Properties

Label 469.1.bl.a.83.1
Level $469$
Weight $1$
Character 469.83
Analytic conductor $0.234$
Analytic rank $0$
Dimension $20$
Projective image $D_{33}$
CM discriminant -7
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [469,1,Mod(6,469)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(469, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([33, 40]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("469.6");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 469 = 7 \cdot 67 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 469.bl (of order \(66\), degree \(20\), 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: \(0.234061490925\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{33}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{33} - \cdots)\)

Embedding invariants

Embedding label 83.1
Root \(0.580057 - 0.814576i\) of defining polynomial
Character \(\chi\) \(=\) 469.83
Dual form 469.1.bl.a.356.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.88431 - 0.363170i) q^{2} +(2.49035 + 0.996987i) q^{4} +(-0.327068 - 0.945001i) q^{7} +(-2.71616 - 1.74557i) q^{8} +(-0.654861 + 0.755750i) q^{9} +O(q^{10})\) \(q+(-1.88431 - 0.363170i) q^{2} +(2.49035 + 0.996987i) q^{4} +(-0.327068 - 0.945001i) q^{7} +(-2.71616 - 1.74557i) q^{8} +(-0.654861 + 0.755750i) q^{9} +(0.462997 - 1.90850i) q^{11} +(0.273100 + 1.89945i) q^{14} +(2.54272 + 2.42448i) q^{16} +(1.50842 - 1.18624i) q^{18} +(-1.56554 + 3.42805i) q^{22} +(0.0552004 - 0.0775182i) q^{23} +(0.841254 - 0.540641i) q^{25} +(0.127639 - 2.67947i) q^{28} +(0.327068 - 0.566498i) q^{29} +(-2.03794 - 2.86188i) q^{32} +(-2.38431 + 1.22920i) q^{36} +(-0.723734 - 1.25354i) q^{37} +(0.223734 - 1.55610i) q^{43} +(3.05578 - 4.29124i) q^{44} +(-0.132167 + 0.126021i) q^{46} +(-0.786053 + 0.618159i) q^{49} +(-1.78153 + 0.713215i) q^{50} +(0.283341 + 1.97068i) q^{53} +(-0.761197 + 3.13770i) q^{56} +(-0.822032 + 0.948676i) q^{58} +(0.928368 + 0.371662i) q^{63} +(1.34125 + 2.93694i) q^{64} +(0.723734 + 0.690079i) q^{67} +(0.437742 + 0.175245i) q^{71} +(3.09792 - 0.909632i) q^{72} +(0.908487 + 2.62490i) q^{74} +(-1.95496 + 0.186677i) q^{77} +(0.0395325 + 0.829889i) q^{79} +(-0.142315 - 0.989821i) q^{81} +(-0.986715 + 2.85093i) q^{86} +(-4.58900 + 4.37560i) q^{88} +(0.214753 - 0.138014i) q^{92} +(1.70566 - 0.879330i) q^{98} +(1.13915 + 1.59971i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} + 3 q^{4} + q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} + 3 q^{4} + q^{7} - 8 q^{8} - 2 q^{9} - q^{11} - 4 q^{14} + 5 q^{16} + 2 q^{18} - 7 q^{22} - q^{23} - 2 q^{25} - 8 q^{28} - q^{29} + 6 q^{32} - 8 q^{36} - q^{37} - 9 q^{43} - 3 q^{44} - 13 q^{46} + q^{49} + 2 q^{50} + 2 q^{53} - 7 q^{56} + 4 q^{58} + q^{63} + 8 q^{64} + q^{67} - q^{71} + 14 q^{72} - 2 q^{74} - q^{77} + 2 q^{79} - 2 q^{81} - 2 q^{86} - 4 q^{88} - 5 q^{92} + 2 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/469\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{33}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.88431 0.363170i −1.88431 0.363170i −0.888835 0.458227i \(-0.848485\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(3\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(4\) 2.49035 + 0.996987i 2.49035 + 0.996987i
\(5\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(6\) 0 0
\(7\) −0.327068 0.945001i −0.327068 0.945001i
\(8\) −2.71616 1.74557i −2.71616 1.74557i
\(9\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(10\) 0 0
\(11\) 0.462997 1.90850i 0.462997 1.90850i 0.0475819 0.998867i \(-0.484848\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(12\) 0 0
\(13\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(14\) 0.273100 + 1.89945i 0.273100 + 1.89945i
\(15\) 0 0
\(16\) 2.54272 + 2.42448i 2.54272 + 2.42448i
\(17\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(18\) 1.50842 1.18624i 1.50842 1.18624i
\(19\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.56554 + 3.42805i −1.56554 + 3.42805i
\(23\) 0.0552004 0.0775182i 0.0552004 0.0775182i −0.786053 0.618159i \(-0.787879\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(24\) 0 0
