Properties

Label 1407.1.cb.a.446.1
Level $1407$
Weight $1$
Character 1407.446
Analytic conductor $0.702$
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] = [1407,1,Mod(101,1407)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1407, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([33, 11, 65]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1407.101");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1407 = 3 \cdot 7 \cdot 67 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1407.cb (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: \(0.702184472775\)
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 446.1
Root \(-0.888835 - 0.458227i\) of defining polynomial
Character \(\chi\) \(=\) 1407.446
Dual form 1407.1.cb.a.836.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.928368 - 0.371662i) q^{3} +(-0.841254 - 0.540641i) q^{4} +(0.327068 - 0.945001i) q^{7} +(0.723734 - 0.690079i) q^{9} +O(q^{10})\) \(q+(0.928368 - 0.371662i) q^{3} +(-0.841254 - 0.540641i) q^{4} +(0.327068 - 0.945001i) q^{7} +(0.723734 - 0.690079i) q^{9} +(-0.981929 - 0.189251i) q^{12} +(-1.88431 - 0.363170i) q^{13} +(0.415415 + 0.909632i) q^{16} +(-0.348311 - 1.18624i) q^{19} +(-0.0475819 - 0.998867i) q^{21} +(0.981929 + 0.189251i) q^{25} +(0.415415 - 0.909632i) q^{27} +(-0.786053 + 0.618159i) q^{28} +(0.544078 + 0.627899i) q^{31} +(-0.981929 + 0.189251i) q^{36} -0.654136 q^{37} +(-1.88431 + 0.363170i) q^{39} +(0.817178 + 1.27155i) q^{43} +(0.723734 + 0.690079i) q^{48} +(-0.786053 - 0.618159i) q^{49} +(1.38884 + 1.32425i) q^{52} +(-0.764241 - 0.971812i) q^{57} +(0.653077 - 1.43004i) q^{61} +(-0.415415 - 0.909632i) q^{63} +(0.142315 - 0.989821i) q^{64} +(0.327068 + 0.945001i) q^{67} +(0.154861 - 0.110276i) q^{73} +(0.981929 - 0.189251i) q^{75} +(-0.348311 + 1.18624i) q^{76} +(0.357685 - 1.85585i) q^{79} +(0.0475819 - 0.998867i) q^{81} +(-0.500000 + 0.866025i) q^{84} +(-0.959493 + 1.66189i) q^{91} +(0.738471 + 0.380708i) q^{93} +(0.928368 + 1.60798i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{3} + 2 q^{4} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + q^{3} + 2 q^{4} - q^{7} + q^{9} - q^{12} + 2 q^{13} - 2 q^{16} - q^{21} + q^{25} - 2 q^{27} + q^{28} + 4 q^{31} - q^{36} + 2 q^{37} + 2 q^{39} + q^{48} + q^{49} + 9 q^{52} - 19 q^{57} - 2 q^{61} + 2 q^{63} + 2 q^{64} - q^{67} - 8 q^{73} + q^{75} + 8 q^{79} + q^{81} - 10 q^{84} - 2 q^{91} - 2 q^{93} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1407\mathbb{Z}\right)^\times\).

