Properties

Label 4020.2.q.m.3781.4
Level $4020$
Weight $2$
Character 4020.3781
Analytic conductor $32.100$
Analytic rank $0$
Dimension $24$
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] = [4020,2,Mod(841,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4020.841");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.q (of order \(3\), 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: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 3781.4
Character \(\chi\) \(=\) 4020.3781
Dual form 4020.2.q.m.841.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +(-1.09486 - 1.89635i) q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +(-1.09486 - 1.89635i) q^{7} +1.00000 q^{9} +(2.56534 + 4.44330i) q^{11} +(3.21617 - 5.57057i) q^{13} +1.00000 q^{15} +(0.653954 - 1.13268i) q^{17} +(-2.76451 + 4.78828i) q^{19} +(-1.09486 - 1.89635i) q^{21} +(-1.52897 + 2.64825i) q^{23} +1.00000 q^{25} +1.00000 q^{27} +(3.69002 + 6.39130i) q^{29} +(1.00060 + 1.73309i) q^{31} +(2.56534 + 4.44330i) q^{33} +(-1.09486 - 1.89635i) q^{35} +(1.40879 - 2.44010i) q^{37} +(3.21617 - 5.57057i) q^{39} +(1.19738 + 2.07392i) q^{41} +9.64370 q^{43} +1.00000 q^{45} +(1.71064 + 2.96291i) q^{47} +(1.10258 - 1.90972i) q^{49} +(0.653954 - 1.13268i) q^{51} -6.09552 q^{53} +(2.56534 + 4.44330i) q^{55} +(-2.76451 + 4.78828i) q^{57} -5.99534 q^{59} +(6.06494 - 10.5048i) q^{61} +(-1.09486 - 1.89635i) q^{63} +(3.21617 - 5.57057i) q^{65} +(-7.43204 - 3.42998i) q^{67} +(-1.52897 + 2.64825i) q^{69} +(7.59393 + 13.1531i) q^{71} +(6.15933 - 10.6683i) q^{73} +1.00000 q^{75} +(5.61736 - 9.72956i) q^{77} +(4.94026 + 8.55679i) q^{79} +1.00000 q^{81} +(6.69164 - 11.5903i) q^{83} +(0.653954 - 1.13268i) q^{85} +(3.69002 + 6.39130i) q^{87} -5.88126 q^{89} -14.0850 q^{91} +(1.00060 + 1.73309i) q^{93} +(-2.76451 + 4.78828i) q^{95} +(7.09796 - 12.2940i) q^{97} +(2.56534 + 4.44330i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 24 q^{3} + 24 q^{5} - 3 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 24 q^{3} + 24 q^{5} - 3 q^{7} + 24 q^{9} - 2 q^{13} + 24 q^{15} - 6 q^{19} - 3 q^{21} + 2 q^{23} + 24 q^{25} + 24 q^{27} + 7 q^{29} - 8 q^{31} - 3 q^{35} - 10 q^{37} - 2 q^{39} + 2 q^{41} + 28 q^{43} + 24 q^{45} - 3 q^{47} - 17 q^{49} + 36 q^{53} - 6 q^{57} - 10 q^{59} + 9 q^{61} - 3 q^{63} - 2 q^{65} - 46 q^{67} + 2 q^{69} - 12 q^{71} + 6 q^{73} + 24 q^{75} - 5 q^{77} + 2 q^{79} + 24 q^{81} + 11 q^{83} + 7 q^{87} + 52 q^{89} - 22 q^{91} - 8 q^{93} - 6 q^{95} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.09486 1.89635i −0.413817 0.716752i 0.581487 0.813556i \(-0.302471\pi\)
−0.995303 + 0.0968041i \(0.969138\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.56534 + 4.44330i 0.773480 + 1.33971i 0.935645 + 0.352943i \(0.114819\pi\)
−0.162165 + 0.986764i \(0.551848\pi\)
\(12\) 0 0
\(13\) 3.21617 5.57057i 0.892005 1.54500i 0.0545377 0.998512i \(-0.482631\pi\)
0.837468 0.546487i \(-0.184035\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0.653954 1.13268i 0.158607 0.274716i −0.775759 0.631029i \(-0.782633\pi\)
0.934367 + 0.356313i \(0.115966\pi\)
\(18\) 0 0
\(19\) −2.76451 + 4.78828i −0.634223 + 1.09851i 0.352456 + 0.935828i \(0.385347\pi\)
−0.986679 + 0.162678i \(0.947987\pi\)
\(20\) 0 0
\(21\) −1.09486 1.89635i −0.238917 0.413817i
\(22\) 0 0
\(23\) −1.52897 + 2.64825i −0.318811 + 0.552197i −0.980240 0.197810i \(-0.936617\pi\)
0.661429 + 0.750008i \(0.269950\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.69002 + 6.39130i 0.685220 + 1.18684i 0.973368 + 0.229250i \(0.0736272\pi\)
−0.288148 + 0.957586i \(0.593039\pi\)
\(30\) 0 0
\(31\) 1.00060 + 1.73309i 0.179713 + 0.311272i 0.941782 0.336223i \(-0.109150\pi\)
−0.762069 + 0.647496i \(0.775816\pi\)
\(32\) 0 0
\(33\) 2.56534 + 4.44330i 0.446569 + 0.773480i
\(34\) 0 0
\(35\) −1.09486 1.89635i −0.185065 0.320541i
\(36\) 0 0
\(37\) 1.40879 2.44010i 0.231604 0.401150i −0.726676 0.686980i \(-0.758936\pi\)
0.958280 + 0.285830i \(0.0922692\pi\)
\(38\) 0 0
\(39\) 3.21617 5.57057i 0.515000 0.892005i
\(40\) 0 0
\(41\) 1.19738 + 2.07392i 0.186999 + 0.323891i 0.944248 0.329234i \(-0.106791\pi\)
−0.757249 + 0.653126i \(0.773457\pi\)
\(42\) 0 0
\(43\) 9.64370 1.47065 0.735325 0.677714i \(-0.237029\pi\)
0.735325 + 0.677714i \(0.237029\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 1.71064 + 2.96291i 0.249522 + 0.432185i 0.963393 0.268092i \(-0.0863931\pi\)
−0.713871 + 0.700277i \(0.753060\pi\)
\(48\) 0 0
\(49\) 1.10258 1.90972i 0.157511 0.272817i
\(50\) 0 0
\(51\) 0.653954 1.13268i 0.0915719 0.158607i
\(52\) 0 0
\(53\) −6.09552 −0.837284 −0.418642 0.908151i \(-0.637494\pi\)
−0.418642 + 0.908151i \(0.637494\pi\)
\(54\) 0 0
\(55\) 2.56534 + 4.44330i 0.345911 + 0.599135i
\(56\) 0 0
\(57\) −2.76451 + 4.78828i −0.366169 + 0.634223i
\(58\) 0 0
\(59\) −5.99534 −0.780527 −0.390264 0.920703i \(-0.627616\pi\)
−0.390264 + 0.920703i \(0.627616\pi\)
\(60\) 0 0
\(61\) 6.06494 10.5048i 0.776536 1.34500i −0.157391 0.987536i \(-0.550308\pi\)
0.933927 0.357464i \(-0.116358\pi\)
\(62\) 0 0
\(63\) −1.09486 1.89635i −0.137939 0.238917i
\(64\) 0 0
\(65\) 3.21617 5.57057i 0.398917 0.690944i
