(432\) 0 0
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 2.00000 3.46410i 0.0954548 0.165333i −0.814344 0.580383i \(-0.802903\pi\)
0.909798 + 0.415051i \(0.136236\pi\)
\(440\) 0 0
\(441\) 33.0000 25.9808i 1.57143 1.23718i
\(442\) 0 0
\(443\) 18.5000 32.0429i 0.878962 1.52241i 0.0264796 0.999649i \(-0.491570\pi\)
0.852482 0.522757i \(-0.175096\pi\)
\(444\) 0 0
\(445\) 6.50000 + 11.2583i 0.308130 + 0.533696i
\(446\) 0 0
\(447\) 27.0000 1.27706
\(448\) 0 0
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) −5.00000 8.66025i −0.235441 0.407795i
\(452\) 0 0
\(453\) −15.0000 + 25.9808i −0.704761 + 1.22068i
\(454\) 0 0
\(455\) −12.0000 10.3923i −0.562569 0.487199i
\(456\) 0 0
\(457\) −14.0000 + 24.2487i −0.654892 + 1.13431i 0.327028 + 0.945015i \(0.393953\pi\)
−0.981921 + 0.189292i \(0.939381\pi\)
\(458\) 0 0
\(459\) −9.00000 15.5885i −0.420084 0.727607i
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) −17.0000 −0.790057 −0.395029 0.918669i \(-0.629265\pi\)
−0.395029 + 0.918669i \(0.629265\pi\)
\(464\) 0 0
\(465\) −3.00000 5.19615i −0.139122 0.240966i
\(466\) 0 0
\(467\) 2.50000 4.33013i 0.115686 0.200374i −0.802368 0.596830i \(-0.796427\pi\)
0.918054 + 0.396456i \(0.129760\pi\)
\(468\) 0 0
\(469\) 7.50000 2.59808i 0.346318 0.119968i
\(470\) 0 0
\(471\) 3.00000 5.19615i 0.138233 0.239426i
\(472\) 0 0
\(473\) 1.00000 + 1.73205i 0.0459800 + 0.0796398i
\(474\) 0 0
\(475\) 0 0
\(476\) 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
\(481\) 24.0000 41.5692i 1.09431 1.89539i
\(482\) 0 0
\(483\) −13.5000 + 70.1481i −0.614271 + 3.19185i
\(484\) 0 0
\(485\) −5.00000 + 8.66025i −0.227038 + 0.393242i
\(486\) 0 0
\(487\) 16.0000 + 27.7128i 0.725029 + 1.25579i 0.958962 + 0.283535i \(0.0915071\pi\)
−0.233933 + 0.972253i \(0.575160\pi\)
\(488\) 0 0
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 0 0
\(493\) −3.00000 5.19615i −0.135113 0.234023i
\(494\) 0 0
\(495\) 6.00000 10.3923i 0.269680 0.467099i
\(496\) 0 0
\(497\) 4.00000 20.7846i 0.179425 0.932317i
\(498\) 0 0
\(499\) −3.00000 + 5.19615i −0.134298 + 0.232612i −0.925329 0.379165i \(-0.876211\pi\)
0.791031 + 0.611776i \(0.209545\pi\)
\(500\) 0 0
\(501\) −13.5000 23.3827i −0.603136 1.04466i
\(502\) 0 0
\(503\) −27.0000 −1.20387 −0.601935 0.798545i \(-0.705603\pi\)
−0.601935 + 0.798545i \(0.705603\pi\)
\(504\) 0 0
\(505\) 3.00000 0.133498
\(506\) 0 0
\(507\) −34.5000 59.7558i −1.53220 2.65385i
\(508\) 0 0
\(509\) 20.5000 35.5070i 0.908647 1.57382i 0.0927004 0.995694i \(-0.470450\pi\)
0.815946 0.578128i \(-0.196217\pi\)
\(510\) 0 0