\(n\) \(337\) \(470\) \(1207\)
\(\chi(n)\) \(e\left(\frac{61}{66}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(3\) 0.928368 0.371662i 0.928368 0.371662i
\(4\) −0.841254 0.540641i −0.841254 0.540641i
\(5\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(6\) 0 0
\(7\) 0.327068 0.945001i 0.327068 0.945001i
\(8\) 0 0
\(9\) 0.723734 0.690079i 0.723734 0.690079i
\(10\) 0 0
\(11\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(12\) −0.981929 0.189251i −0.981929 0.189251i
\(13\) −1.88431 0.363170i −1.88431 0.363170i −0.888835 0.458227i \(-0.848485\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(17\) 0 0 0.998867 0.0475819i \(-0.0151515\pi\)
−0.998867 + 0.0475819i \(0.984848\pi\)
\(18\) 0 0
\(19\) −0.348311 1.18624i −0.348311 1.18624i −0.928368 0.371662i \(-0.878788\pi\)
0.580057 0.814576i \(-0.303030\pi\)
\(20\) 0 0
\(21\) −0.0475819 0.998867i −0.0475819 0.998867i
\(22\) 0 0
\(23\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(24\) 0 0
\(25\) 0.981929 + 0.189251i 0.981929 + 0.189251i
\(26\) 0 0
\(27\) 0.415415 0.909632i 0.415415 0.909632i
\(28\) −0.786053 + 0.618159i −0.786053 + 0.618159i
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0.544078 + 0.627899i 0.544078 + 0.627899i 0.959493 0.281733i \(-0.0909091\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.981929 + 0.189251i −0.981929 + 0.189251i
\(37\) −0.654136 −0.654136 −0.327068 0.945001i \(-0.606061\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(38\) 0 0
\(39\) −1.88431 + 0.363170i −1.88431 + 0.363170i
\(40\) 0 0
\(41\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(42\) 0 0
\(43\) 0.817178 + 1.27155i 0.817178 + 1.27155i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(48\) 0.723734 + 0.690079i 0.723734 + 0.690079i
\(49\) −0.786053 0.618159i −0.786053 0.618159i
\(50\) 0 0
\(51\) 0 0
\(52\) 1.38884 + 1.32425i 1.38884 + 1.32425i
\(53\) 0 0 −0.998867 0.0475819i \(-0.984848\pi\)
0.998867 + 0.0475819i \(0.0151515\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.764241 0.971812i −0.764241 0.971812i
\(58\) 0 0
\(59\) 0 0 −0.945001 0.327068i \(-0.893939\pi\)
0.945001 + 0.327068i \(0.106061\pi\)
\(60\) 0 0
\(61\) 0.653077 1.43004i 0.653077 1.43004i −0.235759 0.971812i \(-0.575758\pi\)
0.888835 0.458227i \(-0.151515\pi\)
\(62\) 0 0
\(63\) −0.415415 0.909632i −0.415415 0.909632i
\(64\) 0.142315 0.989821i 0.142315 0.989821i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.327068 + 0.945001i 0.327068 + 0.945001i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(72\) 0 0
\(73\) 0.154861 0.110276i 0.154861 0.110276i −0.500000 0.866025i \(-0.666667\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(74\) 0 0
\(75\) 0.981929 0.189251i 0.981929 0.189251i
\(76\) −0.348311 + 1.18624i −0.348311 + 1.18624i
\(77\) 0 0
\(78\) 0 0
\(79\) 0.357685 1.85585i 0.357685 1.85585i −0.142315 0.989821i \(-0.545455\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(80\) 0 0
\(81\) 0.0475819 0.998867i 0.0475819 0.998867i
\(82\) 0 0
\(83\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(84\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.371662 0.928368i \(-0.378788\pi\)
−0.371662 + 0.928368i \(0.621212\pi\)
\(90\) 0 0
\(91\) −0.959493 + 1.66189i −0.959493 + 1.66189i
\(92\) 0 0
\(93\) 0.738471 + 0.380708i 0.738471 + 0.380708i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.928368 + 1.60798i 0.928368 + 1.60798i 0.786053 + 0.618159i \(0.212121\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −0.723734 0.690079i −0.723734 0.690079i
\(101\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(102\) 0 0
