Properties

Label 1680.2.k.i.209.20
Level $1680$
Weight $2$
Character 1680.209
Analytic conductor $13.415$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1680,2,Mod(209,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.k (of order \(2\), degree \(1\), 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: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 209.20
Character \(\chi\) \(=\) 1680.209
Dual form 1680.2.k.i.209.19

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.37873 + 1.04838i) q^{3} +(-1.84099 - 1.26915i) q^{5} +(-2.63201 + 0.269297i) q^{7} +(0.801780 + 2.89087i) q^{9} +O(q^{10})\) \(q+(1.37873 + 1.04838i) q^{3} +(-1.84099 - 1.26915i) q^{5} +(-2.63201 + 0.269297i) q^{7} +(0.801780 + 2.89087i) q^{9} -3.01442i q^{11} +4.39041 q^{13} +(-1.20767 - 3.67988i) q^{15} +2.71017i q^{17} +8.23502i q^{19} +(-3.91115 - 2.38807i) q^{21} -3.81803 q^{23} +(1.77850 + 4.67300i) q^{25} +(-1.92531 + 4.82630i) q^{27} -1.17337i q^{29} -1.73412i q^{31} +(3.16027 - 4.15606i) q^{33} +(5.18729 + 2.84465i) q^{35} -4.60838i q^{37} +(6.05318 + 4.60284i) q^{39} -10.6600 q^{41} +9.18515i q^{43} +(2.19289 - 6.33966i) q^{45} +12.0659i q^{47} +(6.85496 - 1.41759i) q^{49} +(-2.84130 + 3.73659i) q^{51} +7.14027 q^{53} +(-3.82575 + 5.54952i) q^{55} +(-8.63346 + 11.3538i) q^{57} -9.11464 q^{59} +13.5606i q^{61} +(-2.88880 - 7.39289i) q^{63} +(-8.08271 - 5.57210i) q^{65} +0.494447i q^{67} +(-5.26402 - 4.00276i) q^{69} +5.15821i q^{71} +4.76038 q^{73} +(-2.44703 + 8.30735i) q^{75} +(0.811774 + 7.93398i) q^{77} -12.1064 q^{79} +(-7.71430 + 4.63569i) q^{81} +8.16745i q^{83} +(3.43962 - 4.98941i) q^{85} +(1.23014 - 1.61776i) q^{87} -2.26858 q^{89} +(-11.5556 + 1.18232i) q^{91} +(1.81803 - 2.39088i) q^{93} +(10.4515 - 15.1606i) q^{95} +2.92728 q^{97} +(8.71430 - 2.41690i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 2 q^{9} + 6 q^{15} - 2 q^{21} - 16 q^{23} + 8 q^{25} + 8 q^{35} + 2 q^{39} - 6 q^{51} - 24 q^{53} - 8 q^{57} + 16 q^{63} - 16 q^{65} - 8 q^{77} - 4 q^{79} + 18 q^{81} - 12 q^{85} - 12 q^{91} - 32 q^{93} - 24 q^{95} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.37873 + 1.04838i 0.796009 + 0.605285i
\(4\) 0 0
\(5\) −1.84099 1.26915i −0.823317 0.567582i
\(6\) 0 0
\(7\) −2.63201 + 0.269297i −0.994806 + 0.101785i
\(8\) 0 0
\(9\) 0.801780 + 2.89087i 0.267260 + 0.963624i
\(10\) 0 0
\(11\) 3.01442i 0.908881i −0.890777 0.454440i \(-0.849839\pi\)
0.890777 0.454440i \(-0.150161\pi\)
\(12\) 0 0
\(13\) 4.39041 1.21768 0.608840 0.793293i \(-0.291635\pi\)
0.608840 + 0.793293i \(0.291635\pi\)
\(14\) 0 0
\(15\) −1.20767 3.67988i −0.311818 0.950142i
\(16\) 0 0
\(17\) 2.71017i 0.657314i 0.944449 + 0.328657i \(0.106596\pi\)
−0.944449 + 0.328657i \(0.893404\pi\)
\(18\) 0 0
\(19\) 8.23502i 1.88924i 0.328163 + 0.944621i \(0.393570\pi\)
−0.328163 + 0.944621i \(0.606430\pi\)
\(20\) 0 0
\(21\) −3.91115 2.38807i −0.853484 0.521120i
\(22\) 0 0
\(23\) −3.81803 −0.796114 −0.398057 0.917361i \(-0.630315\pi\)
−0.398057 + 0.917361i \(0.630315\pi\)
\(24\) 0 0
\(25\) 1.77850 + 4.67300i 0.355701 + 0.934600i
\(26\) 0 0
\(27\) −1.92531 + 4.82630i −0.370526 + 0.928822i
\(28\) 0 0
\(29\) 1.17337i 0.217889i −0.994048 0.108944i \(-0.965253\pi\)
0.994048 0.108944i \(-0.0347471\pi\)
\(30\) 0 0
\(31\) 1.73412i 0.311458i −0.987800 0.155729i \(-0.950227\pi\)
0.987800 0.155729i \(-0.0497726\pi\)
\(32\) 0 0
\(33\) 3.16027 4.15606i 0.550132 0.723477i
\(34\) 0 0
\(35\) 5.18729 + 2.84465i 0.876812 + 0.480833i
\(36\) 0 0
\(37\) 4.60838i 0.757613i −0.925476 0.378806i \(-0.876335\pi\)
0.925476 0.378806i \(-0.123665\pi\)
\(38\) 0 0
\(39\) 6.05318 + 4.60284i 0.969284 + 0.737044i
\(40\) 0 0
\(41\) −10.6600 −1.66482 −0.832408 0.554163i \(-0.813038\pi\)
−0.832408 + 0.554163i \(0.813038\pi\)
\(42\) 0 0
\(43\) 9.18515i 1.40072i 0.713789 + 0.700361i \(0.246978\pi\)
−0.713789 + 0.700361i \(0.753022\pi\)
\(44\) 0 0
\(45\) 2.19289 6.33966i 0.326896 0.945060i
\(46\) 0 0
\(47\) 12.0659i 1.75999i 0.474980 + 0.879997i \(0.342455\pi\)
−0.474980 + 0.879997i \(0.657545\pi\)
\(48\) 0 0
\(49\) 6.85496 1.41759i 0.979280 0.202512i
\(50\) 0 0
\(51\) −2.84130 + 3.73659i −0.397862 + 0.523227i
\(52\) 0 0
\(53\) 7.14027 0.980791 0.490396 0.871500i \(-0.336852\pi\)
0.490396 + 0.871500i \(0.336852\pi\)
\(54\) 0 0
\(55\) −3.82575 + 5.54952i −0.515865 + 0.748297i
\(56\) 0 0
\(57\) −8.63346 + 11.3538i −1.14353 + 1.50385i
\(58\) 0 0
\(59\) −9.11464 −1.18663 −0.593313 0.804972i \(-0.702180\pi\)
−0.593313 + 0.804972i \(0.702180\pi\)
\(60\) 0 0
\(61\) 13.5606i 1.73626i 0.496340 + 0.868128i \(0.334677\pi\)
−0.496340 + 0.868128i \(0.665323\pi\)
\(62\) 0 0
\(63\) −2.88880 7.39289i −0.363954 0.931417i
\(64\) 0 0
\(65\) −8.08271 5.57210i −1.00254 0.691134i
\(66\) 0 0
\(67\) 0.494447i 0.0604063i 0.999544 + 0.0302031i \(0.00961542\pi\)
−0.999544 + 0.0302031i \(0.990385\pi\)
\(68\) 0 0
\(69\) −5.26402 4.00276i −0.633714 0.481876i
\(70\) 0 0
\(71\) 5.15821i 0.612167i 0.952005 + 0.306084i \(0.0990187\pi\)
−0.952005 + 0.306084i \(0.900981\pi\)
\(72\) 0 0
\(73\) 4.76038 0.557160 0.278580 0.960413i \(-0.410136\pi\)
