Properties

Label 280.2.bo.a.73.4
Level $280$
Weight $2$
Character 280.73
Analytic conductor $2.236$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [280,2,Mod(17,280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(280, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("280.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.bo (of order \(12\), degree \(4\), 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: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 73.4
Character \(\chi\) \(=\) 280.73
Dual form 280.2.bo.a.257.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.364260 - 1.35944i) q^{3} +(-0.489612 + 2.18181i) q^{5} +(-0.501338 - 2.59782i) q^{7} +(0.882692 - 0.509622i) q^{9} +O(q^{10})\) \(q+(-0.364260 - 1.35944i) q^{3} +(-0.489612 + 2.18181i) q^{5} +(-0.501338 - 2.59782i) q^{7} +(0.882692 - 0.509622i) q^{9} +(1.86869 - 3.23667i) q^{11} +(4.55026 - 4.55026i) q^{13} +(3.14438 - 0.129149i) q^{15} +(-5.91633 + 1.58528i) q^{17} +(0.616087 + 1.06709i) q^{19} +(-3.34895 + 1.62782i) q^{21} +(0.721250 - 2.69174i) q^{23} +(-4.52056 - 2.13648i) q^{25} +(-3.99986 - 3.99986i) q^{27} +2.82808i q^{29} +(5.94396 + 3.43175i) q^{31} +(-5.08074 - 1.36138i) q^{33} +(5.91340 + 0.178099i) q^{35} +(7.52551 + 2.01646i) q^{37} +(-7.84327 - 4.52831i) q^{39} +5.56966i q^{41} +(-1.95176 - 1.95176i) q^{43} +(0.679721 + 2.17538i) q^{45} +(-2.98797 + 11.1512i) q^{47} +(-6.49732 + 2.60477i) q^{49} +(4.31017 + 7.46543i) q^{51} +(9.44513 - 2.53082i) q^{53} +(6.14686 + 5.66184i) q^{55} +(1.22623 - 1.22623i) q^{57} +(0.916152 - 1.58682i) q^{59} +(-3.43491 + 1.98315i) q^{61} +(-1.76643 - 2.03758i) q^{63} +(7.69992 + 12.1556i) q^{65} +(-2.12646 - 7.93605i) q^{67} -3.92198 q^{69} +0.570063 q^{71} +(-0.546036 - 2.03783i) q^{73} +(-1.25774 + 6.92365i) q^{75} +(-9.34513 - 3.23186i) q^{77} +(-9.47466 + 5.47020i) q^{79} +(-2.45170 + 4.24647i) q^{81} +(-8.80820 + 8.80820i) q^{83} +(-0.562062 - 13.6845i) q^{85} +(3.84459 - 1.03016i) q^{87} +(3.00362 + 5.20243i) q^{89} +(-14.1020 - 9.53952i) q^{91} +(2.50010 - 9.33049i) q^{93} +(-2.62984 + 0.821721i) q^{95} +(-3.70694 - 3.70694i) q^{97} -3.80931i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 4 q^{7} + 4 q^{11} + 8 q^{15} - 4 q^{21} - 4 q^{23} - 8 q^{25} - 36 q^{33} + 24 q^{35} + 8 q^{37} - 16 q^{43} + 48 q^{45} + 24 q^{51} + 16 q^{53} - 96 q^{57} - 36 q^{61} - 68 q^{63} + 12 q^{65} - 16 q^{67} - 64 q^{71} - 48 q^{73} - 48 q^{75} + 4 q^{77} - 40 q^{85} - 12 q^{87} - 80 q^{91} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.364260 1.35944i −0.210306 0.784872i −0.987767 0.155940i \(-0.950159\pi\)
0.777461 0.628931i \(-0.216507\pi\)
\(4\) 0 0
\(5\) −0.489612 + 2.18181i −0.218961 + 0.975734i
\(6\) 0 0
\(7\) −0.501338 2.59782i −0.189488 0.981883i
\(8\) 0 0
\(9\) 0.882692 0.509622i 0.294231 0.169874i
\(10\) 0 0
\(11\) 1.86869 3.23667i 0.563432 0.975893i −0.433762 0.901028i \(-0.642814\pi\)
0.997194 0.0748654i \(-0.0238527\pi\)
\(12\) 0 0
\(13\) 4.55026 4.55026i 1.26201 1.26201i 0.311899 0.950115i \(-0.399035\pi\)
0.950115 0.311899i \(-0.100965\pi\)
\(14\) 0 0
\(15\) 3.14438 0.129149i 0.811874 0.0333461i
\(16\) 0 0
\(17\) −5.91633 + 1.58528i −1.43492 + 0.384486i −0.890752 0.454489i \(-0.849822\pi\)
−0.544169 + 0.838975i \(0.683155\pi\)
\(18\) 0 0
\(19\) 0.616087 + 1.06709i 0.141340 + 0.244808i 0.928001 0.372577i \(-0.121526\pi\)
−0.786661 + 0.617385i \(0.788192\pi\)
\(20\) 0 0
\(21\) −3.34895 + 1.62782i −0.730802 + 0.355219i
\(22\) 0 0
\(23\) 0.721250 2.69174i 0.150391 0.561267i −0.849065 0.528288i \(-0.822834\pi\)
0.999456 0.0329787i \(-0.0104994\pi\)
\(24\) 0 0
\(25\) −4.52056 2.13648i −0.904112 0.427295i
\(26\) 0 0
\(27\) −3.99986 3.99986i −0.769774 0.769774i
\(28\) 0 0
\(29\) 2.82808i 0.525161i 0.964910 + 0.262580i \(0.0845735\pi\)
−0.964910 + 0.262580i \(0.915427\pi\)
\(30\) 0 0
\(31\) 5.94396 + 3.43175i 1.06757 + 0.616360i 0.927516 0.373784i \(-0.121940\pi\)
0.140051 + 0.990144i \(0.455273\pi\)
\(32\) 0 0
\(33\) −5.08074 1.36138i −0.884444 0.236986i
\(34\) 0 0
\(35\) 5.91340 + 0.178099i 0.999547 + 0.0301042i
\(36\) 0 0
\(37\) 7.52551 + 2.01646i 1.23719 + 0.331503i 0.817374 0.576108i \(-0.195429\pi\)
0.419813 + 0.907611i \(0.362096\pi\)
\(38\) 0 0
\(39\) −7.84327 4.52831i −1.25593 0.725110i
\(40\) 0 0
\(41\) 5.56966i 0.869835i 0.900470 + 0.434917i \(0.143222\pi\)
−0.900470 + 0.434917i \(0.856778\pi\)
\(42\) 0 0
\(43\) −1.95176 1.95176i −0.297641 0.297641i 0.542448 0.840089i \(-0.317497\pi\)
−0.840089 + 0.542448i \(0.817497\pi\)
\(44\) 0 0
\(45\) 0.679721 + 2.17538i 0.101327 + 0.324286i
\(46\) 0 0
\(47\) −2.98797 + 11.1512i −0.435840 + 1.62658i 0.303209 + 0.952924i \(0.401942\pi\)
−0.739048 + 0.673652i \(0.764725\pi\)
\(48\) 0 0
\(49\) −6.49732 + 2.60477i −0.928189 + 0.372110i
\(50\) 0 0
\(51\) 4.31017 + 7.46543i 0.603544 + 1.04537i
\(52\) 0 0
\(53\) 9.44513 2.53082i 1.29739 0.347634i 0.456927 0.889504i \(-0.348950\pi\)
0.840462 + 0.541870i \(0.182284\pi\)
\(54\) 0 0
\(55\) 6.14686 + 5.66184i 0.828842 + 0.763442i
\(56\) 0 0
\(57\) 1.22623 1.22623i 0.162418 0.162418i
\(58\) 0 0
\(59\) 0.916152 1.58682i 0.119273 0.206587i −0.800207 0.599724i \(-0.795277\pi\)
0.919480 + 0.393137i \(0.128610\pi\)
\(60\) 0 0
\(61\) −3.43491 + 1.98315i −0.439795 + 0.253916i −0.703511 0.710685i \(-0.748385\pi\)
0.263716 + 0.964600i \(0.415052\pi\)
\(62\) 0 0
\(63\) −1.76643 2.03758i −0.222550 0.256711i
\(64\) 0 0
\(65\) 7.69992 + 12.1556i 0.955058 + 1.50772i
\(66\) 0 0
\(67\) −2.12646 7.93605i −0.259788 0.969542i −0.965364 0.260908i \(-0.915978\pi\)
