Properties

Label 469.1.bl.a.419.1
Level $469$
Weight $1$
Character 469.419
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 419.1
Root \(-0.888835 - 0.458227i\) of defining polynomial
Character \(\chi\) \(=\) 469.419
Dual form 469.1.bl.a.272.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.283341 - 0.0270558i) q^{2} +(-0.902379 + 0.173919i) q^{4} +(0.580057 + 0.814576i) q^{7} +(-0.524075 + 0.153882i) q^{8} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\) \(q+(0.283341 - 0.0270558i) q^{2} +(-0.902379 + 0.173919i) q^{4} +(0.580057 + 0.814576i) q^{7} +(-0.524075 + 0.153882i) q^{8} +(0.415415 - 0.909632i) q^{9} +(1.56499 + 1.23072i) q^{11} +(0.186393 + 0.215109i) q^{14} +(0.708829 - 0.283772i) q^{16} +(0.0930932 - 0.268975i) q^{18} +(0.476723 + 0.306371i) q^{22} +(-1.28656 - 0.663268i) q^{23} +(-0.959493 - 0.281733i) q^{25} +(-0.665101 - 0.634173i) q^{28} +(-0.580057 + 1.00469i) q^{29} +(0.678645 - 0.349866i) q^{32} +(-0.216659 + 0.893081i) q^{36} +(-0.928368 - 1.60798i) q^{37} +(0.428368 - 0.494363i) q^{43} +(-1.62626 - 0.838394i) q^{44} +(-0.382481 - 0.153122i) q^{46} +(-0.327068 + 0.945001i) q^{49} +(-0.279486 - 0.0538665i) q^{50} +(-0.0623191 - 0.0719200i) q^{53} +(-0.429342 - 0.337639i) q^{56} +(-0.137171 + 0.300363i) q^{58} +(0.981929 - 0.189251i) q^{63} +(-0.459493 + 0.295298i) q^{64} +(0.928368 - 0.371662i) q^{67} +(-1.54370 + 0.297523i) q^{71} +(-0.0777324 + 0.540641i) q^{72} +(-0.306550 - 0.430489i) q^{74} +(-0.0947329 + 1.98869i) q^{77} +(1.21769 - 1.16106i) q^{79} +(-0.654861 - 0.755750i) q^{81} +(0.107999 - 0.151663i) q^{86} +(-1.00956 - 0.404166i) q^{88} +(1.27632 + 0.374761i) q^{92} +(-0.0671040 + 0.276606i) q^{98} +(1.76962 - 0.912303i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} + 3 q^{4} + q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/469\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\)
\(\chi(n)\) \(-1\) \(e\left(\frac{32}{33}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.283341 0.0270558i 0.283341 0.0270558i 0.0475819 0.998867i \(-0.484848\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(3\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(4\) −0.902379 + 0.173919i −0.902379 + 0.173919i
\(5\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(6\) 0 0
\(7\) 0.580057 + 0.814576i 0.580057 + 0.814576i
\(8\) −0.524075 + 0.153882i −0.524075 + 0.153882i
\(9\) 0.415415 0.909632i 0.415415 0.909632i
\(10\) 0 0
\(11\) 1.56499 + 1.23072i 1.56499 + 1.23072i 0.841254 + 0.540641i \(0.181818\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(12\) 0 0
\(13\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(14\) 0.186393 + 0.215109i 0.186393 + 0.215109i
\(15\) 0 0
\(16\) 0.708829 0.283772i 0.708829 0.283772i
\(17\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(18\) 0.0930932 0.268975i 0.0930932 0.268975i
\(19\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.476723 + 0.306371i 0.476723 + 0.306371i
\(23\) −1.28656 0.663268i −1.28656 0.663268i −0.327068 0.945001i \(-0.606061\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(24\) 0 0
\(25\) −0.959493 0.281733i −0.959493 0.281733i
\(26\) 0 0
\(27\) 0 0
\(28\) −0.665101 0.634173i −0.665101 0.634173i
\(29\) −0.580057 + 1.00469i −0.580057 + 1.00469i 0.415415 + 0.909632i \(0.363636\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(30\) 0 0
\(31\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(32\) 0.678645 0.349866i 0.678645 0.349866i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.216659 + 0.893081i −0.216659 + 0.893081i
\(37\) −0.928368 1.60798i −0.928368 1.60798i −0.786053 0.618159i \(-0.787879\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(42\) 0 0
\(43\) 0.428368 0.494363i 0.428368 0.494363i −0.500000 0.866025i \(-0.666667\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(44\) −1.62626 0.838394i −1.62626 0.838394i
\(45\) 0 0
\(46\) −0.382481 0.153122i −0.382481 0.153122i
\(47\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(48\) 0 0