\(25\) 0.841254 0.540641i 0.841254 0.540641i
\(26\) 0 0
\(27\) 0 0
\(28\) 0.127639 2.67947i 0.127639 2.67947i
\(29\) 0.327068 0.566498i 0.327068 0.566498i −0.654861 0.755750i \(-0.727273\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(30\) 0 0
\(31\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(32\) −2.03794 2.86188i −2.03794 2.86188i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −2.38431 + 1.22920i −2.38431 + 1.22920i
\(37\) −0.723734 1.25354i −0.723734 1.25354i −0.959493 0.281733i \(-0.909091\pi\)
0.235759 0.971812i \(-0.424242\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(42\) 0 0
\(43\) 0.223734 1.55610i 0.223734 1.55610i −0.500000 0.866025i \(-0.666667\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(44\) 3.05578 4.29124i 3.05578 4.29124i
\(45\) 0 0
\(46\) −0.132167 + 0.126021i −0.132167 + 0.126021i
\(47\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(48\) 0 0
\(49\) −0.786053 + 0.618159i −0.786053 + 0.618159i
\(50\) −1.78153 + 0.713215i −1.78153 + 0.713215i
\(51\) 0 0
\(52\) 0 0
\(53\) 0.283341 + 1.97068i 0.283341 + 1.97068i 0.235759 + 0.971812i \(0.424242\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.761197 + 3.13770i −0.761197 + 3.13770i
\(57\) 0 0
\(58\) −0.822032 + 0.948676i −0.822032 + 0.948676i
\(59\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(60\) 0 0
\(61\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(62\) 0 0
\(63\) 0.928368 + 0.371662i 0.928368 + 0.371662i
\(64\) 1.34125 + 2.93694i 1.34125 + 2.93694i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.723734 + 0.690079i 0.723734 + 0.690079i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.437742 + 0.175245i 0.437742 + 0.175245i 0.580057 0.814576i \(-0.303030\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(72\) 3.09792 0.909632i 3.09792 0.909632i
\(73\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(74\) 0.908487 + 2.62490i 0.908487 + 2.62490i
\(75\) 0 0
\(76\) 0 0
\(77\) −1.95496 + 0.186677i −1.95496 + 0.186677i
\(78\) 0 0
\(79\) 0.0395325 + 0.829889i 0.0395325 + 0.829889i 0.928368 + 0.371662i \(0.121212\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(80\) 0 0
\(81\) −0.142315 0.989821i −0.142315 0.989821i
\(82\) 0 0
\(83\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.986715 + 2.85093i −0.986715 + 2.85093i
\(87\) 0 0
\(88\) −4.58900 + 4.37560i −4.58900 + 4.37560i
\(89\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.214753 0.138014i 0.214753 0.138014i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) 1.70566 0.879330i 1.70566 0.879330i
\(99\) 1.13915 + 1.59971i 1.13915 + 1.59971i
\(100\) 2.63403 0.507668i 2.63403 0.507668i
\(101\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(102\) 0 0
\(103\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.181791 3.81627i 0.181791 3.81627i
\(107\) 1.50842 + 0.442913i 1.50842 + 0.442913i 0.928368 0.371662i \(-0.121212\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(108\) 0 0
\(109\) −1.49547 + 0.961081i −1.49547 + 0.961081i −0.500000 + 0.866025i \(0.666667\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.45949 3.19584i 1.45949 3.19584i
\(113\) −0.723734 + 0.690079i −0.723734 + 0.690079i −0.959493 0.281733i \(-0.909091\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.37931 1.08470i 1.37931 1.08470i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.53917 1.30903i −2.53917 1.30903i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −1.61435 1.03748i −1.61435 1.03748i
\(127\) 0.0930932 + 0.268975i 0.0930932 + 0.268975i 0.981929 0.189251i \(-0.0606061\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(128\) −0.632425 2.60689i −0.632425 2.60689i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.11312 1.56316i −1.11312 1.56316i
\(135\) 0 0
\(136\) 0 0
\(137\) 0.481929 + 1.05528i 0.481929 + 1.05528i 0.981929 + 0.189251i \(0.0606061\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.761197 0.489191i −0.761197 0.489191i
\(143\) 0 0
\(144\) −3.49743 + 0.333964i −3.49743 + 0.333964i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −0.552586 3.84332i −0.552586 3.84332i