\(511\) 35.0000 12.1244i 1.54831 0.536350i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.50000 11.2583i −0.286424 0.496101i
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) −48.0000 −2.10697
\(520\) 0 0
\(521\) −7.00000 12.1244i −0.306676 0.531178i 0.670957 0.741496i \(-0.265883\pi\)
−0.977633 + 0.210318i \(0.932550\pi\)
\(522\) 0 0
\(523\) 2.00000 3.46410i 0.0874539 0.151475i −0.818980 0.573822i \(-0.805460\pi\)
0.906434 + 0.422347i \(0.138794\pi\)
\(524\) 0 0
\(525\) −6.00000 5.19615i −0.261861 0.226779i
\(526\) 0 0
\(527\) 2.00000 3.46410i 0.0871214 0.150899i
\(528\) 0 0
\(529\) −29.0000 50.2295i −1.26087 2.18389i
\(530\) 0 0
\(531\) 48.0000 2.08302
\(532\) 0 0
\(533\) −30.0000 −1.29944
\(534\) 0 0
\(535\) 7.50000 + 12.9904i 0.324253 + 0.561623i
\(536\) 0 0
\(537\) −9.00000 + 15.5885i −0.388379 + 0.672692i
\(538\) 0 0
\(539\) 2.00000 + 13.8564i 0.0861461 + 0.596838i
\(540\) 0 0
\(541\) −16.5000 + 28.5788i −0.709390 + 1.22870i 0.255693 + 0.966758i \(0.417696\pi\)
−0.965084 + 0.261942i \(0.915637\pi\)
\(542\) 0 0
\(543\) −1.50000 2.59808i −0.0643712 0.111494i
\(544\) 0 0
\(545\) −9.00000 −0.385518
\(546\) 0 0
\(547\) 15.0000 0.641354 0.320677 0.947189i \(-0.396090\pi\)
0.320677 + 0.947189i \(0.396090\pi\)
\(548\) 0 0
\(549\) −21.0000 36.3731i −0.896258 1.55236i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000 + 6.92820i 0.340195 + 0.294617i
\(554\) 0 0
\(555\) 12.0000 20.7846i 0.509372 0.882258i
\(556\) 0 0
\(557\) 1.00000 + 1.73205i 0.0423714 + 0.0733893i 0.886433 0.462856i \(-0.153175\pi\)
−0.844062 + 0.536246i \(0.819842\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) −6.50000 11.2583i −0.273942 0.474482i 0.695925 0.718114i \(-0.254994\pi\)
−0.969868 + 0.243632i \(0.921661\pi\)
\(564\) 0 0
\(565\) −2.00000 + 3.46410i −0.0841406 + 0.145736i
\(566\) 0 0
\(567\) 22.5000 7.79423i 0.944911 0.327327i
\(568\) 0 0
\(569\) −15.0000 + 25.9808i −0.628833 + 1.08917i 0.358954 + 0.933355i \(0.383134\pi\)
−0.987786 + 0.155815i \(0.950200\pi\)
\(570\) 0 0
\(571\) 12.0000 + 20.7846i 0.502184 + 0.869809i 0.999997 + 0.00252413i \(0.000803457\pi\)
−0.497812 + 0.867285i \(0.665863\pi\)
\(572\) 0 0
\(573\) −18.0000 −0.751961
\(574\) 0 0
\(575\) 9.00000 0.375326
\(576\) 0 0
\(577\) −11.0000 19.0526i −0.457936 0.793168i 0.540916 0.841077i \(-0.318078\pi\)
−0.998852 + 0.0479084i \(0.984744\pi\)
\(578\) 0 0
\(579\) −3.00000 + 5.19615i −0.124676 + 0.215945i
\(580\) 0 0
\(581\) −0.500000 + 2.59808i −0.0207435 + 0.107786i
\(582\) 0 0
\(583\) −4.00000 + 6.92820i −0.165663 + 0.286937i
\(584\) 0 0
\(585\) −18.0000 31.1769i −0.744208 1.28901i
\(586\) 0 0
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −21.0000 36.3731i −0.863825 1.49619i