\(66\) 0 0
\(67\) −7.43204 3.42998i −0.907968 0.419039i
\(68\) 0 0
\(69\) −1.52897 + 2.64825i −0.184066 + 0.318811i
\(70\) 0 0
\(71\) 7.59393 + 13.1531i 0.901234 + 1.56098i 0.825895 + 0.563824i \(0.190671\pi\)
0.0753390 + 0.997158i \(0.475996\pi\)
\(72\) 0 0
\(73\) 6.15933 10.6683i 0.720895 1.24863i −0.239747 0.970835i \(-0.577064\pi\)
0.960642 0.277791i \(-0.0896022\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 5.61736 9.72956i 0.640158 1.10879i
\(78\) 0 0
\(79\) 4.94026 + 8.55679i 0.555823 + 0.962714i 0.997839 + 0.0657065i \(0.0209301\pi\)
−0.442016 + 0.897007i \(0.645737\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.69164 11.5903i 0.734503 1.27220i −0.220438 0.975401i \(-0.570749\pi\)
0.954941 0.296795i \(-0.0959179\pi\)
\(84\) 0 0
\(85\) 0.653954 1.13268i 0.0709313 0.122857i
\(86\) 0 0
\(87\) 3.69002 + 6.39130i 0.395612 + 0.685220i
\(88\) 0 0
\(89\) −5.88126 −0.623412 −0.311706 0.950179i \(-0.600900\pi\)
−0.311706 + 0.950179i \(0.600900\pi\)
\(90\) 0 0
\(91\) −14.0850 −1.47651
\(92\) 0 0
\(93\) 1.00060 + 1.73309i 0.103757 + 0.179713i
\(94\) 0 0
\(95\) −2.76451 + 4.78828i −0.283633 + 0.491267i
\(96\) 0 0
\(97\) 7.09796 12.2940i 0.720688 1.24827i −0.240036 0.970764i \(-0.577159\pi\)
0.960724 0.277505i \(-0.0895075\pi\)
\(98\) 0 0
\(99\) 2.56534 + 4.44330i 0.257827 + 0.446569i
\(100\) 0 0
\(101\) 8.95422 + 15.5092i 0.890978 + 1.54322i 0.838704 + 0.544587i \(0.183314\pi\)
0.0522739 + 0.998633i \(0.483353\pi\)
\(102\) 0 0
\(103\) −2.17465 3.76661i −0.214275 0.371135i 0.738773 0.673954i \(-0.235405\pi\)
−0.953048 + 0.302819i \(0.902072\pi\)
\(104\) 0 0
\(105\) −1.09486 1.89635i −0.106847 0.185065i
\(106\) 0 0
\(107\) −16.3415 −1.57980 −0.789898 0.613238i \(-0.789867\pi\)
−0.789898 + 0.613238i \(0.789867\pi\)
\(108\) 0 0
\(109\) −11.4267 −1.09448 −0.547240 0.836976i \(-0.684321\pi\)
−0.547240 + 0.836976i \(0.684321\pi\)
\(110\) 0 0
\(111\) 1.40879 2.44010i 0.133717 0.231604i
\(112\) 0 0
\(113\) −6.89488 11.9423i −0.648616 1.12344i −0.983454 0.181160i \(-0.942015\pi\)
0.334838 0.942276i \(-0.391319\pi\)
\(114\) 0 0
\(115\) −1.52897 + 2.64825i −0.142577 + 0.246950i
\(116\) 0 0
\(117\) 3.21617 5.57057i 0.297335 0.515000i
\(118\) 0 0
\(119\) −2.86394 −0.262537
\(120\) 0 0
\(121\) −7.66197 + 13.2709i −0.696542 + 1.20645i
\(122\) 0 0
\(123\) 1.19738 + 2.07392i 0.107964 + 0.186999i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.98026 8.62607i −0.441927 0.765440i 0.555906 0.831245i \(-0.312372\pi\)
−0.997832 + 0.0658058i \(0.979038\pi\)
\(128\) 0 0
\(129\) 9.64370 0.849081
\(130\) 0 0
\(131\) 4.69171 0.409917 0.204958 0.978771i \(-0.434294\pi\)
0.204958 + 0.978771i \(0.434294\pi\)
\(132\) 0 0
\(133\) 12.1070 1.04981
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 10.6501 0.909899 0.454949 0.890517i \(-0.349657\pi\)
0.454949 + 0.890517i \(0.349657\pi\)
\(138\) 0 0
\(139\) 21.2588 1.80315 0.901576 0.432621i \(-0.142411\pi\)
0.901576 + 0.432621i \(0.142411\pi\)
\(140\) 0 0
\(141\) 1.71064 + 2.96291i 0.144062 + 0.249522i
\(142\) 0 0
\(143\) 33.0023 2.75979
\(144\) 0 0
\(145\) 3.69002 + 6.39130i 0.306440 + 0.530769i
\(146\) 0 0
\(147\) 1.10258 1.90972i 0.0909391 0.157511i
\(148\) 0 0
\(149\) 16.8858 1.38334 0.691670 0.722214i \(-0.256875\pi\)
0.691670 + 0.722214i \(0.256875\pi\)
\(150\) 0 0
\(151\) 3.19850 5.53996i 0.260290 0.450835i −0.706029 0.708183i \(-0.749515\pi\)
0.966319 + 0.257348i \(0.0828485\pi\)
\(152\) 0 0
\(153\) 0.653954 1.13268i 0.0528691 0.0915719i
\(154\) 0 0
\(155\) 1.00060 + 1.73309i 0.0803701 + 0.139205i
\(156\) 0 0
\(157\) −11.8207 + 20.4740i −0.943392 + 1.63400i −0.184453 + 0.982841i \(0.559051\pi\)
−0.758939 + 0.651161i \(0.774282\pi\)
\(158\) 0 0
\(159\) −6.09552 −0.483406
\(160\) 0 0
\(161\) 6.69599 0.527718
\(162\) 0 0
\(163\) 9.34983 + 16.1944i 0.732335 + 1.26844i 0.955883 + 0.293749i \(0.0949030\pi\)
−0.223547 + 0.974693i \(0.571764\pi\)
\(164\) 0 0
\(165\) 2.56534 + 4.44330i 0.199712 + 0.345911i
\(166\) 0 0
\(167\) −0.867052 1.50178i −0.0670945 0.116211i 0.830527 0.556979i \(-0.188040\pi\)
−0.897621 + 0.440768i \(0.854706\pi\)
\(168\) 0 0
\(169\) −14.1875 24.5735i −1.09135 1.89027i
\(170\) 0 0
\(171\) −2.76451 + 4.78828i −0.211408 + 0.366169i
\(172\) 0 0
\(173\) −2.87760 + 4.98415i −0.218780 + 0.378938i −0.954435 0.298418i \(-0.903541\pi\)
0.735655 + 0.677356i \(0.236874\pi\)
\(174\) 0 0
\(175\) −1.09486 1.89635i −0.0827634 0.143350i
\(176\) 0 0
\(177\) −5.99534 −0.450638
\(178\) 0 0
\(179\) −3.48707 −0.260636 −0.130318 0.991472i \(-0.541600\pi\)
−0.130318 + 0.991472i \(0.541600\pi\)
\(180\) 0 0
\(181\) 4.44487 + 7.69874i 0.330384 + 0.572243i 0.982587 0.185802i \(-0.0594883\pi\)
−0.652203 + 0.758045i \(0.726155\pi\)
\(182\) 0 0
\(183\) 6.06494 10.5048i 0.448333 0.776536i
\(184\) 0 0
\(185\) 1.40879 2.44010i 0.103577 0.179400i
\(186\) 0 0
\(187\) 6.71047 0.490718
\(188\) 0 0
\(189\) −1.09486 1.89635i −0.0796391 0.137939i
\(190\) 0 0
\(191\) 8.80189 15.2453i 0.636883 1.10311i −0.349230 0.937037i \(-0.613557\pi\)
0.986113 0.166076i \(-0.0531097\pi\)
\(192\) 0 0
\(193\) −2.90313 −0.208972 −0.104486 0.994526i \(-0.533320\pi\)
−0.104486 + 0.994526i \(0.533320\pi\)