\(149\) 0.654861 + 0.755750i 0.654861 + 0.755750i 0.981929 0.189251i \(-0.0606061\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(150\) 0 0
\(151\) −1.21590 + 0.486774i −1.21590 + 0.486774i −0.888835 0.458227i \(-0.848485\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 3.75155 + 0.358230i 3.75155 + 0.358230i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(158\) 0.226900 1.57812i 0.226900 1.57812i
\(159\) 0 0
\(160\) 0 0
\(161\) −0.0913090 0.0268107i −0.0913090 0.0268107i
\(162\) −0.0913090 + 1.91681i −0.0913090 + 1.91681i
\(163\) −0.580057 + 1.00469i −0.580057 + 1.00469i 0.415415 + 0.909632i \(0.363636\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(168\) 0 0
\(169\) 0.580057 + 0.814576i 0.580057 + 0.814576i
\(170\) 0 0
\(171\) 0 0
\(172\) 2.10859 3.65219i 2.10859 3.65219i
\(173\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(174\) 0 0
\(175\) −0.786053 0.618159i −0.786053 0.618159i
\(176\) 5.80439 3.73026i 5.80439 3.73026i
\(177\) 0 0
\(178\) 0 0
\(179\) 0.771316 1.68895i 0.771316 1.68895i 0.0475819 0.998867i \(-0.484848\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(180\) 0 0
\(181\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −0.285247 + 0.114196i −0.285247 + 0.114196i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.469383 + 0.0448206i −0.469383 + 0.0448206i −0.327068 0.945001i \(-0.606061\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(192\) 0 0
\(193\) 0.975950 + 0.627205i 0.975950 + 0.627205i 0.928368 0.371662i \(-0.121212\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.57385 + 0.755750i −2.57385 + 0.755750i
\(197\) 1.34378 + 0.537970i 1.34378 + 0.537970i 0.928368 0.371662i \(-0.121212\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(198\) −1.56554 3.42805i −1.56554 3.42805i
\(199\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(200\) −3.22871 −3.22871
\(201\) 0 0
\(202\) 0 0
\(203\) −0.642315 0.123796i −0.642315 0.123796i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.0224357 + 0.0924813i 0.0224357 + 0.0924813i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0.995472 0.0950560i 0.995472 0.0950560i 0.415415 0.909632i \(-0.363636\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(212\) −1.25912 + 5.19017i −1.25912 + 5.19017i
\(213\) 0 0
\(214\) −2.68148 1.38240i −2.68148 1.38240i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 3.16697 1.26786i 3.16697 1.26786i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(224\) −2.03794 + 2.86188i −2.03794 + 2.86188i
\(225\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(226\) 1.61435 1.03748i 1.61435 1.03748i
\(227\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(228\) 0 0
\(229\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.87723 + 0.967781i −1.87723 + 0.967781i
\(233\) −0.379436 0.532843i −0.379436 0.532843i 0.580057 0.814576i \(-0.303030\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.0475819 + 0.0824143i −0.0475819 + 0.0824143i −0.888835 0.458227i \(-0.848485\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(240\) 0 0
\(241\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(242\) 4.30917 + 3.38877i 4.30917 + 3.38877i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(252\) 1.94142 + 1.85114i 1.94142 + 1.85114i
\(253\) −0.122386 0.141241i −0.122386 0.141241i
\(254\) −0.0777324 0.540641i −0.0777324 0.540641i
\(255\) 0 0
\(256\) 0.0913090 + 1.91681i 0.0913090 + 1.91681i
\(257\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(258\) 0 0
\(259\) −0.947890 + 1.09392i −0.947890 + 1.09392i
\(260\) 0 0
\(261\) 0.213947 + 0.618159i 0.213947 + 0.618159i
\(262\) 0 0
\(263\) −1.78153 + 0.523103i −1.78153 + 0.523103i −0.995472 0.0950560i \(-0.969697\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1.11435 + 2.44009i 1.11435 + 2.44009i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −0.524856 2.16349i −0.524856 2.16349i
\(275\) −0.642315 1.85585i −0.642315 1.85585i
\(276\) 0 0
\(277\) 0.857685 0.989821i 0.857685 0.989821i −0.142315 0.989821i \(-0.545455\pi\)
