Properties

Label 1191.1.z.a.824.1
Level $1191$
Weight $1$
Character 1191.824
Analytic conductor $0.594$
Analytic rank $0$
Dimension $20$
Projective image $D_{66}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1191,1,Mod(65,1191)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1191, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([33, 17]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1191.65");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1191 = 3 \cdot 397 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1191.z (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: \(0.594386430046\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{66}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{66} - \cdots)\)

Embedding invariants

Embedding label 824.1
Root \(0.235759 + 0.971812i\) of defining polynomial
Character \(\chi\) \(=\) 1191.824
Dual form 1191.1.z.a.1019.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.786053 + 0.618159i) q^{3} +(-0.841254 + 0.540641i) q^{4} +(1.68323 - 0.408346i) q^{7} +(0.235759 - 0.971812i) q^{9} +O(q^{10})\) \(q+(-0.786053 + 0.618159i) q^{3} +(-0.841254 + 0.540641i) q^{4} +(1.68323 - 0.408346i) q^{7} +(0.235759 - 0.971812i) q^{9} +(0.327068 - 0.945001i) q^{12} +(-0.173440 + 0.336426i) q^{13} +(0.415415 - 0.909632i) q^{16} +(0.839614 - 1.17907i) q^{19} +(-1.07068 + 1.36148i) q^{21} +(0.888835 + 0.458227i) q^{25} +(0.415415 + 0.909632i) q^{27} +(-1.19525 + 1.25354i) q^{28} +(-0.279486 + 1.94387i) q^{31} +(0.327068 + 0.945001i) q^{36} +(-0.771316 + 0.308788i) q^{37} +(-0.0716321 - 0.371662i) q^{39} +(0.452418 - 0.132842i) q^{43} +(0.235759 + 0.971812i) q^{48} +(1.77767 - 0.916453i) q^{49} +(-0.0359789 - 0.376789i) q^{52} +(0.0688733 + 1.44583i) q^{57} +(0.566514 + 1.09888i) q^{61} -1.73205i q^{63} +(0.142315 + 0.989821i) q^{64} +(-1.56199 + 0.625325i) q^{67} +(1.76962 - 0.168978i) q^{73} +(-0.981929 + 0.189251i) q^{75} +(-0.0688733 + 1.44583i) q^{76} +(-0.786053 - 1.36148i) q^{79} +(-0.888835 - 0.458227i) q^{81} +(0.164642 - 1.72421i) q^{84} +(-0.154560 + 0.637106i) q^{91} +(-0.981929 - 1.70075i) q^{93} +(1.44091 - 1.37391i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{3} + 2 q^{4} + 3 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + q^{3} + 2 q^{4} + 3 q^{7} + q^{9} - q^{12} - 3 q^{13} - 2 q^{16} - q^{19} - 3 q^{21} - q^{25} - 2 q^{27} - 3 q^{28} + 2 q^{31} - q^{36} - 2 q^{37} - 19 q^{39} + 9 q^{43} + q^{48} - 2 q^{49} + 3 q^{52} - q^{57} + 3 q^{61} + 2 q^{64} - 2 q^{67} - q^{73} - q^{75} + q^{76} + q^{79} + q^{81} + 3 q^{84} + 3 q^{91} - q^{93} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1191\mathbb{Z}\right)^\times\).