0.705576 0.708635i \(-0.250689\pi\)
\(68\) 0 0
\(69\) −3.92198 −0.472151
\(70\) 0 0
\(71\) 0.570063 0.0676540 0.0338270 0.999428i \(-0.489230\pi\)
0.0338270 + 0.999428i \(0.489230\pi\)
\(72\) 0 0
\(73\) −0.546036 2.03783i −0.0639087 0.238510i 0.926581 0.376094i \(-0.122733\pi\)
−0.990490 + 0.137584i \(0.956066\pi\)
\(74\) 0 0
\(75\) −1.25774 + 6.92365i −0.145232 + 0.799475i
\(76\) 0 0
\(77\) −9.34513 3.23186i −1.06498 0.368304i
\(78\) 0 0
\(79\) −9.47466 + 5.47020i −1.06598 + 0.615445i −0.927081 0.374861i \(-0.877690\pi\)
−0.138901 + 0.990306i \(0.544357\pi\)
\(80\) 0 0
\(81\) −2.45170 + 4.24647i −0.272411 + 0.471831i
\(82\) 0 0
\(83\) −8.80820 + 8.80820i −0.966826 + 0.966826i −0.999467 0.0326414i \(-0.989608\pi\)
0.0326414 + 0.999467i \(0.489608\pi\)
\(84\) 0 0
\(85\) −0.562062 13.6845i −0.0609642 1.48429i
\(86\) 0 0
\(87\) 3.84459 1.03016i 0.412184 0.110444i
\(88\) 0 0
\(89\) 3.00362 + 5.20243i 0.318383 + 0.551456i 0.980151 0.198253i \(-0.0635267\pi\)
−0.661767 + 0.749709i \(0.730193\pi\)
\(90\) 0 0
\(91\) −14.1020 9.53952i −1.47829 1.00001i
\(92\) 0 0
\(93\) 2.50010 9.33049i 0.259248 0.967527i
\(94\) 0 0
\(95\) −2.62984 + 0.821721i −0.269815 + 0.0843068i
\(96\) 0 0
\(97\) −3.70694 3.70694i −0.376383 0.376383i 0.493413 0.869795i \(-0.335749\pi\)
−0.869795 + 0.493413i \(0.835749\pi\)
\(98\) 0 0
\(99\) 3.80931i 0.382850i
\(100\) 0 0
\(101\) 3.46339 + 1.99959i 0.344620 + 0.198967i 0.662313 0.749227i \(-0.269575\pi\)
−0.317693 + 0.948194i \(0.602908\pi\)
\(102\) 0 0
\(103\) 2.65701 + 0.711944i 0.261803 + 0.0701500i 0.387333 0.921940i \(-0.373396\pi\)
−0.125530 + 0.992090i \(0.540063\pi\)
\(104\) 0 0
\(105\) −1.91190 8.10377i −0.186582 0.790847i
\(106\) 0 0
\(107\) 8.20544 + 2.19864i 0.793250 + 0.212551i 0.632618 0.774464i \(-0.281980\pi\)
0.160632 + 0.987014i \(0.448647\pi\)
\(108\) 0 0
\(109\) 9.96993 + 5.75614i 0.954946 + 0.551339i 0.894614 0.446840i \(-0.147451\pi\)
0.0603324 + 0.998178i \(0.480784\pi\)
\(110\) 0 0
\(111\) 10.9650i 1.04075i
\(112\) 0 0
\(113\) 7.36636 + 7.36636i 0.692969 + 0.692969i 0.962884 0.269915i \(-0.0869957\pi\)
−0.269915 + 0.962884i \(0.586996\pi\)
\(114\) 0 0
\(115\) 5.51973 + 2.89154i 0.514717 + 0.269637i
\(116\) 0 0
\(117\) 1.69756 6.33539i 0.156940 0.585707i
\(118\) 0 0
\(119\) 7.08435 + 14.5748i 0.649421 + 1.33607i
\(120\) 0 0
\(121\) −1.48403 2.57041i −0.134912 0.233674i
\(122\) 0 0
\(123\) 7.57160 2.02880i 0.682708 0.182931i
\(124\) 0 0
\(125\) 6.87470 8.81695i 0.614892 0.788612i
\(126\) 0 0
\(127\) −8.56353 + 8.56353i −0.759890 + 0.759890i −0.976302 0.216412i \(-0.930565\pi\)
0.216412 + 0.976302i \(0.430565\pi\)
\(128\) 0 0
\(129\) −1.94235 + 3.36425i −0.171014 + 0.296206i
\(130\) 0 0
\(131\) 1.01451 0.585728i 0.0886381 0.0511753i −0.455026 0.890478i \(-0.650370\pi\)
0.543664 + 0.839303i \(0.317037\pi\)
\(132\) 0 0
\(133\) 2.46325 2.13546i 0.213591 0.185168i
\(134\) 0 0
\(135\) 10.6853 6.76854i 0.919644 0.582544i
\(136\) 0 0
\(137\) −2.65579 9.91154i −0.226899 0.846800i −0.981635 0.190770i \(-0.938901\pi\)
0.754735 0.656029i \(-0.227765\pi\)
\(138\) 0 0
\(139\) −8.10664 −0.687596 −0.343798 0.939044i \(-0.611714\pi\)
−0.343798 + 0.939044i \(0.611714\pi\)
\(140\) 0 0
\(141\) 16.2478 1.36831
\(142\) 0 0
\(143\) −6.22465 23.2307i −0.520532 1.94265i
\(144\) 0 0
\(145\) −6.17032 1.38466i −0.512417 0.114990i
\(146\) 0 0
\(147\) 5.90774 + 7.88388i 0.487262 + 0.650252i
\(148\) 0 0
\(149\) 5.60127 3.23390i 0.458874 0.264931i −0.252697 0.967546i \(-0.581317\pi\)
0.711571 + 0.702614i \(0.247984\pi\)
\(150\) 0 0
\(151\) 3.00407 5.20320i 0.244468 0.423430i −0.717514 0.696544i \(-0.754720\pi\)
0.961982 + 0.273114i \(0.0880535\pi\)
\(152\) 0 0
\(153\) −4.41441 + 4.41441i −0.356884 + 0.356884i
\(154\) 0 0
\(155\) −10.3976 + 11.2884i −0.835159 + 0.906702i
\(156\) 0 0
\(157\) 6.64043 1.77930i 0.529964 0.142003i 0.0160922 0.999871i \(-0.494877\pi\)
0.513871 + 0.857867i \(0.328211\pi\)
\(158\) 0 0
\(159\) −6.88097 11.9182i −0.545697 0.945174i
\(160\) 0 0
\(161\) −7.35425 0.524203i −0.579596 0.0413130i
\(162\) 0 0
\(163\) −3.38778 + 12.6434i −0.265351 + 0.990305i 0.696683 + 0.717379i \(0.254658\pi\)
−0.962035 + 0.272926i \(0.912008\pi\)
\(164\) 0 0
\(165\) 5.45786 10.4186i 0.424894 0.811091i
\(166\) 0 0
\(167\) −6.33743 6.33743i −0.490405 0.490405i 0.418029 0.908434i \(-0.362721\pi\)
−0.908434 + 0.418029i \(0.862721\pi\)
\(168\) 0 0
\(169\) 28.4097i 2.18536i
\(170\) 0 0
\(171\) 1.08763 + 0.627943i 0.0831731 + 0.0480200i
\(172\) 0 0
\(173\) 1.95448 + 0.523701i 0.148596 + 0.0398162i 0.332350 0.943156i \(-0.392158\pi\)
−0.183754 + 0.982972i \(0.558825\pi\)
\(174\) 0 0
\(175\) −3.28385 + 12.8147i −0.248235 + 0.968700i
\(176\) 0 0
\(177\) −2.49090 0.667435i −0.187228 0.0501675i
\(178\) 0 0
\(179\) 3.54703 + 2.04788i 0.265118 + 0.153066i 0.626667 0.779287i \(-0.284419\pi\)
−0.361549 + 0.932353i \(0.617752\pi\)
\(180\) 0 0
\(181\) 10.6327i 0.790323i −0.918612 0.395162i \(-0.870689\pi\)
0.918612 0.395162i \(-0.129311\pi\)
\(182\) 0 0
\(183\) 3.94716 + 3.94716i 0.291783 + 0.291783i
\(184\) 0 0
\(185\) −8.08409 + 15.4319i −0.594354 + 1.13458i
\(186\) 0 0
\(187\) −5.92479 + 22.1116i −0.433264 + 1.61696i
\(188\) 0 0
\(189\) −8.38563 + 12.3962i −0.609965 + 0.901690i
\(190\) 0 0
\(191\) −6.36951 11.0323i −0.460882 0.798271i 0.538123 0.842866i \(-0.319133\pi\)
−0.999005 + 0.0445955i \(0.985800\pi\)
\(192\) 0 0
\(193\) 11.1345 2.98349i 0.801482 0.214756i 0.165247 0.986252i \(-0.447158\pi\)
0.636234 + 0.771496i \(0.280491\pi\)
\(194\) 0 0
