Properties

Label 469.1.bl.a.55.1
Level $469$
Weight $1$
Character 469.55
Analytic conductor $0.234$
Analytic rank $0$
Dimension $20$
Projective image $D_{33}$
CM discriminant -7
Inner twists $4$

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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 55.1
Root \(-0.327068 - 0.945001i\) of defining polynomial
Character \(\chi\) \(=\) 469.55
Dual form 469.1.bl.a.307.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.56199 + 0.625325i) q^{2} +(1.32503 + 1.26342i) q^{4} +(-0.786053 + 0.618159i) q^{7} +(0.580699 + 1.27155i) q^{8} +(-0.142315 - 0.989821i) q^{9} +O(q^{10})\) \(q+(1.56199 + 0.625325i) q^{2} +(1.32503 + 1.26342i) q^{4} +(-0.786053 + 0.618159i) q^{7} +(0.580699 + 1.27155i) q^{8} +(-0.142315 - 0.989821i) q^{9} +(-1.65033 - 0.850806i) q^{11} +(-1.61435 + 0.474017i) q^{14} +(0.0247953 + 0.520518i) q^{16} +(0.396666 - 1.63508i) q^{18} +(-2.04577 - 2.36094i) q^{22} +(0.651174 + 1.88144i) q^{23} +(0.415415 - 0.909632i) q^{25} +(-1.82254 - 0.174031i) q^{28} +(0.786053 + 1.36148i) q^{29} +(0.170438 - 0.492448i) q^{32} +(1.06199 - 1.49135i) q^{36} +(-0.0475819 + 0.0824143i) q^{37} +(-0.452418 - 0.132842i) q^{43} +(-1.11182 - 3.21241i) q^{44} +(-0.159389 + 3.34598i) q^{46} +(0.235759 - 0.971812i) q^{49} +(1.21769 - 1.16106i) q^{50} +(-1.88431 + 0.553283i) q^{53} +(-1.24248 - 0.640544i) q^{56} +(0.376434 + 2.61816i) q^{58} +(0.723734 + 0.690079i) q^{63} +(0.915415 - 1.05645i) q^{64} +(0.0475819 + 0.998867i) q^{67} +(-1.28656 - 1.22673i) q^{71} +(1.17597 - 0.755750i) q^{72} +(-0.125858 + 0.0989758i) q^{74} +(1.82318 - 0.351390i) q^{77} +(1.30379 - 0.124497i) q^{79} +(-0.959493 + 0.281733i) q^{81} +(-0.623601 - 0.490405i) q^{86} +(0.123498 - 2.59255i) q^{88} +(-1.51422 + 3.31568i) q^{92} +(0.975950 - 1.37053i) q^{98} +(-0.607279 + 1.75462i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} + 3 q^{4} + q^{7} - 8 q^{8} - 2 q^{9}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/469\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\)
\(\chi(n)\) \(-1\) \(e\left(\frac{4}{33}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.56199 + 0.625325i 1.56199 + 0.625325i 0.981929 0.189251i \(-0.0606061\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(3\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(4\) 1.32503 + 1.26342i 1.32503 + 1.26342i
\(5\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(6\) 0 0
\(7\) −0.786053 + 0.618159i −0.786053 + 0.618159i
\(8\) 0.580699 + 1.27155i 0.580699 + 1.27155i
\(9\) −0.142315 0.989821i −0.142315 0.989821i
\(10\) 0 0
\(11\) −1.65033 0.850806i −1.65033 0.850806i −0.995472 0.0950560i \(-0.969697\pi\)
−0.654861 0.755750i \(-0.727273\pi\)
\(12\) 0 0
\(13\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(14\) −1.61435 + 0.474017i −1.61435 + 0.474017i
\(15\) 0 0
\(16\) 0.0247953 + 0.520518i 0.0247953 + 0.520518i
\(17\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(18\) 0.396666 1.63508i 0.396666 1.63508i
\(19\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.04577 2.36094i −2.04577 2.36094i
\(23\) 0.651174 + 1.88144i 0.651174 + 1.88144i 0.415415 + 0.909632i \(0.363636\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(24\) 0 0
\(25\) 0.415415 0.909632i 0.415415 0.909632i
\(26\) 0 0
\(27\) 0 0
\(28\) −1.82254 0.174031i −1.82254 0.174031i
\(29\) 0.786053 + 1.36148i 0.786053 + 1.36148i 0.928368 + 0.371662i \(0.121212\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(30\) 0 0
\(31\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(32\) 0.170438 0.492448i 0.170438 0.492448i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.06199 1.49135i 1.06199 1.49135i
\(37\) −0.0475819 + 0.0824143i −0.0475819 + 0.0824143i −0.888835 0.458227i \(-0.848485\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(42\) 0 0
\(43\) −0.452418 0.132842i −0.452418 0.132842i 0.0475819 0.998867i \(-0.484848\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) −1.11182 3.21241i −1.11182 3.21241i