\(n\) \(398\) \(799\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{66}\right)\)

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 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(3\) −0.786053 + 0.618159i −0.786053 + 0.618159i
\(4\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(5\) 0 0 −0.971812 0.235759i \(-0.924242\pi\)
0.971812 + 0.235759i \(0.0757576\pi\)
\(6\) 0 0
\(7\) 1.68323 0.408346i 1.68323 0.408346i 0.723734 0.690079i \(-0.242424\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(8\) 0 0
\(9\) 0.235759 0.971812i 0.235759 0.971812i
\(10\) 0 0
\(11\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(12\) 0.327068 0.945001i 0.327068 0.945001i
\(13\) −0.173440 + 0.336426i −0.173440 + 0.336426i −0.959493 0.281733i \(-0.909091\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.415415 0.909632i 0.415415 0.909632i
\(17\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(18\) 0 0
\(19\) 0.839614 1.17907i 0.839614 1.17907i −0.142315 0.989821i \(-0.545455\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(20\) 0 0
\(21\) −1.07068 + 1.36148i −1.07068 + 1.36148i
\(22\) 0 0
\(23\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(24\) 0 0
\(25\) 0.888835 + 0.458227i 0.888835 + 0.458227i
\(26\) 0 0
\(27\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(28\) −1.19525 + 1.25354i −1.19525 + 1.25354i
\(29\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(30\) 0 0
\(31\) −0.279486 + 1.94387i −0.279486 + 1.94387i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.327068 + 0.945001i 0.327068 + 0.945001i
\(37\) −0.771316 + 0.308788i −0.771316 + 0.308788i −0.723734 0.690079i \(-0.757576\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(38\) 0 0
\(39\) −0.0716321 0.371662i −0.0716321 0.371662i
\(40\) 0 0
\(41\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) 0.452418 0.132842i 0.452418 0.132842i −0.0475819 0.998867i \(-0.515152\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(48\) 0.235759 + 0.971812i 0.235759 + 0.971812i
\(49\) 1.77767 0.916453i 1.77767 0.916453i
\(50\) 0 0
\(51\) 0 0
\(52\) −0.0359789 0.376789i −0.0359789 0.376789i
\(53\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.0688733 + 1.44583i 0.0688733 + 1.44583i
\(58\) 0 0
\(59\) 0 0 0.618159 0.786053i \(-0.287879\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(60\) 0 0
\(61\) 0.566514 + 1.09888i 0.566514 + 1.09888i 0.981929 + 0.189251i \(0.0606061\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(62\) 0 0
\(63\) 1.73205i 1.73205i
\(64\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(65\) 0 0
\(66\) 0 0
\(67\) −1.56199 + 0.625325i −1.56199 + 0.625325i −0.981929 0.189251i \(-0.939394\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(72\) 0 0
\(73\) 1.76962 0.168978i 1.76962 0.168978i 0.841254 0.540641i \(-0.181818\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(74\) 0 0
\(75\) −0.981929 + 0.189251i −0.981929 + 0.189251i
\(76\) −0.0688733 + 1.44583i −0.0688733 + 1.44583i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.786053 1.36148i −0.786053 1.36148i −0.928368 0.371662i \(-0.878788\pi\)
0.142315 0.989821i \(-0.454545\pi\)
\(80\) 0 0
\(81\) −0.888835 0.458227i −0.888835 0.458227i
\(82\) 0 0
\(83\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(84\) 0.164642 1.72421i 0.164642 1.72421i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.458227 0.888835i \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(90\) 0 0
\(91\) −0.154560 + 0.637106i −0.154560 + 0.637106i
\(92\) 0 0
\(93\) −0.981929 1.70075i −0.981929 1.70075i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.44091 1.37391i 1.44091 1.37391i 0.654861 0.755750i \(-0.272727\pi\)
0.786053 0.618159i \(-0.212121\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −0.995472 + 0.0950560i −0.995472 + 0.0950560i
\(101\) 0 0 −0.945001 0.327068i \(-0.893939\pi\)
0.945001 + 0.327068i \(0.106061\pi\)
\(102\) 0 0
\(103\) −0.735759 0.105786i −0.735759 0.105786i −0.235759 0.971812i \(-0.575758\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(108\) −0.841254 0.540641i −0.841254 0.540641i