0.278580 + 0.960413i \(0.410136\pi\)
\(74\) 0 0
\(75\) −2.44703 + 8.30735i −0.282558 + 0.959250i
\(76\) 0 0
\(77\) 0.811774 + 7.93398i 0.0925102 + 0.904161i
\(78\) 0 0
\(79\) −12.1064 −1.36207 −0.681036 0.732250i \(-0.738470\pi\)
−0.681036 + 0.732250i \(0.738470\pi\)
\(80\) 0 0
\(81\) −7.71430 + 4.63569i −0.857144 + 0.515077i
\(82\) 0 0
\(83\) 8.16745i 0.896494i 0.893910 + 0.448247i \(0.147952\pi\)
−0.893910 + 0.448247i \(0.852048\pi\)
\(84\) 0 0
\(85\) 3.43962 4.98941i 0.373080 0.541177i
\(86\) 0 0
\(87\) 1.23014 1.61776i 0.131885 0.173442i
\(88\) 0 0
\(89\) −2.26858 −0.240469 −0.120235 0.992745i \(-0.538365\pi\)
−0.120235 + 0.992745i \(0.538365\pi\)
\(90\) 0 0
\(91\) −11.5556 + 1.18232i −1.21136 + 0.123941i
\(92\) 0 0
\(93\) 1.81803 2.39088i 0.188521 0.247923i
\(94\) 0 0
\(95\) 10.4515 15.1606i 1.07230 1.55544i
\(96\) 0 0
\(97\) 2.92728 0.297220 0.148610 0.988896i \(-0.452520\pi\)
0.148610 + 0.988896i \(0.452520\pi\)
\(98\) 0 0
\(99\) 8.71430 2.41690i 0.875820 0.242908i
\(100\) 0 0
\(101\) 0.416078 0.0414013 0.0207007 0.999786i \(-0.493410\pi\)
0.0207007 + 0.999786i \(0.493410\pi\)
\(102\) 0 0
\(103\) 10.4552 1.03019 0.515093 0.857134i \(-0.327757\pi\)
0.515093 + 0.857134i \(0.327757\pi\)
\(104\) 0 0
\(105\) 4.16958 + 9.36027i 0.406909 + 0.913469i
\(106\) 0 0
\(107\) 5.93315 0.573579 0.286789 0.957994i \(-0.407412\pi\)
0.286789 + 0.957994i \(0.407412\pi\)
\(108\) 0 0
\(109\) −11.7565 −1.12607 −0.563033 0.826435i \(-0.690365\pi\)
−0.563033 + 0.826435i \(0.690365\pi\)
\(110\) 0 0
\(111\) 4.83135 6.35370i 0.458572 0.603066i
\(112\) 0 0
\(113\) 14.2543 1.34093 0.670465 0.741941i \(-0.266095\pi\)
0.670465 + 0.741941i \(0.266095\pi\)
\(114\) 0 0
\(115\) 7.02896 + 4.84566i 0.655454 + 0.451860i
\(116\) 0 0
\(117\) 3.52014 + 12.6921i 0.325437 + 1.17339i
\(118\) 0 0
\(119\) −0.729842 7.13321i −0.0669045 0.653900i
\(120\) 0 0
\(121\) 1.91329 0.173936
\(122\) 0 0
\(123\) −14.6973 11.1758i −1.32521 1.00769i
\(124\) 0 0
\(125\) 2.65654 10.8601i 0.237608 0.971361i
\(126\) 0 0
\(127\) 15.4193i 1.36824i −0.729371 0.684119i \(-0.760187\pi\)
0.729371 0.684119i \(-0.239813\pi\)
\(128\) 0 0
\(129\) −9.62956 + 12.6638i −0.847836 + 1.11499i
\(130\) 0 0
\(131\) −7.07988 −0.618572 −0.309286 0.950969i \(-0.600090\pi\)
−0.309286 + 0.950969i \(0.600090\pi\)
\(132\) 0 0
\(133\) −2.21767 21.6746i −0.192296 1.87943i
\(134\) 0 0
\(135\) 9.66979 6.44167i 0.832243 0.554411i
\(136\) 0 0
\(137\) 13.6571 1.16681 0.583403 0.812183i \(-0.301721\pi\)
0.583403 + 0.812183i \(0.301721\pi\)
\(138\) 0 0
\(139\) 7.97668i 0.676573i −0.941043 0.338286i \(-0.890153\pi\)
0.941043 0.338286i \(-0.109847\pi\)
\(140\) 0 0
\(141\) −12.6497 + 16.6356i −1.06530 + 1.40097i
\(142\) 0 0
\(143\) 13.2345i 1.10673i
\(144\) 0 0
\(145\) −1.48918 + 2.16016i −0.123670 + 0.179392i
\(146\) 0 0
\(147\) 10.9373 + 5.23217i 0.902093 + 0.431542i
\(148\) 0 0
\(149\) 11.2515i 0.921755i −0.887464 0.460878i \(-0.847535\pi\)
0.887464 0.460878i \(-0.152465\pi\)
\(150\) 0 0
\(151\) 2.87722 0.234145 0.117072 0.993123i \(-0.462649\pi\)
0.117072 + 0.993123i \(0.462649\pi\)
\(152\) 0 0
\(153\) −7.83477 + 2.17296i −0.633404 + 0.175674i
\(154\) 0 0
\(155\) −2.20087 + 3.19251i −0.176778 + 0.256428i
\(156\) 0 0
\(157\) −10.7109 −0.854826 −0.427413 0.904056i \(-0.640575\pi\)
−0.427413 + 0.904056i \(0.640575\pi\)
\(158\) 0 0
\(159\) 9.84448 + 7.48574i 0.780718 + 0.593658i
\(160\) 0 0
\(161\) 10.0491 1.02818i 0.791979 0.0810323i
\(162\) 0 0
\(163\) 13.9669i 1.09397i 0.837141 + 0.546987i \(0.184225\pi\)
−0.837141 + 0.546987i \(0.815775\pi\)
\(164\) 0 0
\(165\) −11.0927 + 3.64041i −0.863566 + 0.283406i
\(166\) 0 0
\(167\) 8.59766i 0.665307i −0.943049 0.332653i \(-0.892056\pi\)
0.943049 0.332653i \(-0.107944\pi\)
\(168\) 0 0
\(169\) 6.27569 0.482745
\(170\) 0 0
\(171\) −23.8064 + 6.60267i −1.82052 + 0.504919i
\(172\) 0 0
\(173\) 8.74412i 0.664803i 0.943138 + 0.332401i \(0.107859\pi\)
−0.943138 + 0.332401i \(0.892141\pi\)
\(174\) 0 0
\(175\) −5.93947 11.8204i −0.448982 0.893541i
\(176\) 0 0
\(177\) −12.5666 9.55565i −0.944564 0.718246i
\(178\) 0 0
\(179\) 17.0676i 1.27569i −0.770165 0.637845i \(-0.779826\pi\)
0.770165 0.637845i \(-0.220174\pi\)
\(180\) 0 0
\(181\) 6.65867i 0.494935i −0.968896 0.247468i \(-0.920402\pi\)
0.968896 0.247468i \(-0.0795984\pi\)
\(182\) 0 0
\(183\) −14.2167 + 18.6964i −1.05093 + 1.38208i
\(184\) 0 0
\(185\) −5.84873 + 8.48399i −0.430008 + 0.623755i
\(186\) 0 0
\(187\) 8.16959 0.597420
\(188\) 0 0
\(189\) 3.76773 13.2214i 0.274062 0.961712i
\(190\) 0 0
\(191\) 13.1256i 0.949733i 0.880058 + 0.474866i \(0.157504\pi\)
−0.880058 + 0.474866i \(0.842496\pi\)
\(192\) 0 0
\(193\) 10.7880i 0.776538i −0.921546 0.388269i \(-0.873073\pi\)
0.921546 0.388269i \(-0.126927\pi\)
\(194\) 0 0
\(195\) −5.30215 16.1562i −0.379695 1.15697i
\(196\) 0 0
\(197\) −15.1781 −1.08139 −0.540696 0.841218i \(-0.681839\pi\)
−0.540696 + 0.841218i \(0.681839\pi\)
\(198\) 0 0
\(199\) 19.8125i 1.40447i −0.711944 0.702236i \(-0.752185\pi\)
0.711944 0.702236i \(-0.247815\pi\)
\(200\) 0 0
\(201\) −0.518370 + 0.681707i −0.0365630 + 0.0480839i