\(103\) 1.83673 + 0.635697i 1.83673 + 0.635697i 0.995472 + 0.0950560i \(0.0303030\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(108\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(109\) −0.344298 + 1.78639i −0.344298 + 1.78639i 0.235759 + 0.971812i \(0.424242\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(110\) 0 0
\(111\) −0.607279 + 0.243118i −0.607279 + 0.243118i
\(112\) 0.995472 0.0950560i 0.995472 0.0950560i
\(113\) 0 0 0.814576 0.580057i \(-0.196970\pi\)
−0.814576 + 0.580057i \(0.803030\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.61435 + 1.03748i −1.61435 + 1.03748i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.654861 0.755750i 0.654861 0.755750i
\(122\) 0 0
\(123\) 0 0
\(124\) −0.118239 0.822373i −0.118239 0.822373i
\(125\) 0 0
\(126\) 0 0
\(127\) −1.42131 + 1.35522i −1.42131 + 1.35522i −0.580057 + 0.814576i \(0.696970\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(128\) 0 0
\(129\) 1.23123 + 0.876756i 1.23123 + 0.876756i
\(130\) 0 0
\(131\) 0 0 0.618159 0.786053i \(-0.287879\pi\)
−0.618159 + 0.786053i \(0.712121\pi\)
\(132\) 0 0
\(133\) −1.23492 0.0588264i −1.23492 0.0588264i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.371662 0.928368i \(-0.621212\pi\)
0.371662 + 0.928368i \(0.378788\pi\)
\(138\) 0 0
\(139\) −0.118239 0.258908i −0.118239 0.258908i 0.841254 0.540641i \(-0.181818\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.928368 + 0.371662i 0.928368 + 0.371662i
\(145\) 0 0
\(146\) 0 0
\(147\) −0.959493 0.281733i −0.959493 0.281733i
\(148\) 0.550294 + 0.353653i 0.550294 + 0.353653i
\(149\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(150\) 0 0
\(151\) 0.0947329 + 1.98869i 0.0947329 + 1.98869i 0.142315 + 0.989821i \(0.454545\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.78153 + 0.713215i 1.78153 + 0.713215i
\(157\) 0.348311 0.442913i 0.348311 0.442913i −0.580057 0.814576i \(-0.696970\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.888835 + 1.53951i 0.888835 + 1.53951i 0.841254 + 0.540641i \(0.181818\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.971812 0.235759i \(-0.924242\pi\)
0.971812 + 0.235759i \(0.0757576\pi\)
\(168\) 0 0
\(169\) 2.49035 + 0.996987i 2.49035 + 0.996987i
\(170\) 0 0
\(171\) −1.07068 0.618159i −1.07068 0.618159i
\(172\) 1.51150i 1.51150i
\(173\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(174\) 0 0
\(175\) 0.500000 0.866025i 0.500000 0.866025i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(180\) 0 0
\(181\) −1.22373 + 1.55610i −1.22373 + 1.55610i −0.500000 + 0.866025i \(0.666667\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(182\) 0 0
\(183\) 0.0748038 1.57033i 0.0748038 1.57033i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −0.723734 0.690079i −0.723734 0.690079i
\(190\) 0 0
\(191\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(192\) −0.235759 0.971812i −0.235759 0.971812i
\(193\) 0.473420 1.36786i 0.473420 1.36786i −0.415415 0.909632i \(-0.636364\pi\)
0.888835 0.458227i \(-0.151515\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.327068 + 0.945001i 0.327068 + 0.945001i
\(197\) 0 0 −0.458227 0.888835i \(-0.651515\pi\)
0.458227 + 0.888835i \(0.348485\pi\)
\(198\) 0 0
\(199\) −1.37859 1.44583i −1.37859 1.44583i −0.723734 0.690079i \(-0.757576\pi\)
−0.654861 0.755750i \(-0.727273\pi\)
\(200\) 0 0
\(201\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −0.452418 1.86489i −0.452418 1.86489i
\(209\) 0 0
\(210\) 0 0
\(211\) −0.911911 + 0.717135i −0.911911 + 0.717135i −0.959493 0.281733i \(-0.909091\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.771316 0.308788i 0.771316 0.308788i
\(218\) 0 0
\(219\) 0.102782 0.159932i 0.102782 0.159932i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.735759 0.105786i 0.735759 0.105786i 0.235759 0.971812i \(-0.424242\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(224\) 0 0