\(195\) 13.7201 14.8954i 0.982514 1.06668i
\(196\) 0 0
\(197\) −4.36293 + 4.36293i −0.310846 + 0.310846i −0.845237 0.534391i \(-0.820541\pi\)
0.534391 + 0.845237i \(0.320541\pi\)
\(198\) 0 0
\(199\) 8.80171 15.2450i 0.623937 1.08069i −0.364809 0.931082i \(-0.618866\pi\)
0.988745 0.149608i \(-0.0478011\pi\)
\(200\) 0 0
\(201\) −10.0140 + 5.78157i −0.706331 + 0.407801i
\(202\) 0 0
\(203\) 7.34683 1.41782i 0.515646 0.0995117i
\(204\) 0 0
\(205\) −12.1519 2.72697i −0.848727 0.190460i
\(206\) 0 0
\(207\) −0.735130 2.74354i −0.0510951 0.190689i
\(208\) 0 0
\(209\) 4.60511 0.318542
\(210\) 0 0
\(211\) 22.0556 1.51837 0.759185 0.650875i \(-0.225598\pi\)
0.759185 + 0.650875i \(0.225598\pi\)
\(212\) 0 0
\(213\) −0.207651 0.774964i −0.0142280 0.0530997i
\(214\) 0 0
\(215\) 5.21398 3.30276i 0.355590 0.225247i
\(216\) 0 0
\(217\) 5.93512 17.1618i 0.402902 1.16502i
\(218\) 0 0
\(219\) −2.57141 + 1.48460i −0.173760 + 0.100320i
\(220\) 0 0
\(221\) −19.7074 + 34.1343i −1.32566 + 2.29612i
\(222\) 0 0
\(223\) −8.37713 + 8.37713i −0.560974 + 0.560974i −0.929584 0.368610i \(-0.879834\pi\)
0.368610 + 0.929584i \(0.379834\pi\)
\(224\) 0 0
\(225\) −5.07906 + 0.417929i −0.338604 + 0.0278620i
\(226\) 0 0
\(227\) 11.3948 3.05323i 0.756300 0.202650i 0.139989 0.990153i \(-0.455293\pi\)
0.616311 + 0.787503i \(0.288626\pi\)
\(228\) 0 0
\(229\) 3.21704 + 5.57208i 0.212588 + 0.368214i 0.952524 0.304464i \(-0.0984774\pi\)
−0.739936 + 0.672678i \(0.765144\pi\)
\(230\) 0 0
\(231\) −0.989449 + 13.8814i −0.0651010 + 0.913326i
\(232\) 0 0
\(233\) −3.93160 + 14.6729i −0.257568 + 0.961256i 0.709076 + 0.705132i \(0.249112\pi\)
−0.966644 + 0.256124i \(0.917554\pi\)
\(234\) 0 0
\(235\) −22.8669 11.9789i −1.49167 0.781420i
\(236\) 0 0
\(237\) 10.8876 + 10.8876i 0.707228 + 0.707228i
\(238\) 0 0
\(239\) 17.0337i 1.10182i 0.834566 + 0.550908i \(0.185718\pi\)
−0.834566 + 0.550908i \(0.814282\pi\)
\(240\) 0 0
\(241\) −0.561340 0.324090i −0.0361591 0.0208765i 0.481811 0.876275i \(-0.339979\pi\)
−0.517971 + 0.855398i \(0.673312\pi\)
\(242\) 0 0
\(243\) −9.72586 2.60604i −0.623914 0.167177i
\(244\) 0 0
\(245\) −2.50195 15.4512i −0.159843 0.987142i
\(246\) 0 0
\(247\) 7.65890 + 2.05220i 0.487324 + 0.130578i
\(248\) 0 0
\(249\) 15.1827 + 8.76572i 0.962163 + 0.555505i
\(250\) 0 0
\(251\) 2.99161i 0.188828i −0.995533 0.0944142i \(-0.969902\pi\)
0.995533 0.0944142i \(-0.0300978\pi\)
\(252\) 0 0
\(253\) −7.36449 7.36449i −0.463002 0.463002i
\(254\) 0 0
\(255\) −18.3984 + 5.74879i −1.15215 + 0.360003i
\(256\) 0 0
\(257\) 1.22356 4.56640i 0.0763237 0.284844i −0.917206 0.398412i \(-0.869561\pi\)
0.993530 + 0.113568i \(0.0362280\pi\)
\(258\) 0 0
\(259\) 1.46556 20.5608i 0.0910652 1.27759i
\(260\) 0 0
\(261\) 1.44125 + 2.49632i 0.0892112 + 0.154518i
\(262\) 0 0
\(263\) −24.0481 + 6.44367i −1.48287 + 0.397334i −0.907323 0.420435i \(-0.861877\pi\)
−0.575547 + 0.817769i \(0.695211\pi\)
\(264\) 0 0
\(265\) 0.897304 + 21.8466i 0.0551210 + 1.34202i
\(266\) 0 0
\(267\) 5.97828 5.97828i 0.365865 0.365865i
\(268\) 0 0
\(269\) 0.153942 0.266636i 0.00938603 0.0162571i −0.861294 0.508107i \(-0.830346\pi\)
0.870680 + 0.491849i \(0.163679\pi\)
\(270\) 0 0
\(271\) 24.2593 14.0061i 1.47365 0.850812i 0.474089 0.880477i \(-0.342777\pi\)
0.999560 + 0.0296653i \(0.00944415\pi\)
\(272\) 0 0
\(273\) −7.83160 + 22.6456i −0.473990 + 1.37057i
\(274\) 0 0
\(275\) −15.3626 + 10.6392i −0.926400 + 0.641565i
\(276\) 0 0
\(277\) 7.32354 + 27.3318i 0.440029 + 1.64221i 0.728737 + 0.684793i \(0.240108\pi\)
−0.288708 + 0.957417i \(0.593226\pi\)
\(278\) 0 0
\(279\) 6.99558 0.418814
\(280\) 0 0
\(281\) −2.60283 −0.155272 −0.0776358 0.996982i \(-0.524737\pi\)
−0.0776358 + 0.996982i \(0.524737\pi\)
\(282\) 0 0
\(283\) 4.67227 + 17.4371i 0.277738 + 1.03653i 0.953985 + 0.299856i \(0.0969386\pi\)
−0.676247 + 0.736675i \(0.736395\pi\)
\(284\) 0 0
\(285\) 2.07502 + 3.27578i 0.122914 + 0.194040i
\(286\) 0 0
\(287\) 14.4690 2.79228i 0.854076 0.164823i
\(288\) 0 0
\(289\) 17.7675 10.2581i 1.04515 0.603415i
\(290\) 0 0
\(291\) −3.68906 + 6.38965i −0.216257 + 0.374568i
\(292\) 0 0
\(293\) −17.2326 + 17.2326i −1.00674 + 1.00674i −0.00676223 + 0.999977i \(0.502152\pi\)
−0.999977 + 0.00676223i \(0.997848\pi\)
\(294\) 0 0
\(295\) 3.01358 + 2.77579i 0.175457 + 0.161613i
\(296\) 0 0
\(297\) −20.4207 + 5.47172i −1.18493 + 0.317502i
\(298\) 0 0
\(299\) −8.96625 15.5300i −0.518532 0.898123i
\(300\) 0 0
\(301\) −4.09183 + 6.04882i −0.235849 + 0.348648i
\(302\) 0 0
\(303\) 1.45674 5.43663i 0.0836876 0.312327i
\(304\) 0 0
\(305\) −2.64507 8.46528i −0.151456 0.484720i
\(306\) 0 0
\(307\) 11.3375 + 11.3375i 0.647063 + 0.647063i 0.952282 0.305219i \(-0.0987298\pi\)
−0.305219 + 0.952282i \(0.598730\pi\)
\(308\) 0 0
\(309\) 3.87137i 0.220235i
\(310\) 0 0
\(311\) 17.9899 + 10.3865i 1.02011 + 0.588962i 0.914136 0.405407i \(-0.132870\pi\)
0.105975 + 0.994369i \(0.466204\pi\)
\(312\) 0 0
\(313\) −7.34380 1.96776i −0.415096 0.111225i 0.0452258 0.998977i \(-0.485599\pi\)
−0.460322 + 0.887752i \(0.652266\pi\)
\(314\) 0 0
\(315\) 5.31047 2.85639i 0.299211 0.160940i
\(316\) 0 0
\(317\) 1.65329 + 0.442998i 0.0928581 + 0.0248812i 0.304949 0.952369i \(-0.401361\pi\)
−0.212091 + 0.977250i \(0.568027\pi\)
\(318\) 0 0
\(319\) 9.15356 + 5.28481i 0.512501 + 0.295892i
\(320\) 0 0
\(321\) 11.9557i 0.667300i
\(322\) 0 0
\(323\) −5.33661 5.33661i −0.296937 0.296937i
\(324\) 0 0
\(325\) −30.2912 + 10.8482i −1.68026 + 0.601750i
\(326\) 0 0
\(327\) 4.19347 15.6502i 0.231899 0.865460i
\(328\) 0 0