\(592\) 0 0
\(593\) 18.0000 31.1769i 0.739171 1.28028i −0.213697 0.976900i \(-0.568551\pi\)
0.952869 0.303383i \(-0.0981160\pi\)
\(594\) 0 0
\(595\) 1.00000 5.19615i 0.0409960 0.213021i
\(596\) 0 0
\(597\) 30.0000 51.9615i 1.22782 2.12664i
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) 18.0000 0.733017
\(604\) 0 0
\(605\) −3.50000 6.06218i −0.142295 0.246463i
\(606\) 0 0
\(607\) 0.500000 0.866025i 0.0202944 0.0351509i −0.855700 0.517472i \(-0.826873\pi\)
0.875994 + 0.482322i \(0.160206\pi\)
\(608\) 0 0
\(609\) 22.5000 7.79423i 0.911746 0.315838i
\(610\) 0 0
\(611\) −24.0000 + 41.5692i −0.970936 + 1.68171i
\(612\) 0 0
\(613\) 3.00000 + 5.19615i 0.121169 + 0.209871i 0.920229 0.391381i \(-0.128002\pi\)
−0.799060 + 0.601251i \(0.794669\pi\)
\(614\) 0 0
\(615\) −15.0000 −0.604858
\(616\) 0 0
\(617\) 20.0000 0.805170 0.402585 0.915383i \(-0.368112\pi\)
0.402585 + 0.915383i \(0.368112\pi\)
\(618\) 0 0
\(619\) −5.00000 8.66025i −0.200967 0.348085i 0.747873 0.663842i \(-0.231075\pi\)
−0.948840 + 0.315757i \(0.897742\pi\)
\(620\) 0 0
\(621\) −40.5000 + 70.1481i −1.62521 + 2.81494i
\(622\) 0 0
\(623\) −26.0000 22.5167i −1.04167 0.902111i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 34.0000 1.35352 0.676759 0.736204i \(-0.263384\pi\)
0.676759 + 0.736204i \(0.263384\pi\)
\(632\) 0 0
\(633\) 6.00000 + 10.3923i 0.238479 + 0.413057i
\(634\) 0 0
\(635\) −8.00000 + 13.8564i −0.317470 + 0.549875i
\(636\) 0 0
\(637\) 39.0000 + 15.5885i 1.54524 + 0.617637i
\(638\) 0 0
\(639\) 24.0000 41.5692i 0.949425 1.64445i
\(640\) 0 0
\(641\) 15.5000 + 26.8468i 0.612213 + 1.06038i 0.990867 + 0.134846i \(0.0430539\pi\)
−0.378653 + 0.925539i \(0.623613\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) 3.00000 0.118125
\(646\) 0 0
\(647\) 0.500000 + 0.866025i 0.0196570 + 0.0340470i 0.875687 0.482880i \(-0.160409\pi\)
−0.856030 + 0.516927i \(0.827076\pi\)
\(648\) 0 0
\(649\) −8.00000 + 13.8564i −0.314027 + 0.543912i
\(650\) 0 0
\(651\) 12.0000 + 10.3923i 0.470317 + 0.407307i
\(652\) 0 0
\(653\) 17.0000 29.4449i 0.665261 1.15227i −0.313953 0.949439i \(-0.601653\pi\)
0.979214 0.202828i \(-0.0650132\pi\)
\(654\) 0 0
\(655\) 2.00000 + 3.46410i 0.0781465 + 0.135354i
\(656\) 0 0
\(657\) 84.0000 3.27715
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) −15.5000 26.8468i −0.602880 1.04422i −0.992383 0.123194i \(-0.960686\pi\)
0.389503 0.921025i \(-0.372647\pi\)
\(662\) 0 0
\(663\) 18.0000 31.1769i 0.699062 1.21081i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −13.5000 + 23.3827i −0.522722 + 0.905381i
\(668\) 0 0