\(194\) 0 0
\(195\) 3.21617 5.57057i 0.230315 0.398917i
\(196\) 0 0
\(197\) 10.2625 + 17.7752i 0.731175 + 1.26643i 0.956381 + 0.292121i \(0.0943610\pi\)
−0.225206 + 0.974311i \(0.572306\pi\)
\(198\) 0 0
\(199\) −2.49651 + 4.32408i −0.176973 + 0.306526i −0.940842 0.338845i \(-0.889964\pi\)
0.763869 + 0.645371i \(0.223297\pi\)
\(200\) 0 0
\(201\) −7.43204 3.42998i −0.524216 0.241932i
\(202\) 0 0
\(203\) 8.08009 13.9951i 0.567111 0.982265i
\(204\) 0 0
\(205\) 1.19738 + 2.07392i 0.0836284 + 0.144849i
\(206\) 0 0
\(207\) −1.52897 + 2.64825i −0.106270 + 0.184066i
\(208\) 0 0
\(209\) −28.3677 −1.96223
\(210\) 0 0
\(211\) 10.4849 18.1604i 0.721810 1.25021i −0.238463 0.971152i \(-0.576644\pi\)
0.960273 0.279061i \(-0.0900231\pi\)
\(212\) 0 0
\(213\) 7.59393 + 13.1531i 0.520327 + 0.901234i
\(214\) 0 0
\(215\) 9.64370 0.657695
\(216\) 0 0
\(217\) 2.19103 3.79497i 0.148737 0.257619i
\(218\) 0 0
\(219\) 6.15933 10.6683i 0.416209 0.720895i
\(220\) 0 0
\(221\) −4.20646 7.28580i −0.282957 0.490096i
\(222\) 0 0
\(223\) 15.0267 1.00626 0.503131 0.864210i \(-0.332182\pi\)
0.503131 + 0.864210i \(0.332182\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −1.70708 2.95674i −0.113303 0.196246i 0.803797 0.594903i \(-0.202810\pi\)
−0.917100 + 0.398657i \(0.869476\pi\)
\(228\) 0 0
\(229\) −7.38564 + 12.7923i −0.488057 + 0.845339i −0.999906 0.0137365i \(-0.995627\pi\)
0.511849 + 0.859075i \(0.328961\pi\)
\(230\) 0 0
\(231\) 5.61736 9.72956i 0.369595 0.640158i
\(232\) 0 0
\(233\) −7.28641 12.6204i −0.477349 0.826792i 0.522314 0.852753i \(-0.325069\pi\)
−0.999663 + 0.0259607i \(0.991736\pi\)
\(234\) 0 0
\(235\) 1.71064 + 2.96291i 0.111590 + 0.193279i
\(236\) 0 0
\(237\) 4.94026 + 8.55679i 0.320905 + 0.555823i
\(238\) 0 0
\(239\) −7.82106 13.5465i −0.505902 0.876249i −0.999977 0.00682886i \(-0.997826\pi\)
0.494074 0.869420i \(-0.335507\pi\)
\(240\) 0 0
\(241\) −27.5691 −1.77588 −0.887942 0.459955i \(-0.847866\pi\)
−0.887942 + 0.459955i \(0.847866\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.10258 1.90972i 0.0704411 0.122008i
\(246\) 0 0
\(247\) 17.7823 + 30.7998i 1.13146 + 1.95975i
\(248\) 0 0
\(249\) 6.69164 11.5903i 0.424065 0.734503i
\(250\) 0 0
\(251\) −1.92360 + 3.33177i −0.121416 + 0.210300i −0.920326 0.391151i \(-0.872077\pi\)
0.798910 + 0.601451i \(0.205410\pi\)
\(252\) 0 0
\(253\) −15.6893 −0.986376
\(254\) 0 0
\(255\) 0.653954 1.13268i 0.0409522 0.0709313i
\(256\) 0 0
\(257\) −7.46624 12.9319i −0.465731 0.806670i 0.533503 0.845798i \(-0.320875\pi\)
−0.999234 + 0.0391282i \(0.987542\pi\)
\(258\) 0 0
\(259\) −6.16971 −0.383367
\(260\) 0 0
\(261\) 3.69002 + 6.39130i 0.228407 + 0.395612i
\(262\) 0 0
\(263\) −11.8497 −0.730683 −0.365342 0.930874i \(-0.619048\pi\)
−0.365342 + 0.930874i \(0.619048\pi\)
\(264\) 0 0
\(265\) −6.09552 −0.374445
\(266\) 0 0
\(267\) −5.88126 −0.359927
\(268\) 0 0
\(269\) 20.3661 1.24174 0.620871 0.783913i \(-0.286779\pi\)
0.620871 + 0.783913i \(0.286779\pi\)
\(270\) 0 0
\(271\) 12.0754 0.733531 0.366765 0.930313i \(-0.380465\pi\)
0.366765 + 0.930313i \(0.380465\pi\)
\(272\) 0 0
\(273\) −14.0850 −0.852462
\(274\) 0 0
\(275\) 2.56534 + 4.44330i 0.154696 + 0.267941i
\(276\) 0 0
\(277\) −15.4329 −0.927272 −0.463636 0.886026i \(-0.653455\pi\)
−0.463636 + 0.886026i \(0.653455\pi\)
\(278\) 0 0
\(279\) 1.00060 + 1.73309i 0.0599044 + 0.103757i
\(280\) 0 0
\(281\) −0.837123 + 1.44994i −0.0499386 + 0.0864962i −0.889914 0.456128i \(-0.849236\pi\)
0.839976 + 0.542624i \(0.182569\pi\)
\(282\) 0 0
\(283\) −6.71825 −0.399358 −0.199679 0.979861i \(-0.563990\pi\)
−0.199679 + 0.979861i \(0.563990\pi\)
\(284\) 0 0
\(285\) −2.76451 + 4.78828i −0.163756 + 0.283633i
\(286\) 0 0
\(287\) 2.62191 4.54128i 0.154766 0.268063i
\(288\) 0 0
\(289\) 7.64469 + 13.2410i 0.449688 + 0.778882i
\(290\) 0 0
\(291\) 7.09796 12.2940i 0.416090 0.720688i
\(292\) 0 0
\(293\) 24.8308 1.45063 0.725315 0.688418i \(-0.241694\pi\)
0.725315 + 0.688418i \(0.241694\pi\)
\(294\) 0 0
\(295\) −5.99534 −0.349062
\(296\) 0 0
\(297\) 2.56534 + 4.44330i 0.148856 + 0.257827i
\(298\) 0 0
\(299\) 9.83483 + 17.0344i 0.568763 + 0.985126i
\(300\) 0 0
\(301\) −10.5585 18.2878i −0.608580 1.05409i
\(302\) 0 0
\(303\) 8.95422 + 15.5092i 0.514407 + 0.890978i
\(304\) 0 0
\(305\) 6.06494 10.5048i 0.347278 0.601502i
\(306\) 0 0
\(307\) −4.87572 + 8.44500i −0.278272 + 0.481982i −0.970955 0.239260i \(-0.923095\pi\)
0.692683 + 0.721242i \(0.256428\pi\)
\(308\) 0 0
\(309\) −2.17465 3.76661i −0.123712 0.214275i
\(310\) 0 0
\(311\) −2.57755 −0.146159 −0.0730797 0.997326i \(-0.523283\pi\)
−0.0730797 + 0.997326i \(0.523283\pi\)
\(312\) 0 0
\(313\) −2.12885 −0.120329 −0.0601647 0.998188i \(-0.519163\pi\)
−0.0601647 + 0.998188i \(0.519163\pi\)
\(314\) 0 0
\(315\) −1.09486 1.89635i −0.0616882 0.106847i
\(316\) 0 0
\(317\) −12.0111 + 20.8038i −0.674608 + 1.16846i 0.301975 + 0.953316i \(0.402354\pi\)
−0.976583 + 0.215140i \(0.930979\pi\)
\(318\) 0 0
\(319\) −18.9323 + 32.7918i −1.06001 + 1.83599i
\(320\) 0 0
\(321\) −16.3415 −0.912096
\(322\) 0 0
\(323\) 3.61573 + 6.26263i 0.201185 + 0.348462i
\(324\) 0 0
\(325\) 3.21617 5.57057i 0.178401 0.309000i