\(225\) 0.841254 0.540641i 0.841254 0.540641i
\(226\) 0 0
\(227\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(228\) 0.117519 + 1.23072i 0.117519 + 1.23072i
\(229\) −0.857685 0.989821i −0.857685 0.989821i 0.142315 0.989821i \(-0.454545\pi\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −0.357685 1.85585i −0.357685 1.85585i
\(238\) 0 0
\(239\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) 0 0
\(241\) 0.0359789 + 0.376789i 0.0359789 + 0.376789i 0.995472 + 0.0950560i \(0.0303030\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(242\) 0 0
\(243\) −0.327068 0.945001i −0.327068 0.945001i
\(244\) −1.32254 + 0.849945i −1.32254 + 0.849945i
\(245\) 0 0
\(246\) 0 0
\(247\) 0.225518 + 2.36173i 0.225518 + 2.36173i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(252\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(257\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(258\) 0 0
\(259\) −0.213947 + 0.618159i −0.213947 + 0.618159i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0.235759 0.971812i 0.235759 0.971812i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) −0.370638 0.291473i −0.370638 0.291473i 0.415415 0.909632i \(-0.363636\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(272\) 0 0
\(273\) −0.273100 + 1.89945i −0.273100 + 1.89945i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −0.0224357 0.0924813i −0.0224357 0.0924813i 0.959493 0.281733i \(-0.0909091\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(278\) 0 0
\(279\) 0.827068 + 0.0789754i 0.827068 + 0.0789754i
\(280\) 0 0
\(281\) 0 0 0.189251 0.981929i \(-0.439394\pi\)
−0.189251 + 0.981929i \(0.560606\pi\)
\(282\) 0 0
\(283\) −1.12424 + 1.17907i −1.12424 + 1.17907i −0.142315 + 0.989821i \(0.545455\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.995472 0.0950560i 0.995472 0.0950560i
\(290\) 0 0
\(291\) 1.45949 + 1.14776i 1.45949 + 1.14776i
\(292\) −0.189897 + 0.00904590i −0.189897 + 0.00904590i
\(293\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −0.928368 0.371662i −0.928368 0.371662i
\(301\) 1.46889 0.356349i 1.46889 0.356349i
\(302\) 0 0
\(303\) 0 0
\(304\) 0.934347 0.809616i 0.934347 0.809616i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.02181 + 0.353653i 1.02181 + 0.353653i 0.786053 0.618159i \(-0.212121\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(308\) 0 0
\(309\) 1.94142 0.0924813i 1.94142 0.0924813i
\(310\) 0 0
\(311\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(312\) 0 0
\(313\) −1.34378 0.537970i −1.34378 0.537970i −0.415415 0.909632i \(-0.636364\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −1.30425 + 1.36786i −1.30425 + 1.36786i
\(317\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.580057 + 0.814576i −0.580057 + 0.814576i
\(325\) −1.78153 0.713215i −1.78153 0.713215i
\(326\) 0 0
\(327\) 0.344298 + 1.78639i 0.344298 + 1.78639i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.97740 0.0941952i −1.97740 0.0941952i −0.981929 0.189251i \(-0.939394\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(332\) 0 0
\(333\) −0.473420 + 0.451405i −0.473420 + 0.451405i
\(334\) 0 0
\(335\) 0 0
\(336\) 0.888835 0.458227i 0.888835 0.458227i
\(337\) 0.367833 0.0892353i 0.367833 0.0892353i −0.0475819 0.998867i \(-0.515152\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(348\) 0 0
\(349\) 0.746170 1.16106i 0.746170 1.16106i −0.235759 0.971812i \(-0.575758\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(350\) 0 0
\(351\) −1.11312 + 1.56316i −1.11312 + 1.56316i
\(352\) 0 0
\(353\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(360\) 0 0
\(361\) −0.444587 + 0.285719i −0.444587 + 0.285719i
\(362\) 0 0
\(363\) 0.327068 0.945001i 0.327068 0.945001i
\(364\) 1.70566 0.879330i 1.70566 0.879330i