\(669\) 24.0000 + 41.5692i 0.927894 + 1.60716i
\(670\) 0 0
\(671\) 14.0000 0.540464
\(672\) 0 0
\(673\) 16.0000 0.616755 0.308377 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) 0 0
\(675\) −4.50000 7.79423i −0.173205 0.300000i
\(676\) 0 0
\(677\) −3.00000 + 5.19615i −0.115299 + 0.199704i −0.917899 0.396813i \(-0.870116\pi\)
0.802600 + 0.596518i \(0.203449\pi\)
\(678\) 0 0
\(679\) 5.00000 25.9808i 0.191882 0.997050i
\(680\) 0 0
\(681\) −6.00000 + 10.3923i −0.229920 + 0.398234i
\(682\) 0 0
\(683\) 14.5000 + 25.1147i 0.554827 + 0.960989i 0.997917 + 0.0645115i \(0.0205489\pi\)
−0.443090 + 0.896477i \(0.646118\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) −42.0000 −1.60240
\(688\) 0 0
\(689\) 12.0000 + 20.7846i 0.457164 + 0.791831i
\(690\) 0 0
\(691\) −13.0000 + 22.5167i −0.494543 + 0.856574i −0.999980 0.00628943i \(-0.997998\pi\)
0.505437 + 0.862864i \(0.331331\pi\)
\(692\) 0 0
\(693\) −6.00000 + 31.1769i −0.227921 + 1.18431i
\(694\) 0 0
\(695\) −5.00000 + 8.66025i −0.189661 + 0.328502i
\(696\) 0 0
\(697\) −5.00000 8.66025i −0.189389 0.328031i
\(698\) 0 0
\(699\) −54.0000 −2.04247
\(700\) 0 0
\(701\) −29.0000 −1.09531 −0.547657 0.836703i \(-0.684480\pi\)
−0.547657 + 0.836703i \(0.684480\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −12.0000 + 20.7846i −0.451946 + 0.782794i
\(706\) 0 0
\(707\) −7.50000 + 2.59808i −0.282067 + 0.0977107i
\(708\) 0 0
\(709\) −15.5000 + 26.8468i −0.582115 + 1.00825i 0.413114 + 0.910679i \(0.364441\pi\)
−0.995228 + 0.0975728i \(0.968892\pi\)
\(710\) 0 0
\(711\) 12.0000 + 20.7846i 0.450035 + 0.779484i
\(712\) 0 0
\(713\) −18.0000 −0.674105
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 0 0
\(717\) −39.0000 67.5500i −1.45648 2.52270i
\(718\) 0 0
\(719\) −3.00000 + 5.19615i −0.111881 + 0.193784i −0.916529 0.399969i \(-0.869021\pi\)
0.804648 + 0.593753i \(0.202354\pi\)
\(720\) 0 0
\(721\) 26.0000 + 22.5167i 0.968291 + 0.838564i
\(722\) 0 0
\(723\) −15.0000 + 25.9808i −0.557856 + 0.966235i
\(724\) 0 0
\(725\) −1.50000 2.59808i −0.0557086 0.0964901i
\(726\) 0 0
\(727\) −3.00000 −0.111264 −0.0556319 0.998451i \(-0.517717\pi\)
−0.0556319 + 0.998451i \(0.517717\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 1.00000 + 1.73205i 0.0369863 + 0.0640622i
\(732\) 0 0
\(733\) −17.0000 + 29.4449i −0.627909 + 1.08757i 0.360061 + 0.932929i \(0.382756\pi\)
−0.987971 + 0.154642i \(0.950578\pi\)
\(734\) 0 0
\(735\) 19.5000 + 7.79423i 0.719268 + 0.287494i
\(736\) 0 0
\(737\) −3.00000 + 5.19615i −0.110506 + 0.191403i
\(738\) 0 0
\(739\) 5.00000 + 8.66025i 0.183928 + 0.318573i 0.943215 0.332184i \(-0.107785\pi\)