\(202\) 0 0
\(203\) 0.315985 + 3.08832i 0.0221778 + 0.216757i
\(204\) 0 0
\(205\) 19.6250 + 13.5292i 1.37067 + 0.944920i
\(206\) 0 0
\(207\) −3.06122 11.0374i −0.212769 0.767155i
\(208\) 0 0
\(209\) 24.8238 1.71710
\(210\) 0 0
\(211\) −5.58950 −0.384797 −0.192399 0.981317i \(-0.561627\pi\)
−0.192399 + 0.981317i \(0.561627\pi\)
\(212\) 0 0
\(213\) −5.40779 + 7.11177i −0.370536 + 0.487291i
\(214\) 0 0
\(215\) 11.6573 16.9098i 0.795025 1.15324i
\(216\) 0 0
\(217\) 0.466995 + 4.56423i 0.0317017 + 0.309840i
\(218\) 0 0
\(219\) 6.56326 + 4.99070i 0.443504 + 0.337241i
\(220\) 0 0
\(221\) 11.8988i 0.800398i
\(222\) 0 0
\(223\) −2.13609 −0.143043 −0.0715216 0.997439i \(-0.522785\pi\)
−0.0715216 + 0.997439i \(0.522785\pi\)
\(224\) 0 0
\(225\) −12.0831 + 8.88815i −0.805539 + 0.592543i
\(226\) 0 0
\(227\) 3.92566i 0.260555i 0.991478 + 0.130277i \(0.0415868\pi\)
−0.991478 + 0.130277i \(0.958413\pi\)
\(228\) 0 0
\(229\) 15.0457i 0.994251i 0.867679 + 0.497126i \(0.165611\pi\)
−0.867679 + 0.497126i \(0.834389\pi\)
\(230\) 0 0
\(231\) −7.19864 + 11.7898i −0.473636 + 0.775715i
\(232\) 0 0
\(233\) 9.72492 0.637101 0.318550 0.947906i \(-0.396804\pi\)
0.318550 + 0.947906i \(0.396804\pi\)
\(234\) 0 0
\(235\) 15.3135 22.2132i 0.998941 1.44903i
\(236\) 0 0
\(237\) −16.6914 12.6921i −1.08422 0.824441i
\(238\) 0 0
\(239\) 2.52992i 0.163647i 0.996647 + 0.0818233i \(0.0260743\pi\)
−0.996647 + 0.0818233i \(0.973926\pi\)
\(240\) 0 0
\(241\) 9.87223i 0.635926i −0.948103 0.317963i \(-0.897001\pi\)
0.948103 0.317963i \(-0.102999\pi\)
\(242\) 0 0
\(243\) −15.4959 1.69620i −0.994062 0.108811i
\(244\) 0 0
\(245\) −14.4191 6.09022i −0.921200 0.389090i
\(246\) 0 0
\(247\) 36.1551i 2.30049i
\(248\) 0 0
\(249\) −8.56263 + 11.2607i −0.542635 + 0.713617i
\(250\) 0 0
\(251\) 17.2182 1.08680 0.543402 0.839473i \(-0.317136\pi\)
0.543402 + 0.839473i \(0.317136\pi\)
\(252\) 0 0
\(253\) 11.5091i 0.723573i
\(254\) 0 0
\(255\) 9.97312 3.27299i 0.624541 0.204962i
\(256\) 0 0
\(257\) 6.40758i 0.399694i 0.979827 + 0.199847i \(0.0640445\pi\)
−0.979827 + 0.199847i \(0.935955\pi\)
\(258\) 0 0
\(259\) 1.24102 + 12.1293i 0.0771135 + 0.753678i
\(260\) 0 0
\(261\) 3.39206 0.940783i 0.209963 0.0582330i
\(262\) 0 0
\(263\) 3.69040 0.227560 0.113780 0.993506i \(-0.463704\pi\)
0.113780 + 0.993506i \(0.463704\pi\)
\(264\) 0 0
\(265\) −13.1452 9.06209i −0.807502 0.556680i
\(266\) 0 0
\(267\) −3.12776 2.37835i −0.191416 0.145553i
\(268\) 0 0
\(269\) 17.4759 1.06553 0.532763 0.846264i \(-0.321154\pi\)
0.532763 + 0.846264i \(0.321154\pi\)
\(270\) 0 0
\(271\) 3.60083i 0.218735i −0.994001 0.109367i \(-0.965118\pi\)
0.994001 0.109367i \(-0.0348825\pi\)
\(272\) 0 0
\(273\) −17.1716 10.4846i −1.03927 0.634557i
\(274\) 0 0
\(275\) 14.0864 5.36115i 0.849440 0.323290i
\(276\) 0 0
\(277\) 9.39015i 0.564199i 0.959385 + 0.282100i \(0.0910309\pi\)
−0.959385 + 0.282100i \(0.908969\pi\)
\(278\) 0 0
\(279\) 5.01313 1.39039i 0.300128 0.0832402i
\(280\) 0 0
\(281\) 2.80219i 0.167165i 0.996501 + 0.0835824i \(0.0266362\pi\)
−0.996501 + 0.0835824i \(0.973364\pi\)
\(282\) 0 0
\(283\) 11.8686 0.705518 0.352759 0.935714i \(-0.385244\pi\)
0.352759 + 0.935714i \(0.385244\pi\)
\(284\) 0 0
\(285\) 30.3039 9.94516i 1.79505 0.589100i
\(286\) 0 0
\(287\) 28.0573 2.87072i 1.65617 0.169453i
\(288\) 0 0
\(289\) 9.65496 0.567939
\(290\) 0 0
\(291\) 4.03592 + 3.06891i 0.236590 + 0.179903i
\(292\) 0 0
\(293\) 15.5145i 0.906366i 0.891418 + 0.453183i \(0.149712\pi\)
−0.891418 + 0.453183i \(0.850288\pi\)
\(294\) 0 0
\(295\) 16.7800 + 11.5679i 0.976968 + 0.673507i
\(296\) 0 0
\(297\) 14.5485 + 5.80369i 0.844189 + 0.336764i
\(298\) 0 0
\(299\) −16.7627 −0.969412
\(300\) 0 0
\(301\) −2.47353 24.1754i −0.142572 1.39345i
\(302\) 0 0
\(303\) 0.573658 + 0.436210i 0.0329558 + 0.0250596i
\(304\) 0 0
\(305\) 17.2105 24.9650i 0.985468 1.42949i
\(306\) 0 0
\(307\) −4.98662 −0.284601 −0.142301 0.989823i \(-0.545450\pi\)
−0.142301 + 0.989823i \(0.545450\pi\)
\(308\) 0 0
\(309\) 14.4149 + 10.9611i 0.820037 + 0.623556i
\(310\) 0 0
\(311\) 23.2945 1.32091 0.660455 0.750865i \(-0.270363\pi\)
0.660455 + 0.750865i \(0.270363\pi\)
\(312\) 0 0
\(313\) 7.16856 0.405191 0.202595 0.979263i \(-0.435062\pi\)
0.202595 + 0.979263i \(0.435062\pi\)
\(314\) 0 0
\(315\) −4.06445 + 17.2766i −0.229006 + 0.973425i
\(316\) 0 0
\(317\) 2.91003 0.163444 0.0817218 0.996655i \(-0.473958\pi\)
0.0817218 + 0.996655i \(0.473958\pi\)
\(318\) 0 0
\(319\) −3.53702 −0.198035
\(320\) 0 0
\(321\) 8.18019 + 6.22022i 0.456574 + 0.347179i
\(322\) 0 0
\(323\) −22.3183 −1.24182
\(324\) 0 0
\(325\) 7.80836 + 20.5164i 0.433130 + 1.13804i
\(326\) 0 0
\(327\) −16.2090 12.3253i −0.896358 0.681591i
\(328\) 0 0
\(329\) −3.24932 31.7576i −0.179141 1.75085i
\(330\) 0 0
\(331\) −29.4323 −1.61775 −0.808874 0.587982i \(-0.799922\pi\)
−0.808874 + 0.587982i \(0.799922\pi\)
\(332\) 0 0
\(333\) 13.3222 3.69491i 0.730054 0.202480i
\(334\) 0 0
\(335\) 0.627528 0.910273i 0.0342855 0.0497335i
\(336\) 0 0
\(337\) 15.8598i 0.863940i 0.901888 + 0.431970i \(0.142181\pi\)
−0.901888 + 0.431970i \(0.857819\pi\)