\(49\) −0.327068 + 0.945001i −0.327068 + 0.945001i
\(50\) −0.279486 0.0538665i −0.279486 0.0538665i
\(51\) 0 0
\(52\) 0 0
\(53\) −0.0623191 0.0719200i −0.0623191 0.0719200i 0.723734 0.690079i \(-0.242424\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.429342 0.337639i −0.429342 0.337639i
\(57\) 0 0
\(58\) −0.137171 + 0.300363i −0.137171 + 0.300363i
\(59\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(60\) 0 0
\(61\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(62\) 0 0
\(63\) 0.981929 0.189251i 0.981929 0.189251i
\(64\) −0.459493 + 0.295298i −0.459493 + 0.295298i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.928368 0.371662i 0.928368 0.371662i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.54370 + 0.297523i −1.54370 + 0.297523i −0.888835 0.458227i \(-0.848485\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(72\) −0.0777324 + 0.540641i −0.0777324 + 0.540641i
\(73\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(74\) −0.306550 0.430489i −0.306550 0.430489i
\(75\) 0 0
\(76\) 0 0
\(77\) −0.0947329 + 1.98869i −0.0947329 + 1.98869i
\(78\) 0 0
\(79\) 1.21769 1.16106i 1.21769 1.16106i 0.235759 0.971812i \(-0.424242\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(80\) 0 0
\(81\) −0.654861 0.755750i −0.654861 0.755750i
\(82\) 0 0
\(83\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.107999 0.151663i 0.107999 0.151663i
\(87\) 0 0
\(88\) −1.00956 0.404166i −1.00956 0.404166i
\(89\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.27632 + 0.374761i 1.27632 + 0.374761i
\(93\) 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 \(0.333333\pi\)
\(98\) −0.0671040 + 0.276606i −0.0671040 + 0.276606i
\(99\) 1.76962 0.912303i 1.76962 0.912303i
\(100\) 0.914825 + 0.0873552i 0.914825 + 0.0873552i
\(101\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(102\) 0 0
\(103\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −0.0196034 0.0186918i −0.0196034 0.0186918i
\(107\) 0.0930932 + 0.647478i 0.0930932 + 0.647478i 0.981929 + 0.189251i \(0.0606061\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(108\) 0 0
\(109\) −0.452418 0.132842i −0.452418 0.132842i 0.0475819 0.998867i \(-0.484848\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.642315 + 0.412791i 0.642315 + 0.412791i
\(113\) −0.928368 0.371662i −0.928368 0.371662i −0.142315 0.989821i \(-0.545455\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.348696 1.00749i 0.348696 1.00749i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.698756 + 2.88031i 0.698756 + 2.88031i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0.273100 0.0801894i 0.273100 0.0801894i
\(127\) −0.759713 1.06687i −0.759713 1.06687i −0.995472 0.0950560i \(-0.969697\pi\)
0.235759 0.971812i \(-0.424242\pi\)
\(128\) −0.722372 + 0.568079i −0.722372 + 0.568079i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0.252989 0.130425i 0.252989 0.130425i
\(135\) 0 0
\(136\) 0 0
\(137\) −1.49547 + 0.961081i −1.49547 + 0.961081i −0.500000 + 0.866025i \(0.666667\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(138\) 0 0
\(139\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.429342 + 0.126066i −0.429342 + 0.126066i
\(143\) 0 0
\(144\) 0.0363298 0.762656i 0.0363298 0.762656i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 1.11740 + 1.28955i 1.11740 + 1.28955i
\(149\) −0.415415 0.909632i −0.415415 0.909632i −0.995472 0.0950560i \(-0.969697\pi\)
0.580057 0.814576i \(-0.303030\pi\)
\(150\) 0 0
\(151\) 0.815816 + 0.157236i 0.815816 + 0.157236i 0.580057 0.814576i \(-0.303030\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0.0269638 + 0.566040i 0.0269638 + 0.566040i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(158\) 0.313607 0.361922i 0.313607 0.361922i
\(159\) 0 0
\(160\) 0 0
\(161\) −0.205996 1.43273i −0.205996 1.43273i
\(162\) −0.205996 0.196417i −0.205996 0.196417i
\(163\) 0.888835 1.53951i 0.888835 1.53951i 0.0475819 0.998867i \(-0.484848\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(168\) 0 0