\(329\) 30.4669 + 2.17165i 1.67969 + 0.119727i
\(330\) 0 0
\(331\) −13.0087 22.5317i −0.715023 1.23846i −0.962951 0.269678i \(-0.913083\pi\)
0.247928 0.968779i \(-0.420250\pi\)
\(332\) 0 0
\(333\) 7.67034 2.05526i 0.420332 0.112628i
\(334\) 0 0
\(335\) 18.3561 0.753938i 1.00290 0.0411921i
\(336\) 0 0
\(337\) −22.0521 + 22.0521i −1.20125 + 1.20125i −0.227468 + 0.973785i \(0.573045\pi\)
−0.973785 + 0.227468i \(0.926955\pi\)
\(338\) 0 0
\(339\) 7.33083 12.6974i 0.398156 0.689627i
\(340\) 0 0
\(341\) 22.2149 12.8258i 1.20300 0.694554i
\(342\) 0 0
\(343\) 10.0241 + 15.5730i 0.541249 + 0.840862i
\(344\) 0 0
\(345\) 1.92025 8.55700i 0.103383 0.460693i
\(346\) 0 0
\(347\) −5.71867 21.3424i −0.306994 1.14572i −0.931215 0.364470i \(-0.881250\pi\)
0.624221 0.781248i \(-0.285417\pi\)
\(348\) 0 0
\(349\) −21.6246 −1.15754 −0.578770 0.815491i \(-0.696467\pi\)
−0.578770 + 0.815491i \(0.696467\pi\)
\(350\) 0 0
\(351\) −36.4008 −1.94293
\(352\) 0 0
\(353\) −0.979086 3.65400i −0.0521115 0.194483i 0.934963 0.354746i \(-0.115433\pi\)
−0.987074 + 0.160263i \(0.948766\pi\)
\(354\) 0 0
\(355\) −0.279109 + 1.24377i −0.0148136 + 0.0660123i
\(356\) 0 0
\(357\) 17.2330 14.9397i 0.912066 0.790695i
\(358\) 0 0
\(359\) −3.68781 + 2.12916i −0.194635 + 0.112373i −0.594151 0.804354i \(-0.702512\pi\)
0.399516 + 0.916726i \(0.369178\pi\)
\(360\) 0 0
\(361\) 8.74087 15.1396i 0.460046 0.796823i
\(362\) 0 0
\(363\) −2.95374 + 2.95374i −0.155031 + 0.155031i
\(364\) 0 0
\(365\) 4.71351 0.193598i 0.246716 0.0101334i
\(366\) 0 0
\(367\) −22.7211 + 6.08811i −1.18603 + 0.317797i −0.797317 0.603561i \(-0.793748\pi\)
−0.388716 + 0.921357i \(0.627081\pi\)
\(368\) 0 0
\(369\) 2.83842 + 4.91629i 0.147762 + 0.255932i
\(370\) 0 0
\(371\) −11.3098 23.2679i −0.587176 1.20801i
\(372\) 0 0
\(373\) −2.02436 + 7.55500i −0.104817 + 0.391183i −0.998324 0.0578658i \(-0.981570\pi\)
0.893507 + 0.449049i \(0.148237\pi\)
\(374\) 0 0
\(375\) −14.4903 6.13406i −0.748274 0.316761i
\(376\) 0 0
\(377\) 12.8685 + 12.8685i 0.662760 + 0.662760i
\(378\) 0 0
\(379\) 24.6449i 1.26592i 0.774184 + 0.632961i \(0.218161\pi\)
−0.774184 + 0.632961i \(0.781839\pi\)
\(380\) 0 0
\(381\) 14.7609 + 8.52223i 0.756225 + 0.436607i
\(382\) 0 0
\(383\) 22.6611 + 6.07201i 1.15793 + 0.310265i 0.786137 0.618052i \(-0.212078\pi\)
0.371789 + 0.928317i \(0.378745\pi\)
\(384\) 0 0
\(385\) 11.6268 18.8069i 0.592555 0.958489i
\(386\) 0 0
\(387\) −2.71747 0.728143i −0.138137 0.0370136i
\(388\) 0 0
\(389\) −20.5933 11.8895i −1.04412 0.602823i −0.123123 0.992391i \(-0.539291\pi\)
−0.920998 + 0.389568i \(0.872624\pi\)
\(390\) 0 0
\(391\) 17.0686i 0.863198i
\(392\) 0 0
\(393\) −1.16581 1.16581i −0.0588071 0.0588071i
\(394\) 0 0
\(395\) −7.29601 23.3501i −0.367102 1.17487i
\(396\) 0 0
\(397\) 5.98647 22.3418i 0.300452 1.12130i −0.636338 0.771410i \(-0.719552\pi\)
0.936790 0.349892i \(-0.113782\pi\)
\(398\) 0 0
\(399\) −3.80028 2.57077i −0.190252 0.128699i
\(400\) 0 0
\(401\) −18.8264 32.6083i −0.940147 1.62838i −0.765187 0.643808i \(-0.777354\pi\)
−0.174960 0.984576i \(-0.555980\pi\)
\(402\) 0 0
\(403\) 42.6619 11.4312i 2.12514 0.569430i
\(404\) 0 0
\(405\) −8.06461 7.42827i −0.400733 0.369114i
\(406\) 0 0
\(407\) 20.5895 20.5895i 1.02058 1.02058i
\(408\) 0 0
\(409\) 3.10540 5.37871i 0.153552 0.265960i −0.778979 0.627050i \(-0.784262\pi\)
0.932531 + 0.361090i \(0.117595\pi\)
\(410\) 0 0
\(411\) −12.5067 + 7.22076i −0.616911 + 0.356174i
\(412\) 0 0
\(413\) −4.58158 1.58446i −0.225445 0.0779663i
\(414\) 0 0
\(415\) −14.9052 23.5304i −0.731667 1.15506i
\(416\) 0 0
\(417\) 2.95292 + 11.0205i 0.144605 + 0.539674i
\(418\) 0 0
\(419\) 12.9069 0.630541 0.315270 0.949002i \(-0.397905\pi\)
0.315270 + 0.949002i \(0.397905\pi\)
\(420\) 0 0
\(421\) −25.0983 −1.22322 −0.611608 0.791161i \(-0.709477\pi\)
−0.611608 + 0.791161i \(0.709477\pi\)
\(422\) 0 0
\(423\) 3.04547 + 11.3658i 0.148076 + 0.552626i
\(424\) 0 0
\(425\) 30.1321 + 5.47376i 1.46162 + 0.265516i
\(426\) 0 0
\(427\) 6.87390 + 7.92904i 0.332651 + 0.383713i
\(428\) 0 0
\(429\) −29.3133 + 16.9241i −1.41526 + 0.817101i
\(430\) 0 0
\(431\) −14.1918 + 24.5809i −0.683593 + 1.18402i 0.290283 + 0.956941i \(0.406250\pi\)
−0.973877 + 0.227078i \(0.927083\pi\)
\(432\) 0 0
\(433\) 20.7667 20.7667i 0.997984 0.997984i −0.00201350 0.999998i \(-0.500641\pi\)
0.999998 + 0.00201350i \(0.000640917\pi\)
\(434\) 0 0
\(435\) 0.365243 + 8.89254i 0.0175121 + 0.426365i
\(436\) 0 0
\(437\) 3.31669 0.888706i 0.158659 0.0425125i
\(438\) 0 0
\(439\) 2.84512 + 4.92790i 0.135790 + 0.235196i 0.925899 0.377771i \(-0.123309\pi\)
−0.790109 + 0.612967i \(0.789976\pi\)
\(440\) 0 0
\(441\) −4.40768 + 5.61039i −0.209890 + 0.267161i
\(442\) 0 0
\(443\) −3.38845 + 12.6459i −0.160990 + 0.600824i 0.837527 + 0.546395i \(0.184000\pi\)
−0.998518 + 0.0544285i \(0.982666\pi\)
\(444\) 0 0
\(445\) −12.8213 + 4.00616i −0.607788 + 0.189910i
\(446\) 0 0
\(447\) −6.43660 6.43660i −0.304441 0.304441i
\(448\) 0 0
\(449\) 11.4035i 0.538163i −0.963117 0.269081i \(-0.913280\pi\)
0.963117 0.269081i \(-0.0867200\pi\)
\(450\) 0 0
\(451\) 18.0272 + 10.4080i 0.848866 + 0.490093i
\(452\) 0 0
\(453\) −8.16769 2.18853i −0.383751 0.102826i
\(454\) 0 0
\(455\) 27.7179 26.0971i 1.29943 1.22345i
\(456\) 0 0
\(457\) 2.45668 + 0.658264i 0.114918 + 0.0307923i 0.315820 0.948819i \(-0.397720\pi\)
−0.200901 + 0.979611i \(0.564387\pi\)
\(458\) 0 0
\(459\) 30.0054 + 17.3236i 1.40053 + 0.808597i
\(460\) 0 0
\(461\) 28.3975i 1.32260i −0.750121 0.661301i \(-0.770005\pi\)
0.750121 0.661301i \(-0.229995\pi\)
\(462\) 0 0