1.00000 \(0\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.49547 0.770969i −1.49547 0.770969i −0.500000 0.866025i \(-0.666667\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(282\) 0 0
\(283\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(284\) 0.915415 + 0.872846i 0.915415 + 0.872846i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 3.49743 + 0.333964i 3.49743 + 0.333964i
\(289\) 0.723734 0.690079i 0.723734 0.690079i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.222372 + 4.66816i −0.222372 + 4.66816i
\(297\) 0 0
\(298\) −0.959493 1.66189i −0.959493 1.66189i
\(299\) 0 0
\(300\) 0 0
\(301\) −1.54370 + 0.297523i −1.54370 + 0.297523i
\(302\) 2.46792 0.475652i 2.46792 0.475652i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(308\) −5.05467 1.48418i −5.05467 1.48418i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(312\) 0 0
\(313\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −0.728939 + 2.10613i −0.728939 + 2.10613i
\(317\) 0.223734 0.175946i 0.223734 0.175946i −0.500000 0.866025i \(-0.666667\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(318\) 0 0
\(319\) −0.929730 0.886496i −0.929730 0.886496i
\(320\) 0 0
\(321\) 0 0
\(322\) 0.162317 + 0.0836804i 0.162317 + 0.0836804i
\(323\) 0 0
\(324\) 0.632425 2.60689i 0.632425 2.60689i
\(325\) 0 0
\(326\) 1.45788 1.68248i 1.45788 1.68248i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.07701 + 0.431171i 1.07701 + 0.431171i 0.841254 0.540641i \(-0.181818\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(332\) 0 0
\(333\) 1.42131 + 0.273935i 1.42131 + 0.273935i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.82318 + 0.351390i 1.82318 + 0.351390i 0.981929 0.189251i \(-0.0606061\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(338\) −0.797176 1.74557i −0.797176 1.74557i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(344\) −3.32399 + 3.83609i −3.32399 + 3.83609i
\(345\) 0 0
\(346\) 0 0
\(347\) −0.0845850 1.77566i −0.0845850 1.77566i −0.500000 0.866025i \(-0.666667\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(348\) 0 0
\(349\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(350\) 1.25667 + 1.45027i 1.25667 + 1.45027i
\(351\) 0 0
\(352\) −6.40545 + 2.56436i −6.40545 + 2.56436i
\(353\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −2.06677 + 2.90237i −2.06677 + 2.90237i
\(359\) 0.252989 1.75958i 0.252989 1.75958i −0.327068 0.945001i \(-0.606061\pi\)
0.580057 0.814576i \(-0.303030\pi\)
\(360\) 0 0
\(361\) −0.786053 0.618159i −0.786053 0.618159i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(368\) 0.328301 0.0632748i 0.328301 0.0632748i
\(369\) 0 0
\(370\) 0 0
\(371\) 1.76962 0.912303i 1.76962 0.912303i
\(372\) 0 0
\(373\) −0.928368 + 1.60798i −0.928368 + 1.60798i −0.142315 + 0.989821i \(0.545455\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.975950 1.37053i 0.975950 1.37053i 0.0475819 0.998867i \(-0.484848\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0.900739 + 0.0860101i 0.900739 + 0.0860101i
\(383\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.61121 1.53628i −1.61121 1.53628i
\(387\) 1.02951 + 1.18812i 1.02951 + 1.18812i
\(388\) 0 0
\(389\) 1.76962 + 0.912303i 1.76962 + 0.912303i 0.928368 + 0.371662i \(0.121212\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 3.21409 0.306908i 3.21409 0.306908i
\(393\) 0 0
\(394\) −2.33673 1.50172i −2.33673 1.50172i
\(395\) 0 0
\(396\) 1.24199 + 5.11956i 1.24199 + 5.11956i
\(397\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 3.44985 + 0.664903i 3.44985 + 0.664903i
\(401\) 0.830830 0.830830 0.415415 0.909632i \(-0.363636\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 1.16536 + 0.466540i 1.16536 + 0.466540i
\(407\) −2.72747 + 0.800859i −2.72747 + 0.800859i
\(408\) 0 0
\(409\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −0.00868932 0.182411i −0.00868932 0.182411i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(420\) 0 0
\(421\) 0.581419 1.67990i 0.581419 1.67990i −0.142315 0.989821i \(-0.545455\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(422\) −1.91030 0.182411i −1.91030 0.182411i