\(338\) 0 0
\(339\) 19.6528 + 14.9440i 1.06739 + 0.811645i
\(340\) 0 0
\(341\) −5.22737 −0.283078
\(342\) 0 0
\(343\) −17.6606 + 5.57712i −0.953581 + 0.301136i
\(344\) 0 0
\(345\) 4.61091 + 14.0499i 0.248243 + 0.756421i
\(346\) 0 0
\(347\) 11.0724 0.594400 0.297200 0.954815i \(-0.403947\pi\)
0.297200 + 0.954815i \(0.403947\pi\)
\(348\) 0 0
\(349\) 13.9422i 0.746306i −0.927770 0.373153i \(-0.878277\pi\)
0.927770 0.373153i \(-0.121723\pi\)
\(350\) 0 0
\(351\) −8.45290 + 21.1894i −0.451182 + 1.13101i
\(352\) 0 0
\(353\) 31.2855i 1.66516i −0.553904 0.832580i \(-0.686863\pi\)
0.553904 0.832580i \(-0.313137\pi\)
\(354\) 0 0
\(355\) 6.54656 9.49623i 0.347455 0.504008i
\(356\) 0 0
\(357\) 6.47209 10.5999i 0.342539 0.561006i
\(358\) 0 0
\(359\) 2.58454i 0.136407i 0.997671 + 0.0682033i \(0.0217267\pi\)
−0.997671 + 0.0682033i \(0.978273\pi\)
\(360\) 0 0
\(361\) −48.8155 −2.56924
\(362\) 0 0
\(363\) 2.63791 + 2.00586i 0.138454 + 0.105281i
\(364\) 0 0
\(365\) −8.76381 6.04164i −0.458719 0.316234i
\(366\) 0 0
\(367\) −25.8803 −1.35094 −0.675470 0.737388i \(-0.736059\pi\)
−0.675470 + 0.737388i \(0.736059\pi\)
\(368\) 0 0
\(369\) −8.54700 30.8168i −0.444939 1.60426i
\(370\) 0 0
\(371\) −18.7933 + 1.92285i −0.975697 + 0.0998296i
\(372\) 0 0
\(373\) 5.37044i 0.278071i 0.990287 + 0.139035i \(0.0444002\pi\)
−0.990287 + 0.139035i \(0.955600\pi\)
\(374\) 0 0
\(375\) 15.0483 12.1881i 0.777088 0.629392i
\(376\) 0 0
\(377\) 5.15157i 0.265319i
\(378\) 0 0
\(379\) 33.6626 1.72913 0.864565 0.502521i \(-0.167594\pi\)
0.864565 + 0.502521i \(0.167594\pi\)
\(380\) 0 0
\(381\) 16.1653 21.2590i 0.828174 1.08913i
\(382\) 0 0
\(383\) 5.70589i 0.291557i −0.989317 0.145779i \(-0.953431\pi\)
0.989317 0.145779i \(-0.0465687\pi\)
\(384\) 0 0
\(385\) 8.57496 15.6367i 0.437020 0.796918i
\(386\) 0 0
\(387\) −26.5531 + 7.36447i −1.34977 + 0.374357i
\(388\) 0 0
\(389\) 30.8578i 1.56455i 0.622930 + 0.782277i \(0.285942\pi\)
−0.622930 + 0.782277i \(0.714058\pi\)
\(390\) 0 0
\(391\) 10.3475i 0.523297i
\(392\) 0 0
\(393\) −9.76122 7.42243i −0.492388 0.374412i
\(394\) 0 0
\(395\) 22.2877 + 15.3648i 1.12142 + 0.773088i
\(396\) 0 0
\(397\) −22.7345 −1.14101 −0.570505 0.821294i \(-0.693253\pi\)
−0.570505 + 0.821294i \(0.693253\pi\)
\(398\) 0 0
\(399\) 19.6658 32.2084i 0.984522 1.61244i
\(400\) 0 0
\(401\) 13.7170i 0.684992i 0.939519 + 0.342496i \(0.111272\pi\)
−0.939519 + 0.342496i \(0.888728\pi\)
\(402\) 0 0
\(403\) 7.61351i 0.379256i
\(404\) 0 0
\(405\) 20.0854 + 1.25635i 0.998049 + 0.0624286i
\(406\) 0 0
\(407\) −13.8916 −0.688580
\(408\) 0 0
\(409\) 12.6202i 0.624030i −0.950077 0.312015i \(-0.898996\pi\)
0.950077 0.312015i \(-0.101004\pi\)
\(410\) 0 0
\(411\) 18.8294 + 14.3179i 0.928788 + 0.706250i
\(412\) 0 0
\(413\) 23.9898 2.45455i 1.18046 0.120780i
\(414\) 0 0
\(415\) 10.3657 15.0362i 0.508834 0.738099i
\(416\) 0 0
\(417\) 8.36262 10.9977i 0.409519 0.538558i
\(418\) 0 0
\(419\) −23.6389 −1.15484 −0.577418 0.816449i \(-0.695940\pi\)
−0.577418 + 0.816449i \(0.695940\pi\)
\(420\) 0 0
\(421\) −8.42017 −0.410374 −0.205187 0.978723i \(-0.565780\pi\)
−0.205187 + 0.978723i \(0.565780\pi\)
\(422\) 0 0
\(423\) −34.8810 + 9.67421i −1.69597 + 0.470376i
\(424\) 0 0
\(425\) −12.6646 + 4.82006i −0.614325 + 0.233807i
\(426\) 0 0
\(427\) −3.65183 35.6916i −0.176725 1.72724i
\(428\) 0 0
\(429\) 13.8749 18.2468i 0.669885 0.880964i
\(430\) 0 0
\(431\) 18.5310i 0.892606i −0.894882 0.446303i \(-0.852740\pi\)
0.894882 0.446303i \(-0.147260\pi\)
\(432\) 0 0
\(433\) 9.12614 0.438574 0.219287 0.975660i \(-0.429627\pi\)
0.219287 + 0.975660i \(0.429627\pi\)
\(434\) 0 0
\(435\) −4.31786 + 1.41704i −0.207025 + 0.0679418i
\(436\) 0 0
\(437\) 31.4415i 1.50405i
\(438\) 0 0
\(439\) 16.9640i 0.809645i −0.914395 0.404823i \(-0.867333\pi\)
0.914395 0.404823i \(-0.132667\pi\)
\(440\) 0 0
\(441\) 9.59423 + 18.6802i 0.456868 + 0.889534i
\(442\) 0 0
\(443\) 4.34739 0.206551 0.103275 0.994653i \(-0.467068\pi\)
0.103275 + 0.994653i \(0.467068\pi\)
\(444\) 0 0
\(445\) 4.17644 + 2.87918i 0.197982 + 0.136486i
\(446\) 0 0
\(447\) 11.7959 15.5127i 0.557925 0.733725i
\(448\) 0 0
\(449\) 15.9781i 0.754053i 0.926202 + 0.377027i \(0.123053\pi\)
−0.926202 + 0.377027i \(0.876947\pi\)
\(450\) 0 0
\(451\) 32.1338i 1.51312i
\(452\) 0 0
\(453\) 3.96690 + 3.01643i 0.186381 + 0.141724i
\(454\) 0 0
\(455\) 22.7743 + 12.4892i 1.06768 + 0.585501i
\(456\) 0 0
\(457\) 16.4839i 0.771085i −0.922690 0.385542i \(-0.874014\pi\)
0.922690 0.385542i \(-0.125986\pi\)
\(458\) 0 0
\(459\) −13.0801 5.21792i −0.610527 0.243552i
\(460\) 0 0
\(461\) −20.8741 −0.972203 −0.486102 0.873902i \(-0.661581\pi\)
−0.486102 + 0.873902i \(0.661581\pi\)
\(462\) 0 0
\(463\) 9.46270i 0.439769i −0.975526 0.219885i \(-0.929432\pi\)
0.975526 0.219885i \(-0.0705681\pi\)
\(464\) 0 0
\(465\) −6.38137 + 2.09424i −0.295929 + 0.0971182i
\(466\) 0 0
\(467\) 19.1519i 0.886245i 0.896461 + 0.443122i \(0.146129\pi\)
−0.896461 + 0.443122i \(0.853871\pi\)
\(468\) 0 0
\(469\) −0.133153 1.30139i −0.00614844 0.0600926i
\(470\) 0 0
\(471\) −14.7675 11.2292i −0.680449 0.517414i