−0.759287 + 0.650756i \(0.774452\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.00000 0.110059 0.0550297 0.998485i \(-0.482475\pi\)
0.0550297 + 0.998485i \(0.482475\pi\)
\(744\) 0 0
\(745\) 4.50000 + 7.79423i 0.164867 + 0.285558i
\(746\) 0 0
\(747\) −3.00000 + 5.19615i −0.109764 + 0.190117i
\(748\) 0 0
\(749\) −30.0000 25.9808i −1.09618 0.949316i
\(750\) 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
\(753\) 45.0000 + 77.9423i 1.63989 + 2.84037i
\(754\) 0 0
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) −27.0000 46.7654i −0.980038 1.69748i
\(760\) 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
\(763\) 22.5000 7.79423i 0.814555 0.282170i
\(764\) 0 0
\(765\) 6.00000 10.3923i 0.216930 0.375735i
\(766\) 0 0
\(767\) 24.0000 + 41.5692i 0.866590 + 1.50098i
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) 0 0
\(773\) 9.00000 + 15.5885i 0.323708 + 0.560678i 0.981250 0.192740i \(-0.0617373\pi\)
−0.657542 + 0.753418i \(0.728404\pi\)
\(774\) 0 0
\(775\) 1.00000 1.73205i 0.0359211 0.0622171i
\(776\) 0 0
\(777\) −12.0000 + 62.3538i −0.430498 + 2.23693i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 8.00000 + 13.8564i 0.286263 + 0.495821i
\(782\) 0 0
\(783\) 27.0000 0.964901
\(784\) 0 0
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) −5.50000 9.52628i −0.196054 0.339575i 0.751192 0.660084i \(-0.229479\pi\)
−0.947245 + 0.320509i \(0.896146\pi\)
\(788\) 0 0
\(789\) 25.5000 44.1673i 0.907824 1.57240i
\(790\) 0 0
\(791\) 2.00000 10.3923i 0.0711118 0.369508i
\(792\) 0 0
\(793\) 21.0000 36.3731i 0.745732 1.29165i
\(794\) 0 0
\(795\) 6.00000 + 10.3923i 0.212798 + 0.368577i
\(796\) 0 0
\(797\) 36.0000 1.27519 0.637593 0.770374i \(-0.279930\pi\)
0.637593 + 0.770374i \(0.279930\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −39.0000 67.5500i −1.37800 2.38676i
\(802\) 0 0
\(803\) −14.0000 + 24.2487i −0.494049 + 0.855718i
\(804\) 0 0
\(805\) −22.5000 + 7.79423i −0.793021 + 0.274710i
\(806\) 0 0
\(807\) −13.5000 + 23.3827i −0.475223 + 0.823110i
\(808\) 0 0
\(809\) −22.5000 38.9711i −0.791058 1.37015i −0.925312 0.379206i \(-0.876197\pi\)
0.134255 0.990947i \(-0.457136\pi\)
\(810\) 0 0
\(811\) −50.0000 −1.75574 −0.877869 0.478901i \(-0.841035\pi\)
−0.877869 + 0.478901i \(0.841035\pi\)
\(812\) 0 0
\(813\) 72.0000 2.52515
\(814\) 0 0
\(815\) −4.00000 6.92820i −0.140114 0.242684i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 72.0000 + 62.3538i 2.51588 + 2.17882i
\(820\) 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
\(823\) 9.50000 + 16.4545i 0.331149 + 0.573567i 0.982737 0.185006i \(-0.0592303\pi\)
−0.651588 + 0.758573i \(0.725897\pi\)