\(109\) 0.340611 0.850806i 0.340611 0.850806i −0.654861 0.755750i \(-0.727273\pi\)
0.995472 0.0950560i \(-0.0303030\pi\)
\(110\) 0 0
\(111\) 0.415415 0.719520i 0.415415 0.719520i
\(112\) 0.327793 1.70075i 0.327793 1.70075i
\(113\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.286053 + 0.247866i 0.286053 + 0.247866i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.786053 0.618159i −0.786053 0.618159i
\(122\) 0 0
\(123\) 0 0
\(124\) −0.815816 1.78639i −0.815816 1.78639i
\(125\) 0 0
\(126\) 0 0
\(127\) −1.70566 + 0.879330i −1.70566 + 0.879330i −0.723734 + 0.690079i \(0.757576\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(128\) 0 0
\(129\) −0.273507 + 0.384087i −0.273507 + 0.384087i
\(130\) 0 0
\(131\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(132\) 0 0
\(133\) 0.931791 2.32750i 0.931791 2.32750i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(138\) 0 0
\(139\) −1.30425 0.451405i −1.30425 0.451405i −0.415415 0.909632i \(-0.636364\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.786053 0.618159i −0.786053 0.618159i
\(145\) 0 0
\(146\) 0 0
\(147\) −0.830830 + 1.81926i −0.830830 + 1.81926i
\(148\) 0.481929 0.676774i 0.481929 0.676774i
\(149\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(150\) 0 0
\(151\) −0.481929 1.05528i −0.481929 1.05528i −0.981929 0.189251i \(-0.939394\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0.261197 + 0.273935i 0.261197 + 0.273935i
\(157\) 0.190112i 0.190112i 0.995472 + 0.0950560i \(0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.0395325 0.0865641i 0.0395325 0.0865641i −0.888835 0.458227i \(-0.848485\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(168\) 0 0
\(169\) 0.496956 + 0.697876i 0.496956 + 0.697876i
\(170\) 0 0
\(171\) −0.947890 1.09392i −0.947890 1.09392i
\(172\) −0.308779 + 0.356349i −0.308779 + 0.356349i
\(173\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(174\) 0 0
\(175\) 1.68323 + 0.408346i 1.68323 + 0.408346i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(180\) 0 0
\(181\) −0.512546 + 1.74557i −0.512546 + 1.74557i 0.142315 + 0.989821i \(0.454545\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(182\) 0 0
\(183\) −1.12459 0.513585i −1.12459 0.513585i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.07068 + 1.36148i 1.07068 + 1.36148i
\(190\) 0 0
\(191\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(192\) −0.723734 0.690079i −0.723734 0.690079i
\(193\) 0.880786 + 0.627205i 0.880786 + 0.627205i 0.928368 0.371662i \(-0.121212\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(197\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(198\) 0 0
\(199\) 0.425839 1.45027i 0.425839 1.45027i −0.415415 0.909632i \(-0.636364\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(200\) 0 0
\(201\) 0.841254 1.45709i 0.841254 1.45709i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.233975 + 0.297523i 0.233975 + 0.297523i
\(209\) 0 0
\(210\) 0 0
\(211\) 1.36611 1.43273i 1.36611 1.43273i 0.580057 0.814576i \(-0.303030\pi\)
0.786053 0.618159i \(-0.212121\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.323333 + 3.38610i 0.323333 + 3.38610i
\(218\) 0 0
\(219\) −1.28656 + 1.22673i −1.28656 + 1.22673i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.374650 + 1.94387i 0.374650 + 1.94387i 0.327068 + 0.945001i \(0.393939\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(224\) 0 0
\(225\) 0.654861 0.755750i 0.654861 0.755750i
\(226\) 0 0
\(227\) 0 0 −0.971812 0.235759i \(-0.924242\pi\)
0.971812 + 0.235759i \(0.0757576\pi\)
\(228\) −0.839614 1.17907i −0.839614 1.17907i
\(229\) 0.279486 0.0538665i 0.279486 0.0538665i −0.0475819 0.998867i \(-0.515152\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.998867 0.0475819i \(-0.0151515\pi\)
−0.998867 + 0.0475819i \(0.984848\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.45949 + 0.584293i 1.45949 + 0.584293i
\(238\) 0 0
\(239\) 0 0 0.690079 0.723734i \(-0.257576\pi\)
−0.690079 + 0.723734i \(0.742424\pi\)