\(169\) −0.888835 + 0.458227i −0.888835 + 0.458227i
\(170\) 0 0
\(171\) 0 0
\(172\) −0.300571 + 0.520604i −0.300571 + 0.520604i
\(173\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(174\) 0 0
\(175\) −0.327068 0.945001i −0.327068 0.945001i
\(176\) 1.45855 + 0.428269i 1.45855 + 0.428269i
\(177\) 0 0
\(178\) 0 0
\(179\) 1.65210 + 1.06174i 1.65210 + 1.06174i 0.928368 + 0.371662i \(0.121212\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(180\) 0 0
\(181\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0.776320 + 0.149623i 0.776320 + 0.149623i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.0748038 + 1.57033i −0.0748038 + 1.57033i 0.580057 + 0.814576i \(0.303030\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(192\) 0 0
\(193\) 1.70566 0.500828i 1.70566 0.500828i 0.723734 0.690079i \(-0.242424\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.130785 0.909632i 0.130785 0.909632i
\(197\) 1.82318 0.351390i 1.82318 0.351390i 0.841254 0.540641i \(-0.181818\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(198\) 0.476723 0.306371i 0.476723 0.306371i
\(199\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(200\) 0.546200 0.546200
\(201\) 0 0
\(202\) 0 0
\(203\) −1.15486 + 0.110276i −1.15486 + 0.110276i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.13779 + 0.894765i −1.13779 + 0.894765i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.0475819 + 0.998867i −0.0475819 + 0.998867i 0.841254 + 0.540641i \(0.181818\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(212\) 0.0687437 + 0.0540606i 0.0687437 + 0.0540606i
\(213\) 0 0
\(214\) 0.0438951 + 0.180938i 0.0438951 + 0.180938i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.131783 0.0253990i −0.131783 0.0253990i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(224\) 0.678645 + 0.349866i 0.678645 + 0.349866i
\(225\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(226\) −0.273100 0.0801894i −0.273100 0.0801894i
\(227\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(228\) 0 0
\(229\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.149390 0.615793i 0.149390 0.615793i
\(233\) −1.03115 + 0.531595i −1.03115 + 0.531595i −0.888835 0.458227i \(-0.848485\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.723734 + 1.25354i −0.723734 + 1.25354i 0.235759 + 0.971812i \(0.424242\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(240\) 0 0
\(241\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(242\) 0.275915 + 0.797204i 0.275915 + 0.797204i
\(243\) 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 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(252\) −0.853157 + 0.341553i −0.853157 + 0.341553i
\(253\) −1.19715 2.62140i −1.19715 2.62140i
\(254\) −0.244123 0.281733i −0.244123 0.281733i
\(255\) 0 0
\(256\) 0.205996 0.196417i 0.205996 0.196417i
\(257\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(258\) 0 0
\(259\) 0.771316 1.68895i 0.771316 1.68895i
\(260\) 0 0
\(261\) 0.672932 + 0.945001i 0.672932 + 0.945001i
\(262\) 0 0
\(263\) −0.279486 + 1.94387i −0.279486 + 1.94387i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −0.773100 + 0.496841i −0.773100 + 0.496841i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −0.397725 + 0.312775i −0.397725 + 0.312775i
\(275\) −1.15486 1.62177i −1.15486 1.62177i
\(276\) 0 0
\(277\) 0.345139 0.755750i 0.345139 0.755750i −0.654861 0.755750i \(-0.727273\pi\)
1.00000 \(0\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.452418 1.86489i −0.452418 1.86489i −0.500000 0.866025i \(-0.666667\pi\)
0.0475819 0.998867i \(-0.484848\pi\)
\(282\) 0 0
\(283\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(284\) 1.34125 0.536957i 1.34125 0.536957i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.0363298 0.762656i −0.0363298 0.762656i
\(289\) 0.928368 + 0.371662i 0.928368 + 0.371662i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.733975 + 0.699843i 0.733975 + 0.699843i
\(297\) 0 0
\(298\) −0.142315 0.246497i −0.142315 0.246497i
\(299\) 0 0
\(300\) 0 0