\(365\) 0 0
\(366\) 0 0
\(367\) −0.165101 1.14831i −0.165101 1.14831i −0.888835 0.458227i \(-0.848485\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −0.415415 0.719520i −0.415415 0.719520i
\(373\) 0.487975 0.281733i 0.487975 0.281733i −0.235759 0.971812i \(-0.575758\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.512953 1.28129i −0.512953 1.28129i −0.928368 0.371662i \(-0.878788\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(380\) 0 0
\(381\) −0.815816 + 1.78639i −0.815816 + 1.78639i
\(382\) 0 0
\(383\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.46889 + 0.356349i 1.46889 + 0.356349i
\(388\) 0.0883470 1.85463i 0.0883470 1.85463i
\(389\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.0871144 0.912303i 0.0871144 0.912303i −0.841254 0.540641i \(-0.818182\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(398\) 0 0
\(399\) −1.16832 + 0.404360i −1.16832 + 0.404360i
\(400\) 0.235759 + 0.971812i 0.235759 + 0.971812i
\(401\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(402\) 0 0
\(403\) −0.797176 1.38075i −0.797176 1.38075i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.91030 0.560914i −1.91030 0.560914i −0.981929 0.189251i \(-0.939394\pi\)
−0.928368 0.371662i \(-0.878788\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.20147 1.52779i −1.20147 1.52779i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −0.205996 0.196417i −0.205996 0.196417i
\(418\) 0 0
\(419\) 0 0 0.458227 0.888835i \(-0.348485\pi\)
−0.458227 + 0.888835i \(0.651515\pi\)
\(420\) 0 0
\(421\) 0.947890 + 0.903811i 0.947890 + 0.903811i 0.995472 0.0950560i \(-0.0303030\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.13779 1.08488i −1.13779 1.08488i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 1.00000 1.00000
\(433\) 1.92837 0.371662i 1.92837 0.371662i 0.928368 0.371662i \(-0.121212\pi\)
1.00000 \(0\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.25544 1.31666i 1.25544 1.31666i
\(437\) 0 0
\(438\) 0 0
\(439\) 0.916453i 0.916453i 0.888835 + 0.458227i \(0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(440\) 0 0
\(441\) −0.995472 + 0.0950560i −0.995472 + 0.0950560i
\(442\) 0 0
\(443\) 0 0 0.458227 0.888835i \(-0.348485\pi\)
−0.458227 + 0.888835i \(0.651515\pi\)
\(444\) 0.642315 + 0.123796i 0.642315 + 0.123796i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.888835 0.458227i −0.888835 0.458227i
\(449\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0.827068 + 1.81103i 0.827068 + 1.81103i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.462997 0.0892353i −0.462997 0.0892353i −0.0475819 0.998867i \(-0.515152\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(462\) 0 0
\(463\) 0.131192 + 1.37391i 0.131192 + 1.37391i 0.786053 + 0.618159i \(0.212121\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(468\) 1.91899 1.91899
\(469\) 1.00000 1.00000
\(470\) 0 0
\(471\) 0.158746 0.540641i 0.158746 0.540641i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −0.117519 1.23072i −0.117519 1.23072i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.371662 0.928368i \(-0.378788\pi\)
−0.371662 + 0.928368i \(0.621212\pi\)
\(480\) 0 0
\(481\) 1.23259 + 0.237563i 1.23259 + 0.237563i
\(482\) 0 0
\(483\) 0 0
\(484\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(485\) 0 0
\(486\) 0 0
\(487\) 0.495472 + 0.961081i 0.495472 + 0.961081i 0.995472 + 0.0950560i \(0.0303030\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(488\) 0 0
\(489\) 1.39734 + 1.09888i 1.39734 + 1.09888i
\(490\) 0 0
\(491\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −0.345139 + 0.755750i −0.345139 + 0.755750i
\(497\) 0 0
\(498\) 0 0
\(499\) 0.378502i 0.378502i 0.981929 + 0.189251i \(0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.68251 2.68251