\(326\) 0 0
\(327\) −11.4267 −0.631898
\(328\) 0 0
\(329\) 3.74581 6.48793i 0.206513 0.357691i
\(330\) 0 0
\(331\) −1.23470 2.13857i −0.0678654 0.117546i 0.830096 0.557620i \(-0.188286\pi\)
−0.897961 + 0.440074i \(0.854952\pi\)
\(332\) 0 0
\(333\) 1.40879 2.44010i 0.0772014 0.133717i
\(334\) 0 0
\(335\) −7.43204 3.42998i −0.406056 0.187400i
\(336\) 0 0
\(337\) −1.16405 + 2.01620i −0.0634101 + 0.109829i −0.895988 0.444079i \(-0.853531\pi\)
0.832578 + 0.553909i \(0.186864\pi\)
\(338\) 0 0
\(339\) −6.89488 11.9423i −0.374479 0.648616i
\(340\) 0 0
\(341\) −5.13376 + 8.89194i −0.278009 + 0.481525i
\(342\) 0 0
\(343\) −20.1567 −1.08836
\(344\) 0 0
\(345\) −1.52897 + 2.64825i −0.0823167 + 0.142577i
\(346\) 0 0
\(347\) −13.8506 23.9900i −0.743541 1.28785i −0.950873 0.309580i \(-0.899812\pi\)
0.207332 0.978271i \(-0.433522\pi\)
\(348\) 0 0
\(349\) −25.4323 −1.36136 −0.680680 0.732581i \(-0.738316\pi\)
−0.680680 + 0.732581i \(0.738316\pi\)
\(350\) 0 0
\(351\) 3.21617 5.57057i 0.171667 0.297335i
\(352\) 0 0
\(353\) −8.32428 + 14.4181i −0.443057 + 0.767397i −0.997915 0.0645483i \(-0.979439\pi\)
0.554858 + 0.831945i \(0.312773\pi\)
\(354\) 0 0
\(355\) 7.59393 + 13.1531i 0.403044 + 0.698093i
\(356\) 0 0
\(357\) −2.86394 −0.151576
\(358\) 0 0
\(359\) 11.0289 0.582085 0.291043 0.956710i \(-0.405998\pi\)
0.291043 + 0.956710i \(0.405998\pi\)
\(360\) 0 0
\(361\) −5.78507 10.0200i −0.304478 0.527371i
\(362\) 0 0
\(363\) −7.66197 + 13.2709i −0.402149 + 0.696542i
\(364\) 0 0
\(365\) 6.15933 10.6683i 0.322394 0.558403i
\(366\) 0 0
\(367\) 2.56468 + 4.44215i 0.133875 + 0.231878i 0.925167 0.379560i \(-0.123925\pi\)
−0.791292 + 0.611438i \(0.790591\pi\)
\(368\) 0 0
\(369\) 1.19738 + 2.07392i 0.0623329 + 0.107964i
\(370\) 0 0
\(371\) 6.67372 + 11.5592i 0.346482 + 0.600125i
\(372\) 0 0
\(373\) −0.633359 1.09701i −0.0327941 0.0568010i 0.849163 0.528132i \(-0.177107\pi\)
−0.881957 + 0.471331i \(0.843774\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 47.4710 2.44488
\(378\) 0 0
\(379\) −9.97798 + 17.2824i −0.512535 + 0.887736i 0.487360 + 0.873201i \(0.337960\pi\)
−0.999894 + 0.0145347i \(0.995373\pi\)
\(380\) 0 0
\(381\) −4.98026 8.62607i −0.255147 0.441927i
\(382\) 0 0
\(383\) 5.84522 10.1242i 0.298677 0.517324i −0.677157 0.735839i \(-0.736788\pi\)
0.975834 + 0.218515i \(0.0701214\pi\)
\(384\) 0 0
\(385\) 5.61736 9.72956i 0.286287 0.495864i
\(386\) 0 0
\(387\) 9.64370 0.490217
\(388\) 0 0
\(389\) 7.84799 13.5931i 0.397909 0.689198i −0.595559 0.803312i \(-0.703069\pi\)
0.993468 + 0.114113i \(0.0364027\pi\)
\(390\) 0 0
\(391\) 1.99975 + 3.46366i 0.101132 + 0.175165i
\(392\) 0 0
\(393\) 4.69171 0.236666
\(394\) 0 0
\(395\) 4.94026 + 8.55679i 0.248572 + 0.430539i
\(396\) 0 0
\(397\) −32.6028 −1.63629 −0.818144 0.575013i \(-0.804997\pi\)
−0.818144 + 0.575013i \(0.804997\pi\)
\(398\) 0 0
\(399\) 12.1070 0.606107
\(400\) 0 0
\(401\) −12.0617 −0.602334 −0.301167 0.953571i \(-0.597376\pi\)
−0.301167 + 0.953571i \(0.597376\pi\)
\(402\) 0 0
\(403\) 12.8724 0.641220
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 14.4562 0.716565
\(408\) 0 0
\(409\) 7.02900 + 12.1746i 0.347562 + 0.601995i 0.985816 0.167831i \(-0.0536763\pi\)
−0.638254 + 0.769826i \(0.720343\pi\)
\(410\) 0 0
\(411\) 10.6501 0.525330
\(412\) 0 0
\(413\) 6.56404 + 11.3693i 0.322995 + 0.559444i
\(414\) 0 0
\(415\) 6.69164 11.5903i 0.328480 0.568943i
\(416\) 0 0
\(417\) 21.2588 1.04105
\(418\) 0 0
\(419\) −10.3224 + 17.8789i −0.504281 + 0.873440i 0.495707 + 0.868490i \(0.334909\pi\)
−0.999988 + 0.00494998i \(0.998424\pi\)
\(420\) 0 0
\(421\) 15.1937 26.3163i 0.740498 1.28258i −0.211771 0.977319i \(-0.567923\pi\)
0.952269 0.305261i \(-0.0987436\pi\)
\(422\) 0 0
\(423\) 1.71064 + 2.96291i 0.0831741 + 0.144062i
\(424\) 0 0
\(425\) 0.653954 1.13268i 0.0317214 0.0549431i
\(426\) 0 0
\(427\) −26.5610 −1.28538
\(428\) 0 0
\(429\) 33.0023 1.59337
\(430\) 0 0
\(431\) −11.4033 19.7512i −0.549280 0.951380i −0.998324 0.0578709i \(-0.981569\pi\)
0.449044 0.893509i \(-0.351765\pi\)
\(432\) 0 0
\(433\) 9.30756 + 16.1212i 0.447293 + 0.774734i 0.998209 0.0598268i \(-0.0190548\pi\)
−0.550916 + 0.834561i \(0.685721\pi\)
\(434\) 0 0
\(435\) 3.69002 + 6.39130i 0.176923 + 0.306440i
\(436\) 0 0
\(437\) −8.45369 14.6422i −0.404395 0.700432i
\(438\) 0 0
\(439\) 2.03543 3.52548i 0.0971460 0.168262i −0.813356 0.581766i \(-0.802362\pi\)
0.910502 + 0.413504i \(0.135695\pi\)
\(440\) 0 0
\(441\) 1.10258 1.90972i 0.0525037 0.0909391i
\(442\) 0 0
\(443\) 3.55314 + 6.15422i 0.168815 + 0.292396i 0.938003 0.346626i \(-0.112673\pi\)
−0.769189 + 0.639022i \(0.779339\pi\)
\(444\) 0 0
\(445\) −5.88126 −0.278799
\(446\) 0 0
\(447\) 16.8858 0.798672
\(448\) 0 0
\(449\) −12.9189 22.3761i −0.609679 1.05599i −0.991293 0.131673i \(-0.957965\pi\)
0.381615 0.924322i \(-0.375368\pi\)
\(450\) 0 0
\(451\) −6.14336 + 10.6406i −0.289280 + 0.501047i
\(452\) 0 0
\(453\) 3.19850 5.53996i 0.150278 0.260290i
\(454\) 0 0
\(455\) −14.0850 −0.660314
\(456\) 0 0
\(457\) −16.3814 28.3735i −0.766292 1.32726i −0.939561 0.342381i \(-0.888767\pi\)
0.173270 0.984874i \(-0.444567\pi\)
\(458\) 0 0