\(463\) −8.29144 8.29144i −0.385336 0.385336i 0.487684 0.873020i \(-0.337842\pi\)
−0.873020 + 0.487684i \(0.837842\pi\)
\(464\) 0 0
\(465\) 19.1333 + 10.0230i 0.887283 + 0.464808i
\(466\) 0 0
\(467\) 0.448458 1.67367i 0.0207522 0.0774481i −0.954773 0.297335i \(-0.903902\pi\)
0.975525 + 0.219887i \(0.0705688\pi\)
\(468\) 0 0
\(469\) −19.5503 + 9.50279i −0.902751 + 0.438798i
\(470\) 0 0
\(471\) −4.83768 8.37912i −0.222909 0.386089i
\(472\) 0 0
\(473\) −9.96446 + 2.66997i −0.458166 + 0.122765i
\(474\) 0 0
\(475\) −0.505238 6.14012i −0.0231819 0.281728i
\(476\) 0 0
\(477\) 7.04738 7.04738i 0.322677 0.322677i
\(478\) 0 0
\(479\) 10.7131 18.5557i 0.489496 0.847831i −0.510431 0.859919i \(-0.670514\pi\)
0.999927 + 0.0120872i \(0.00384756\pi\)
\(480\) 0 0
\(481\) 43.4184 25.0676i 1.97971 1.14299i
\(482\) 0 0
\(483\) 1.96624 + 10.1886i 0.0894669 + 0.463597i
\(484\) 0 0
\(485\) 9.90279 6.27287i 0.449663 0.284836i
\(486\) 0 0
\(487\) 10.0631 + 37.5562i 0.456005 + 1.70183i 0.685117 + 0.728433i \(0.259751\pi\)
−0.229112 + 0.973400i \(0.573582\pi\)
\(488\) 0 0
\(489\) 18.4219 0.833067
\(490\) 0 0
\(491\) −5.54467 −0.250228 −0.125114 0.992142i \(-0.539930\pi\)
−0.125114 + 0.992142i \(0.539930\pi\)
\(492\) 0 0
\(493\) −4.48329 16.7318i −0.201917 0.753565i
\(494\) 0 0
\(495\) 8.31118 + 1.86508i 0.373560 + 0.0838292i
\(496\) 0 0
\(497\) −0.285794 1.48092i −0.0128196 0.0664283i
\(498\) 0 0
\(499\) −4.28573 + 2.47437i −0.191856 + 0.110768i −0.592851 0.805312i \(-0.701998\pi\)
0.400995 + 0.916080i \(0.368664\pi\)
\(500\) 0 0
\(501\) −6.30686 + 10.9238i −0.281770 + 0.488040i
\(502\) 0 0
\(503\) −14.3450 + 14.3450i −0.639614 + 0.639614i −0.950460 0.310846i \(-0.899388\pi\)
0.310846 + 0.950460i \(0.399388\pi\)
\(504\) 0 0
\(505\) −6.05844 + 6.57743i −0.269597 + 0.292692i
\(506\) 0 0
\(507\) −38.6212 + 10.3485i −1.71523 + 0.459594i
\(508\) 0 0
\(509\) −17.6679 30.6017i −0.783116 1.35640i −0.930118 0.367260i \(-0.880296\pi\)
0.147002 0.989136i \(-0.453038\pi\)
\(510\) 0 0
\(511\) −5.02017 + 2.44015i −0.222079 + 0.107946i
\(512\) 0 0
\(513\) 1.80396 6.73249i 0.0796470 0.297247i
\(514\) 0 0
\(515\) −2.85423 + 5.44851i −0.125772 + 0.240090i
\(516\) 0 0
\(517\) 30.5093 + 30.5093i 1.34180 + 1.34180i
\(518\) 0 0
\(519\) 2.84775i 0.125002i
\(520\) 0 0
\(521\) −19.0681 11.0090i −0.835388 0.482312i 0.0203058 0.999794i \(-0.493536\pi\)
−0.855694 + 0.517482i \(0.826869\pi\)
\(522\) 0 0
\(523\) −25.4125 6.80926i −1.11121 0.297748i −0.343889 0.939010i \(-0.611744\pi\)
−0.767322 + 0.641262i \(0.778411\pi\)
\(524\) 0 0
\(525\) 18.6169 0.203700i 0.812510 0.00889020i
\(526\) 0 0
\(527\) −40.6067 10.8805i −1.76886 0.473964i
\(528\) 0 0
\(529\) 13.1933 + 7.61716i 0.573622 + 0.331181i
\(530\) 0 0
\(531\) 1.86757i 0.0810454i
\(532\) 0 0
\(533\) 25.3434 + 25.3434i 1.09774 + 1.09774i
\(534\) 0 0
\(535\) −8.81449 + 16.8262i −0.381084 + 0.727460i
\(536\) 0 0
\(537\) 1.49192 5.56793i 0.0643812 0.240274i
\(538\) 0 0
\(539\) −3.71071 + 25.8972i −0.159831 + 1.11547i
\(540\) 0 0
\(541\) 3.33567 + 5.77755i 0.143412 + 0.248396i 0.928779 0.370633i \(-0.120859\pi\)
−0.785368 + 0.619030i \(0.787526\pi\)
\(542\) 0 0
\(543\) −14.4545 + 3.87307i −0.620302 + 0.166209i
\(544\) 0 0
\(545\) −17.4402 + 18.9342i −0.747056 + 0.811052i
\(546\) 0 0
\(547\) 8.18996 8.18996i 0.350177 0.350177i −0.509998 0.860175i \(-0.670354\pi\)
0.860175 + 0.509998i \(0.170354\pi\)
\(548\) 0 0
\(549\) −2.02131 + 3.50101i −0.0862674 + 0.149420i
\(550\) 0 0
\(551\) −3.01782 + 1.74234i −0.128564 + 0.0742262i
\(552\) 0 0
\(553\) 18.9606 + 21.8710i 0.806286 + 0.930050i
\(554\) 0 0
\(555\) 23.9235 + 5.36858i 1.01549 + 0.227884i
\(556\) 0 0
\(557\) −9.76062 36.4271i −0.413571 1.54347i −0.787681 0.616083i \(-0.788719\pi\)
0.374111 0.927384i \(-0.377948\pi\)
\(558\) 0 0
\(559\) −17.7621 −0.751255
\(560\) 0 0
\(561\) 32.2175 1.36023
\(562\) 0 0
\(563\) −9.97457 37.2256i −0.420378 1.56887i −0.773814 0.633413i \(-0.781653\pi\)
0.353436 0.935459i \(-0.385013\pi\)
\(564\) 0 0
\(565\) −19.6786 + 12.4653i −0.827886 + 0.524420i
\(566\) 0 0
\(567\) 12.2607 + 4.24016i 0.514901 + 0.178070i
\(568\) 0 0
\(569\) −10.4114 + 6.01105i −0.436470 + 0.251996i −0.702099 0.712079i \(-0.747754\pi\)
0.265629 + 0.964075i \(0.414420\pi\)
\(570\) 0 0
\(571\) −15.7498 + 27.2794i −0.659107 + 1.14161i 0.321740 + 0.946828i \(0.395732\pi\)
−0.980847 + 0.194779i \(0.937601\pi\)
\(572\) 0 0
\(573\) −12.6776 + 12.6776i −0.529614 + 0.529614i
\(574\) 0 0
\(575\) −9.01130 + 10.6273i −0.375797 + 0.443187i
\(576\) 0 0
\(577\) 11.7414 3.14609i 0.488800 0.130973i −0.00599739 0.999982i \(-0.501909\pi\)
0.494797 + 0.869009i \(0.335242\pi\)
\(578\) 0 0
\(579\) −8.11174 14.0499i −0.337112 0.583896i
\(580\) 0 0
\(581\) 27.2980 + 18.4662i 1.13251 + 0.766108i
\(582\) 0 0
\(583\) 9.45864 35.3001i 0.391737 1.46198i
\(584\) 0 0
\(585\) 12.9914 + 6.80563i 0.537130 + 0.281378i
\(586\) 0 0
\(587\) −19.1797 19.1797i −0.791631 0.791631i 0.190128 0.981759i \(-0.439110\pi\)
−0.981759 + 0.190128i \(0.939110\pi\)
\(588\) 0 0
\(589\) 8.45702i 0.348465i
\(590\) 0 0
\(591\) 7.52038 + 4.34189i 0.309347 + 0.178602i
\(592\) 0 0
\(593\) −29.4676 7.89583i −1.21009 0.324243i −0.403293 0.915071i \(-0.632134\pi\)
−0.806797 + 0.590828i \(0.798801\pi\)
\(594\) 0 0
\(595\) −35.2680 + 8.32068i −1.44585 + 0.341115i
\(596\) 0 0
\(597\) −23.9307 6.41223i −0.979420 0.262435i
\(598\) 0 0
\(599\) 13.8406 + 7.99090i 0.565513 + 0.326499i 0.755355 0.655315i \(-0.227464\pi\)
−0.189842 + 0.981815i \(0.560798\pi\)
\(600\) 0 0