\(301\) 0.651174 + 0.0621796i 0.651174 + 0.0621796i
\(302\) 0.235408 + 0.0224787i 0.235408 + 0.0224787i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(308\) −0.260386 1.81103i −0.260386 1.81103i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(312\) 0 0
\(313\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −0.896884 + 1.25950i −0.896884 + 1.25950i
\(317\) 0.428368 1.23769i 0.428368 1.23769i −0.500000 0.866025i \(-0.666667\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(318\) 0 0
\(319\) −2.14427 + 0.858437i −2.14427 + 0.858437i
\(320\) 0 0
\(321\) 0 0
\(322\) −0.0971309 0.400379i −0.0971309 0.400379i
\(323\) 0 0
\(324\) 0.722372 + 0.568079i 0.722372 + 0.568079i
\(325\) 0 0
\(326\) 0.210191 0.460254i 0.210191 0.460254i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.74555 + 0.336426i −1.74555 + 0.336426i −0.959493 0.281733i \(-0.909091\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(332\) 0 0
\(333\) −1.84833 + 0.176494i −1.84833 + 0.176494i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.95496 + 0.186677i −1.95496 + 0.186677i −0.995472 0.0950560i \(-0.969697\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(338\) −0.239446 + 0.153882i −0.239446 + 0.153882i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(344\) −0.148423 + 0.325002i −0.148423 + 0.325002i
\(345\) 0 0
\(346\) 0 0
\(347\) 0.341254 0.325385i 0.341254 0.325385i −0.500000 0.866025i \(-0.666667\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(348\) 0 0
\(349\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(350\) −0.118239 0.258908i −0.118239 0.258908i
\(351\) 0 0
\(352\) 1.49266 + 0.287686i 1.49266 + 0.287686i
\(353\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0.496834 + 0.256136i 0.496834 + 0.256136i
\(359\) −0.308779 + 0.356349i −0.308779 + 0.356349i −0.888835 0.458227i \(-0.848485\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(360\) 0 0
\(361\) −0.327068 0.945001i −0.327068 0.945001i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(368\) −1.10017 0.105053i −1.10017 0.105053i
\(369\) 0 0
\(370\) 0 0
\(371\) 0.0224357 0.0924813i 0.0224357 0.0924813i
\(372\) 0 0
\(373\) −0.981929 + 1.70075i −0.981929 + 1.70075i −0.327068 + 0.945001i \(0.606061\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.70566 + 0.879330i 1.70566 + 0.879330i 0.981929 + 0.189251i \(0.0606061\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0.0212914 + 0.446961i 0.0212914 + 0.446961i
\(383\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.469734 0.188053i 0.469734 0.188053i
\(387\) −0.271738 0.595023i −0.271738 0.595023i
\(388\) 0 0
\(389\) 0.0224357 + 0.0924813i 0.0224357 + 0.0924813i 0.981929 0.189251i \(-0.0606061\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.0259893 0.545582i 0.0259893 0.545582i
\(393\) 0 0
\(394\) 0.507075 0.148891i 0.507075 0.148891i
\(395\) 0 0
\(396\) −1.43820 + 1.13101i −1.43820 + 1.13101i
\(397\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.760064 + 0.0725773i −0.760064 + 0.0725773i
\(401\) 1.68251 1.68251 0.841254 0.540641i \(-0.181818\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −0.324236 + 0.0624913i −0.324236 + 0.0624913i
\(407\) 0.526089 3.65903i 0.526089 3.65903i
\(408\) 0 0
\(409\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −0.298173 + 0.284307i −0.298173 + 0.284307i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(420\) 0 0
\(421\) 0.273507 0.384087i 0.273507 0.384087i −0.654861 0.755750i \(-0.727273\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(422\) 0.0135432 + 0.284307i 0.0135432 + 0.284307i
\(423\) 0 0
\(424\) 0.0437271 + 0.0281017i 0.0437271 + 0.0281017i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −0.196614 0.568079i −0.196614 0.568079i
\(429\) 0 0
\(430\) 0 0
\(431\) 0.995472 1.72421i 0.995472 1.72421i 0.415415 0.909632i \(-0.363636\pi\)
0.580057 0.814576i \(-0.303030\pi\)
\(432\) 0 0
\(433\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.431356 + 0.0411895i 0.431356 + 0.0411895i