\(824\) 0 0
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) −23.0000 −0.799788 −0.399894 0.916561i \(-0.630953\pi\)
−0.399894 + 0.916561i \(0.630953\pi\)
\(828\) 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
\(831\) −27.0000 + 46.7654i −0.936620 + 1.62227i
\(832\) 0 0
\(833\) 2.00000 + 13.8564i 0.0692959 + 0.480096i
\(834\) 0 0
\(835\) 4.50000 7.79423i 0.155729 0.269730i
\(836\) 0 0
\(837\) 9.00000 + 15.5885i 0.311086 + 0.538816i
\(838\) 0 0
\(839\) −14.0000 −0.483334 −0.241667 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 33.0000 + 57.1577i 1.13658 + 1.96861i
\(844\) 0 0
\(845\) 11.5000 19.9186i 0.395612 0.685220i
\(846\) 0 0
\(847\) 14.0000 + 12.1244i 0.481046 + 0.416598i
\(848\) 0 0
\(849\) −6.00000 + 10.3923i −0.205919 + 0.356663i
\(850\) 0 0
\(851\) −36.0000 62.3538i −1.23406 2.13746i
\(852\) 0 0
\(853\) −40.0000 −1.36957 −0.684787 0.728743i \(-0.740105\pi\)
−0.684787 + 0.728743i \(0.740105\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 27.0000 + 46.7654i 0.922302 + 1.59747i 0.795843 + 0.605503i \(0.207028\pi\)
0.126459 + 0.991972i \(0.459639\pi\)
\(858\) 0 0
\(859\) −18.0000 + 31.1769i −0.614152 + 1.06374i 0.376381 + 0.926465i \(0.377169\pi\)
−0.990533 + 0.137277i \(0.956165\pi\)
\(860\) 0 0
\(861\) 37.5000 12.9904i 1.27800 0.442711i
\(862\) 0 0
\(863\) 18.5000 32.0429i 0.629747 1.09075i −0.357855 0.933777i \(-0.616492\pi\)
0.987602 0.156977i \(-0.0501749\pi\)
\(864\) 0 0
\(865\) −8.00000 13.8564i −0.272008 0.471132i
\(866\) 0 0
\(867\) −39.0000 −1.32451
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 9.00000 + 15.5885i 0.304953 + 0.528195i
\(872\) 0 0
\(873\) 30.0000 51.9615i 1.01535 1.75863i
\(874\) 0 0
\(875\) 0.500000 2.59808i 0.0169031 0.0878310i
\(876\) 0 0
\(877\) −16.0000 + 27.7128i −0.540282 + 0.935795i 0.458606 + 0.888640i \(0.348349\pi\)
−0.998888 + 0.0471555i \(0.984984\pi\)
\(878\) 0 0
\(879\) −6.00000 10.3923i −0.202375 0.350524i
\(880\) 0 0
\(881\) −31.0000 −1.04442 −0.522208 0.852818i \(-0.674892\pi\)
−0.522208 + 0.852818i \(0.674892\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 12.0000 + 20.7846i 0.403376 + 0.698667i
\(886\) 0 0
\(887\) 26.5000 45.8993i 0.889783 1.54115i 0.0496513 0.998767i \(-0.484189\pi\)
0.840132 0.542383i \(-0.182478\pi\)
\(888\) 0 0
\(889\) 8.00000 41.5692i 0.268311 1.39419i
\(890\) 0 0
\(891\) −9.00000 + 15.5885i −0.301511 + 0.522233i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −6.00000 −0.200558
\(896\) 0 0
\(897\) −162.000 −5.40902
\(898\) 0 0
\(899\) 3.00000 + 5.19615i 0.100056 + 0.173301i
\(900\) 0 0
\(901\) −4.00000 + 6.92820i −0.133259 + 0.230812i