\(240\) 0 0
\(241\) −1.25544 1.31666i −1.25544 1.31666i −0.928368 0.371662i \(-0.878788\pi\)
−0.327068 0.945001i \(-0.606061\pi\)
\(242\) 0 0
\(243\) 0.981929 0.189251i 0.981929 0.189251i
\(244\) −1.07068 0.618159i −1.07068 0.618159i
\(245\) 0 0
\(246\) 0 0
\(247\) 0.251049 + 0.486967i 0.251049 + 0.486967i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(252\) 0.936417 + 1.45709i 0.936417 + 1.45709i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.654861 0.755750i −0.654861 0.755750i
\(257\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(258\) 0 0
\(259\) −1.17221 + 0.834725i −1.17221 + 0.834725i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0.975950 1.37053i 0.975950 1.37053i
\(269\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(270\) 0 0
\(271\) 1.11312 0.326842i 1.11312 0.326842i 0.327068 0.945001i \(-0.393939\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(272\) 0 0
\(273\) −0.272340 0.596342i −0.272340 0.596342i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.273100 + 0.0801894i 0.273100 + 0.0801894i 0.415415 0.909632i \(-0.363636\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(278\) 0 0
\(279\) 1.82318 + 0.729892i 1.82318 + 0.729892i
\(280\) 0 0
\(281\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(282\) 0 0
\(283\) −1.88786 + 0.653395i −1.88786 + 0.653395i −0.928368 + 0.371662i \(0.878788\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.841254 0.540641i −0.841254 0.540641i
\(290\) 0 0
\(291\) −0.283341 + 1.97068i −0.283341 + 1.97068i
\(292\) −1.39734 + 1.09888i −1.39734 + 1.09888i
\(293\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0.723734 0.690079i 0.723734 0.690079i
\(301\) 0.707277 0.408346i 0.707277 0.408346i
\(302\) 0 0
\(303\) 0 0
\(304\) −0.723734 1.25354i −0.723734 1.25354i
\(305\) 0 0
\(306\) 0 0
\(307\) −1.82318 0.351390i −1.82318 0.351390i −0.841254 0.540641i \(-0.818182\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(308\) 0 0
\(309\) 0.643738 0.371662i 0.643738 0.371662i
\(310\) 0 0
\(311\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(312\) 0 0
\(313\) −0.179656 + 1.88144i −0.179656 + 1.88144i 0.235759 + 0.971812i \(0.424242\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.39734 + 0.720381i 1.39734 + 0.720381i
\(317\) 0 0 0.618159 0.786053i \(-0.287879\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.995472 0.0950560i 0.995472 0.0950560i
\(325\) −0.308319 + 0.219553i −0.308319 + 0.219553i
\(326\) 0 0
\(327\) 0.258195 + 0.879330i 0.258195 + 0.879330i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.62731 + 1.15880i −1.62731 + 1.15880i −0.786053 + 0.618159i \(0.787879\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(332\) 0 0
\(333\) 0.118239 + 0.822373i 0.118239 + 0.822373i
\(334\) 0 0
\(335\) 0 0
\(336\) 0.793672 + 1.53951i 0.793672 + 1.53951i
\(337\) 1.05080 1.63508i 1.05080 1.63508i 0.327068 0.945001i \(-0.393939\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.30900 1.13425i 1.30900 1.13425i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.189251 0.981929i \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(348\) 0 0
\(349\) −0.452418 1.86489i −0.452418 1.86489i −0.500000 0.866025i \(-0.666667\pi\)
0.0475819 0.998867i \(-0.484848\pi\)
\(350\) 0 0
\(351\) −0.378074 0.0180099i −0.378074 0.0180099i
\(352\) 0 0
\(353\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.945001 0.327068i \(-0.893939\pi\)
0.945001 + 0.327068i \(0.106061\pi\)
\(360\) 0 0
\(361\) −0.358193 1.03493i −0.358193 1.03493i
\(362\) 0 0
\(363\) 1.00000 1.00000
\(364\) −0.214421 0.619529i −0.214421 0.619529i
\(365\) 0 0
\(366\) 0 0
\(367\) −0.0623191 0.0719200i −0.0623191 0.0719200i 0.723734 0.690079i \(-0.242424\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 1.74555 + 0.899892i 1.74555 + 0.899892i
\(373\) −1.99547 0.0950560i −1.99547 0.0950560i −0.995472 0.0950560i \(-0.969697\pi\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.88786 + 0.0899299i −1.88786 + 0.0899299i −0.959493 0.281733i \(-0.909091\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(380\) 0 0