\(459\) 0.653954 1.13268i 0.0305240 0.0528691i
\(460\) 0 0
\(461\) 21.3170 0.992830 0.496415 0.868085i \(-0.334649\pi\)
0.496415 + 0.868085i \(0.334649\pi\)
\(462\) 0 0
\(463\) −2.59481 + 4.49434i −0.120591 + 0.208870i −0.920001 0.391916i \(-0.871812\pi\)
0.799410 + 0.600786i \(0.205146\pi\)
\(464\) 0 0
\(465\) 1.00060 + 1.73309i 0.0464017 + 0.0803701i
\(466\) 0 0
\(467\) −4.84248 + 8.38742i −0.224083 + 0.388124i −0.956044 0.293223i \(-0.905272\pi\)
0.731961 + 0.681347i \(0.238605\pi\)
\(468\) 0 0
\(469\) 1.63259 + 17.8491i 0.0753859 + 0.824193i
\(470\) 0 0
\(471\) −11.8207 + 20.4740i −0.544668 + 0.943392i
\(472\) 0 0
\(473\) 24.7394 + 42.8499i 1.13752 + 1.97024i
\(474\) 0 0
\(475\) −2.76451 + 4.78828i −0.126845 + 0.219701i
\(476\) 0 0
\(477\) −6.09552 −0.279095
\(478\) 0 0
\(479\) −12.9993 + 22.5155i −0.593954 + 1.02876i 0.399740 + 0.916629i \(0.369100\pi\)
−0.993694 + 0.112129i \(0.964233\pi\)
\(480\) 0 0
\(481\) −9.06184 15.6956i −0.413184 0.715656i
\(482\) 0 0
\(483\) 6.69599 0.304678
\(484\) 0 0
\(485\) 7.09796 12.2940i 0.322302 0.558243i
\(486\) 0 0
\(487\) −4.18670 + 7.25157i −0.189717 + 0.328600i −0.945156 0.326619i \(-0.894090\pi\)
0.755439 + 0.655219i \(0.227424\pi\)
\(488\) 0 0
\(489\) 9.34983 + 16.1944i 0.422814 + 0.732335i
\(490\) 0 0
\(491\) 12.0276 0.542796 0.271398 0.962467i \(-0.412514\pi\)
0.271398 + 0.962467i \(0.412514\pi\)
\(492\) 0 0
\(493\) 9.65242 0.434723
\(494\) 0 0
\(495\) 2.56534 + 4.44330i 0.115304 + 0.199712i
\(496\) 0 0
\(497\) 16.6285 28.8015i 0.745891 1.29192i
\(498\) 0 0
\(499\) 18.1760 31.4817i 0.813668 1.40931i −0.0966125 0.995322i \(-0.530801\pi\)
0.910280 0.413992i \(-0.135866\pi\)
\(500\) 0 0
\(501\) −0.867052 1.50178i −0.0387370 0.0670945i
\(502\) 0 0
\(503\) 17.1626 + 29.7265i 0.765242 + 1.32544i 0.940118 + 0.340848i \(0.110714\pi\)
−0.174876 + 0.984590i \(0.555953\pi\)
\(504\) 0 0
\(505\) 8.95422 + 15.5092i 0.398458 + 0.690149i
\(506\) 0 0
\(507\) −14.1875 24.5735i −0.630090 1.09135i
\(508\) 0 0
\(509\) 0.296174 0.0131277 0.00656385 0.999978i \(-0.497911\pi\)
0.00656385 + 0.999978i \(0.497911\pi\)
\(510\) 0 0
\(511\) −26.9743 −1.19327
\(512\) 0 0
\(513\) −2.76451 + 4.78828i −0.122056 + 0.211408i
\(514\) 0 0
\(515\) −2.17465 3.76661i −0.0958266 0.165977i
\(516\) 0 0
\(517\) −8.77675 + 15.2018i −0.386001 + 0.668573i
\(518\) 0 0
\(519\) −2.87760 + 4.98415i −0.126313 + 0.218780i
\(520\) 0 0
\(521\) −4.54903 −0.199297 −0.0996484 0.995023i \(-0.531772\pi\)
−0.0996484 + 0.995023i \(0.531772\pi\)
\(522\) 0 0
\(523\) 0.473267 0.819723i 0.0206945 0.0358440i −0.855493 0.517815i \(-0.826746\pi\)
0.876187 + 0.481971i \(0.160079\pi\)
\(524\) 0 0
\(525\) −1.09486 1.89635i −0.0477835 0.0827634i
\(526\) 0 0
\(527\) 2.61739 0.114015
\(528\) 0 0
\(529\) 6.82453 + 11.8204i 0.296719 + 0.513932i
\(530\) 0 0
\(531\) −5.99534 −0.260176
\(532\) 0 0
\(533\) 15.4039 0.667215
\(534\) 0 0
\(535\) −16.3415 −0.706506
\(536\) 0 0
\(537\) −3.48707 −0.150478
\(538\) 0 0
\(539\) 11.3140 0.487327
\(540\) 0 0
\(541\) −9.61021 −0.413175 −0.206588 0.978428i \(-0.566236\pi\)
−0.206588 + 0.978428i \(0.566236\pi\)
\(542\) 0 0
\(543\) 4.44487 + 7.69874i 0.190748 + 0.330384i
\(544\) 0 0
\(545\) −11.4267 −0.489466
\(546\) 0 0
\(547\) −4.39368 7.61007i −0.187860 0.325383i 0.756676 0.653790i \(-0.226822\pi\)
−0.944537 + 0.328406i \(0.893488\pi\)
\(548\) 0 0
\(549\) 6.06494 10.5048i 0.258845 0.448333i
\(550\) 0 0
\(551\) −40.8045 −1.73833
\(552\) 0 0
\(553\) 10.8178 18.7369i 0.460018 0.796774i
\(554\) 0 0
\(555\) 1.40879 2.44010i 0.0598000 0.103577i
\(556\) 0 0
\(557\) 15.5966 + 27.0140i 0.660848 + 1.14462i 0.980393 + 0.197051i \(0.0631364\pi\)
−0.319546 + 0.947571i \(0.603530\pi\)
\(558\) 0 0
\(559\) 31.0158 53.7209i 1.31183 2.27215i
\(560\) 0 0
\(561\) 6.71047 0.283316
\(562\) 0 0
\(563\) 15.1080 0.636728 0.318364 0.947969i \(-0.396867\pi\)
0.318364 + 0.947969i \(0.396867\pi\)
\(564\) 0 0
\(565\) −6.89488 11.9423i −0.290070 0.502416i
\(566\) 0 0
\(567\) −1.09486 1.89635i −0.0459797 0.0796391i
\(568\) 0 0
\(569\) −5.46892 9.47245i −0.229269 0.397106i 0.728323 0.685234i \(-0.240300\pi\)
−0.957592 + 0.288129i \(0.906967\pi\)
\(570\) 0 0
\(571\) −18.2021 31.5269i −0.761733 1.31936i −0.941956 0.335735i \(-0.891015\pi\)
0.180223 0.983626i \(-0.442318\pi\)
\(572\) 0 0
\(573\) 8.80189 15.2453i 0.367704 0.636883i
\(574\) 0 0
\(575\) −1.52897 + 2.64825i −0.0637623 + 0.110439i
\(576\) 0 0
\(577\) −20.2555 35.0836i −0.843249 1.46055i −0.887134 0.461513i \(-0.847307\pi\)
0.0438849 0.999037i \(-0.486027\pi\)
\(578\) 0 0
\(579\) −2.90313 −0.120650
\(580\) 0 0
\(581\) −29.3055 −1.21580
\(582\) 0 0
\(583\) −15.6371 27.0842i −0.647622 1.12171i
\(584\) 0 0
\(585\) 3.21617 5.57057i 0.132972 0.230315i
\(586\) 0 0
\(587\) −17.7514 + 30.7463i −0.732677 + 1.26903i 0.223058 + 0.974805i \(0.428396\pi\)
−0.955735 + 0.294229i \(0.904937\pi\)
\(588\) 0 0
\(589\) −11.0647 −0.455913
\(590\) 0 0
\(591\) 10.2625 + 17.7752i 0.422144 + 0.731175i
\(592\) 0 0
\(593\) 9.99456 17.3111i 0.410427 0.710881i −0.584509 0.811387i \(-0.698713\pi\)
0.994936 + 0.100506i \(0.0320462\pi\)
\(594\) 0 0
\(595\) −2.86394 −0.117410
\(596\) 0 0