\(902\) 0 0
\(903\) −7.50000 + 2.59808i −0.249584 + 0.0864586i
\(904\) 0 0
\(905\) 0.500000 0.866025i 0.0166206 0.0287877i
\(906\) 0 0
\(907\) −18.5000 32.0429i −0.614282 1.06397i −0.990510 0.137441i \(-0.956112\pi\)
0.376228 0.926527i \(-0.377221\pi\)
\(908\) 0 0
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) −54.0000 −1.78910 −0.894550 0.446968i \(-0.852504\pi\)
−0.894550 + 0.446968i \(0.852504\pi\)
\(912\) 0 0
\(913\) −1.00000 1.73205i −0.0330952 0.0573225i
\(914\) 0 0
\(915\) 10.5000 18.1865i 0.347119 0.601228i
\(916\) 0 0
\(917\) −8.00000 6.92820i −0.264183 0.228789i
\(918\) 0 0
\(919\) 16.0000 27.7128i 0.527791 0.914161i −0.471684 0.881768i \(-0.656354\pi\)
0.999475 0.0323936i \(-0.0103130\pi\)
\(920\) 0 0
\(921\) 1.50000 + 2.59808i 0.0494267 + 0.0856095i
\(922\) 0 0
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) 39.0000 + 67.5500i 1.28093 + 2.21863i
\(928\) 0 0
\(929\) 2.50000 4.33013i 0.0820223 0.142067i −0.822096 0.569349i \(-0.807195\pi\)
0.904118 + 0.427282i \(0.140529\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 36.0000 62.3538i 1.17859 2.04137i
\(934\) 0 0
\(935\) 2.00000 + 3.46410i 0.0654070 + 0.113288i
\(936\) 0 0
\(937\) −52.0000 −1.69877 −0.849383 0.527777i \(-0.823026\pi\)
−0.849383 + 0.527777i \(0.823026\pi\)
\(938\) 0 0
\(939\) −12.0000 −0.391605
\(940\) 0 0
\(941\) −19.0000 32.9090i −0.619382 1.07280i −0.989599 0.143856i \(-0.954050\pi\)
0.370216 0.928946i \(-0.379284\pi\)
\(942\) 0 0
\(943\) −22.5000 + 38.9711i −0.732701 + 1.26908i
\(944\) 0 0
\(945\) 18.0000 + 15.5885i 0.585540 + 0.507093i
\(946\) 0 0
\(947\) 14.5000 25.1147i 0.471187 0.816119i −0.528270 0.849076i \(-0.677159\pi\)
0.999457 + 0.0329571i \(0.0104925\pi\)
\(948\) 0 0
\(949\) 42.0000 + 72.7461i 1.36338 + 2.36144i
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) −60.0000 −1.94359 −0.971795 0.235826i \(-0.924220\pi\)
−0.971795 + 0.235826i \(0.924220\pi\)
\(954\) 0 0
\(955\) −3.00000 5.19615i −0.0970777 0.168144i
\(956\) 0 0
\(957\) −9.00000 + 15.5885i −0.290929 + 0.503903i
\(958\) 0 0
\(959\) −30.0000 + 10.3923i −0.968751 + 0.335585i
\(960\) 0 0
\(961\) 13.5000 23.3827i 0.435484 0.754280i
\(962\) 0 0
\(963\) −45.0000 77.9423i −1.45010 2.51166i
\(964\) 0 0
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) −37.0000 −1.18984 −0.594920 0.803785i \(-0.702816\pi\)
−0.594920 + 0.803785i \(0.702816\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 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 25.9808i 0.160293 0.832905i
\(974\) 0 0
\(975\) 9.00000 15.5885i 0.288231 0.499230i
\(976\) 0 0