\(45\) 0 0
\(46\) −0.159389 + 3.34598i −0.159389 + 3.34598i
\(47\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(48\) 0 0
\(49\) 0.235759 0.971812i 0.235759 0.971812i
\(50\) 1.21769 1.16106i 1.21769 1.16106i
\(51\) 0 0
\(52\) 0 0
\(53\) −1.88431 + 0.553283i −1.88431 + 0.553283i −0.888835 + 0.458227i \(0.848485\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.24248 0.640544i −1.24248 0.640544i
\(57\) 0 0
\(58\) 0.376434 + 2.61816i 0.376434 + 2.61816i
\(59\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(60\) 0 0
\(61\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(62\) 0 0
\(63\) 0.723734 + 0.690079i 0.723734 + 0.690079i
\(64\) 0.915415 1.05645i 0.915415 1.05645i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.0475819 + 0.998867i 0.0475819 + 0.998867i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.28656 1.22673i −1.28656 1.22673i −0.959493 0.281733i \(-0.909091\pi\)
−0.327068 0.945001i \(-0.606061\pi\)
\(72\) 1.17597 0.755750i 1.17597 0.755750i
\(73\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(74\) −0.125858 + 0.0989758i −0.125858 + 0.0989758i
\(75\) 0 0
\(76\) 0 0
\(77\) 1.82318 0.351390i 1.82318 0.351390i
\(78\) 0 0
\(79\) 1.30379 0.124497i 1.30379 0.124497i 0.580057 0.814576i \(-0.303030\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(80\) 0 0
\(81\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(82\) 0 0
\(83\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.623601 0.490405i −0.623601 0.490405i
\(87\) 0 0
\(88\) 0.123498 2.59255i 0.123498 2.59255i
\(89\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.51422 + 3.31568i −1.51422 + 3.31568i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 0.975950 1.37053i 0.975950 1.37053i
\(99\) −0.607279 + 1.75462i −0.607279 + 1.75462i
\(100\) 1.69968 0.680451i 1.69968 0.680451i
\(101\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(102\) 0 0
\(103\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −3.28924 0.314085i −3.28924 0.314085i
\(107\) 0.396666 + 0.254922i 0.396666 + 0.254922i 0.723734 0.690079i \(-0.242424\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(108\) 0 0
\(109\) 0.481929 1.05528i 0.481929 1.05528i −0.500000 0.866025i \(-0.666667\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.341254 0.393828i −0.341254 0.393828i
\(113\) −0.0475819 + 0.998867i −0.0475819 + 0.998867i 0.841254 + 0.540641i \(0.181818\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.678575 + 2.79713i −0.678575 + 2.79713i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.41967 + 1.99365i 1.41967 + 1.99365i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0.698939 + 1.53046i 0.698939 + 1.53046i
\(127\) 1.50842 1.18624i 1.50842 1.18624i 0.580057 0.814576i \(-0.303030\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(128\) 1.62731 0.838935i 1.62731 0.838935i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −0.550294 + 1.58997i −0.550294 + 1.58997i
\(135\) 0 0
\(136\) 0 0
\(137\) 0.428368 0.494363i 0.428368 0.494363i −0.500000 0.866025i \(-0.666667\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(138\) 0 0
\(139\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.24248 2.72066i −1.24248 2.72066i
\(143\) 0 0
\(144\) 0.511691 0.0986204i 0.511691 0.0986204i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −0.167171 + 0.0490859i −0.167171 + 0.0490859i
\(149\) 0.142315 0.989821i 0.142315 0.989821i −0.786053 0.618159i \(-0.787879\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(150\) 0 0
\(151\) −0.205996 + 0.196417i −0.205996 + 0.196417i −0.786053 0.618159i \(-0.787879\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 3.06752 + 0.591215i 3.06752 + 0.591215i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(158\) 2.11435 + 0.620830i 2.11435 + 0.620830i
\(159\) 0 0
\(160\) 0 0
\(161\) −1.67489 1.07639i −1.67489 1.07639i
\(162\) −1.67489 0.159932i −1.67489 0.159932i