\(508\) 1.92837 0.371662i 1.92837 0.371662i
\(509\) 0 0 0.814576 0.580057i \(-0.196970\pi\)
−0.814576 + 0.580057i \(0.803030\pi\)
\(510\) 0 0
\(511\) −0.0535608 0.182411i −0.0535608 0.182411i
\(512\) 0 0
\(513\) −1.22373 0.175946i −1.22373 0.175946i
\(514\) 0 0
\(515\) 0 0
\(516\) −0.561767 1.40323i −0.561767 1.40323i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(522\) 0 0
\(523\) −1.04305 0.903811i −1.04305 0.903811i −0.0475819 0.998867i \(-0.515152\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(524\) 0 0
\(525\) 0.142315 0.989821i 0.142315 0.989821i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.959493 0.281733i −0.959493 0.281733i
\(530\) 0 0
\(531\) 0 0
\(532\) 1.00707 + 0.717135i 1.00707 + 0.717135i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.189897 1.98869i 0.189897 1.98869i 0.0475819 0.998867i \(-0.484848\pi\)
0.142315 0.989821i \(-0.454545\pi\)
\(542\) 0 0
\(543\) −0.557730 + 1.89945i −0.557730 + 1.89945i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.0871144 0.912303i 0.0871144 0.912303i −0.841254 0.540641i \(-0.818182\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(548\) 0 0
\(549\) −0.514186 1.48564i −0.514186 1.48564i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.63679 0.945001i −1.63679 0.945001i
\(554\) 0 0
\(555\) 0 0
\(556\) −0.0405070 + 0.281733i −0.0405070 + 0.281733i
\(557\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(558\) 0 0
\(559\) −1.07802 2.69277i −1.07802 2.69277i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.928368 0.371662i −0.928368 0.371662i
\(568\) 0 0
\(569\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(570\) 0 0
\(571\) −0.264241 1.83784i −0.264241 1.83784i −0.500000 0.866025i \(-0.666667\pi\)
0.235759 0.971812i \(-0.424242\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.580057 0.814576i −0.580057 0.814576i
\(577\) −1.67489 + 1.07639i −1.67489 + 1.07639i −0.786053 + 0.618159i \(0.787879\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(578\) 0 0
\(579\) −0.0688733 1.44583i −0.0688733 1.44583i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(588\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(589\) 0.555330 0.864110i 0.555330 0.864110i
\(590\) 0 0
\(591\) 0 0
\(592\) −0.271738 0.595023i −0.271738 0.595023i
\(593\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.81720 0.829889i −1.81720 0.829889i
\(598\) 0 0
\(599\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(600\) 0 0
\(601\) 1.58323 0.384087i 1.58323 0.384087i 0.654861 0.755750i \(-0.272727\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(602\) 0 0
\(603\) 0.888835 + 0.458227i 0.888835 + 0.458227i
\(604\) 0.995472 1.72421i 0.995472 1.72421i
\(605\) 0 0
\(606\) 0 0
\(607\) −1.37859 0.0656706i −1.37859 0.0656706i −0.654861 0.755750i \(-0.727273\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0.0883470 + 0.0353688i 0.0883470 + 0.0353688i 0.415415 0.909632i \(-0.363636\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(618\) 0 0
\(619\) −0.172932 + 1.81103i −0.172932 + 1.81103i 0.327068 + 0.945001i \(0.393939\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −1.11312 1.56316i −1.11312 1.56316i
\(625\) 0.928368 + 0.371662i 0.928368 + 0.371662i
\(626\) 0 0
\(627\) 0 0
\(628\) −0.532475 + 0.184291i −0.532475 + 0.184291i
\(629\) 0 0
\(630\) 0 0
\(631\) −1.87076 0.647478i −1.87076 0.647478i −0.981929 0.189251i \(-0.939394\pi\)
−0.888835 0.458227i \(-0.848485\pi\)
\(632\) 0 0
\(633\) −0.580057 + 1.00469i −0.580057 + 1.00469i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.25667 + 1.45027i 1.25667 + 1.45027i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −0.833635 0.380708i −0.833635 0.380708i −0.0475819 0.998867i \(-0.515152\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.601300 0.573338i 0.601300 0.573338i