\(472\) 0 0
\(473\) 27.6879 1.27309
\(474\) 0 0
\(475\) −38.4822 + 14.6460i −1.76569 + 0.672005i
\(476\) 0 0
\(477\) 5.72492 + 20.6416i 0.262126 + 0.945114i
\(478\) 0 0
\(479\) 22.9310 1.04774 0.523871 0.851798i \(-0.324487\pi\)
0.523871 + 0.851798i \(0.324487\pi\)
\(480\) 0 0
\(481\) 20.2327i 0.922530i
\(482\) 0 0
\(483\) 14.9329 + 9.11772i 0.679470 + 0.414871i
\(484\) 0 0
\(485\) −5.38909 3.71516i −0.244706 0.168697i
\(486\) 0 0
\(487\) 31.0733i 1.40807i −0.710167 0.704033i \(-0.751381\pi\)
0.710167 0.704033i \(-0.248619\pi\)
\(488\) 0 0
\(489\) −14.6427 + 19.2566i −0.662166 + 0.870812i
\(490\) 0 0
\(491\) 5.50641i 0.248501i 0.992251 + 0.124250i \(0.0396526\pi\)
−0.992251 + 0.124250i \(0.960347\pi\)
\(492\) 0 0
\(493\) 3.18003 0.143221
\(494\) 0 0
\(495\) −19.1104 6.61028i −0.858947 0.297110i
\(496\) 0 0
\(497\) −1.38909 13.5765i −0.0623093 0.608988i
\(498\) 0 0
\(499\) 22.6482 1.01387 0.506937 0.861983i \(-0.330778\pi\)
0.506937 + 0.861983i \(0.330778\pi\)
\(500\) 0 0
\(501\) 9.01365 11.8538i 0.402700 0.529590i
\(502\) 0 0
\(503\) 3.10345i 0.138376i 0.997604 + 0.0691881i \(0.0220409\pi\)
−0.997604 + 0.0691881i \(0.977959\pi\)
\(504\) 0 0
\(505\) −0.765997 0.528067i −0.0340864 0.0234987i
\(506\) 0 0
\(507\) 8.65246 + 6.57933i 0.384269 + 0.292198i
\(508\) 0 0
\(509\) 22.9387 1.01674 0.508371 0.861138i \(-0.330248\pi\)
0.508371 + 0.861138i \(0.330248\pi\)
\(510\) 0 0
\(511\) −12.5294 + 1.28196i −0.554266 + 0.0567104i
\(512\) 0 0
\(513\) −39.7447 15.8550i −1.75477 0.700014i
\(514\) 0 0
\(515\) −19.2480 13.2693i −0.848170 0.584715i
\(516\) 0 0
\(517\) 36.3717 1.59962
\(518\) 0 0
\(519\) −9.16719 + 12.0558i −0.402395 + 0.529189i
\(520\) 0 0
\(521\) 17.6343 0.772572 0.386286 0.922379i \(-0.373758\pi\)
0.386286 + 0.922379i \(0.373758\pi\)
\(522\) 0 0
\(523\) 31.8408 1.39230 0.696151 0.717895i \(-0.254894\pi\)
0.696151 + 0.717895i \(0.254894\pi\)
\(524\) 0 0
\(525\) 4.20345 22.5240i 0.183454 0.983028i
\(526\) 0 0
\(527\) 4.69978 0.204725
\(528\) 0 0
\(529\) −8.42266 −0.366203
\(530\) 0 0
\(531\) −7.30794 26.3493i −0.317138 1.14346i
\(532\) 0 0
\(533\) −46.8019 −2.02721
\(534\) 0 0
\(535\) −10.9229 7.53007i −0.472237 0.325553i
\(536\) 0 0
\(537\) 17.8934 23.5315i 0.772156 1.01546i
\(538\) 0 0
\(539\) −4.27320 20.6637i −0.184060 0.890049i
\(540\) 0 0
\(541\) −29.4219 −1.26494 −0.632472 0.774583i \(-0.717960\pi\)
−0.632472 + 0.774583i \(0.717960\pi\)
\(542\) 0 0
\(543\) 6.98085 9.18050i 0.299577 0.393973i
\(544\) 0 0
\(545\) 21.6436 + 14.9207i 0.927108 + 0.639135i
\(546\) 0 0
\(547\) 20.7168i 0.885784i 0.896575 + 0.442892i \(0.146048\pi\)
−0.896575 + 0.442892i \(0.853952\pi\)
\(548\) 0 0
\(549\) −39.2020 + 10.8726i −1.67310 + 0.464032i
\(550\) 0 0
\(551\) 9.66270 0.411645
\(552\) 0 0
\(553\) 31.8641 3.26021i 1.35500 0.138638i
\(554\) 0 0
\(555\) −16.9583 + 5.56539i −0.719840 + 0.236238i
\(556\) 0 0
\(557\) −8.76381 −0.371335 −0.185667 0.982613i \(-0.559445\pi\)
−0.185667 + 0.982613i \(0.559445\pi\)
\(558\) 0 0
\(559\) 40.3265i 1.70563i
\(560\) 0 0
\(561\) 11.2636 + 8.56487i 0.475551 + 0.361609i
\(562\) 0 0
\(563\) 1.85064i 0.0779952i −0.999239 0.0389976i \(-0.987584\pi\)
0.999239 0.0389976i \(-0.0124165\pi\)
\(564\) 0 0
\(565\) −26.2420 18.0909i −1.10401 0.761088i
\(566\) 0 0
\(567\) 19.0557 14.2786i 0.800266 0.599646i
\(568\) 0 0
\(569\) 11.8626i 0.497307i −0.968593 0.248653i \(-0.920012\pi\)
0.968593 0.248653i \(-0.0799880\pi\)
\(570\) 0 0
\(571\) −29.9166 −1.25197 −0.625985 0.779835i \(-0.715303\pi\)
−0.625985 + 0.779835i \(0.715303\pi\)
\(572\) 0 0
\(573\) −13.7606 + 18.0966i −0.574859 + 0.755996i
\(574\) 0 0
\(575\) −6.79038 17.8416i −0.283178 0.744048i
\(576\) 0 0
\(577\) −30.5778 −1.27297 −0.636486 0.771288i \(-0.719613\pi\)
−0.636486 + 0.771288i \(0.719613\pi\)
\(578\) 0 0
\(579\) 11.3100 14.8737i 0.470027 0.618131i
\(580\) 0 0
\(581\) −2.19947 21.4968i −0.0912495 0.891838i
\(582\) 0 0
\(583\) 21.5237i 0.891422i
\(584\) 0 0
\(585\) 9.62767 27.8337i 0.398055 1.15078i
\(586\) 0 0
\(587\) 39.6667i 1.63722i −0.574351 0.818609i \(-0.694745\pi\)
0.574351 0.818609i \(-0.305255\pi\)
\(588\) 0 0
\(589\) 14.2805 0.588419
\(590\) 0 0
\(591\) −20.9264 15.9124i −0.860797 0.654550i
\(592\) 0 0
\(593\) 23.8526i 0.979509i −0.871860 0.489755i \(-0.837086\pi\)
0.871860 0.489755i \(-0.162914\pi\)
\(594\) 0 0
\(595\) −7.70949 + 14.0585i −0.316058 + 0.576341i
\(596\) 0 0
\(597\) 20.7711 27.3161i 0.850106 1.11797i
\(598\) 0 0
\(599\) 37.7489i 1.54238i 0.636605 + 0.771190i \(0.280338\pi\)
−0.636605 + 0.771190i \(0.719662\pi\)
\(600\) 0 0
\(601\) 43.9087i 1.79107i 0.444986 + 0.895537i \(0.353209\pi\)
−0.444986 + 0.895537i \(0.646791\pi\)
\(602\) 0 0
\(603\) −1.42938 + 0.396438i −0.0582090 + 0.0161442i
\(604\) 0 0
\(605\) −3.52235 2.42826i −0.143204 0.0987227i
\(606\) 0 0
\(607\) −4.62053 −0.187542 −0.0937709 0.995594i \(-0.529892\pi\)
−0.0937709 + 0.995594i \(0.529892\pi\)
\(608\) 0 0
\(609\) −2.80209 + 4.58922i −0.113546 + 0.185965i
\(610\) 0 0
\(611\) 52.9743i 2.14311i
\(612\) 0 0
\(613\) 28.2008i 1.13902i 0.821985 + 0.569510i \(0.192867\pi\)