\(977\) 3.00000 + 5.19615i 0.0959785 + 0.166240i 0.910017 0.414572i \(-0.136069\pi\)
−0.814038 + 0.580812i \(0.802735\pi\)
\(978\) 0 0
\(979\) 26.0000 0.830964
\(980\) 0 0
\(981\) 54.0000 1.72409
\(982\) 0 0
\(983\) −9.50000 16.4545i −0.303003 0.524816i 0.673812 0.738903i \(-0.264656\pi\)
−0.976815 + 0.214087i \(0.931323\pi\)
\(984\) 0 0
\(985\) 7.00000 12.1244i 0.223039 0.386314i
\(986\) 0 0
\(987\) 12.0000 62.3538i 0.381964 1.98474i
\(988\) 0 0
\(989\) 4.50000 7.79423i 0.143092 0.247842i
\(990\) 0 0
\(991\) −19.0000 32.9090i −0.603555 1.04539i −0.992278 0.124033i \(-0.960417\pi\)
0.388723 0.921355i \(-0.372916\pi\)
\(992\) 0 0
\(993\) −30.0000 −0.952021
\(994\) 0 0
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) 20.0000 + 34.6410i 0.633406 + 1.09709i 0.986850 + 0.161636i \(0.0516771\pi\)
−0.353444 + 0.935456i \(0.614990\pi\)
\(998\) 0 0
\(999\) −36.0000 + 62.3538i −1.13899 + 1.97279i
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.a.401.1 2
4.3 odd 2 140.2.i.b.121.1 yes 2
7.2 even 3 3920.2.a.bi.1.1 1
7.4 even 3 inner 560.2.q.a.81.1 2
7.5 odd 6 3920.2.a.d.1.1 1
12.11 even 2 1260.2.s.b.541.1 2
20.3 even 4 700.2.r.c.149.1 4
20.7 even 4 700.2.r.c.149.2 4
20.19 odd 2 700.2.i.a.401.1 2
28.3 even 6 980.2.i.a.361.1 2
28.11 odd 6 140.2.i.b.81.1 2
28.19 even 6 980.2.a.i.1.1 1
28.23 odd 6 980.2.a.a.1.1 1
28.27 even 2 980.2.i.a.961.1 2
84.11 even 6 1260.2.s.b.361.1 2
84.23 even 6 8820.2.a.w.1.1 1
84.47 odd 6 8820.2.a.k.1.1 1
140.19 even 6 4900.2.a.a.1.1 1
140.23 even 12 4900.2.e.c.2549.1 2
140.39 odd 6 700.2.i.a.501.1 2
140.47 odd 12 4900.2.e.b.2549.1 2
140.67 even 12 700.2.r.c.249.1 4
140.79 odd 6 4900.2.a.v.1.1 1
140.103 odd 12 4900.2.e.b.2549.2 2
140.107 even 12 4900.2.e.c.2549.2 2
140.123 even 12 700.2.r.c.249.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.i.b.81.1 2 28.11 odd 6
140.2.i.b.121.1 yes 2 4.3 odd 2
560.2.q.a.81.1 2 7.4 even 3 inner
560.2.q.a.401.1 2 1.1 even 1 trivial
700.2.i.a.401.1 2 20.19 odd 2
700.2.i.a.501.1 2 140.39 odd 6
700.2.r.c.149.1 4 20.3 even 4
700.2.r.c.149.2 4 20.7 even 4
700.2.r.c.249.1 4 140.67 even 12
700.2.r.c.249.2 4 140.123 even 12
980.2.a.a.1.1 1 28.23 odd 6
980.2.a.i.1.1 1 28.19 even 6
980.2.i.a.361.1 2 28.3 even 6
980.2.i.a.961.1 2 28.27 even 2
1260.2.s.b.361.1 2 84.11 even 6
1260.2.s.b.541.1 2 12.11 even 2
3920.2.a.d.1.1 1 7.5 odd 6
3920.2.a.bi.1.1 1 7.2 even 3
4900.2.a.a.1.1 1 140.19 even 6
4900.2.a.v.1.1 1 140.79 odd 6
4900.2.e.b.2549.1 2 140.47 odd 12
4900.2.e.b.2549.2 2 140.103 odd 12
4900.2.e.c.2549.1 2 140.23 even 12
4900.2.e.c.2549.2 2 140.107 even 12
8820.2.a.k.1.1 1 84.47 odd 6
8820.2.a.w.1.1 1 84.23 even 6