\(437\) 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
\(441\) 0.723734 + 0.690079i 0.723734 + 0.690079i
\(442\) 0 0
\(443\) 0.428368 + 1.23769i 0.428368 + 1.23769i 0.928368 + 0.371662i \(0.121212\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.507075 0.203002i −0.507075 0.203002i
\(449\) −0.0623191 1.30824i −0.0623191 1.30824i −0.786053 0.618159i \(-0.787879\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(450\) −0.165101 + 0.231852i −0.165101 + 0.231852i
\(451\) 0 0
\(452\) 0.902379 + 0.173919i 0.902379 + 0.173919i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.13779 + 1.08488i −1.13779 + 1.08488i −0.142315 + 0.989821i \(0.545455\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(462\) 0 0
\(463\) −0.370638 + 0.291473i −0.370638 + 0.291473i −0.786053 0.618159i \(-0.787879\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(464\) −0.126058 + 0.876756i −0.126058 + 0.876756i
\(465\) 0 0
\(466\) −0.277784 + 0.178521i −0.277784 + 0.178521i
\(467\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(468\) 0 0
\(469\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.27881 0.246471i 1.27881 0.246471i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.0913090 + 0.0268107i −0.0913090 + 0.0268107i
\(478\) −0.171148 + 0.374761i −0.171148 + 0.374761i
\(479\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −1.13148 2.47760i −1.13148 2.47760i
\(485\) 0 0
\(486\) 0 0
\(487\) −0.607279 + 1.75462i −0.607279 + 1.75462i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.41542 + 0.909632i 1.41542 + 0.909632i 1.00000 \(0\)
0.415415 + 0.909632i \(0.363636\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.13779 1.08488i −1.13779 1.08488i
\(498\) 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 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(504\) −0.485482 + 0.250283i −0.485482 + 0.250283i
\(505\) 0 0
\(506\) −0.410127 0.710361i −0.410127 0.710361i
\(507\) 0 0
\(508\) 0.871098 + 0.830590i 0.871098 + 0.830590i
\(509\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.654861 0.755750i 0.654861 0.755750i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0.172850 0.499416i 0.172850 0.499416i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(522\) 0.216237 + 0.249551i 0.216237 + 0.249551i
\(523\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −0.0265970 + 0.558339i −0.0265970 + 0.558339i
\(527\) 0 0
\(528\) 0 0
\(529\) 0.635257 + 0.892094i 0.635257 + 0.892094i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −0.429342 + 0.337639i −0.429342 + 0.337639i
\(537\) 0 0
\(538\) 0 0
\(539\) −1.67489 + 1.07639i −1.67489 + 1.07639i
\(540\) 0 0
\(541\) −0.165101 + 1.14831i −0.165101 + 1.14831i 0.723734 + 0.690079i \(0.242424\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.13779 0.894765i −1.13779 0.894765i −0.142315 0.989821i \(-0.545455\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(548\) 1.18233 1.12735i 1.18233 1.12735i
\(549\) 0 0
\(550\) −0.371098 0.428269i −0.371098 0.428269i
\(551\) 0 0
\(552\) 0 0
\(553\) 1.65210 + 0.318417i 1.65210 + 0.318417i
\(554\) 0.0773447 0.223473i 0.0773447 0.223473i
\(555\) 0 0
\(556\) 0 0
\(557\) 1.56199 + 0.625325i 1.56199 + 0.625325i 0.981929 0.189251i \(-0.0606061\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −0.178645 0.516160i −0.178645 0.516160i
\(563\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.235759 0.971812i 0.235759 0.971812i
\(568\) 0.763230 0.393472i 0.763230 0.393472i
\(569\) 0.283341 + 0.0270558i 0.283341 + 0.0270558i 0.235759 0.971812i \(-0.424242\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(570\) 0 0
\(571\) 1.76962 0.912303i 1.76962 0.912303i 0.841254 0.540641i \(-0.181818\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.04758 + 0.998867i 1.04758 + 0.998867i
\(576\) 0.0777324 + 0.540641i 0.0777324 + 0.540641i
\(577\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(578\) 0.273100 + 0.0801894i 0.273100 + 0.0801894i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.00901515 0.189251i −0.00901515 0.189251i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.11435 0.876337i −1.11435 0.876337i