−0.821985 + 0.569510i \(0.807133\pi\)
\(614\) 0 0
\(615\) 12.8738 + 39.2277i 0.519120 + 1.58181i
\(616\) 0 0
\(617\) 7.84004 0.315628 0.157814 0.987469i \(-0.449555\pi\)
0.157814 + 0.987469i \(0.449555\pi\)
\(618\) 0 0
\(619\) 9.24487i 0.371583i 0.982589 + 0.185791i \(0.0594849\pi\)
−0.982589 + 0.185791i \(0.940515\pi\)
\(620\) 0 0
\(621\) 7.35089 18.4270i 0.294981 0.739448i
\(622\) 0 0
\(623\) 5.97094 0.610923i 0.239220 0.0244761i
\(624\) 0 0
\(625\) −18.6738 + 16.6219i −0.746954 + 0.664876i
\(626\) 0 0
\(627\) 34.2252 + 26.0249i 1.36682 + 1.03933i
\(628\) 0 0
\(629\) 12.4895 0.497989
\(630\) 0 0
\(631\) 34.6577 1.37970 0.689851 0.723951i \(-0.257676\pi\)
0.689851 + 0.723951i \(0.257676\pi\)
\(632\) 0 0
\(633\) −7.70640 5.85995i −0.306302 0.232912i
\(634\) 0 0
\(635\) −19.5694 + 28.3867i −0.776587 + 1.12649i
\(636\) 0 0
\(637\) 30.0961 6.22378i 1.19245 0.246595i
\(638\) 0 0
\(639\) −14.9117 + 4.13575i −0.589899 + 0.163608i
\(640\) 0 0
\(641\) 23.1085i 0.912732i −0.889792 0.456366i \(-0.849151\pi\)
0.889792 0.456366i \(-0.150849\pi\)
\(642\) 0 0
\(643\) −6.11558 −0.241175 −0.120588 0.992703i \(-0.538478\pi\)
−0.120588 + 0.992703i \(0.538478\pi\)
\(644\) 0 0
\(645\) 33.8003 11.0926i 1.33088 0.436771i
\(646\) 0 0
\(647\) 22.7956i 0.896188i −0.893987 0.448094i \(-0.852103\pi\)
0.893987 0.448094i \(-0.147897\pi\)
\(648\) 0 0
\(649\) 27.4753i 1.07850i
\(650\) 0 0
\(651\) −4.14121 + 6.78242i −0.162307 + 0.265824i
\(652\) 0 0
\(653\) −17.0044 −0.665432 −0.332716 0.943027i \(-0.607965\pi\)
−0.332716 + 0.943027i \(0.607965\pi\)
\(654\) 0 0
\(655\) 13.0340 + 8.98544i 0.509280 + 0.351090i
\(656\) 0 0
\(657\) 3.81678 + 13.7616i 0.148907 + 0.536893i
\(658\) 0 0
\(659\) 13.4782i 0.525038i 0.964927 + 0.262519i \(0.0845532\pi\)
−0.964927 + 0.262519i \(0.915447\pi\)
\(660\) 0 0
\(661\) 29.4321i 1.14478i 0.819983 + 0.572388i \(0.193983\pi\)
−0.819983 + 0.572388i \(0.806017\pi\)
\(662\) 0 0
\(663\) −12.4745 + 16.4052i −0.484469 + 0.637124i
\(664\) 0 0
\(665\) −23.4257 + 42.7174i −0.908411 + 1.65651i
\(666\) 0 0
\(667\) 4.47995i 0.173464i
\(668\) 0 0
\(669\) −2.94509 2.23944i −0.113864 0.0865819i
\(670\) 0 0
\(671\) 40.8773 1.57805
\(672\) 0 0
\(673\) 4.23323i 0.163179i 0.996666 + 0.0815895i \(0.0259996\pi\)
−0.996666 + 0.0815895i \(0.974000\pi\)
\(674\) 0 0
\(675\) −25.9775 0.413376i −0.999873 0.0159108i
\(676\) 0 0
\(677\) 18.8318i 0.723764i 0.932224 + 0.361882i \(0.117866\pi\)
−0.932224 + 0.361882i \(0.882134\pi\)
\(678\) 0 0
\(679\) −7.70462 + 0.788307i −0.295676 + 0.0302525i
\(680\) 0 0
\(681\) −4.11560 + 5.41241i −0.157710 + 0.207404i
\(682\) 0 0
\(683\) 44.2515 1.69324 0.846618 0.532202i \(-0.178635\pi\)
0.846618 + 0.532202i \(0.178635\pi\)
\(684\) 0 0
\(685\) −25.1426 17.3330i −0.960651 0.662259i
\(686\) 0 0
\(687\) −15.7737 + 20.7440i −0.601805 + 0.791433i
\(688\) 0 0
\(689\) 31.3487 1.19429
\(690\) 0 0
\(691\) 50.4812i 1.92039i −0.279322 0.960197i \(-0.590110\pi\)
0.279322 0.960197i \(-0.409890\pi\)
\(692\) 0 0
\(693\) −22.2853 + 8.70804i −0.846547 + 0.330791i
\(694\) 0 0
\(695\) −10.1236 + 14.6850i −0.384011 + 0.557034i
\(696\) 0 0
\(697\) 28.8905i 1.09431i
\(698\) 0 0
\(699\) 13.4080 + 10.1955i 0.507138 + 0.385628i
\(700\) 0 0
\(701\) 44.6207i 1.68530i 0.538462 + 0.842650i \(0.319006\pi\)
−0.538462 + 0.842650i \(0.680994\pi\)
\(702\) 0 0
\(703\) 37.9501 1.43131
\(704\) 0 0
\(705\) 44.4011 14.5716i 1.67224 0.548798i
\(706\) 0 0
\(707\) −1.09512 + 0.112049i −0.0411863 + 0.00421403i
\(708\) 0 0
\(709\) −0.166857 −0.00626645 −0.00313323 0.999995i \(-0.500997\pi\)
−0.00313323 + 0.999995i \(0.500997\pi\)
\(710\) 0 0
\(711\) −9.70664 34.9979i −0.364027 1.31253i
\(712\) 0 0
\(713\) 6.62093i 0.247956i
\(714\) 0 0
\(715\) −16.7966 + 24.3646i −0.628158 + 0.911186i
\(716\) 0 0
\(717\) −2.65232 + 3.48806i −0.0990528 + 0.130264i
\(718\) 0 0
\(719\) −32.9774 −1.22985 −0.614925 0.788586i \(-0.710814\pi\)
−0.614925 + 0.788586i \(0.710814\pi\)
\(720\) 0 0
\(721\) −27.5183 + 2.81557i −1.02484 + 0.104857i
\(722\) 0 0
\(723\) 10.3499 13.6111i 0.384916 0.506203i
\(724\) 0 0
\(725\) 5.48315 2.08684i 0.203639 0.0775033i
\(726\) 0 0
\(727\) 35.8146 1.32829 0.664144 0.747604i \(-0.268796\pi\)
0.664144 + 0.747604i \(0.268796\pi\)
\(728\) 0 0
\(729\) −19.5864 18.5843i −0.725421 0.688306i
\(730\) 0 0
\(731\) −24.8933 −0.920713
\(732\) 0 0
\(733\) −14.7825 −0.546003 −0.273002 0.962014i \(-0.588016\pi\)
−0.273002 + 0.962014i \(0.588016\pi\)
\(734\) 0 0
\(735\) −13.4951 23.5135i −0.497773 0.867307i
\(736\) 0 0
\(737\) 1.49047 0.0549021
\(738\) 0 0
\(739\) 44.5145 1.63749 0.818746 0.574156i \(-0.194670\pi\)
0.818746 + 0.574156i \(0.194670\pi\)
\(740\) 0 0
\(741\) −37.9044 + 49.8480i −1.39245 + 1.83121i
\(742\) 0 0
\(743\) 9.42880 0.345909 0.172955 0.984930i \(-0.444669\pi\)
0.172955 + 0.984930i \(0.444669\pi\)
\(744\) 0 0
\(745\) −14.2798 + 20.7138i −0.523172 + 0.758897i
\(746\) 0 0
\(747\) −23.6111 + 6.54850i −0.863884 + 0.239597i
\(748\) 0 0
\(749\) −15.6161 + 1.59778i −0.570600 + 0.0583816i
\(750\) 0 0
\(751\) 23.8428 0.870037 0.435019 0.900421i \(-0.356742\pi\)