\(381\) 0.797176 1.74557i 0.797176 1.74557i
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.0224357 0.470984i −0.0224357 0.470984i
\(388\) −0.469383 + 1.93482i −0.469383 + 1.93482i
\(389\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.00000 1.00000
\(398\) 0 0
\(399\) 0.706328 + 2.40553i 0.706328 + 2.40553i
\(400\) 0.786053 0.618159i 0.786053 0.618159i
\(401\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(402\) 0 0
\(403\) −0.605495 0.431171i −0.605495 0.431171i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.651174 1.88144i 0.651174 1.88144i 0.235759 0.971812i \(-0.424242\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.676152 0.308788i 0.676152 0.308788i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.30425 0.451405i 1.30425 0.451405i
\(418\) 0 0
\(419\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(420\) 0 0
\(421\) −0.915415 0.0436066i −0.915415 0.0436066i −0.415415 0.909632i \(-0.636364\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.40230 + 1.61834i 1.40230 + 1.61834i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 1.00000 1.00000
\(433\) 0.213947 + 0.618159i 0.213947 + 0.618159i 1.00000 \(0\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.173440 + 0.899892i 0.173440 + 0.899892i
\(437\) 0 0
\(438\) 0 0
\(439\) 1.21590 + 0.486774i 1.21590 + 0.486774i 0.888835 0.458227i \(-0.151515\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(440\) 0 0
\(441\) −0.471518 1.94362i −0.471518 1.94362i
\(442\) 0 0
\(443\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(444\) 0.0395325 + 0.829889i 0.0395325 + 0.829889i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0.643738 + 1.60798i 0.643738 + 1.60798i
\(449\) 0 0 −0.0950560 0.995472i \(-0.530303\pi\)
0.0950560 + 0.995472i \(0.469697\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.03115 + 0.531595i 1.03115 + 0.531595i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.880786 + 1.37053i −0.880786 + 1.37053i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(462\) 0 0
\(463\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(468\) −0.374650 0.0538665i −0.374650 0.0538665i
\(469\) −2.37383 + 1.69039i −2.37383 + 1.69039i
\(470\) 0 0
\(471\) −0.117519 0.149438i −0.117519 0.149438i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.28656 0.663268i 1.28656 0.663268i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(480\) 0 0
\(481\) 0.0298924 0.313047i 0.0298924 0.313047i
\(482\) 0 0
\(483\) 0 0
\(484\) 0.995472 + 0.0950560i 0.995472 + 0.0950560i
\(485\) 0 0
\(486\) 0 0
\(487\) −1.88431 0.363170i −1.88431 0.363170i −0.888835 0.458227i \(-0.848485\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(488\) 0 0
\(489\) 0.0224357 + 0.0924813i 0.0224357 + 0.0924813i
\(490\) 0 0
\(491\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.65210 + 1.06174i 1.65210 + 1.06174i
\(497\) 0 0
\(498\) 0 0
\(499\) −1.23576 + 0.971812i −1.23576 + 0.971812i −0.235759 + 0.971812i \(0.575758\pi\)
−1.00000 \(1.00000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.822032 0.241370i −0.822032 0.241370i
\(508\) 0.959493 1.66189i 0.959493 1.66189i
\(509\) 0 0 0.189251 0.981929i \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(510\) 0 0
\(511\) 2.90967 1.00705i 2.90967 1.00705i
\(512\) 0 0
\(513\) 1.42131 + 0.273935i 1.42131 + 0.273935i
\(514\) 0 0
\(515\) 0 0
\(516\) 0.0224357 0.470984i 0.0224357 0.470984i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(522\) 0 0
\(523\) 0.273100 0.0801894i 0.273100 0.0801894i −0.142315 0.989821i \(-0.545455\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(524\) 0 0
\(525\) −1.57553 + 0.719520i −1.57553 + 0.719520i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.786053 0.618159i −0.786053 0.618159i
\(530\) 0 0
\(531\) 0 0
\(532\) 0.474470 + 2.46178i 0.474470 + 2.46178i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.39734 1.09888i −1.39734 1.09888i −0.981929 0.189251i \(-0.939394\pi\)
−0.415415 0.909632i \(-0.636364\pi\)
\(542\) 0 0