\(163\) 0.327068 + 0.566498i 0.327068 + 0.566498i 0.981929 0.189251i \(-0.0606061\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(168\) 0 0
\(169\) −0.327068 + 0.945001i −0.327068 + 0.945001i
\(170\) 0 0
\(171\) 0 0
\(172\) −0.431635 0.747613i −0.431635 0.747613i
\(173\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(174\) 0 0
\(175\) 0.235759 + 0.971812i 0.235759 + 0.971812i
\(176\) 0.401939 0.880124i 0.401939 0.880124i
\(177\) 0 0
\(178\) 0 0
\(179\) −0.947890 1.09392i −0.947890 1.09392i −0.995472 0.0950560i \(-0.969697\pi\)
0.0475819 0.998867i \(-0.484848\pi\)
\(180\) 0 0
\(181\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2.01422 + 1.92056i −2.01422 + 1.92056i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.74555 + 0.336426i −1.74555 + 0.336426i −0.959493 0.281733i \(-0.909091\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(192\) 0 0
\(193\) −0.271738 0.595023i −0.271738 0.595023i 0.723734 0.690079i \(-0.242424\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.54019 0.989821i 1.54019 0.989821i
\(197\) 0.0688733 + 0.0656706i 0.0688733 + 0.0656706i 0.723734 0.690079i \(-0.242424\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(198\) −2.04577 + 2.36094i −2.04577 + 2.36094i
\(199\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(200\) 1.39788 1.39788
\(201\) 0 0
\(202\) 0 0
\(203\) −1.45949 0.584293i −1.45949 0.584293i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.76962 0.912303i 1.76962 0.912303i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.981929 + 0.189251i −0.981929 + 0.189251i −0.654861 0.755750i \(-0.727273\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(212\) −3.19580 1.64755i −3.19580 1.64755i
\(213\) 0 0
\(214\) 0.460178 + 0.646229i 0.460178 + 0.646229i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 1.41266 1.34697i 1.41266 1.34697i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(224\) 0.170438 + 0.492448i 0.170438 + 0.492448i
\(225\) −0.959493 0.281733i −0.959493 0.281733i
\(226\) −0.698939 + 1.53046i −0.698939 + 1.53046i
\(227\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(228\) 0 0
\(229\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.27474 + 1.79012i −1.27474 + 1.79012i
\(233\) 0.514186 1.48564i 0.514186 1.48564i −0.327068 0.945001i \(-0.606061\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.995472 + 1.72421i 0.995472 + 1.72421i 0.580057 + 0.814576i \(0.303030\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(240\) 0 0
\(241\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(242\) 0.970828 + 4.00181i 0.970828 + 4.00181i
\(243\) 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 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(252\) 0.0871144 + 1.82876i 0.0871144 + 1.82876i
\(253\) 0.526089 3.65903i 0.526089 3.65903i
\(254\) 3.09792 0.909632i 3.09792 0.909632i
\(255\) 0 0
\(256\) 1.67489 0.159932i 1.67489 0.159932i
\(257\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(258\) 0 0
\(259\) −0.0135432 0.0941952i −0.0135432 0.0941952i
\(260\) 0 0
\(261\) 1.23576 0.971812i 1.23576 0.971812i
\(262\) 0 0
\(263\) 1.21769 0.782560i 1.21769 0.782560i 0.235759 0.971812i \(-0.424242\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.19894 + 1.38365i −1.19894 + 1.38365i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0.978242 0.504319i 0.978242 0.504319i
\(275\) −1.45949 + 1.14776i −1.45949 + 1.14776i
\(276\) 0 0
\(277\) 0.0405070 + 0.281733i 0.0405070 + 0.281733i 1.00000 \(0\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.481929 + 0.676774i 0.481929 + 0.676774i 0.981929 0.189251i \(-0.0606061\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 0 0
\(283\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(284\) −0.154861 3.25093i −0.154861 3.25093i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.511691 0.0986204i −0.511691 0.0986204i
\(289\) 0.0475819 0.998867i 0.0475819 0.998867i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.132425 0.0126451i −0.132425 0.0126451i