\(597\) −2.49651 + 4.32408i −0.102175 + 0.176973i
\(598\) 0 0
\(599\) −8.07834 13.9921i −0.330072 0.571701i 0.652454 0.757829i \(-0.273740\pi\)
−0.982526 + 0.186127i \(0.940406\pi\)
\(600\) 0 0
\(601\) −5.94057 + 10.2894i −0.242321 + 0.419712i −0.961375 0.275242i \(-0.911242\pi\)
0.719054 + 0.694954i \(0.244575\pi\)
\(602\) 0 0
\(603\) −7.43204 3.42998i −0.302656 0.139680i
\(604\) 0 0
\(605\) −7.66197 + 13.2709i −0.311503 + 0.539539i
\(606\) 0 0
\(607\) −7.60851 13.1783i −0.308820 0.534892i 0.669285 0.743006i \(-0.266601\pi\)
−0.978104 + 0.208114i \(0.933267\pi\)
\(608\) 0 0
\(609\) 8.08009 13.9951i 0.327422 0.567111i
\(610\) 0 0
\(611\) 22.0068 0.890301
\(612\) 0 0
\(613\) −15.4993 + 26.8456i −0.626012 + 1.08428i 0.362333 + 0.932049i \(0.381980\pi\)
−0.988344 + 0.152235i \(0.951353\pi\)
\(614\) 0 0
\(615\) 1.19738 + 2.07392i 0.0482829 + 0.0836284i
\(616\) 0 0
\(617\) 20.9244 0.842386 0.421193 0.906971i \(-0.361612\pi\)
0.421193 + 0.906971i \(0.361612\pi\)
\(618\) 0 0
\(619\) 15.0689 26.1002i 0.605672 1.04905i −0.386273 0.922384i \(-0.626238\pi\)
0.991945 0.126670i \(-0.0404289\pi\)
\(620\) 0 0
\(621\) −1.52897 + 2.64825i −0.0613553 + 0.106270i
\(622\) 0 0
\(623\) 6.43914 + 11.1529i 0.257979 + 0.446832i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −28.3677 −1.13290
\(628\) 0 0
\(629\) −1.84257 3.19143i −0.0734682 0.127251i
\(630\) 0 0
\(631\) −2.87033 + 4.97155i −0.114266 + 0.197914i −0.917486 0.397768i \(-0.869785\pi\)
0.803220 + 0.595682i \(0.203118\pi\)
\(632\) 0 0
\(633\) 10.4849 18.1604i 0.416737 0.721810i
\(634\) 0 0
\(635\) −4.98026 8.62607i −0.197636 0.342315i
\(636\) 0 0
\(637\) −7.09216 12.2840i −0.281002 0.486709i
\(638\) 0 0
\(639\) 7.59393 + 13.1531i 0.300411 + 0.520327i
\(640\) 0 0
\(641\) −17.6342 30.5434i −0.696511 1.20639i −0.969669 0.244423i \(-0.921402\pi\)
0.273158 0.961969i \(-0.411932\pi\)
\(642\) 0 0
\(643\) −25.5756 −1.00861 −0.504303 0.863527i \(-0.668250\pi\)
−0.504303 + 0.863527i \(0.668250\pi\)
\(644\) 0 0
\(645\) 9.64370 0.379720
\(646\) 0 0
\(647\) −20.4109 + 35.3528i −0.802437 + 1.38986i 0.115571 + 0.993299i \(0.463130\pi\)
−0.918008 + 0.396562i \(0.870203\pi\)
\(648\) 0 0
\(649\) −15.3801 26.6391i −0.603722 1.04568i
\(650\) 0 0
\(651\) 2.19103 3.79497i 0.0858731 0.148737i
\(652\) 0 0
\(653\) −8.93268 + 15.4719i −0.349563 + 0.605461i −0.986172 0.165726i \(-0.947003\pi\)
0.636609 + 0.771187i \(0.280337\pi\)
\(654\) 0 0
\(655\) 4.69171 0.183320
\(656\) 0 0
\(657\) 6.15933 10.6683i 0.240298 0.416209i
\(658\) 0 0
\(659\) −14.0009 24.2502i −0.545396 0.944654i −0.998582 0.0532380i \(-0.983046\pi\)
0.453185 0.891416i \(-0.350288\pi\)
\(660\) 0 0
\(661\) −4.51742 −0.175707 −0.0878536 0.996133i \(-0.528001\pi\)
−0.0878536 + 0.996133i \(0.528001\pi\)
\(662\) 0 0
\(663\) −4.20646 7.28580i −0.163365 0.282957i
\(664\) 0 0
\(665\) 12.1070 0.469489
\(666\) 0 0
\(667\) −22.5677 −0.873823
\(668\) 0 0
\(669\) 15.0267 0.580965
\(670\) 0 0
\(671\) 62.2346 2.40254
\(672\) 0 0
\(673\) 5.75306 0.221764 0.110882 0.993834i \(-0.464632\pi\)
0.110882 + 0.993834i \(0.464632\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −20.8807 36.1665i −0.802511 1.38999i −0.917959 0.396676i \(-0.870164\pi\)
0.115447 0.993314i \(-0.463170\pi\)
\(678\) 0 0
\(679\) −31.0850 −1.19293
\(680\) 0 0
\(681\) −1.70708 2.95674i −0.0654153 0.113303i
\(682\) 0 0
\(683\) −17.0221 + 29.4831i −0.651332 + 1.12814i 0.331468 + 0.943466i \(0.392456\pi\)
−0.982800 + 0.184673i \(0.940877\pi\)
\(684\) 0 0
\(685\) 10.6501 0.406919
\(686\) 0 0
\(687\) −7.38564 + 12.7923i −0.281780 + 0.488057i
\(688\) 0 0
\(689\) −19.6042 + 33.9555i −0.746862 + 1.29360i
\(690\) 0 0
\(691\) 23.5182 + 40.7347i 0.894674 + 1.54962i 0.834207 + 0.551451i \(0.185926\pi\)
0.0604666 + 0.998170i \(0.480741\pi\)
\(692\) 0 0
\(693\) 5.61736 9.72956i 0.213386 0.369595i
\(694\) 0 0
\(695\) 21.2588 0.806394
\(696\) 0 0
\(697\) 3.13212 0.118637
\(698\) 0 0
\(699\) −7.28641 12.6204i −0.275597 0.477349i
\(700\) 0 0
\(701\) 3.32651 + 5.76168i 0.125640 + 0.217616i 0.921983 0.387230i \(-0.126568\pi\)
−0.796343 + 0.604846i \(0.793235\pi\)
\(702\) 0 0
\(703\) 7.78926 + 13.4914i 0.293777 + 0.508838i
\(704\) 0 0
\(705\) 1.71064 + 2.96291i 0.0644264 + 0.111590i
\(706\) 0 0
\(707\) 19.6072 33.9606i 0.737404 1.27722i
\(708\) 0 0
\(709\) 18.2579 31.6236i 0.685689 1.18765i −0.287531 0.957771i \(-0.592834\pi\)
0.973220 0.229877i \(-0.0738323\pi\)
\(710\) 0 0
\(711\) 4.94026 + 8.55679i 0.185274 + 0.320905i
\(712\) 0 0
\(713\) −6.11953 −0.229178
\(714\) 0 0
\(715\) 33.0023 1.23422
\(716\) 0 0
\(717\) −7.82106 13.5465i −0.292083 0.505902i
\(718\) 0 0
\(719\) −4.88473 + 8.46060i −0.182170 + 0.315527i −0.942619 0.333870i \(-0.891645\pi\)
0.760449 + 0.649397i \(0.224979\pi\)
\(720\) 0 0
\(721\) −4.76187 + 8.24779i −0.177341 + 0.307164i
\(722\) 0 0
\(723\) −27.5691 −1.02531
\(724\) 0 0
\(725\) 3.69002 + 6.39130i 0.137044 + 0.237367i
\(726\) 0 0
\(727\) −24.7491 + 42.8668i −0.917895 + 1.58984i −0.115288 + 0.993332i \(0.536779\pi\)
−0.802607 + 0.596508i \(0.796554\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.30654 10.9232i 0.233256 0.404011i
\(732\) 0 0