\(601\) 35.9834i 1.46779i −0.679260 0.733897i \(-0.737699\pi\)
0.679260 0.733897i \(-0.262301\pi\)
\(602\) 0 0
\(603\) −5.92139 5.92139i −0.241138 0.241138i
\(604\) 0 0
\(605\) 6.33473 1.97936i 0.257544 0.0804723i
\(606\) 0 0
\(607\) −0.457120 + 1.70600i −0.0185539 + 0.0692442i −0.974582 0.224031i \(-0.928078\pi\)
0.956028 + 0.293275i \(0.0947451\pi\)
\(608\) 0 0
\(609\) −4.60360 9.47110i −0.186547 0.383788i
\(610\) 0 0
\(611\) 37.1450 + 64.3371i 1.50273 + 2.60280i
\(612\) 0 0
\(613\) 3.65567 0.979533i 0.147651 0.0395629i −0.184237 0.982882i \(-0.558981\pi\)
0.331887 + 0.943319i \(0.392315\pi\)
\(614\) 0 0
\(615\) 0.719316 + 17.5131i 0.0290056 + 0.706196i
\(616\) 0 0
\(617\) −2.94869 + 2.94869i −0.118710 + 0.118710i −0.763966 0.645256i \(-0.776751\pi\)
0.645256 + 0.763966i \(0.276751\pi\)
\(618\) 0 0
\(619\) −15.6267 + 27.0662i −0.628088 + 1.08788i 0.359847 + 0.933011i \(0.382829\pi\)
−0.987935 + 0.154869i \(0.950504\pi\)
\(620\) 0 0
\(621\) −13.6515 + 7.88169i −0.547816 + 0.316282i
\(622\) 0 0
\(623\) 12.0091 10.4110i 0.481136 0.417110i
\(624\) 0 0
\(625\) 15.8709 + 19.3161i 0.634838 + 0.772645i
\(626\) 0 0
\(627\) −1.67746 6.26036i −0.0669912 0.250015i
\(628\) 0 0
\(629\) −47.7201 −1.90272
\(630\) 0 0
\(631\) −19.8658 −0.790844 −0.395422 0.918500i \(-0.629402\pi\)
−0.395422 + 0.918500i \(0.629402\pi\)
\(632\) 0 0
\(633\) −8.03397 29.9832i −0.319322 1.19173i
\(634\) 0 0
\(635\) −14.4912 22.8768i −0.575064 0.907836i
\(636\) 0 0
\(637\) −17.7121 + 41.4169i −0.701779 + 1.64100i
\(638\) 0 0
\(639\) 0.503189 0.290517i 0.0199059 0.0114927i
\(640\) 0 0
\(641\) −20.8732 + 36.1535i −0.824443 + 1.42798i 0.0779013 + 0.996961i \(0.475178\pi\)
−0.902344 + 0.431016i \(0.858155\pi\)
\(642\) 0 0
\(643\) 4.58193 4.58193i 0.180694 0.180694i −0.610964 0.791658i \(-0.709218\pi\)
0.791658 + 0.610964i \(0.209218\pi\)
\(644\) 0 0
\(645\) −6.38915 5.88501i −0.251572 0.231722i
\(646\) 0 0
\(647\) 8.45072 2.26436i 0.332232 0.0890213i −0.0888468 0.996045i \(-0.528318\pi\)
0.421079 + 0.907024i \(0.361652\pi\)
\(648\) 0 0
\(649\) −3.42401 5.93056i −0.134404 0.232795i
\(650\) 0 0
\(651\) −25.4923 1.81707i −0.999123 0.0712165i
\(652\) 0 0
\(653\) −0.895030 + 3.34030i −0.0350252 + 0.130716i −0.981225 0.192868i \(-0.938221\pi\)
0.946199 + 0.323584i \(0.104888\pi\)
\(654\) 0 0
\(655\) 0.781229 + 2.50024i 0.0305251 + 0.0976926i
\(656\) 0 0
\(657\) −1.52051 1.52051i −0.0593206 0.0593206i
\(658\) 0 0
\(659\) 1.86861i 0.0727907i −0.999337 0.0363953i \(-0.988412\pi\)
0.999337 0.0363953i \(-0.0115876\pi\)
\(660\) 0 0
\(661\) 40.7798 + 23.5442i 1.58615 + 0.915764i 0.993933 + 0.109984i \(0.0350801\pi\)
0.592216 + 0.805779i \(0.298253\pi\)
\(662\) 0 0
\(663\) 53.5820 + 14.3573i 2.08095 + 0.557590i
\(664\) 0 0
\(665\) 3.45312 + 6.41987i 0.133906 + 0.248952i
\(666\) 0 0
\(667\) 7.61246 + 2.03975i 0.294755 + 0.0789795i
\(668\) 0 0
\(669\) 14.4396 + 8.33672i 0.558268 + 0.322316i
\(670\) 0 0
\(671\) 14.8236i 0.572257i
\(672\) 0 0
\(673\) −3.20914 3.20914i −0.123703 0.123703i 0.642545 0.766248i \(-0.277879\pi\)
−0.766248 + 0.642545i \(0.777879\pi\)
\(674\) 0 0
\(675\) 9.53601 + 26.6272i 0.367041 + 1.02488i
\(676\) 0 0
\(677\) −4.04037 + 15.0789i −0.155284 + 0.579528i 0.843797 + 0.536663i \(0.180315\pi\)
−0.999081 + 0.0428651i \(0.986351\pi\)
\(678\) 0 0
\(679\) −7.77153 + 11.4884i −0.298244 + 0.440884i
\(680\) 0 0
\(681\) −8.30135 14.3784i −0.318109 0.550980i
\(682\) 0 0
\(683\) 22.6701 6.07444i 0.867448 0.232432i 0.202464 0.979290i \(-0.435105\pi\)
0.664984 + 0.746858i \(0.268438\pi\)
\(684\) 0 0
\(685\) 22.9254 0.941613i 0.875933 0.0359772i
\(686\) 0 0
\(687\) 6.40306 6.40306i 0.244292 0.244292i
\(688\) 0 0
\(689\) 31.4619 54.4937i 1.19860 2.07604i
\(690\) 0 0
\(691\) 12.8955 7.44520i 0.490566 0.283229i −0.234243 0.972178i \(-0.575261\pi\)
0.724809 + 0.688949i \(0.241928\pi\)
\(692\) 0 0
\(693\) −9.89590 + 1.90975i −0.375914 + 0.0725455i
\(694\) 0 0
\(695\) 3.96910 17.6871i 0.150557 0.670910i
\(696\) 0 0
\(697\) −8.82945 32.9520i −0.334439 1.24814i
\(698\) 0 0
\(699\) 21.3791 0.808630
\(700\) 0 0
\(701\) 23.9673 0.905232 0.452616 0.891705i \(-0.350491\pi\)
0.452616 + 0.891705i \(0.350491\pi\)
\(702\) 0 0
\(703\) 2.48462 + 9.27274i 0.0937093 + 0.349728i
\(704\) 0 0
\(705\) −7.95512 + 35.4496i −0.299607 + 1.33511i
\(706\) 0 0
\(707\) 3.45824 9.99973i 0.130061 0.376079i
\(708\) 0 0
\(709\) −0.893033 + 0.515593i −0.0335386 + 0.0193635i −0.516676 0.856181i \(-0.672831\pi\)
0.483137 + 0.875545i \(0.339497\pi\)
\(710\) 0 0
\(711\) −5.57547 + 9.65699i −0.209096 + 0.362166i
\(712\) 0 0
\(713\) 13.5245 13.5245i 0.506495 0.506495i
\(714\) 0 0
\(715\) 53.7326 2.20696i 2.00949 0.0825356i
\(716\) 0 0
\(717\) 23.1562 6.20468i 0.864784 0.231718i
\(718\) 0 0
\(719\) 12.0529 + 20.8762i 0.449497 + 0.778552i 0.998353 0.0573650i \(-0.0182699\pi\)
−0.548856 + 0.835917i \(0.684937\pi\)
\(720\) 0 0
\(721\) 0.517440 7.25936i 0.0192705 0.270353i
\(722\) 0 0
\(723\) −0.236106 + 0.881160i −0.00878088 + 0.0327707i
\(724\) 0 0
\(725\) 6.04212 12.7845i 0.224399 0.474804i
\(726\) 0 0
\(727\) 0.650788 + 0.650788i 0.0241364 + 0.0241364i 0.719072 0.694936i \(-0.244567\pi\)
−0.694936 + 0.719072i \(0.744567\pi\)
\(728\) 0 0
\(729\) 28.8812i 1.06967i
\(730\) 0 0
\(731\) 14.6414 + 8.45320i 0.541531 + 0.312653i
\(732\) 0 0
\(733\) −13.2928 3.56180i −0.490981 0.131558i 0.00483005 0.999988i \(-0.498463\pi\)
−0.495811 + 0.868430i \(0.665129\pi\)
\(734\) 0 0
\(735\) −20.0936 + 9.02950i −0.741164 + 0.333058i
\(736\) 0 0
\(737\) −29.6601 7.94739i −1.09254 0.292746i