\(297\) 0 0
\(298\) 0.841254 1.45709i 0.841254 1.45709i
\(299\) 0 0
\(300\) 0 0
\(301\) 0.437742 0.175245i 0.437742 0.175245i
\(302\) −0.444587 + 0.177986i −0.444587 + 0.177986i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(308\) 2.85973 + 1.83784i 2.85973 + 1.83784i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(312\) 0 0
\(313\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.88486 + 1.48227i 1.88486 + 1.48227i
\(317\) −0.452418 + 1.86489i −0.452418 + 1.86489i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0 0
\(319\) −0.138891 2.91568i −0.138891 2.91568i
\(320\) 0 0
\(321\) 0 0
\(322\) −1.94306 2.72865i −1.94306 2.72865i
\(323\) 0 0
\(324\) −1.62731 0.838935i −1.62731 0.838935i
\(325\) 0 0
\(326\) 0.156630 + 1.08939i 0.156630 + 1.08939i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.473420 0.451405i −0.473420 0.451405i 0.415415 0.909632i \(-0.363636\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(332\) 0 0
\(333\) 0.0883470 + 0.0353688i 0.0883470 + 0.0353688i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.34378 + 0.537970i 1.34378 + 0.537970i 0.928368 0.371662i \(-0.121212\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(338\) −1.10181 + 1.27155i −1.10181 + 1.27155i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(344\) −0.0938031 0.652415i −0.0938031 0.652415i
\(345\) 0 0
\(346\) 0 0
\(347\) −1.15486 + 0.110276i −1.15486 + 0.110276i −0.654861 0.755750i \(-0.727273\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(350\) −0.239446 + 1.66538i −0.239446 + 1.66538i
\(351\) 0 0
\(352\) −0.700257 + 0.667694i −0.700257 + 0.667694i
\(353\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −0.796533 2.30143i −0.796533 2.30143i
\(359\) −1.11312 0.326842i −1.11312 0.326842i −0.327068 0.945001i \(-0.606061\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(360\) 0 0
\(361\) 0.235759 + 0.971812i 0.235759 + 0.971812i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(368\) −0.963180 + 0.385599i −0.963180 + 0.385599i
\(369\) 0 0
\(370\) 0 0
\(371\) 1.13915 1.59971i 1.13915 1.59971i
\(372\) 0 0
\(373\) −0.723734 1.25354i −0.723734 1.25354i −0.959493 0.281733i \(-0.909091\pi\)
0.235759 0.971812i \(-0.424242\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.271738 0.785135i −0.271738 0.785135i −0.995472 0.0950560i \(-0.969697\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2.93689 0.566040i −2.93689 0.566040i
\(383\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.0523681 1.09934i −0.0523681 1.09934i
\(387\) −0.0671040 + 0.466718i −0.0671040 + 0.466718i
\(388\) 0 0
\(389\) 1.13915 + 1.59971i 1.13915 + 1.59971i 0.723734 + 0.690079i \(0.242424\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.37262 0.264550i 1.37262 0.264550i
\(393\) 0 0
\(394\) 0.0665137 + 0.145645i 0.0665137 + 0.145645i
\(395\) 0 0
\(396\) −3.02148 + 1.55768i −3.02148 + 1.55768i
\(397\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.483780 + 0.193676i 0.483780 + 0.193676i
\(401\) −1.30972 −1.30972 −0.654861 0.755750i \(-0.727273\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −1.91433 1.82531i −1.91433 1.82531i
\(407\) 0.148645 0.0955280i 0.148645 0.0955280i
\(408\) 0 0
\(409\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 3.33461 0.318417i 3.33461 0.318417i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(420\) 0 0
\(421\) −0.911911 0.717135i −0.911911 0.717135i 0.0475819 0.998867i \(-0.484848\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(422\) −1.65210 0.318417i −1.65210 0.318417i
\(423\) 0 0
\(424\) −1.79774 2.07471i −1.79774 2.07471i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0.203523 + 0.838935i 0.203523 + 0.838935i
\(429\) 0 0
\(430\) 0 0
\(431\) −0.928368 1.60798i −0.928368 1.60798i −0.786053 0.618159i \(-0.787879\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(432\) 0 0
\(433\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.97183 0.789400i 1.97183 0.789400i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) −0.995472 0.0950560i −0.995472 0.0950560i