\(652\) 0.0845850 1.77566i 0.0845850 1.77566i
\(653\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.0359789 0.186677i 0.0359789 0.186677i
\(658\) 0 0
\(659\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(660\) 0 0
\(661\) 0.396666 + 1.63508i 0.396666 + 1.63508i 0.723734 + 0.690079i \(0.242424\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0.643738 0.371662i 0.643738 0.371662i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.61257 + 0.231852i 1.61257 + 0.231852i 0.888835 0.458227i \(-0.151515\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(674\) 0 0
\(675\) 0.580057 0.814576i 0.580057 0.814576i
\(676\) −1.55601 2.18511i −1.55601 2.18511i
\(677\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(678\) 0 0
\(679\) 1.82318 0.351390i 1.82318 0.351390i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.945001 0.327068i \(-0.106061\pi\)
−0.945001 + 0.327068i \(0.893939\pi\)
\(684\) 0.566514 + 1.09888i 0.566514 + 1.09888i
\(685\) 0 0
\(686\) 0 0
\(687\) −1.16413 0.600149i −1.16413 0.600149i
\(688\) −0.817178 + 1.27155i −0.817178 + 1.27155i
\(689\) 0 0
\(690\) 0 0
\(691\) −0.0359789 0.376789i −0.0359789 0.376789i −0.995472 0.0950560i \(-0.969697\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −0.888835 + 0.458227i −0.888835 + 0.458227i
\(701\) 0 0 −0.189251 0.981929i \(-0.560606\pi\)
0.189251 + 0.981929i \(0.439394\pi\)
\(702\) 0 0
\(703\) 0.227843 + 0.775961i 0.227843 + 0.775961i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.186393 + 0.215109i 0.186393 + 0.215109i 0.841254 0.540641i \(-0.181818\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(710\) 0 0
\(711\) −1.02181 1.58997i −1.02181 1.58997i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(720\) 0 0
\(721\) 1.20147 1.52779i 1.20147 1.52779i
\(722\) 0 0
\(723\) 0.173440 + 0.336426i 0.173440 + 0.336426i
\(724\) 1.87076 0.647478i 1.87076 0.647478i
\(725\) 0 0
\(726\) 0 0
\(727\) −1.39734 + 1.09888i −1.39734 + 1.09888i −0.415415 + 0.909632i \(0.636364\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(728\) 0 0
\(729\) −0.654861 0.755750i −0.654861 0.755750i
\(730\) 0 0
\(731\) 0 0
\(732\) −0.911911 + 1.28060i −0.911911 + 1.28060i
\(733\) −0.0224357 + 0.470984i −0.0224357 + 0.470984i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0.746170 + 0.782560i 0.746170 + 0.782560i 0.981929 0.189251i \(-0.0606061\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(740\) 0 0
\(741\) 1.08713 + 2.10874i 1.08713 + 2.10874i
\(742\) 0 0
\(743\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0.0845850 0.0436066i 0.0845850 0.0436066i −0.415415 0.909632i \(-0.636364\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0.235759 + 0.971812i 0.235759 + 0.971812i
\(757\) 0.117519 0.149438i 0.117519 0.149438i −0.723734 0.690079i \(-0.757576\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.998867 0.0475819i \(-0.0151515\pi\)
−0.998867 + 0.0475819i \(0.984848\pi\)
\(762\) 0 0
\(763\) 1.57553 + 0.909632i 1.57553 + 0.909632i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.327068 + 0.945001i −0.327068 + 0.945001i
\(769\) −1.21590 0.486774i −1.21590 0.486774i −0.327068 0.945001i \(-0.606061\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.13779 + 0.894765i −1.13779 + 0.894765i
\(773\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(774\) 0 0
\(775\) 0.415415 + 0.719520i 0.415415 + 0.719520i
\(776\) 0 0
\(777\) 0.0311250 + 0.653395i 0.0311250 + 0.653395i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.235759 0.971812i 0.235759 0.971812i
\(785\) 0 0
\(786\) 0 0