\(593\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.533064 + 0.748584i 0.533064 + 0.748584i
\(597\) 0 0
\(598\) 0 0
\(599\) 0.0934441 0.0180099i 0.0934441 0.0180099i −0.142315 0.989821i \(-0.545455\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(600\) 0 0
\(601\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(602\) 0.186186 0.186186
\(603\) 0.0475819 0.998867i 0.0475819 0.998867i
\(604\) −0.763521 −0.763521
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −0.0748038 + 1.57033i −0.0748038 + 1.57033i 0.580057 + 0.814576i \(0.303030\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −0.256377 1.05680i −0.256377 1.05680i
\(617\) 0.428368 + 0.494363i 0.428368 + 0.494363i 0.928368 0.371662i \(-0.121212\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(618\) 0 0
\(619\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0.839614 + 0.800570i 0.839614 + 0.800570i 0.981929 0.189251i \(-0.0606061\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(632\) −0.459493 + 0.795865i −0.459493 + 0.795865i
\(633\) 0 0
\(634\) 0.0878875 0.362277i 0.0878875 0.362277i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −0.584334 + 0.301245i −0.584334 + 0.301245i
\(639\) −0.370638 + 1.52779i −0.370638 + 1.52779i
\(640\) 0 0
\(641\) 0.654861 1.13425i 0.654861 1.13425i −0.327068 0.945001i \(-0.606061\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(642\) 0 0
\(643\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(644\) 0.435067 + 1.25704i 0.435067 + 1.25704i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(648\) 0.459493 + 0.295298i 0.459493 + 0.295298i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −0.534316 + 1.54380i −0.534316 + 1.54380i
\(653\) 1.65210 + 0.318417i 1.65210 + 0.318417i 0.928368 0.371662i \(-0.121212\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −0.370638 0.291473i −0.370638 0.291473i 0.415415 0.909632i \(-0.363636\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(660\) 0 0
\(661\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(662\) −0.485482 + 0.142550i −0.485482 + 0.142550i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −0.518932 + 0.100016i −0.518932 + 0.100016i
\(667\) 1.41266 0.907859i 1.41266 0.907859i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.56199 1.00383i 1.56199 1.00383i 0.580057 0.814576i \(-0.303030\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(674\) −0.548871 + 0.105786i −0.548871 + 0.105786i
\(675\) 0 0
\(676\) 0.722372 0.568079i 0.722372 0.568079i
\(677\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −0.370638 1.52779i −0.370638 1.52779i −0.786053 0.618159i \(-0.787879\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.264241 + 0.105786i −0.264241 + 0.105786i
\(687\) 0 0
\(688\) 0.163353 0.471977i 0.163353 0.471977i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(692\) 0 0
\(693\) 1.76962 + 0.912303i 1.76962 + 0.912303i
\(694\) 0.0878875 0.101428i 0.0878875 0.101428i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.459493 + 0.795865i 0.459493 + 0.795865i
\(701\) −0.452418 + 1.86489i −0.452418 + 1.86489i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.08253 0.103369i −1.08253 0.103369i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.205996 0.196417i −0.205996 0.196417i 0.580057 0.814576i \(-0.303030\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(710\) 0 0
\(711\) −0.550294 1.58997i −0.550294 1.58997i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −1.67548 0.670761i −1.67548 0.670761i
\(717\) 0 0
\(718\) −0.0778483 + 0.109323i −0.0778483 + 0.109323i
\(719\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −0.118239 0.258908i −0.118239 0.258908i
\(723\) 0 0
\(724\) 0 0
\(725\) 0.839614 0.800570i 0.839614 0.800570i
\(726\) 0 0
\(727\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(728\) 0 0
\(729\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −1.10517 −1.10517