\(442\) 0 0
\(443\) −0.452418 1.86489i −0.452418 1.86489i −0.500000 0.866025i \(-0.666667\pi\)
0.0475819 0.998867i \(-0.484848\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.0665137 + 1.39629i −0.0665137 + 1.39629i
\(449\) −1.88431 0.363170i −1.88431 0.363170i −0.888835 0.458227i \(-0.848485\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(450\) −1.32254 1.04006i −1.32254 1.04006i
\(451\) 0 0
\(452\) −1.32503 + 1.26342i −1.32503 + 1.26342i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.76962 0.168978i 1.76962 0.168978i 0.841254 0.540641i \(-0.181818\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(462\) 0 0
\(463\) −1.03115 + 0.531595i −1.03115 + 0.531595i −0.888835 0.458227i \(-0.848485\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(464\) −0.689187 + 0.442913i −0.689187 + 0.442913i
\(465\) 0 0
\(466\) 1.73216 1.99902i 1.73216 1.99902i
\(467\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(468\) 0 0
\(469\) −0.654861 0.755750i −0.654861 0.755750i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.633618 + 0.604153i 0.633618 + 0.604153i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.815816 + 1.78639i 0.815816 + 1.78639i
\(478\) 0.476723 + 3.31568i 0.476723 + 3.31568i
\(479\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −0.637698 + 4.43529i −0.637698 + 4.43529i
\(485\) 0 0
\(486\) 0 0
\(487\) 0.0224357 0.0924813i 0.0224357 0.0924813i −0.959493 0.281733i \(-0.909091\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0.857685 + 0.989821i 0.857685 + 0.989821i 1.00000 \(0\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.76962 + 0.168978i 1.76962 + 0.168978i
\(498\) 0 0
\(499\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(504\) −0.457201 + 1.32100i −0.457201 + 1.32100i
\(505\) 0 0
\(506\) 3.10983 5.38638i 3.10983 5.38638i
\(507\) 0 0
\(508\) 3.49743 + 0.333964i 3.49743 + 0.333964i
\(509\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0.0377483 0.155600i 0.0377483 0.155600i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(522\) 2.53794 0.745205i 2.53794 0.745205i
\(523\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 2.39136 0.460898i 2.39136 0.460898i
\(527\) 0 0
\(528\) 0 0
\(529\) −2.32975 + 1.83214i −2.32975 + 1.83214i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −1.24248 + 0.640544i −1.24248 + 0.640544i
\(537\) 0 0
\(538\) 0 0
\(539\) −1.21590 + 1.40323i −1.21590 + 1.40323i
\(540\) 0 0
\(541\) −1.32254 + 0.849945i −1.32254 + 0.849945i −0.995472 0.0950560i \(-0.969697\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.76962 + 0.912303i 1.76962 + 0.912303i 0.928368 + 0.371662i \(0.121212\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(548\) 1.19219 0.113840i 1.19219 0.113840i
\(549\) 0 0
\(550\) −2.99743 + 0.880124i −2.99743 + 0.880124i
\(551\) 0 0
\(552\) 0 0
\(553\) −0.947890 + 0.903811i −0.947890 + 0.903811i
\(554\) −0.112903 + 0.465392i −0.112903 + 0.465392i
\(555\) 0 0
\(556\) 0 0
\(557\) −0.0623191 + 1.30824i −0.0623191 + 1.30824i 0.723734 + 0.690079i \(0.242424\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.329562 + 1.35847i 0.329562 + 1.35847i
\(563\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.580057 0.814576i 0.580057 0.814576i
\(568\) 0.812753 2.34829i 0.812753 2.34829i
\(569\) 1.56199 0.625325i 1.56199 0.625325i 0.580057 0.814576i \(-0.303030\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(570\) 0 0
\(571\) −0.607279 + 1.75462i −0.607279 + 1.75462i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.98193 + 0.189251i 1.98193 + 0.189251i
\(576\) −1.17597 0.755750i −1.17597 0.755750i
\(577\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(578\) 0.698939 1.53046i 0.698939 1.53046i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 3.58047 + 0.690079i 3.58047 + 0.690079i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.0440780 0.0227238i −0.0440780 0.0227238i