\(423\) 0 0
\(424\) 2.67036 5.84727i 2.67036 5.84727i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 3.31493 + 2.60689i 3.31493 + 2.60689i
\(429\) 0 0
\(430\) 0 0
\(431\) −0.981929 + 1.70075i −0.981929 + 1.70075i −0.327068 + 0.945001i \(0.606061\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(432\) 0 0
\(433\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.68244 + 0.902466i −4.68244 + 0.902466i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) 0.0475819 0.998867i 0.0475819 0.998867i
\(442\) 0 0
\(443\) 0.223734 + 0.175946i 0.223734 + 0.175946i 0.723734 0.690079i \(-0.242424\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 2.33673 2.22806i 2.33673 2.22806i
\(449\) 0.283341 + 0.0270558i 0.283341 + 0.0270558i 0.235759 0.971812i \(-0.424242\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(450\) 0.627639 1.81344i 0.627639 1.81344i
\(451\) 0 0
\(452\) −2.49035 + 0.996987i −2.49035 + 0.996987i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.0224357 + 0.470984i 0.0224357 + 0.470984i 0.981929 + 0.189251i \(0.0606061\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(462\) 0 0
\(463\) −0.419102 1.72756i −0.419102 1.72756i −0.654861 0.755750i \(-0.727273\pi\)
0.235759 0.971812i \(-0.424242\pi\)
\(464\) 2.20511 0.647478i 2.20511 0.647478i
\(465\) 0 0
\(466\) 0.521461 + 1.14184i 0.521461 + 1.14184i
\(467\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(468\) 0 0
\(469\) 0.415415 0.909632i 0.415415 0.909632i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.86624 1.14747i −2.86624 1.14747i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.67489 1.07639i −1.67489 1.07639i
\(478\) 0.119589 0.138014i 0.119589 0.138014i
\(479\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −5.01834 5.79147i −5.01834 5.79147i
\(485\) 0 0
\(486\) 0 0
\(487\) −1.13779 + 0.894765i −1.13779 + 0.894765i −0.995472 0.0950560i \(-0.969697\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0.345139 0.755750i 0.345139 0.755750i −0.654861 0.755750i \(-0.727273\pi\)
1.00000 \(0\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.0224357 0.470984i 0.0224357 0.470984i
\(498\) 0 0
\(499\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(504\) −1.87283 2.63003i −1.87283 2.63003i
\(505\) 0 0
\(506\) 0.179318 + 0.310588i 0.179318 + 0.310588i
\(507\) 0 0
\(508\) −0.0363298 + 0.762656i −0.0363298 + 0.762656i
\(509\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.142315 0.989821i 0.142315 0.989821i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 2.18340 1.71704i 2.18340 1.71704i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(522\) −0.178645 1.24250i −0.178645 1.24250i
\(523\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 3.54692 0.338689i 3.54692 0.338689i
\(527\) 0 0
\(528\) 0 0
\(529\) 0.324106 + 0.936443i 0.324106 + 0.936443i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −0.761197 3.13770i −0.761197 3.13770i
\(537\) 0 0
\(538\) 0 0
\(539\) 0.815816 + 1.78639i 0.815816 + 1.78639i
\(540\) 0 0
\(541\) 0.627639 0.184291i 0.627639 0.184291i 0.0475819 0.998867i \(-0.484848\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.0224357 0.0924813i 0.0224357 0.0924813i −0.959493 0.281733i \(-0.909091\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(548\) 0.148076 + 3.10849i 0.148076 + 3.10849i
\(549\) 0 0
\(550\) 0.536330 + 3.73026i 0.536330 + 3.73026i
\(551\) 0 0
\(552\) 0 0
\(553\) 0.771316 0.308788i 0.771316 0.308788i
\(554\) −1.97562 + 1.55364i −1.97562 + 1.55364i
\(555\) 0 0
\(556\) 0 0
\(557\) 0.601300 0.573338i 0.601300 0.573338i −0.327068 0.945001i \(-0.606061\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 2.53794 + 1.99585i 2.53794 + 1.99585i
\(563\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.888835 + 0.458227i −0.888835 + 0.458227i
\(568\) −0.883075 1.24010i −0.883075 1.24010i
\(569\) −1.88431 + 0.363170i −1.88431 + 0.363170i −0.995472 0.0950560i \(-0.969697\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(570\) 0 0
\(571\) 1.13915 + 1.59971i 1.13915 + 1.59971i 0.723734 + 0.690079i \(0.242424\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.00452808 0.0950560i 0.00452808 0.0950560i