\(543\) −0.676152 1.68895i −0.676152 1.68895i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.632425 1.22673i −0.632425 1.22673i −0.959493 0.281733i \(-0.909091\pi\)
0.327068 0.945001i \(-0.393939\pi\)
\(548\) 0 0
\(549\) 1.20147 0.291473i 1.20147 0.291473i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.87906 1.97070i −1.87906 1.97070i
\(554\) 0 0
\(555\) 0 0
\(556\) 1.34125 0.325385i 1.34125 0.325385i
\(557\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(558\) 0 0
\(559\) −0.0337758 + 0.175245i −0.0337758 + 0.175245i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.68323 0.408346i −1.68323 0.408346i
\(568\) 0 0
\(569\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(570\) 0 0
\(571\) −0.357685 1.85585i −0.357685 1.85585i −0.500000 0.866025i \(-0.666667\pi\)
0.142315 0.989821i \(-0.454545\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.995472 + 0.0950560i 0.995472 + 0.0950560i
\(577\) 0.188177 + 1.97068i 0.188177 + 1.97068i 0.235759 + 0.971812i \(0.424242\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(578\) 0 0
\(579\) −1.08006 + 0.0514495i −1.08006 + 0.0514495i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(588\) −0.284630 1.97964i −0.284630 1.97964i
\(589\) 2.05730 + 1.96163i 2.05730 + 1.96163i
\(590\) 0 0
\(591\) 0 0
\(592\) −0.0395325 + 0.829889i −0.0395325 + 0.829889i
\(593\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.561767 + 1.40323i 0.561767 + 1.40323i
\(598\) 0 0
\(599\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(600\) 0 0
\(601\) −1.23123 0.876756i −1.23123 0.876756i −0.235759 0.971812i \(-0.575758\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(602\) 0 0
\(603\) 0.239446 + 1.66538i 0.239446 + 1.66538i
\(604\) 0.975950 + 0.627205i 0.975950 + 0.627205i
\(605\) 0 0
\(606\) 0 0
\(607\) 0.154861 + 0.110276i 0.154861 + 0.110276i 0.654861 0.755750i \(-0.272727\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −0.158746 + 0.540641i −0.158746 + 0.540641i 0.841254 + 0.540641i \(0.181818\pi\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −1.05080 0.254922i −1.05080 0.254922i −0.327068 0.945001i \(-0.606061\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −0.367833 0.0892353i −0.367833 0.0892353i
\(625\) 0.580057 + 0.814576i 0.580057 + 0.814576i
\(626\) 0 0
\(627\) 0 0
\(628\) −0.102782 0.159932i −0.102782 0.159932i
\(629\) 0 0
\(630\) 0 0
\(631\) −0.544078 + 1.19136i −0.544078 + 1.19136i 0.415415 + 0.909632i \(0.363636\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(632\) 0 0
\(633\) −0.188177 + 1.97068i −0.188177 + 1.97068i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.757005i 0.757005i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(642\) 0 0
\(643\) 0.827068 + 1.81103i 0.827068 + 1.81103i 0.500000 + 0.866025i \(0.333333\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −2.34730 2.46178i −2.34730 2.46178i
\(652\) 0.0135432 + 0.0941952i 0.0135432 + 0.0941952i
\(653\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.252989 1.75958i 0.252989 1.75958i
\(658\) 0 0
\(659\) 0 0 −0.189251 0.981929i \(-0.560606\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(660\) 0 0
\(661\) 0.459493 1.14776i 0.459493 1.14776i −0.500000 0.866025i \(-0.666667\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1.49611 1.29639i −1.49611 1.29639i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.223734 0.175946i −0.223734 0.175946i 0.500000 0.866025i \(-0.333333\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(674\) 0 0
\(675\) −0.0475819 + 0.998867i −0.0475819 + 0.998867i
\(676\) −0.795366 0.318417i −0.795366 0.318417i
\(677\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(678\) 0 0
\(679\) 1.86435 2.90099i 1.86435 2.90099i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) 1.38884 + 0.407799i 1.38884 + 0.407799i
\(685\) 0 0
\(686\) 0 0
\(687\) −0.186393 + 0.215109i −0.186393 + 0.215109i
\(688\) 0.0671040 0.466718i 0.0671040 0.466718i
\(689\) 0 0
\(690\) 0 0