\(738\) 0 0
\(739\) −12.8236 7.40373i −0.471725 0.272351i 0.245236 0.969463i \(-0.421134\pi\)
−0.716962 + 0.697113i \(0.754468\pi\)
\(740\) 0 0
\(741\) 11.1593i 0.409948i
\(742\) 0 0
\(743\) −32.2779 32.2779i −1.18416 1.18416i −0.978656 0.205505i \(-0.934116\pi\)
−0.205505 0.978656i \(-0.565884\pi\)
\(744\) 0 0
\(745\) 4.31329 + 13.8043i 0.158027 + 0.505749i
\(746\) 0 0
\(747\) −3.28607 + 12.2638i −0.120231 + 0.448708i
\(748\) 0 0
\(749\) 1.59797 22.4185i 0.0583885 0.819154i
\(750\) 0 0
\(751\) −0.115708 0.200412i −0.00422223 0.00731312i 0.863907 0.503652i \(-0.168011\pi\)
−0.868129 + 0.496339i \(0.834677\pi\)
\(752\) 0 0
\(753\) −4.06690 + 1.08972i −0.148206 + 0.0397117i
\(754\) 0 0
\(755\) 9.88155 + 9.10184i 0.359626 + 0.331250i
\(756\) 0 0
\(757\) −33.8884 + 33.8884i −1.23169 + 1.23169i −0.268382 + 0.963313i \(0.586489\pi\)
−0.963313 + 0.268382i \(0.913511\pi\)
\(758\) 0 0
\(759\) −7.32897 + 12.6942i −0.266025 + 0.460769i
\(760\) 0 0
\(761\) −17.9623 + 10.3705i −0.651132 + 0.375931i −0.788890 0.614535i \(-0.789344\pi\)
0.137758 + 0.990466i \(0.456010\pi\)
\(762\) 0 0
\(763\) 9.95511 28.7858i 0.360399 1.04212i
\(764\) 0 0
\(765\) −7.47004 11.7927i −0.270080 0.426367i
\(766\) 0 0
\(767\) −3.05172 11.3892i −0.110191 0.411239i
\(768\) 0 0
\(769\) −52.0182 −1.87583 −0.937913 0.346872i \(-0.887244\pi\)
−0.937913 + 0.346872i \(0.887244\pi\)
\(770\) 0 0
\(771\) −6.65342 −0.239617
\(772\) 0 0
\(773\) −12.8398 47.9189i −0.461817 1.72352i −0.667232 0.744850i \(-0.732521\pi\)
0.205415 0.978675i \(-0.434146\pi\)
\(774\) 0 0
\(775\) −19.5382 28.2126i −0.701833 1.01342i
\(776\) 0 0
\(777\) −28.4850 + 5.49716i −1.02189 + 0.197210i
\(778\) 0 0
\(779\) −5.94335 + 3.43139i −0.212943 + 0.122942i
\(780\) 0 0
\(781\) 1.06527 1.84511i 0.0381184 0.0660230i
\(782\) 0 0
\(783\) 11.3119 11.3119i 0.404255 0.404255i
\(784\) 0 0
\(785\) 0.630852 + 15.3593i 0.0225161 + 0.548197i
\(786\) 0 0
\(787\) −21.9988 + 5.89457i −0.784174 + 0.210119i −0.628624 0.777709i \(-0.716382\pi\)
−0.155550 + 0.987828i \(0.549715\pi\)
\(788\) 0 0
\(789\) 17.5195 + 30.3447i 0.623712 + 1.08030i
\(790\) 0 0
\(791\) 15.4434 22.8295i 0.549105 0.811724i
\(792\) 0 0
\(793\) −6.60590 + 24.6535i −0.234582 + 0.875473i
\(794\) 0 0
\(795\) 29.3722 9.17767i 1.04172 0.325498i
\(796\) 0 0
\(797\) 19.2056 + 19.2056i 0.680298 + 0.680298i 0.960067 0.279770i \(-0.0902581\pi\)
−0.279770 + 0.960067i \(0.590258\pi\)
\(798\) 0 0
\(799\) 70.7112i 2.50158i
\(800\) 0 0
\(801\) 5.30255 + 3.06143i 0.187356 + 0.108170i
\(802\) 0 0
\(803\) −7.61617 2.04075i −0.268769 0.0720164i
\(804\) 0 0
\(805\) 4.74444 15.7889i 0.167219 0.556485i
\(806\) 0 0
\(807\) −0.418550 0.112150i −0.0147337 0.00394787i
\(808\) 0 0
\(809\) −31.8924 18.4131i −1.12128 0.647370i −0.179550 0.983749i \(-0.557464\pi\)
−0.941727 + 0.336379i \(0.890798\pi\)
\(810\) 0 0
\(811\) 15.8918i 0.558036i −0.960286 0.279018i \(-0.909991\pi\)
0.960286 0.279018i \(-0.0900089\pi\)
\(812\) 0 0
\(813\) −27.8772 27.8772i −0.977695 0.977695i
\(814\) 0 0
\(815\) −25.9267 13.5818i −0.908172 0.475751i
\(816\) 0 0
\(817\) 0.880259 3.28517i 0.0307963 0.114934i
\(818\) 0 0
\(819\) −17.3092 1.23378i −0.604834 0.0431119i
\(820\) 0 0
\(821\) 2.11251 + 3.65898i 0.0737272 + 0.127699i 0.900532 0.434790i \(-0.143177\pi\)
−0.826805 + 0.562489i \(0.809844\pi\)
\(822\) 0 0
\(823\) 50.1654 13.4418i 1.74866 0.468551i 0.764319 0.644839i \(-0.223075\pi\)
0.984339 + 0.176287i \(0.0564087\pi\)
\(824\) 0 0
\(825\) 20.0592 + 17.0091i 0.698373 + 0.592180i
\(826\) 0 0
\(827\) 8.41389 8.41389i 0.292580 0.292580i −0.545519 0.838099i \(-0.683667\pi\)
0.838099 + 0.545519i \(0.183667\pi\)
\(828\) 0 0
\(829\) −15.4011 + 26.6755i −0.534903 + 0.926479i 0.464265 + 0.885696i \(0.346318\pi\)
−0.999168 + 0.0407825i \(0.987015\pi\)
\(830\) 0 0
\(831\) 34.4882 19.9118i 1.19638 0.690732i
\(832\) 0 0
\(833\) 34.3110 25.7107i 1.18881 0.890825i
\(834\) 0 0
\(835\) 16.9299 10.7242i 0.585884 0.371125i
\(836\) 0 0
\(837\) −10.0485 37.5015i −0.347327 1.29624i
\(838\) 0 0
\(839\) −34.0803 −1.17658 −0.588292 0.808649i \(-0.700199\pi\)
−0.588292 + 0.808649i \(0.700199\pi\)
\(840\) 0 0
\(841\) 21.0020 0.724206
\(842\) 0 0
\(843\) 0.948106 + 3.53838i 0.0326545 + 0.121868i
\(844\) 0 0
\(845\) 61.9844 + 13.9097i 2.13233 + 0.478509i
\(846\) 0 0
\(847\) −5.93346 + 5.14388i −0.203876 + 0.176746i
\(848\) 0 0
\(849\) 22.0028 12.7033i 0.755134 0.435977i
\(850\) 0 0
\(851\) 10.8556 18.8024i 0.372124 0.644537i
\(852\) 0 0
\(853\) 31.8639 31.8639i 1.09100 1.09100i 0.0955790 0.995422i \(-0.469530\pi\)
0.995422 0.0955790i \(-0.0304703\pi\)
\(854\) 0 0
\(855\) −1.90257 + 2.06555i −0.0650664 + 0.0706403i
\(856\) 0 0
\(857\) −4.50274 + 1.20650i −0.153811 + 0.0412134i −0.334903 0.942253i \(-0.608703\pi\)
0.181092 + 0.983466i \(0.442037\pi\)
\(858\) 0 0
\(859\) −5.08567 8.80865i −0.173521 0.300547i 0.766127 0.642689i \(-0.222181\pi\)
−0.939648 + 0.342142i \(0.888848\pi\)
\(860\) 0 0
\(861\) −9.06640 18.6525i −0.308982 0.635677i
\(862\) 0 0
\(863\) −6.63845 + 24.7750i −0.225976 + 0.843352i 0.756036 + 0.654530i \(0.227134\pi\)
−0.982011 + 0.188822i \(0.939533\pi\)
\(864\) 0 0
\(865\) −2.09955 + 4.00788i −0.0713868 + 0.136272i
\(866\) 0 0
\(867\) −20.4172 20.4172i −0.693403 0.693403i
\(868\) 0 0
\(869\) 40.8885i 1.38705i
\(870\) 0 0
\(871\) −45.7870 26.4351i −1.55143 0.895720i
\(872\) 0 0
\(873\) −5.16123 1.38295i −0.174681 0.0468056i
\(874\) 0 0
\(875\) −26.3514 13.4389i −0.890839 0.454319i
\(876\) 0 0
\(877\) 34.3421 + 9.20193i 1.15965 + 0.310727i 0.786826 0.617175i \(-0.211723\pi\)