0.435019 + 0.900421i \(0.356742\pi\)
\(752\) 0 0
\(753\) 23.7392 + 18.0513i 0.865105 + 0.657826i
\(754\) 0 0
\(755\) −5.29694 3.65163i −0.192775 0.132896i
\(756\) 0 0
\(757\) 35.2632i 1.28166i 0.767682 + 0.640831i \(0.221410\pi\)
−0.767682 + 0.640831i \(0.778590\pi\)
\(758\) 0 0
\(759\) −12.0660 + 15.8680i −0.437968 + 0.575970i
\(760\) 0 0
\(761\) −52.0087 −1.88531 −0.942657 0.333764i \(-0.891681\pi\)
−0.942657 + 0.333764i \(0.891681\pi\)
\(762\) 0 0
\(763\) 30.9431 3.16598i 1.12022 0.114616i
\(764\) 0 0
\(765\) 17.1816 + 5.94311i 0.621201 + 0.214873i
\(766\) 0 0
\(767\) −40.0170 −1.44493
\(768\) 0 0
\(769\) 28.0310i 1.01082i −0.862879 0.505411i \(-0.831341\pi\)
0.862879 0.505411i \(-0.168659\pi\)
\(770\) 0 0
\(771\) −6.71761 + 8.83431i −0.241929 + 0.318160i
\(772\) 0 0
\(773\) 46.6884i 1.67927i 0.543155 + 0.839633i \(0.317230\pi\)
−0.543155 + 0.839633i \(0.682770\pi\)
\(774\) 0 0
\(775\) 8.10356 3.08415i 0.291088 0.110786i
\(776\) 0 0
\(777\) −11.0051 + 18.0241i −0.394807 + 0.646610i
\(778\) 0 0
\(779\) 87.7855i 3.14524i
\(780\) 0 0
\(781\) 15.5490 0.556387
\(782\) 0 0
\(783\) 5.66303 + 2.25910i 0.202380 + 0.0807336i
\(784\) 0 0
\(785\) 19.7188 + 13.5938i 0.703793 + 0.485184i
\(786\) 0 0
\(787\) 16.1183 0.574554 0.287277 0.957847i \(-0.407250\pi\)
0.287277 + 0.957847i \(0.407250\pi\)
\(788\) 0 0
\(789\) 5.08806 + 3.86896i 0.181140 + 0.137739i
\(790\) 0 0
\(791\) −37.5174 + 3.83864i −1.33397 + 0.136486i
\(792\) 0 0
\(793\) 59.5366i 2.11421i
\(794\) 0 0
\(795\) −8.62307 26.2753i −0.305829 0.931891i
\(796\) 0 0
\(797\) 35.4492i 1.25567i −0.778345 0.627837i \(-0.783940\pi\)
0.778345 0.627837i \(-0.216060\pi\)
\(798\) 0 0
\(799\) −32.7007 −1.15687
\(800\) 0 0
\(801\) −1.81891 6.55819i −0.0642679 0.231722i
\(802\) 0 0
\(803\) 14.3498i 0.506392i
\(804\) 0 0
\(805\) −19.8052 10.8609i −0.698042 0.382798i
\(806\) 0 0
\(807\) 24.0946 + 18.3215i 0.848168 + 0.644947i
\(808\) 0 0
\(809\) 31.8061i 1.11824i −0.829086 0.559121i \(-0.811139\pi\)
0.829086 0.559121i \(-0.188861\pi\)
\(810\) 0 0
\(811\) 40.0124i 1.40503i −0.711671 0.702513i \(-0.752061\pi\)
0.711671 0.702513i \(-0.247939\pi\)
\(812\) 0 0
\(813\) 3.77505 4.96456i 0.132397 0.174115i
\(814\) 0 0
\(815\) 17.7261 25.7130i 0.620920 0.900686i
\(816\) 0 0
\(817\) −75.6398 −2.64630
\(818\) 0 0
\(819\) −12.6830 32.4578i −0.443180 1.13417i
\(820\) 0 0
\(821\) 46.2101i 1.61274i 0.591409 + 0.806371i \(0.298572\pi\)
−0.591409 + 0.806371i \(0.701428\pi\)
\(822\) 0 0
\(823\) 12.6943i 0.442494i 0.975218 + 0.221247i \(0.0710128\pi\)
−0.975218 + 0.221247i \(0.928987\pi\)
\(824\) 0 0
\(825\) 25.0418 + 7.37636i 0.871844 + 0.256812i
\(826\) 0 0
\(827\) −14.8569 −0.516626 −0.258313 0.966061i \(-0.583167\pi\)
−0.258313 + 0.966061i \(0.583167\pi\)
\(828\) 0 0
\(829\) 5.46791i 0.189908i −0.995482 0.0949542i \(-0.969730\pi\)
0.995482 0.0949542i \(-0.0302705\pi\)
\(830\) 0 0
\(831\) −9.84448 + 12.9465i −0.341501 + 0.449108i
\(832\) 0 0
\(833\) 3.84191 + 18.5781i 0.133114 + 0.643694i
\(834\) 0 0
\(835\) −10.9117 + 15.8282i −0.377616 + 0.547758i
\(836\) 0 0
\(837\) 8.36940 + 3.33873i 0.289289 + 0.115403i
\(838\) 0 0
\(839\) −47.5202 −1.64058 −0.820290 0.571948i \(-0.806188\pi\)
−0.820290 + 0.571948i \(0.806188\pi\)
\(840\) 0 0
\(841\) 27.6232 0.952524
\(842\) 0 0
\(843\) −2.93777 + 3.86346i −0.101182 + 0.133065i
\(844\) 0 0
\(845\) −11.5535 7.96480i −0.397452 0.273998i
\(846\) 0 0
\(847\) −5.03580 + 0.515244i −0.173032 + 0.0177040i
\(848\) 0 0
\(849\) 16.3636 + 12.4429i 0.561598 + 0.427039i
\(850\) 0 0
\(851\) 17.5949i 0.603146i
\(852\) 0 0
\(853\) 31.8095 1.08914 0.544569 0.838716i \(-0.316693\pi\)
0.544569 + 0.838716i \(0.316693\pi\)
\(854\) 0 0
\(855\) 52.2072 + 18.0585i 1.78545 + 0.617587i
\(856\) 0 0
\(857\) 47.8666i 1.63509i 0.575864 + 0.817545i \(0.304666\pi\)
−0.575864 + 0.817545i \(0.695334\pi\)
\(858\) 0 0
\(859\) 3.56261i 0.121555i −0.998151 0.0607774i \(-0.980642\pi\)
0.998151 0.0607774i \(-0.0193580\pi\)
\(860\) 0 0
\(861\) 41.6930 + 25.4569i 1.42089 + 0.867569i
\(862\) 0 0
\(863\) −22.3349 −0.760288 −0.380144 0.924927i \(-0.624126\pi\)
−0.380144 + 0.924927i \(0.624126\pi\)
\(864\) 0 0
\(865\) 11.0976 16.0978i 0.377330 0.547343i
\(866\) 0 0
\(867\) 13.3116 + 10.1221i 0.452084 + 0.343765i
\(868\) 0 0
\(869\) 36.4936i 1.23796i
\(870\) 0 0
\(871\) 2.17082i 0.0735555i
\(872\) 0 0
\(873\) 2.34703 + 8.46239i 0.0794350 + 0.286408i
\(874\) 0 0
\(875\) −4.06742 + 29.2994i −0.137504 + 0.990501i
\(876\) 0 0
\(877\) 18.6566i 0.629989i −0.949094 0.314994i \(-0.897997\pi\)
0.949094 0.314994i \(-0.102003\pi\)
\(878\) 0 0
\(879\) −16.2651 + 21.3903i −0.548610 + 0.721475i
\(880\) 0 0
\(881\) −26.8648 −0.905097 −0.452549 0.891740i \(-0.649485\pi\)
−0.452549 + 0.891740i \(0.649485\pi\)
\(882\) 0 0
\(883\) 12.0169i 0.404400i −0.979344 0.202200i \(-0.935191\pi\)
0.979344 0.202200i \(-0.0648092\pi\)
\(884\) 0 0
\(885\) 11.0075 + 33.5408i 0.370011 + 1.12746i
\(886\) 0 0
\(887\) 13.9283i 0.467666i −0.972277 0.233833i \(-0.924873\pi\)
0.972277 0.233833i \(-0.0751268\pi\)
\(888\) 0 0
\(889\) 4.15236 + 40.5836i 0.139266 + 1.36113i