\(576\) −3.09792 0.909632i −3.09792 0.909632i
\(577\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(578\) −1.61435 + 1.03748i −1.61435 + 1.03748i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 3.89223 + 0.371662i 3.89223 + 0.371662i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.19894 4.94209i 1.19894 4.94209i
\(593\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.877362 + 2.53497i 0.877362 + 2.53497i
\(597\) 0 0
\(598\) 0 0
\(599\) −1.84833 0.739959i −1.84833 0.739959i −0.959493 0.281733i \(-0.909091\pi\)
−0.888835 0.458227i \(-0.848485\pi\)
\(600\) 0 0
\(601\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(602\) 3.01685 3.01685
\(603\) −0.995472 + 0.0950560i −0.995472 + 0.0950560i
\(604\) −3.51334 −3.51334
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −0.469383 + 0.0448206i −0.469383 + 0.0448206i −0.327068 0.945001i \(-0.606061\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 5.63586 + 2.90549i 5.63586 + 2.90549i
\(617\) 0.223734 + 1.55610i 0.223734 + 1.55610i 0.723734 + 0.690079i \(0.242424\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(618\) 0 0
\(619\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.415415 0.909632i 0.415415 0.909632i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −0.0311250 + 0.653395i −0.0311250 + 0.653395i 0.928368 + 0.371662i \(0.121212\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(632\) 1.34125 2.32312i 1.34125 2.32312i
\(633\) 0 0
\(634\) −0.485482 + 0.250283i −0.485482 + 0.250283i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 1.42995 + 2.00808i 1.42995 + 2.00808i
\(639\) −0.419102 + 0.216062i −0.419102 + 0.216062i
\(640\) 0 0
\(641\) 0.142315 0.246497i 0.142315 0.246497i −0.786053 0.618159i \(-0.787879\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(642\) 0 0
\(643\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(644\) −0.200662 0.157802i −0.200662 0.157802i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(648\) −1.34125 + 2.93694i −1.34125 + 2.93694i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −2.44621 + 1.92372i −2.44621 + 1.92372i
\(653\) 0.771316 0.308788i 0.771316 0.308788i 0.0475819 0.998867i \(-0.484848\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −0.419102 + 1.72756i −0.419102 + 1.72756i 0.235759 + 0.971812i \(0.424242\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(660\) 0 0
\(661\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(662\) −1.87283 1.20360i −1.87283 1.20360i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −2.57870 1.03236i −2.57870 1.03236i
\(667\) −0.0258596 0.0566247i −0.0258596 0.0566247i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.601300 + 1.31666i 0.601300 + 1.31666i 0.928368 + 0.371662i \(0.121212\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(674\) −3.30782 1.32425i −3.30782 1.32425i
\(675\) 0 0
\(676\) 0.632425 + 2.60689i 0.632425 + 2.60689i
\(677\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −0.419102 0.216062i −0.419102 0.216062i 0.235759 0.971812i \(-0.424242\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.38884 1.32425i −1.38884 1.32425i
\(687\) 0 0
\(688\) 4.34164 3.41430i 4.34164 3.41430i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(692\) 0 0
\(693\) 1.13915 1.59971i 1.13915 1.59971i
\(694\) −0.485482 + 3.37660i −0.485482 + 3.37660i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.34125 2.32312i −1.34125 2.32312i
\(701\) −1.49547 + 0.770969i −1.49547 + 0.770969i −0.995472 0.0950560i \(-0.969697\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 6.22614 1.19999i 6.22614 1.19999i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.0913090 + 1.91681i −0.0913090 + 1.91681i 0.235759 + 0.971812i \(0.424242\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(710\) 0 0
\(711\) −0.653077 0.513585i −0.653077 0.513585i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 3.60471 3.43708i 3.60471 3.43708i
\(717\) 0 0
\(718\) −1.11574 + 3.22371i −1.11574 + 3.22371i