\(733\) −17.3823 30.1070i −0.642030 1.11203i −0.984979 0.172675i \(-0.944759\pi\)
0.342949 0.939354i \(-0.388574\pi\)
\(734\) 0 0
\(735\) 1.10258 1.90972i 0.0406692 0.0704411i
\(736\) 0 0
\(737\) −3.82529 41.8219i −0.140906 1.54053i
\(738\) 0 0
\(739\) −16.8296 + 29.1497i −0.619086 + 1.07229i 0.370567 + 0.928806i \(0.379163\pi\)
−0.989653 + 0.143482i \(0.954170\pi\)
\(740\) 0 0
\(741\) 17.7823 + 30.7998i 0.653249 + 1.13146i
\(742\) 0 0
\(743\) 20.0164 34.6694i 0.734330 1.27190i −0.220687 0.975345i \(-0.570830\pi\)
0.955017 0.296552i \(-0.0958367\pi\)
\(744\) 0 0
\(745\) 16.8858 0.618648
\(746\) 0 0
\(747\) 6.69164 11.5903i 0.244834 0.424065i
\(748\) 0 0
\(749\) 17.8916 + 30.9892i 0.653746 + 1.13232i
\(750\) 0 0
\(751\) −29.2614 −1.06776 −0.533881 0.845559i \(-0.679267\pi\)
−0.533881 + 0.845559i \(0.679267\pi\)
\(752\) 0 0
\(753\) −1.92360 + 3.33177i −0.0700998 + 0.121416i
\(754\) 0 0
\(755\) 3.19850 5.53996i 0.116405 0.201620i
\(756\) 0 0
\(757\) 1.71468 + 2.96992i 0.0623212 + 0.107943i 0.895503 0.445056i \(-0.146816\pi\)
−0.833181 + 0.553000i \(0.813483\pi\)
\(758\) 0 0
\(759\) −15.6893 −0.569485
\(760\) 0 0
\(761\) −34.8831 −1.26451 −0.632256 0.774759i \(-0.717871\pi\)
−0.632256 + 0.774759i \(0.717871\pi\)
\(762\) 0 0
\(763\) 12.5106 + 21.6690i 0.452914 + 0.784470i
\(764\) 0 0
\(765\) 0.653954 1.13268i 0.0236438 0.0409522i
\(766\) 0 0
\(767\) −19.2820 + 33.3975i −0.696234 + 1.20591i
\(768\) 0 0
\(769\) 2.46613 + 4.27146i 0.0889309 + 0.154033i 0.907059 0.421002i \(-0.138322\pi\)
−0.818129 + 0.575035i \(0.804988\pi\)
\(770\) 0 0
\(771\) −7.46624 12.9319i −0.268890 0.465731i
\(772\) 0 0
\(773\) −9.91770 17.1780i −0.356715 0.617848i 0.630695 0.776031i \(-0.282770\pi\)
−0.987410 + 0.158183i \(0.949437\pi\)
\(774\) 0 0
\(775\) 1.00060 + 1.73309i 0.0359426 + 0.0622544i
\(776\) 0 0
\(777\) −6.16971 −0.221337
\(778\) 0 0
\(779\) −13.2407 −0.474396
\(780\) 0 0
\(781\) −38.9621 + 67.4843i −1.39417 + 2.41478i
\(782\) 0 0
\(783\) 3.69002 + 6.39130i 0.131871 + 0.228407i
\(784\) 0 0
\(785\) −11.8207 + 20.4740i −0.421898 + 0.730748i
\(786\) 0 0
\(787\) −8.97185 + 15.5397i −0.319812 + 0.553930i −0.980449 0.196775i \(-0.936953\pi\)
0.660637 + 0.750706i \(0.270286\pi\)
\(788\) 0 0
\(789\) −11.8497 −0.421860
\(790\) 0 0
\(791\) −15.0978 + 26.1502i −0.536817 + 0.929794i
\(792\) 0 0
\(793\) −39.0118 67.5704i −1.38535 2.39949i
\(794\) 0 0
\(795\) −6.09552 −0.216186
\(796\) 0 0
\(797\) −10.8259 18.7510i −0.383473 0.664194i 0.608083 0.793873i \(-0.291939\pi\)
−0.991556 + 0.129679i \(0.958605\pi\)
\(798\) 0 0
\(799\) 4.47472 0.158304
\(800\) 0 0
\(801\) −5.88126 −0.207804
\(802\) 0 0
\(803\) 63.2031 2.23039
\(804\) 0 0
\(805\) 6.69599 0.236003
\(806\) 0 0
\(807\) 20.3661 0.716920
\(808\) 0 0
\(809\) −17.7650 −0.624583 −0.312291 0.949986i \(-0.601097\pi\)
−0.312291 + 0.949986i \(0.601097\pi\)
\(810\) 0 0
\(811\) 19.1863 + 33.2317i 0.673724 + 1.16692i 0.976840 + 0.213970i \(0.0686396\pi\)
−0.303116 + 0.952954i \(0.598027\pi\)
\(812\) 0 0
\(813\) 12.0754 0.423504
\(814\) 0 0
\(815\) 9.34983 + 16.1944i 0.327510 + 0.567265i
\(816\) 0 0
\(817\) −26.6602 + 46.1767i −0.932721 + 1.61552i
\(818\) 0 0
\(819\) −14.0850 −0.492169
\(820\) 0 0
\(821\) −27.0148 + 46.7910i −0.942823 + 1.63302i −0.182770 + 0.983156i \(0.558506\pi\)
−0.760053 + 0.649862i \(0.774827\pi\)
\(822\) 0 0
\(823\) 11.6125 20.1134i 0.404786 0.701109i −0.589511 0.807760i \(-0.700680\pi\)
0.994296 + 0.106651i \(0.0340128\pi\)
\(824\) 0 0
\(825\) 2.56534 + 4.44330i 0.0893138 + 0.154696i
\(826\) 0 0
\(827\) −20.9642 + 36.3111i −0.728997 + 1.26266i 0.228310 + 0.973588i \(0.426680\pi\)
−0.957308 + 0.289072i \(0.906653\pi\)
\(828\) 0 0
\(829\) −4.33054 −0.150406 −0.0752029 0.997168i \(-0.523960\pi\)
−0.0752029 + 0.997168i \(0.523960\pi\)
\(830\) 0 0
\(831\) −15.4329 −0.535361
\(832\) 0 0
\(833\) −1.44207 2.49774i −0.0499648 0.0865415i
\(834\) 0 0
\(835\) −0.867052 1.50178i −0.0300056 0.0519712i
\(836\) 0 0
\(837\) 1.00060 + 1.73309i 0.0345858 + 0.0599044i
\(838\) 0 0
\(839\) −1.82418 3.15958i −0.0629778 0.109081i 0.832817 0.553548i \(-0.186726\pi\)
−0.895795 + 0.444467i \(0.853393\pi\)
\(840\) 0 0
\(841\) −12.7325 + 22.0534i −0.439052 + 0.760461i
\(842\) 0 0
\(843\) −0.837123 + 1.44994i −0.0288321 + 0.0499386i
\(844\) 0 0
\(845\) −14.1875 24.5735i −0.488065 0.845354i
\(846\) 0 0
\(847\) 33.5550 1.15296
\(848\) 0 0
\(849\) −6.71825 −0.230570
\(850\) 0 0
\(851\) 4.30799 + 7.46166i 0.147676 + 0.255782i
\(852\) 0 0
\(853\) −3.74019 + 6.47820i −0.128062 + 0.221809i −0.922926 0.384978i \(-0.874209\pi\)
0.794864 + 0.606788i \(0.207542\pi\)
\(854\) 0 0
\(855\) −2.76451 + 4.78828i −0.0945444 + 0.163756i
\(856\) 0 0
\(857\) −56.4449 −1.92812 −0.964061 0.265682i \(-0.914403\pi\)
−0.964061 + 0.265682i \(0.914403\pi\)
\(858\) 0 0
\(859\) −20.5849 35.6542i −0.702349 1.21650i −0.967640 0.252336i \(-0.918801\pi\)
0.265291 0.964168i \(-0.414532\pi\)
\(860\) 0 0
\(861\) 2.62191 4.54128i 0.0893545 0.154766i
\(862\) 0 0
\(863\) −17.3023 −0.588978 −0.294489 0.955655i \(-0.595150\pi\)
−0.294489 + 0.955655i \(0.595150\pi\)
\(864\) 0 0
\(865\) −2.87760 + 4.98415i −0.0978413 + 0.169466i