\(890\) 0 0
\(891\) 13.9739 + 23.2541i 0.468143 + 0.779042i
\(892\) 0 0
\(893\) −99.3629 −3.32505
\(894\) 0 0
\(895\) −21.6613 + 31.4213i −0.724059 + 1.05030i
\(896\) 0 0
\(897\) −23.1112 17.5738i −0.771661 0.586771i
\(898\) 0 0
\(899\) −2.03477 −0.0678632
\(900\) 0 0
\(901\) 19.3514i 0.644687i
\(902\) 0 0
\(903\) 21.9348 35.9245i 0.729944 1.19549i
\(904\) 0 0
\(905\) −8.45087 + 12.2586i −0.280916 + 0.407488i
\(906\) 0 0
\(907\) 23.7097i 0.787269i 0.919267 + 0.393634i \(0.128782\pi\)
−0.919267 + 0.393634i \(0.871218\pi\)
\(908\) 0 0
\(909\) 0.333603 + 1.20283i 0.0110649 + 0.0398953i
\(910\) 0 0
\(911\) 0.247204i 0.00819025i 0.999992 + 0.00409512i \(0.00130352\pi\)
−0.999992 + 0.00409512i \(0.998696\pi\)
\(912\) 0 0
\(913\) 24.6201 0.814807
\(914\) 0 0
\(915\) 49.9014 16.3767i 1.64969 0.541397i
\(916\) 0 0
\(917\) 18.6343 1.90659i 0.615359 0.0629612i
\(918\) 0 0
\(919\) 14.4956 0.478165 0.239083 0.970999i \(-0.423153\pi\)
0.239083 + 0.970999i \(0.423153\pi\)
\(920\) 0 0
\(921\) −6.87519 5.22790i −0.226545 0.172265i
\(922\) 0 0
\(923\) 22.6467i 0.745424i
\(924\) 0 0
\(925\) 21.5349 8.19602i 0.708065 0.269483i
\(926\) 0 0
\(927\) 8.38281 + 30.2248i 0.275328 + 0.992713i
\(928\) 0 0
\(929\) 3.97297 0.130349 0.0651745 0.997874i \(-0.479240\pi\)
0.0651745 + 0.997874i \(0.479240\pi\)
\(930\) 0 0
\(931\) 11.6738 + 56.4507i 0.382595 + 1.85010i
\(932\) 0 0
\(933\) 32.1168 + 24.4216i 1.05146 + 0.799527i
\(934\) 0 0
\(935\) −15.0402 10.3685i −0.491866 0.339085i
\(936\) 0 0
\(937\) 32.7664 1.07043 0.535216 0.844715i \(-0.320230\pi\)
0.535216 + 0.844715i \(0.320230\pi\)
\(938\) 0 0
\(939\) 9.88349 + 7.51540i 0.322535 + 0.245256i
\(940\) 0 0
\(941\) 34.4787 1.12397 0.561987 0.827146i \(-0.310037\pi\)
0.561987 + 0.827146i \(0.310037\pi\)
\(942\) 0 0
\(943\) 40.7003 1.32538
\(944\) 0 0
\(945\) −23.7163 + 19.5586i −0.771490 + 0.636241i
\(946\) 0 0
\(947\) −43.1115 −1.40093 −0.700467 0.713684i \(-0.747025\pi\)
−0.700467 + 0.713684i \(0.747025\pi\)
\(948\) 0 0
\(949\) 20.9000 0.678443
\(950\) 0 0
\(951\) 4.01214 + 3.05083i 0.130103 + 0.0989300i
\(952\) 0 0
\(953\) −53.6918 −1.73925 −0.869624 0.493714i \(-0.835639\pi\)
−0.869624 + 0.493714i \(0.835639\pi\)
\(954\) 0 0
\(955\) 16.6584 24.1641i 0.539052 0.781931i
\(956\) 0 0
\(957\) −4.87659 3.70816i −0.157638 0.119868i
\(958\) 0 0
\(959\) −35.9457 + 3.67782i −1.16075 + 0.118763i
\(960\) 0 0
\(961\) 27.9928 0.902994
\(962\) 0 0
\(963\) 4.75708 + 17.1520i 0.153295 + 0.552715i
\(964\) 0 0
\(965\) −13.6916 + 19.8606i −0.440749 + 0.639337i
\(966\) 0 0
\(967\) 1.49312i 0.0480155i −0.999712 0.0240077i \(-0.992357\pi\)
0.999712 0.0240077i \(-0.00764263\pi\)
\(968\) 0 0
\(969\) −30.7709 23.3982i −0.988503 0.751658i
\(970\) 0 0
\(971\) 3.95684 0.126981 0.0634905 0.997982i \(-0.479777\pi\)
0.0634905 + 0.997982i \(0.479777\pi\)
\(972\) 0 0
\(973\) 2.14810 + 20.9947i 0.0688648 + 0.673059i
\(974\) 0 0
\(975\) −10.7434 + 36.4727i −0.344066 + 1.16806i
\(976\) 0 0
\(977\) 47.6921 1.52580 0.762902 0.646514i \(-0.223774\pi\)
0.762902 + 0.646514i \(0.223774\pi\)
\(978\) 0 0
\(979\) 6.83846i 0.218558i
\(980\) 0 0
\(981\) −9.42610 33.9865i −0.300952 1.08510i
\(982\) 0 0
\(983\) 13.0582i 0.416491i 0.978077 + 0.208245i \(0.0667752\pi\)
−0.978077 + 0.208245i \(0.933225\pi\)
\(984\) 0 0
\(985\) 27.9427 + 19.2633i 0.890328 + 0.613779i
\(986\) 0 0
\(987\) 28.8142 47.1916i 0.917168 1.50213i
\(988\) 0 0
\(989\) 35.0691i 1.11513i
\(990\) 0 0
\(991\) −12.6624 −0.402236 −0.201118 0.979567i \(-0.564457\pi\)
−0.201118 + 0.979567i \(0.564457\pi\)
\(992\) 0 0
\(993\) −40.5792 30.8564i −1.28774 0.979198i
\(994\) 0 0
\(995\) −25.1451 + 36.4747i −0.797153 + 1.15633i
\(996\) 0 0
\(997\) 22.6851 0.718444 0.359222 0.933252i \(-0.383042\pi\)
0.359222 + 0.933252i \(0.383042\pi\)
\(998\) 0 0
\(999\) 22.2414 + 8.87256i 0.703687 + 0.280715i
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 1680.2.k.i.209.20 24
3.2 odd 2 1680.2.k.h.209.19 24
4.3 odd 2 840.2.k.a.209.5 24
5.4 even 2 1680.2.k.h.209.5 24
7.6 odd 2 inner 1680.2.k.i.209.5 24
12.11 even 2 840.2.k.b.209.6 yes 24
15.14 odd 2 inner 1680.2.k.i.209.6 24
20.19 odd 2 840.2.k.b.209.20 yes 24
21.20 even 2 1680.2.k.h.209.6 24
28.27 even 2 840.2.k.a.209.20 yes 24
35.34 odd 2 1680.2.k.h.209.20 24
60.59 even 2 840.2.k.a.209.19 yes 24
84.83 odd 2 840.2.k.b.209.19 yes 24
105.104 even 2 inner 1680.2.k.i.209.19 24
140.139 even 2 840.2.k.b.209.5 yes 24
420.419 odd 2 840.2.k.a.209.6 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.k.a.209.5 24 4.3 odd 2
840.2.k.a.209.6 yes 24 420.419 odd 2
840.2.k.a.209.19 yes 24 60.59 even 2
840.2.k.a.209.20 yes 24 28.27 even 2
840.2.k.b.209.5 yes 24 140.139 even 2
840.2.k.b.209.6 yes 24 12.11 even 2
840.2.k.b.209.19 yes 24 84.83 odd 2
840.2.k.b.209.20 yes 24 20.19 odd 2
1680.2.k.h.209.5 24 5.4 even 2
1680.2.k.h.209.6 24 21.20 even 2
1680.2.k.h.209.19 24 3.2 odd 2
1680.2.k.h.209.20 24 35.34 odd 2
1680.2.k.i.209.5 24 7.6 odd 2 inner
1680.2.k.i.209.6 24 15.14 odd 2 inner
1680.2.k.i.209.19 24 105.104 even 2 inner
1680.2.k.i.209.20 24 1.1 even 1 trivial