\(719\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.25667 + 1.45027i 1.25667 + 1.45027i
\(723\) 0 0
\(724\) 0 0
\(725\) −0.0311250 0.653395i −0.0311250 0.653395i
\(726\) 0 0
\(727\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(728\) 0 0
\(729\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −0.334343 −0.334343
\(737\) 1.65210 1.06174i 1.65210 1.06174i
\(738\) 0 0
\(739\) −1.95496 0.376789i −1.95496 0.376789i −0.995472 0.0950560i \(-0.969697\pi\)
−0.959493 0.281733i \(-0.909091\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −3.66583 + 1.07639i −3.66583 + 1.07639i
\(743\) 0.195876 + 0.807410i 0.195876 + 0.807410i 0.981929 + 0.189251i \(0.0606061\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.33330 2.69277i 2.33330 2.69277i
\(747\) 0 0
\(748\) 0 0
\(749\) −0.0748038 1.57033i −0.0748038 1.57033i
\(750\) 0 0
\(751\) −0.118239 0.822373i −0.118239 0.822373i −0.959493 0.281733i \(-0.909091\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.67489 0.159932i −1.67489 0.159932i −0.786053 0.618159i \(-0.787879\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(758\) −2.33673 + 2.22806i −2.33673 + 2.22806i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(762\) 0 0
\(763\) 1.39734 + 1.09888i 1.39734 + 1.09888i
\(764\) −1.21361 0.356349i −1.21361 0.356349i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.80514 + 2.53497i 1.80514 + 2.53497i
\(773\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(774\) −1.50842 2.61267i −1.50842 2.61267i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −3.00319 2.36173i −3.00319 2.36173i
\(779\) 0 0
\(780\) 0 0
\(781\) 0.537129 0.754292i 0.537129 0.754292i
\(782\) 0 0
\(783\) 0 0
\(784\) −3.49743 0.333964i −3.49743 0.333964i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(788\) 2.81015 + 2.67947i 2.81015 + 2.67947i
\(789\) 0 0
\(790\) 0 0
\(791\) 0.888835 + 0.458227i 0.888835 + 0.458227i
\(792\) −0.301704 6.33354i −0.301704 6.33354i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −3.26167 1.30578i −3.26167 1.30578i
\(801\) 0 0
\(802\) −1.56554 0.301733i −1.56554 0.301733i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.50842 0.442913i 1.50842 0.442913i 0.580057 0.814576i \(-0.303030\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(810\) 0 0
\(811\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(812\) −1.47617 0.948676i −1.47617 0.948676i
\(813\) 0 0
\(814\) 5.43025 0.518526i 5.43025 0.518526i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.07701 0.431171i 1.07701 0.431171i 0.235759 0.971812i \(-0.424242\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(822\) 0 0
\(823\) 0.327068 0.945001i 0.327068 0.945001i −0.654861 0.755750i \(-0.727273\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.07701 1.51245i 1.07701 1.51245i 0.235759 0.971812i \(-0.424242\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(828\) −0.0363298 + 0.252679i −0.0363298 + 0.252679i
\(829\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(840\) 0 0
\(841\) 0.286053 + 0.495458i 0.286053 + 0.495458i
\(842\) −1.70566 + 2.95429i −1.70566 + 2.95429i
\(843\) 0 0
\(844\) 2.57385 + 0.755750i 2.57385 + 0.755750i
\(845\) 0 0
\(846\) 0 0
\(847\) −0.406556 + 2.82766i −0.406556 + 2.82766i
\(848\) −4.05742 + 5.69784i −4.05742 + 5.69784i
\(849\) 0 0
\(850\) 0 0
\(851\) −0.137123 0.0130936i −0.137123 0.0130936i
\(852\) 0 0
\(853\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −3.32399 3.83609i −3.32399 3.83609i
\(857\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(858\) 0 0
\(859\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2.46792 2.84813i 2.46792 2.84813i
\(863\) −1.67489 1.07639i −1.67489 1.07639i −0.888835 0.458227i \(-0.848485\pi\)
−0.786053 0.618159i \(-0.787879\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.60215 + 0.308788i 1.60215 + 0.308788i
\(870\) 0 0
\(871\) 0 0
\(872\) 5.73958 5.73958
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −0.452418 1.86489i −0.452418 1.86489i −0.500000 0.866025i \(-0.666667\pi\)
0.0475819 0.998867i \(-0.484848\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(882\) −0.452418 + 1.86489i −0.452418 + 1.86489i