\(866\) 0 0
\(867\) 7.64469 + 13.2410i 0.259627 + 0.449688i
\(868\) 0 0
\(869\) −25.3469 + 43.9022i −0.859836 + 1.48928i
\(870\) 0 0
\(871\) −43.0097 + 30.3693i −1.45733 + 1.02903i
\(872\) 0 0
\(873\) 7.09796 12.2940i 0.240229 0.416090i
\(874\) 0 0
\(875\) −1.09486 1.89635i −0.0370129 0.0641082i
\(876\) 0 0
\(877\) 1.95341 3.38340i 0.0659618 0.114249i −0.831158 0.556036i \(-0.812322\pi\)
0.897120 + 0.441786i \(0.145655\pi\)
\(878\) 0 0
\(879\) 24.8308 0.837521
\(880\) 0 0
\(881\) 19.2478 33.3382i 0.648476 1.12319i −0.335011 0.942214i \(-0.608740\pi\)
0.983487 0.180979i \(-0.0579265\pi\)
\(882\) 0 0
\(883\) 24.3758 + 42.2201i 0.820311 + 1.42082i 0.905451 + 0.424451i \(0.139533\pi\)
−0.0851400 + 0.996369i \(0.527134\pi\)
\(884\) 0 0
\(885\) −5.99534 −0.201531
\(886\) 0 0
\(887\) −1.46384 + 2.53545i −0.0491510 + 0.0851321i −0.889554 0.456830i \(-0.848985\pi\)
0.840403 + 0.541962i \(0.182318\pi\)
\(888\) 0 0
\(889\) −10.9053 + 18.8886i −0.365754 + 0.633504i
\(890\) 0 0
\(891\) 2.56534 + 4.44330i 0.0859422 + 0.148856i
\(892\) 0 0
\(893\) −18.9163 −0.633011
\(894\) 0 0
\(895\) −3.48707 −0.116560
\(896\) 0 0
\(897\) 9.83483 + 17.0344i 0.328375 + 0.568763i
\(898\) 0 0
\(899\) −7.38447 + 12.7903i −0.246286 + 0.426580i
\(900\) 0 0
\(901\) −3.98619 + 6.90428i −0.132799 + 0.230015i
\(902\) 0 0
\(903\) −10.5585 18.2878i −0.351364 0.608580i
\(904\) 0 0
\(905\) 4.44487 + 7.69874i 0.147752 + 0.255915i
\(906\) 0 0
\(907\) −12.5020 21.6541i −0.415122 0.719013i 0.580319 0.814389i \(-0.302928\pi\)
−0.995441 + 0.0953763i \(0.969595\pi\)
\(908\) 0 0
\(909\) 8.95422 + 15.5092i 0.296993 + 0.514407i
\(910\) 0 0
\(911\) −41.0407 −1.35974 −0.679870 0.733332i \(-0.737964\pi\)
−0.679870 + 0.733332i \(0.737964\pi\)
\(912\) 0 0
\(913\) 68.6654 2.27249
\(914\) 0 0
\(915\) 6.06494 10.5048i 0.200501 0.347278i
\(916\) 0 0
\(917\) −5.13675 8.89711i −0.169630 0.293809i
\(918\) 0 0
\(919\) −9.76458 + 16.9127i −0.322104 + 0.557900i −0.980922 0.194402i \(-0.937723\pi\)
0.658818 + 0.752302i \(0.271057\pi\)
\(920\) 0 0
\(921\) −4.87572 + 8.44500i −0.160661 + 0.278272i
\(922\) 0 0
\(923\) 97.6935 3.21562
\(924\) 0 0
\(925\) 1.40879 2.44010i 0.0463208 0.0802301i
\(926\) 0 0
\(927\) −2.17465 3.76661i −0.0714250 0.123712i
\(928\) 0 0
\(929\) 42.6080 1.39792 0.698962 0.715159i \(-0.253646\pi\)
0.698962 + 0.715159i \(0.253646\pi\)
\(930\) 0 0
\(931\) 6.09618 + 10.5589i 0.199794 + 0.346054i
\(932\) 0 0
\(933\) −2.57755 −0.0843852
\(934\) 0 0
\(935\) 6.71047 0.219456
\(936\) 0 0
\(937\) 26.8048 0.875675 0.437838 0.899054i \(-0.355744\pi\)
0.437838 + 0.899054i \(0.355744\pi\)
\(938\) 0 0
\(939\) −2.12885 −0.0694722
\(940\) 0 0
\(941\) 16.0030 0.521682 0.260841 0.965382i \(-0.416000\pi\)
0.260841 + 0.965382i \(0.416000\pi\)
\(942\) 0 0
\(943\) −7.32298 −0.238469
\(944\) 0 0
\(945\) −1.09486 1.89635i −0.0356157 0.0616882i
\(946\) 0 0
\(947\) −55.5867 −1.80633 −0.903163 0.429297i \(-0.858761\pi\)
−0.903163 + 0.429297i \(0.858761\pi\)
\(948\) 0 0
\(949\) −39.6189 68.6220i −1.28608 2.22756i
\(950\) 0 0
\(951\) −12.0111 + 20.8038i −0.389485 + 0.674608i
\(952\) 0 0
\(953\) 10.3582 0.335534 0.167767 0.985827i \(-0.446344\pi\)
0.167767 + 0.985827i \(0.446344\pi\)
\(954\) 0 0
\(955\) 8.80189 15.2453i 0.284823 0.493327i
\(956\) 0 0
\(957\) −18.9323 + 32.7918i −0.611996 + 1.06001i
\(958\) 0 0
\(959\) −11.6603 20.1963i −0.376532 0.652172i
\(960\) 0 0
\(961\) 13.4976 23.3785i 0.435406 0.754146i
\(962\) 0 0
\(963\) −16.3415 −0.526599
\(964\) 0 0
\(965\) −2.90313 −0.0934551
\(966\) 0 0
\(967\) 11.7759 + 20.3965i 0.378687 + 0.655906i 0.990872 0.134809i \(-0.0430422\pi\)
−0.612184 + 0.790715i \(0.709709\pi\)
\(968\) 0 0
\(969\) 3.61573 + 6.26263i 0.116154 + 0.201185i
\(970\) 0 0
\(971\) −2.29652 3.97770i −0.0736990 0.127650i 0.826821 0.562465i \(-0.190147\pi\)
−0.900520 + 0.434815i \(0.856814\pi\)
\(972\) 0 0
\(973\) −23.2754 40.3142i −0.746175 1.29241i
\(974\) 0 0
\(975\) 3.21617 5.57057i 0.103000 0.178401i
\(976\) 0 0
\(977\) 16.1599 27.9898i 0.517001 0.895472i −0.482804 0.875728i \(-0.660382\pi\)
0.999805 0.0197434i \(-0.00628492\pi\)
\(978\) 0 0
\(979\) −15.0875 26.1322i −0.482197 0.835190i
\(980\) 0 0
\(981\) −11.4267 −0.364826
\(982\) 0 0
\(983\) 37.3018 1.18974 0.594872 0.803820i \(-0.297203\pi\)
0.594872 + 0.803820i \(0.297203\pi\)
\(984\) 0 0
\(985\) 10.2625 + 17.7752i 0.326991 + 0.566366i
\(986\) 0 0
\(987\) 3.74581 6.48793i 0.119230 0.206513i
\(988\) 0 0
\(989\) −14.7449 + 25.5389i −0.468860 + 0.812089i
\(990\) 0 0
\(991\) −60.3838 −1.91815 −0.959077 0.283146i \(-0.908622\pi\)
−0.959077 + 0.283146i \(0.908622\pi\)
\(992\) 0 0
\(993\) −1.23470 2.13857i −0.0391821 0.0678654i
\(994\) 0 0
\(995\) −2.49651 + 4.32408i −0.0791446 + 0.137083i
\(996\) 0 0
\(997\) −35.4117 −1.12150 −0.560750 0.827985i \(-0.689487\pi\)
−0.560750 + 0.827985i \(0.689487\pi\)
\(998\) 0 0
\(999\) 1.40879 2.44010i 0.0445723 0.0772014i
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 4020.2.q.m.3781.4 yes 24
67.37 even 3 inner 4020.2.q.m.841.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4020.2.q.m.841.4 24 67.37 even 3 inner
4020.2.q.m.3781.4 yes 24 1.1 even 1 trivial