0.372824 + 0.927902i \(0.378390\pi\)
\(878\) 0 0
\(879\) 29.7038 + 17.1495i 1.00188 + 0.578438i
\(880\) 0 0
\(881\) 6.87670i 0.231682i 0.993268 + 0.115841i \(0.0369563\pi\)
−0.993268 + 0.115841i \(0.963044\pi\)
\(882\) 0 0
\(883\) 3.32125 + 3.32125i 0.111769 + 0.111769i 0.760779 0.649011i \(-0.224817\pi\)
−0.649011 + 0.760779i \(0.724817\pi\)
\(884\) 0 0
\(885\) 2.67579 5.10788i 0.0899457 0.171700i
\(886\) 0 0
\(887\) −0.237821 + 0.887559i −0.00798524 + 0.0298013i −0.969804 0.243888i \(-0.921577\pi\)
0.961818 + 0.273689i \(0.0882439\pi\)
\(888\) 0 0
\(889\) 26.5397 + 17.9533i 0.890113 + 0.602133i
\(890\) 0 0
\(891\) 9.16296 + 15.8707i 0.306971 + 0.531689i
\(892\) 0 0
\(893\) −13.7403 + 3.68169i −0.459801 + 0.123203i
\(894\) 0 0
\(895\) −6.20475 + 6.73628i −0.207402 + 0.225169i
\(896\) 0 0
\(897\) −17.8460 + 17.8460i −0.595861 + 0.595861i
\(898\) 0 0
\(899\) −9.70525 + 16.8100i −0.323688 + 0.560644i
\(900\) 0 0
\(901\) −51.8685 + 29.9463i −1.72799 + 0.997656i
\(902\) 0 0
\(903\) 9.71348 + 3.35925i 0.323245 + 0.111789i
\(904\) 0 0
\(905\) 23.1985 + 5.20590i 0.771145 + 0.173050i
\(906\) 0 0
\(907\) −6.47600 24.1687i −0.215032 0.802510i −0.986155 0.165824i \(-0.946972\pi\)
0.771123 0.636686i \(-0.219695\pi\)
\(908\) 0 0
\(909\) 4.07614 0.135197
\(910\) 0 0
\(911\) 32.2439 1.06829 0.534144 0.845394i \(-0.320634\pi\)
0.534144 + 0.845394i \(0.320634\pi\)
\(912\) 0 0
\(913\) 12.0494 + 44.9691i 0.398778 + 1.48826i
\(914\) 0 0
\(915\) −10.5445 + 6.67937i −0.348591 + 0.220813i
\(916\) 0 0
\(917\) −2.03023 2.34186i −0.0670440 0.0773352i
\(918\) 0 0
\(919\) 41.7517 24.1054i 1.37726 0.795163i 0.385433 0.922736i \(-0.374052\pi\)
0.991829 + 0.127573i \(0.0407187\pi\)
\(920\) 0 0
\(921\) 11.2828 19.5423i 0.371780 0.643942i
\(922\) 0 0
\(923\) 2.59393 2.59393i 0.0853803 0.0853803i
\(924\) 0 0
\(925\) −29.7114 25.1936i −0.976906 0.828360i
\(926\) 0 0
\(927\) 2.70815 0.725645i 0.0889472 0.0238333i
\(928\) 0 0
\(929\) 18.1318 + 31.4051i 0.594884 + 1.03037i 0.993563 + 0.113279i \(0.0361354\pi\)
−0.398679 + 0.917090i \(0.630531\pi\)
\(930\) 0 0
\(931\) −6.78245 5.32848i −0.222286 0.174634i
\(932\) 0 0
\(933\) 7.56674 28.2395i 0.247724 0.924518i
\(934\) 0 0
\(935\) −45.3424 23.7529i −1.48286 0.776801i
\(936\) 0 0
\(937\) 0.255843 + 0.255843i 0.00835802 + 0.00835802i 0.711273 0.702915i \(-0.248119\pi\)
−0.702915 + 0.711273i \(0.748119\pi\)
\(938\) 0 0
\(939\) 10.7002i 0.349188i
\(940\) 0 0
\(941\) −20.3365 11.7413i −0.662949 0.382754i 0.130450 0.991455i \(-0.458358\pi\)
−0.793400 + 0.608701i \(0.791691\pi\)
\(942\) 0 0
\(943\) 14.9921 + 4.01712i 0.488210 + 0.130815i
\(944\) 0 0
\(945\) −22.9404 24.3651i −0.746251 0.792598i
\(946\) 0 0
\(947\) 40.2893 + 10.7955i 1.30923 + 0.350806i 0.844932 0.534874i \(-0.179641\pi\)
0.464296 + 0.885680i \(0.346308\pi\)
\(948\) 0 0
\(949\) −11.7573 6.78806i −0.381657 0.220350i
\(950\) 0 0
\(951\) 2.40891i 0.0781143i
\(952\) 0 0
\(953\) 5.97702 + 5.97702i 0.193615 + 0.193615i 0.797256 0.603641i \(-0.206284\pi\)
−0.603641 + 0.797256i \(0.706284\pi\)
\(954\) 0 0
\(955\) 27.1890 8.49549i 0.879814 0.274908i
\(956\) 0 0
\(957\) 3.85009 14.3687i 0.124456 0.464475i
\(958\) 0 0
\(959\) −24.4169 + 11.8683i −0.788464 + 0.383247i
\(960\) 0 0
\(961\) 8.05379 + 13.9496i 0.259800 + 0.449986i
\(962\) 0 0
\(963\) 8.36335 2.24095i 0.269505 0.0722137i
\(964\) 0 0
\(965\) 1.05780 + 25.7542i 0.0340518 + 0.829056i
\(966\) 0 0
\(967\) 34.9020 34.9020i 1.12237 1.12237i 0.130988 0.991384i \(-0.458185\pi\)
0.991384 0.130988i \(-0.0418148\pi\)
\(968\) 0 0
\(969\) −5.31088 + 9.19871i −0.170610 + 0.295505i
\(970\) 0 0
\(971\) −40.5558 + 23.4149i −1.30150 + 0.751419i −0.980661 0.195716i \(-0.937297\pi\)
−0.320836 + 0.947135i \(0.603964\pi\)
\(972\) 0 0
\(973\) 4.06417 + 21.0596i 0.130291 + 0.675139i
\(974\) 0 0
\(975\) 25.7813 + 37.2275i 0.825664 + 1.19223i
\(976\) 0 0
\(977\) 1.29406 + 4.82951i 0.0414008 + 0.154510i 0.983532 0.180735i \(-0.0578477\pi\)
−0.942131 + 0.335245i \(0.891181\pi\)
\(978\) 0 0
\(979\) 22.4514 0.717550
\(980\) 0 0
\(981\) 11.7338 0.374633
\(982\) 0 0
\(983\) 13.9719 + 52.1438i 0.445634 + 1.66313i 0.714258 + 0.699883i \(0.246764\pi\)
−0.268624 + 0.963245i \(0.586569\pi\)
\(984\) 0 0
\(985\) −7.38293 11.6552i −0.235240 0.371366i
\(986\) 0 0
\(987\) −8.14565 42.2089i −0.259279 1.34352i
\(988\) 0 0
\(989\) −6.66135 + 3.84593i −0.211819 + 0.122294i
\(990\) 0 0
\(991\) 0.608356 1.05370i 0.0193251 0.0334720i −0.856201 0.516643i \(-0.827182\pi\)
0.875526 + 0.483171i \(0.160515\pi\)
\(992\) 0 0
\(993\) −25.8919 + 25.8919i −0.821656 + 0.821656i
\(994\) 0 0
\(995\) 28.9522 + 26.6678i 0.917848 + 0.845425i
\(996\) 0 0
\(997\) −19.8462 + 5.31777i −0.628536 + 0.168416i −0.559005 0.829164i \(-0.688817\pi\)
−0.0695305 + 0.997580i \(0.522150\pi\)
\(998\) 0 0
\(999\) −22.0355 38.1665i −0.697171 1.20754i
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 280.2.bo.a.73.4 yes 48
4.3 odd 2 560.2.ci.e.353.9 48
5.2 odd 4 inner 280.2.bo.a.17.4 48
7.5 odd 6 inner 280.2.bo.a.33.4 yes 48
20.7 even 4 560.2.ci.e.17.9 48
28.19 even 6 560.2.ci.e.33.9 48
35.12 even 12 inner 280.2.bo.a.257.4 yes 48
140.47 odd 12 560.2.ci.e.257.9 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.bo.a.17.4 48 5.2 odd 4 inner
280.2.bo.a.33.4 yes 48 7.5 odd 6 inner
280.2.bo.a.73.4 yes 48 1.1 even 1 trivial
280.2.bo.a.257.4 yes 48 35.12 even 12 inner
560.2.ci.e.17.9 48 20.7 even 4
560.2.ci.e.33.9 48 28.19 even 6
560.2.ci.e.257.9 48 140.47 odd 12
560.2.ci.e.353.9 48 4.3 odd 2