\(691\) 1.61257 + 0.231852i 1.61257 + 0.231852i 0.888835 0.458227i \(-0.151515\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.63679 + 0.566498i −1.63679 + 0.566498i
\(701\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(702\) 0 0
\(703\) −0.283524 + 1.16870i −0.283524 + 1.16870i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.327068 + 0.566498i −0.327068 + 0.566498i −0.981929 0.189251i \(-0.939394\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(710\) 0 0
\(711\) −1.50842 + 0.442913i −1.50842 + 0.442913i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(720\) 0 0
\(721\) −1.28165 + 0.122382i −1.28165 + 0.122382i
\(722\) 0 0
\(723\) 1.80075 + 0.258908i 1.80075 + 0.258908i
\(724\) −0.512546 1.74557i −0.512546 1.74557i
\(725\) 0 0
\(726\) 0 0
\(727\) −1.65033 + 0.660694i −1.65033 + 0.660694i −0.995472 0.0950560i \(-0.969697\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(728\) 0 0
\(729\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(730\) 0 0
\(731\) 0 0
\(732\) 1.22373 0.175946i 1.22373 0.175946i
\(733\) 0.866049 + 1.67990i 0.866049 + 1.67990i 0.723734 + 0.690079i \(0.242424\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0.668404 1.04006i 0.668404 1.04006i −0.327068 0.945001i \(-0.606061\pi\)
0.995472 0.0950560i \(-0.0303030\pi\)
\(740\) 0 0
\(741\) −0.498360 0.227594i −0.498360 0.227594i
\(742\) 0 0
\(743\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.50842 0.442913i 1.50842 0.442913i 0.580057 0.814576i \(-0.303030\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −1.63679 0.566498i −1.63679 0.566498i
\(757\) 1.84833 0.739959i 1.84833 0.739959i 0.888835 0.458227i \(-0.151515\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(762\) 0 0
\(763\) 0.225902 1.57119i 0.225902 1.57119i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0.981929 + 0.189251i 0.981929 + 0.189251i
\(769\) 0.0845850 + 0.0436066i 0.0845850 + 0.0436066i 0.500000 0.866025i \(-0.333333\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.08006 0.0514495i −1.08006 0.0514495i
\(773\) 0 0 0.618159 0.786053i \(-0.287879\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(774\) 0 0
\(775\) −1.13915 + 1.59971i −1.13915 + 1.59971i
\(776\) 0 0
\(777\) 0.405424 1.38075i 0.405424 1.38075i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.0951638 1.99773i −0.0951638 1.99773i
\(785\) 0 0
\(786\) 0 0
\(787\) 0.547582 0.132842i 0.547582 0.132842i 0.0475819 0.998867i \(-0.484848\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.467949 −0.467949
\(794\) 0 0
\(795\) 0 0
\(796\) 0.425839 + 1.45027i 0.425839 + 1.45027i
\(797\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0.0800569 + 1.68060i 0.0800569 + 1.68060i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(810\) 0 0
\(811\) 0.0535608 0.182411i 0.0535608 0.182411i −0.928368 0.371662i \(-0.878788\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(812\) 0 0
\(813\) −0.672932 + 0.945001i −0.672932 + 0.945001i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.223226 0.644970i 0.223226 0.644970i
\(818\) 0 0
\(819\) 0.582708 + 0.300407i 0.582708 + 0.300407i
\(820\) 0 0
\(821\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(822\) 0 0
\(823\) −1.34378 1.28129i −1.34378 1.28129i −0.928368 0.371662i \(-0.878788\pi\)
−0.415415 0.909632i \(-0.636364\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(828\) 0 0
\(829\) 1.16011 1.16011 0.580057 0.814576i \(-0.303030\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(830\) 0 0
\(831\) −0.264241 + 0.105786i −0.264241 + 0.105786i
\(832\) −0.357685 0.123796i −0.357685 0.123796i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.88431 + 0.553283i −1.88431 + 0.553283i
\(838\) 0 0
\(839\) 0 0 0.458227 0.888835i \(-0.348485\pi\)
−0.458227 + 0.888835i \(0.651515\pi\)
\(840\) 0 0
\(841\) 0.0475819 + 0.998867i 0.0475819 + 0.998867i
\(842\) 0 0
\(843\) 0 0
\(844\) −0.374650 + 1.94387i −0.374650 + 1.94387i
\(845\) 0 0
\(846\) 0 0
\(847\) −1.57553 0.719520i −1.57553 0.719520i
\(848\) 0 0