\(593\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.43913 1.13174i 1.43913 1.13174i
\(597\) 0 0
\(598\) 0 0
\(599\) 1.42131 + 1.35522i 1.42131 + 1.35522i 0.841254 + 0.540641i \(0.181818\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(600\) 0 0
\(601\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(602\) 0.793332 0.793332
\(603\) 0.981929 0.189251i 0.981929 0.189251i
\(604\) −0.521109 −0.521109
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.74555 + 0.336426i −1.74555 + 0.336426i −0.959493 0.281733i \(-0.909091\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 1.50553 + 2.11422i 1.50553 + 2.11422i
\(617\) −0.452418 + 0.132842i −0.452418 + 0.132842i −0.500000 0.866025i \(-0.666667\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(618\) 0 0
\(619\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.654861 0.755750i −0.654861 0.755750i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.56499 + 0.149438i 1.56499 + 0.149438i 0.841254 0.540641i \(-0.181818\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(632\) 0.915415 + 1.58555i 0.915415 + 1.58555i
\(633\) 0 0
\(634\) −1.87283 + 2.63003i −1.87283 + 2.63003i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 1.60630 4.64110i 1.60630 4.64110i
\(639\) −1.03115 + 1.44805i −1.03115 + 1.44805i
\(640\) 0 0
\(641\) 0.959493 + 1.66189i 0.959493 + 1.66189i 0.723734 + 0.690079i \(0.242424\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(642\) 0 0
\(643\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(644\) −0.859360 3.54233i −0.859360 3.54233i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(648\) −0.915415 1.05645i −0.915415 1.05645i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −0.282348 + 1.16385i −0.282348 + 1.16385i
\(653\) −0.947890 + 0.903811i −0.947890 + 0.903811i −0.995472 0.0950560i \(-0.969697\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.03115 0.531595i −1.03115 0.531595i −0.142315 0.989821i \(-0.545455\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(660\) 0 0
\(661\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(662\) −0.457201 1.00113i −0.457201 1.00113i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.115880 + 0.110491i 0.115880 + 0.110491i
\(667\) −2.04970 + 2.36548i −2.04970 + 2.36548i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.0623191 + 0.0719200i −0.0623191 + 0.0719200i −0.786053 0.618159i \(-0.787879\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(674\) 1.76256 + 1.68060i 1.76256 + 1.68060i
\(675\) 0 0
\(676\) −1.62731 + 0.838935i −1.62731 + 0.838935i
\(677\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.03115 1.44805i −1.03115 1.44805i −0.888835 0.458227i \(-0.848485\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.0800569 + 1.68060i 0.0800569 + 1.68060i
\(687\) 0 0
\(688\) 0.0579288 0.238786i 0.0579288 0.238786i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(692\) 0 0
\(693\) −0.607279 1.75462i −0.607279 1.75462i
\(694\) −1.87283 0.549914i −1.87283 0.549914i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −0.915415 + 1.58555i −0.915415 + 1.58555i
\(701\) 0.481929 0.676774i 0.481929 0.676774i −0.500000 0.866025i \(-0.666667\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −2.40957 + 0.964646i −2.40957 + 0.964646i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.67489 0.159932i −1.67489 0.159932i −0.786053 0.618159i \(-0.787879\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(710\) 0 0
\(711\) −0.308779 1.27280i −0.308779 1.27280i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.126095 2.64707i 0.126095 2.64707i
\(717\) 0 0
\(718\) −1.53430 1.20658i −1.53430 1.20658i
\(719\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −0.239446 + 1.66538i −0.239446 + 1.66538i
\(723\) 0 0
\(724\) 0 0
\(725\) 1.56499 0.149438i 1.56499 0.149438i
\(726\) 0 0
\(727\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(728\) 0 0
\(729\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 1.03750 1.03750