\(737\) 1.91030 + 0.560914i 1.91030 + 0.560914i
\(738\) 0 0
\(739\) −0.0947329 + 0.00904590i −0.0947329 + 0.00904590i −0.142315 0.989821i \(-0.545455\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.00385480 0.0268107i 0.00385480 0.0268107i
\(743\) −1.32254 + 1.04006i −1.32254 + 1.04006i −0.327068 + 0.945001i \(0.606061\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −0.232205 + 0.508459i −0.232205 + 0.508459i
\(747\) 0 0
\(748\) 0 0
\(749\) −0.473420 + 0.451405i −0.473420 + 0.451405i
\(750\) 0 0
\(751\) −1.10181 1.27155i −1.10181 1.27155i −0.959493 0.281733i \(-0.909091\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.0913090 1.91681i −0.0913090 1.91681i −0.327068 0.945001i \(-0.606061\pi\)
0.235759 0.971812i \(-0.424242\pi\)
\(758\) 0.507075 + 0.203002i 0.507075 + 0.203002i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(762\) 0 0
\(763\) −0.154218 0.445585i −0.154218 0.445585i
\(764\) −0.205608 1.43004i −0.205608 1.43004i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.45205 + 0.748584i −1.45205 + 0.748584i
\(773\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(774\) −0.0930932 0.161242i −0.0930932 0.161242i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0.00885911 + 0.0255967i 0.00885911 + 0.0255967i
\(779\) 0 0
\(780\) 0 0
\(781\) −2.78203 1.43424i −2.78203 1.43424i
\(782\) 0 0
\(783\) 0 0
\(784\) 0.0363298 + 0.762656i 0.0363298 + 0.762656i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(788\) −1.58409 + 0.634173i −1.58409 + 0.634173i
\(789\) 0 0
\(790\) 0 0
\(791\) −0.235759 0.971812i −0.235759 0.971812i
\(792\) −0.787028 + 0.750429i −0.787028 + 0.750429i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.749723 + 0.144497i −0.749723 + 0.144497i
\(801\) 0 0
\(802\) 0.476723 0.0455215i 0.476723 0.0455215i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.0930932 0.647478i 0.0930932 0.647478i −0.888835 0.458227i \(-0.848485\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(810\) 0 0
\(811\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(812\) 1.02294 0.300363i 1.02294 0.300363i
\(813\) 0 0
\(814\) 0.0500647 1.05099i 0.0500647 1.05099i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.74555 0.336426i −1.74555 0.336426i −0.786053 0.618159i \(-0.787879\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(822\) 0 0
\(823\) −0.580057 + 0.814576i −0.580057 + 0.814576i −0.995472 0.0950560i \(-0.969697\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.74555 0.899892i −1.74555 0.899892i −0.959493 0.281733i \(-0.909091\pi\)
−0.786053 0.618159i \(-0.787879\pi\)
\(828\) 0.871098 1.00530i 0.871098 1.00530i
\(829\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(830\) 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 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(840\) 0 0
\(841\) −0.172932 0.299527i −0.172932 0.299527i
\(842\) 0.0671040 0.116228i 0.0671040 0.116228i
\(843\) 0 0
\(844\) −0.130785 0.909632i −0.130785 0.909632i
\(845\) 0 0
\(846\) 0 0
\(847\) −1.94091 + 2.23993i −1.94091 + 2.23993i
\(848\) −0.0645824 0.0332946i −0.0645824 0.0332946i
\(849\) 0 0
\(850\) 0 0
\(851\) 0.127880 + 2.68452i 0.127880 + 2.68452i
\(852\) 0 0
\(853\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −0.148423 0.325002i −0.148423 0.325002i
\(857\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(858\) 0 0
\(859\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0.235408 0.515472i 0.235408 0.515472i
\(863\) −0.0913090 + 0.0268107i −0.0913090 + 0.0268107i −0.327068 0.945001i \(-0.606061\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 3.33461 0.318417i 3.33461 0.318417i
\(870\) 0 0
\(871\) 0 0
\(872\) 0.257543 0.257543
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.223734 0.175946i 0.223734 0.175946i −0.500000 0.866025i \(-0.666667\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(882\) 0.223734 + 0.175946i 0.223734 + 0.175946i
\(883\) 0.0688733 0.0656706i 0.0688733 0.0656706i −0.654861 0.755750i \(-0.727273\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.154861 + 0.339098i 0.154861 + 0.339098i