\(849\) 1.08006 1.68060i 1.08006 1.68060i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0.117519 0.149438i 0.117519 0.149438i −0.723734 0.690079i \(-0.757576\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0.759713 0.876756i 0.759713 0.876756i −0.235759 0.971812i \(-0.575758\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.995472 0.0950560i 0.995472 0.0950560i
\(868\) −2.10267 2.67376i −2.10267 2.67376i
\(869\) 0 0
\(870\) 0 0
\(871\) 0.0605348 0.633949i 0.0605348 0.633949i
\(872\) 0 0
\(873\) −0.995472 1.72421i −0.995472 1.72421i
\(874\) 0 0
\(875\) 0 0
\(876\) 0.419102 1.72756i 0.419102 1.72756i
\(877\) −0.959493 + 0.281733i −0.959493 + 0.281733i −0.723734 0.690079i \(-0.757576\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(882\) 0 0
\(883\) 0.495472 + 0.961081i 0.495472 + 0.961081i 0.995472 + 0.0950560i \(0.0303030\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(888\) 0 0
\(889\) −2.51195 + 2.17661i −2.51195 + 2.17661i
\(890\) 0 0
\(891\) 0 0
\(892\) −1.36611 1.43273i −1.36611 1.43273i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(901\) 0 0
\(902\) 0 0
\(903\) −0.303534 + 0.758192i −0.303534 + 0.758192i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.80075 0.822373i 1.80075 0.822373i 0.841254 0.540641i \(-0.181818\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(912\) 1.34378 + 0.537970i 1.34378 + 0.537970i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −0.205996 + 0.196417i −0.205996 + 0.196417i
\(917\) 0 0
\(918\) 0 0
\(919\) −1.49611 1.29639i −1.49611 1.29639i −0.841254 0.540641i \(-0.818182\pi\)
−0.654861 0.755750i \(-0.727273\pi\)
\(920\) 0 0
\(921\) 1.65033 0.850806i 1.65033 0.850806i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −0.827068 0.0789754i −0.827068 0.0789754i
\(926\) 0 0
\(927\) −0.276266 + 0.690079i −0.276266 + 0.690079i
\(928\) 0 0
\(929\) 0 0 −0.189251 0.981929i \(-0.560606\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(930\) 0 0
\(931\) 0.411992 2.86547i 0.411992 2.86547i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.746170 + 0.782560i 0.746170 + 0.782560i 0.981929 0.189251i \(-0.0606061\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(938\) 0 0
\(939\) −1.02181 1.58997i −1.02181 1.58997i
\(940\) 0 0
\(941\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(948\) −1.54370 + 0.297523i −1.54370 + 0.297523i
\(949\) −0.250074 + 0.624655i −0.250074 + 0.624655i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.971812 0.235759i \(-0.0757576\pi\)
−0.971812 + 0.235759i \(0.924242\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.74102 0.804835i −2.74102 0.804835i
\(962\) 0 0
\(963\) 0 0
\(964\) 1.76798 + 0.428908i 1.76798 + 0.428908i
\(965\) 0 0
\(966\) 0 0
\(967\) 0.0947329 0.00904590i 0.0947329 0.00904590i −0.0475819 0.998867i \(-0.515152\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −0.723734 + 0.690079i −0.723734 + 0.690079i
\(973\) −2.37968 0.227232i −2.37968 0.227232i
\(974\) 0 0
\(975\) 0.106636 0.363170i 0.106636 0.363170i
\(976\) 1.23492 0.0588264i 1.23492 0.0588264i
\(977\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −0.746521 0.531595i −0.746521 0.531595i
\(982\) 0 0
\(983\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −0.474470 0.273935i −0.474470 0.273935i
\(989\) 0 0
\(990\) 0 0
\(991\) 0.722372 + 1.80440i 0.722372 + 1.80440i 0.580057 + 0.814576i \(0.303030\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(992\) 0 0
\(993\) 0.562827 1.91681i 0.562827 1.91681i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.57553 + 0.909632i 1.57553 + 0.909632i 0.995472 + 0.0950560i \(0.0303030\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(998\) 0 0
\(999\) −0.601300 0.573338i −0.601300 0.573338i
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 1191.1.z.a.824.1 20
3.2 odd 2 CM 1191.1.z.a.824.1 20
397.225 even 66 inner 1191.1.z.a.1019.1 yes 20
1191.1019 odd 66 inner 1191.1.z.a.1019.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1191.1.z.a.824.1 20 1.1 even 1 trivial
1191.1.z.a.824.1 20 3.2 odd 2 CM
1191.1.z.a.1019.1 yes 20 397.225 even 66 inner
1191.1.z.a.1019.1 yes 20 1191.1019 odd 66 inner