\(737\) 0.771316 1.68895i 0.771316 1.68895i
\(738\) 0 0
\(739\) 1.82318 + 0.729892i 1.82318 + 0.729892i 0.981929 + 0.189251i \(0.0606061\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.77967 1.78639i 2.77967 1.78639i
\(743\) 1.16413 0.600149i 1.16413 0.600149i 0.235759 0.971812i \(-0.424242\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −0.346590 2.41059i −0.346590 2.41059i
\(747\) 0 0
\(748\) 0 0
\(749\) −0.469383 + 0.0448206i −0.469383 + 0.0448206i
\(750\) 0 0
\(751\) 1.25667 0.368991i 1.25667 0.368991i 0.415415 0.909632i \(-0.363636\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.815816 + 0.157236i 0.815816 + 0.157236i 0.580057 0.814576i \(-0.303030\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(758\) 0.0665137 1.39629i 0.0665137 1.39629i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(762\) 0 0
\(763\) 0.273507 + 1.12741i 0.273507 + 1.12741i
\(764\) −2.73795 1.75958i −2.73795 1.75958i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.391700 1.13174i 0.391700 1.13174i
\(773\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(774\) −0.396666 + 0.687046i −0.396666 + 0.687046i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0.778996 + 3.21106i 0.778996 + 3.21106i
\(779\) 0 0
\(780\) 0 0
\(781\) 1.07954 + 3.11913i 1.07954 + 3.11913i
\(782\) 0 0
\(783\) 0 0
\(784\) 0.511691 + 0.0986204i 0.511691 + 0.0986204i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(788\) 0.00829014 + 0.174031i 0.00829014 + 0.174031i
\(789\) 0 0
\(790\) 0 0
\(791\) −0.580057 0.814576i −0.580057 0.814576i
\(792\) −2.58374 + 0.246717i −2.58374 + 0.246717i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.377144 0.359606i −0.377144 0.359606i
\(801\) 0 0
\(802\) −2.04577 0.819001i −2.04577 0.819001i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.396666 0.254922i 0.396666 0.254922i −0.327068 0.945001i \(-0.606061\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(810\) 0 0
\(811\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(812\) −1.19567 2.61816i −1.19567 2.61816i
\(813\) 0 0
\(814\) 0.291917 0.0562623i 0.291917 0.0562623i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.473420 + 0.451405i −0.473420 + 0.451405i −0.888835 0.458227i \(-0.848485\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(822\) 0 0
\(823\) 0.786053 + 0.618159i 0.786053 + 0.618159i 0.928368 0.371662i \(-0.121212\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −0.473420 1.36786i −0.473420 1.36786i −0.888835 0.458227i \(-0.848485\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(828\) 3.49743 + 1.02694i 3.49743 + 1.02694i
\(829\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(830\) 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 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(840\) 0 0
\(841\) −0.735759 + 1.27437i −0.735759 + 1.27437i
\(842\) −0.975950 1.69039i −0.975950 1.69039i
\(843\) 0 0
\(844\) −1.54019 0.989821i −1.54019 0.989821i
\(845\) 0 0
\(846\) 0 0
\(847\) −2.34833 0.689531i −2.34833 0.689531i
\(848\) −0.334716 0.967098i −0.334716 0.967098i
\(849\) 0 0
\(850\) 0 0
\(851\) −0.186042 0.0358566i −0.186042 0.0358566i
\(852\) 0 0
\(853\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −0.0938031 + 0.652415i −0.0938031 + 0.652415i
\(857\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(858\) 0 0
\(859\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −0.444587 3.09217i −0.444587 3.09217i
\(863\) 0.815816 + 1.78639i 0.815816 + 1.78639i 0.580057 + 0.814576i \(0.303030\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2.25761 0.903811i −2.25761 0.903811i
\(870\) 0 0
\(871\) 0 0
\(872\) 1.62170 1.62170
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.49547 + 0.770969i −1.49547 + 0.770969i −0.995472 0.0950560i \(-0.969697\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(882\) −1.49547 0.770969i −1.49547 0.770969i