\(883\) −0.0947329 1.98869i −0.0947329 1.98869i −0.142315 0.989821i \(-0.545455\pi\)
0.0475819 0.998867i \(-0.484848\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.357685 0.412791i −0.357685 0.412791i
\(887\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(888\) 0 0
\(889\) 0.223734 0.175946i 0.223734 0.175946i
\(890\) 0 0
\(891\) −1.95496 0.186677i −1.95496 0.186677i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −2.25667 + 1.45027i −2.25667 + 1.45027i
\(897\) 0 0
\(898\) −0.524075 0.153882i −0.524075 0.153882i
\(899\) 0 0
\(900\) −1.34125 + 2.32312i −1.34125 + 2.32312i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 3.17036 0.611037i 3.17036 0.611037i
\(905\) 0 0
\(906\) 0 0
\(907\) −1.74555 + 0.899892i −1.74555 + 0.899892i −0.786053 + 0.618159i \(0.787879\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.11312 0.326842i −1.11312 0.326842i −0.327068 0.945001i \(-0.606061\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0.128772 0.895626i 0.128772 0.895626i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −0.607279 + 1.75462i −0.607279 + 1.75462i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.28656 0.663268i −1.28656 0.663268i
\(926\) 0.162317 + 3.40746i 0.162317 + 3.40746i
\(927\) 0 0
\(928\) −2.28779 + 0.218458i −2.28779 + 0.218458i
\(929\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.413692 1.70526i −0.413692 1.70526i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) −1.11312 + 1.56316i −1.11312 + 1.56316i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 4.98414 + 3.20311i 4.98414 + 3.20311i
\(947\) 1.25667 1.45027i 1.25667 1.45027i 0.415415 0.909632i \(-0.363636\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.30379 + 1.50465i 1.30379 + 1.50465i 0.723734 + 0.690079i \(0.242424\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(954\) 2.76509 + 2.63651i 2.76509 + 2.63651i
\(955\) 0 0
\(956\) −0.200662 + 0.157802i −0.200662 + 0.157802i
\(957\) 0 0
\(958\) 0 0
\(959\) 0.839614 0.800570i 0.839614 0.800570i
\(960\) 0 0
\(961\) 0.580057 0.814576i 0.580057 0.814576i
\(962\) 0 0
\(963\) −1.32254 + 0.849945i −1.32254 + 0.849945i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −0.723734 + 1.25354i −0.723734 + 1.25354i 0.235759 + 0.971812i \(0.424242\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(968\) 4.61178 + 7.98784i 4.61178 + 7.98784i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 2.46889 1.27280i 2.46889 1.27280i
\(975\) 0 0
\(976\) 0 0
\(977\) −0.0475819 + 0.998867i −0.0475819 + 0.998867i 0.841254 + 0.540641i \(0.181818\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0.252989 1.75958i 0.252989 1.75958i
\(982\) −0.924814 + 1.29872i −0.924814 + 1.29872i
\(983\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.108276 0.103241i −0.108276 0.103241i
\(990\) 0 0
\(991\) 0.186393 + 1.29639i 0.186393 + 1.29639i 0.841254 + 0.540641i \(0.181818\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −0.213323 + 0.879330i −0.213323 + 0.879330i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(998\) −0.627639 1.81344i −0.627639 1.81344i
\(999\) 0 0
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 469.1.bl.a.83.1 20
7.2 even 3 3283.1.cd.a.619.1 20
7.3 odd 6 3283.1.bw.a.2763.1 20
7.4 even 3 3283.1.bw.a.2763.1 20
7.5 odd 6 3283.1.cd.a.619.1 20
7.6 odd 2 CM 469.1.bl.a.83.1 20
67.21 even 33 inner 469.1.bl.a.356.1 yes 20
469.88 even 33 3283.1.cd.a.2567.1 20
469.222 odd 66 3283.1.bw.a.423.1 20
469.289 even 33 3283.1.bw.a.423.1 20
469.356 odd 66 inner 469.1.bl.a.356.1 yes 20
469.423 odd 66 3283.1.cd.a.2567.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
469.1.bl.a.83.1 20 1.1 even 1 trivial
469.1.bl.a.83.1 20 7.6 odd 2 CM
469.1.bl.a.356.1 yes 20 67.21 even 33 inner
469.1.bl.a.356.1 yes 20 469.356 odd 66 inner
3283.1.bw.a.423.1 20 469.222 odd 66
3283.1.bw.a.423.1 20 469.289 even 33
3283.1.bw.a.2763.1 20 7.3 odd 6
3283.1.bw.a.2763.1 20 7.4 even 3
3283.1.cd.a.619.1 20 7.2 even 3
3283.1.cd.a.619.1 20 7.5 odd 6
3283.1.cd.a.2567.1 20 469.88 even 33
3283.1.cd.a.2567.1 20 469.423 odd 66