\(887\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(888\) 0 0
\(889\) 0.428368 1.23769i 0.428368 1.23769i
\(890\) 0 0
\(891\) −0.0947329 1.98869i −0.0947329 1.98869i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −0.881761 0.258908i −0.881761 0.258908i
\(897\) 0 0
\(898\) −0.0530529 0.368991i −0.0530529 0.368991i
\(899\) 0 0
\(900\) 0.459493 0.795865i 0.459493 0.795865i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0.543727 + 0.0519196i 0.543727 + 0.0519196i
\(905\) 0 0
\(906\) 0 0
\(907\) −0.469383 + 1.93482i −0.469383 + 1.93482i −0.142315 + 0.989821i \(0.545455\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0.252989 + 1.75958i 0.252989 + 1.75958i 0.580057 + 0.814576i \(0.303030\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.293029 + 0.338174i −0.293029 + 0.338174i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.13915 1.59971i 1.13915 1.59971i 0.415415 0.909632i \(-0.363636\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0.437742 + 1.80440i 0.437742 + 1.80440i
\(926\) −0.0971309 + 0.0926141i −0.0971309 + 0.0926141i
\(927\) 0 0
\(928\) −0.0421467 + 0.884768i −0.0421467 + 0.884768i
\(929\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0.838033 0.659037i 0.838033 0.659037i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0.252989 + 0.130425i 0.252989 + 0.130425i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0.355671 0.104435i 0.355671 0.104435i
\(947\) −0.118239 + 0.258908i −0.118239 + 0.258908i −0.959493 0.281733i \(-0.909091\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.0395325 + 0.0865641i 0.0395325 + 0.0865641i 0.928368 0.371662i \(-0.121212\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(954\) −0.0251462 + 0.0100670i −0.0251462 + 0.0100670i
\(955\) 0 0
\(956\) 0.435067 1.25704i 0.435067 1.25704i
\(957\) 0 0
\(958\) 0 0
\(959\) −1.65033 0.660694i −1.65033 0.660694i
\(960\) 0 0
\(961\) −0.888835 0.458227i −0.888835 0.458227i
\(962\) 0 0
\(963\) 0.627639 + 0.184291i 0.627639 + 0.184291i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −0.928368 + 1.60798i −0.928368 + 1.60798i −0.142315 + 0.989821i \(0.545455\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(968\) −0.809430 1.40197i −0.809430 1.40197i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −0.124594 + 0.513585i −0.124594 + 0.513585i
\(975\) 0 0
\(976\) 0 0
\(977\) −0.723734 0.690079i −0.723734 0.690079i 0.235759 0.971812i \(-0.424242\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −0.308779 + 0.356349i −0.308779 + 0.356349i
\(982\) 0.425656 + 0.219441i 0.425656 + 0.219441i
\(983\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.879017 + 0.351905i −0.879017 + 0.351905i
\(990\) 0 0
\(991\) −0.544078 0.627899i −0.544078 0.627899i 0.415415 0.909632i \(-0.363636\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −0.351734 0.276606i −0.351734 0.276606i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(998\) 0.165101 + 0.231852i 0.165101 + 0.231852i
\(999\) 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.419.1 yes 20
7.2 even 3 3283.1.cd.a.1893.1 20
7.3 odd 6 3283.1.bw.a.754.1 20
7.4 even 3 3283.1.bw.a.754.1 20
7.5 odd 6 3283.1.cd.a.1893.1 20
7.6 odd 2 CM 469.1.bl.a.419.1 yes 20
67.4 even 33 inner 469.1.bl.a.272.1 20
469.4 even 33 3283.1.cd.a.607.1 20
469.138 odd 66 3283.1.bw.a.1746.1 20
469.205 even 33 3283.1.bw.a.1746.1 20
469.272 odd 66 inner 469.1.bl.a.272.1 20
469.339 odd 66 3283.1.cd.a.607.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
469.1.bl.a.272.1 20 67.4 even 33 inner
469.1.bl.a.272.1 20 469.272 odd 66 inner
469.1.bl.a.419.1 yes 20 1.1 even 1 trivial
469.1.bl.a.419.1 yes 20 7.6 odd 2 CM
3283.1.bw.a.754.1 20 7.3 odd 6
3283.1.bw.a.754.1 20 7.4 even 3
3283.1.bw.a.1746.1 20 469.138 odd 66
3283.1.bw.a.1746.1 20 469.205 even 33
3283.1.cd.a.607.1 20 469.4 even 33
3283.1.cd.a.607.1 20 469.339 odd 66
3283.1.cd.a.1893.1 20 7.2 even 3
3283.1.cd.a.1893.1 20 7.5 odd 6