\(883\) −1.95496 + 0.186677i −1.95496 + 0.186677i −0.995472 0.0950560i \(-0.969697\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.459493 3.19584i 0.459493 3.19584i
\(887\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(888\) 0 0
\(889\) −0.452418 + 1.86489i −0.452418 + 1.86489i
\(890\) 0 0
\(891\) 1.82318 + 0.351390i 1.82318 + 0.351390i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −0.760554 + 1.66538i −0.760554 + 1.66538i
\(897\) 0 0
\(898\) −2.71616 1.74557i −2.71616 1.74557i
\(899\) 0 0
\(900\) −0.915415 1.58555i −0.915415 1.58555i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −1.29774 + 0.519539i −1.29774 + 0.519539i
\(905\) 0 0
\(906\) 0 0
\(907\) 1.07701 1.51245i 1.07701 1.51245i 0.235759 0.971812i \(-0.424242\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −0.550294 0.353653i −0.550294 0.353653i 0.235759 0.971812i \(-0.424242\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 2.86979 + 0.842646i 2.86979 + 0.842646i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −1.13779 0.894765i −1.13779 0.894765i −0.142315 0.989821i \(-0.545455\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0.0552004 + 0.0775182i 0.0552004 + 0.0775182i
\(926\) −1.94306 + 0.185540i −1.94306 + 0.185540i
\(927\) 0 0
\(928\) 0.804433 0.155042i 0.804433 0.155042i
\(929\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.55830 1.31889i 2.55830 1.31889i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) −0.550294 1.58997i −0.550294 1.58997i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0.611910 + 1.33990i 0.611910 + 1.33990i
\(947\) −0.239446 1.66538i −0.239446 1.66538i −0.654861 0.755750i \(-0.727273\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −0.279486 + 1.94387i −0.279486 + 1.94387i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(954\) 0.157220 + 3.30046i 0.157220 + 3.30046i
\(955\) 0 0
\(956\) −0.859360 + 3.54233i −0.859360 + 3.54233i
\(957\) 0 0
\(958\) 0 0
\(959\) −0.0311250 + 0.653395i −0.0311250 + 0.653395i
\(960\) 0 0
\(961\) −0.327068 0.945001i −0.327068 0.945001i
\(962\) 0 0
\(963\) 0.195876 0.428908i 0.195876 0.428908i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −0.0475819 0.0824143i −0.0475819 0.0824143i 0.841254 0.540641i \(-0.181818\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(968\) −1.71063 + 2.96290i −1.71063 + 2.96290i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0.0928751 0.130425i 0.0928751 0.130425i
\(975\) 0 0
\(976\) 0 0
\(977\) 0.995472 + 0.0950560i 0.995472 + 0.0950560i 0.580057 0.814576i \(-0.303030\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.11312 0.326842i −1.11312 0.326842i
\(982\) 0.720732 + 2.08242i 0.720732 + 2.08242i
\(983\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.0446683 0.937702i −0.0446683 0.937702i
\(990\) 0 0
\(991\) 0.273100 0.0801894i 0.273100 0.0801894i −0.142315 0.989821i \(-0.545455\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 2.65846 + 1.37053i 2.65846 + 1.37053i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(998\) 1.32254 1.04006i 1.32254 1.04006i
\(999\) 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.55.1 20
7.2 even 3 3283.1.bw.a.1060.1 20
7.3 odd 6 3283.1.cd.a.3204.1 20
7.4 even 3 3283.1.cd.a.3204.1 20
7.5 odd 6 3283.1.bw.a.1060.1 20
7.6 odd 2 CM 469.1.bl.a.55.1 20
67.39 even 33 inner 469.1.bl.a.307.1 yes 20
469.39 even 33 3283.1.bw.a.2518.1 20
469.173 odd 66 3283.1.cd.a.374.1 20
469.240 even 33 3283.1.cd.a.374.1 20
469.307 odd 66 inner 469.1.bl.a.307.1 yes 20
469.374 odd 66 3283.1.bw.a.2518.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
469.1.bl.a.55.1 20 1.1 even 1 trivial
469.1.bl.a.55.1 20 7.6 odd 2 CM
469.1.bl.a.307.1 yes 20 67.39 even 33 inner
469.1.bl.a.307.1 yes 20 469.307 odd 66 inner
3283.1.bw.a.1060.1 20 7.2 even 3
3283.1.bw.a.1060.1 20 7.5 odd 6
3283.1.bw.a.2518.1 20 469.39 even 33
3283.1.bw.a.2518.1 20 469.374 odd 66
3283.1.cd.a.374.1 20 469.173 odd 66
3283.1.cd.a.374.1 20 469.240 even 33
3283.1.cd.a.3204.1 20 7.3 odd 6
3283.1.cd.a.3204.1 20 7.4 even 3