\(787\) 0.0623191 + 1.30824i 0.0623191 + 1.30824i 0.786053 + 0.618159i \(0.212121\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.74994 + 2.45745i −1.74994 + 2.45745i
\(794\) 0 0
\(795\) 0 0
\(796\) 0.378074 + 1.96163i 0.378074 + 1.96163i
\(797\) 0 0 −0.690079 0.723734i \(-0.742424\pi\)
0.690079 + 0.723734i \(0.257576\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −0.142315 0.989821i −0.142315 0.989821i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.814576 0.580057i \(-0.803030\pi\)
0.814576 + 0.580057i \(0.196970\pi\)
\(810\) 0 0
\(811\) −1.42131 + 1.35522i −1.42131 + 1.35522i −0.580057 + 0.814576i \(0.696970\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(812\) 0 0
\(813\) −0.452418 0.132842i −0.452418 0.132842i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.22373 1.41226i 1.22373 1.41226i
\(818\) 0 0
\(819\) 0.452418 + 1.86489i 0.452418 + 1.86489i
\(820\) 0 0
\(821\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(822\) 0 0
\(823\) −0.111165 + 0.458227i −0.111165 + 0.458227i 0.888835 + 0.458227i \(0.151515\pi\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(828\) 0 0
\(829\) 0.378074 1.96163i 0.378074 1.96163i 0.142315 0.989821i \(-0.454545\pi\)
0.235759 0.971812i \(-0.424242\pi\)
\(830\) 0 0
\(831\) −0.0552004 0.0775182i −0.0552004 0.0775182i
\(832\) −0.627639 + 1.81344i −0.627639 + 1.81344i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.797176 0.234072i 0.797176 0.234072i
\(838\) 0 0
\(839\) 0 0 0.371662 0.928368i \(-0.378788\pi\)
−0.371662 + 0.928368i \(0.621212\pi\)
\(840\) 0 0
\(841\) −0.500000 0.866025i −0.500000 0.866025i
\(842\) 0 0
\(843\) 0 0
\(844\) 1.15486 0.110276i 1.15486 0.110276i
\(845\) 0 0
\(846\) 0 0
\(847\) −0.500000 0.866025i −0.500000 0.866025i
\(848\) 0 0
\(849\) −0.605495 + 1.51245i −0.605495 + 1.51245i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −0.378074 0.0180099i −0.378074 0.0180099i −0.142315 0.989821i \(-0.545455\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(858\) 0 0
\(859\) −0.357685 + 1.85585i −0.357685 + 1.85585i 0.142315 + 0.989821i \(0.454545\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.888835 0.458227i 0.888835 0.458227i
\(868\) −0.815816 0.157236i −0.815816 0.157236i
\(869\) 0 0
\(870\) 0 0
\(871\) −0.273100 1.89945i −0.273100 1.89945i
\(872\) 0 0
\(873\) 1.78153 + 0.523103i 1.78153 + 0.523103i
\(874\) 0 0
\(875\) 0 0
\(876\) −0.172932 + 0.0789754i −0.172932 + 0.0789754i
\(877\) −0.601300 + 1.31666i −0.601300 + 1.31666i 0.327068 + 0.945001i \(0.393939\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.618159 0.786053i \(-0.712121\pi\)
0.618159 + 0.786053i \(0.287879\pi\)
\(882\) 0 0
\(883\) 0.702443 0.243118i 0.702443 0.243118i 0.0475819 0.998867i \(-0.484848\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(888\) 0 0
\(889\) 0.815816 + 1.78639i 0.815816 + 1.78639i
\(890\) 0 0
\(891\) 0 0
\(892\) −0.676152 0.308788i −0.676152 0.308788i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −1.00000 −1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) 1.23123 0.876756i 1.23123 0.876756i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0.759713 + 0.876756i 0.759713 + 0.876756i 0.995472 0.0950560i \(-0.0303030\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(912\) 0.566514 1.09888i 0.566514 1.09888i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0.186393 + 1.29639i 0.186393 + 1.29639i
\(917\) 0 0
\(918\) 0 0
\(919\) 0.0535608 + 0.182411i 0.0535608 + 0.182411i 0.981929 0.189251i \(-0.0606061\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(920\) 0 0
\(921\) 1.08006 0.0514495i 1.08006 0.0514495i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −0.642315 0.123796i −0.642315 0.123796i
\(926\) 0 0
\(927\) 1.76798 0.807410i 1.76798 0.807410i
\(928\) 0 0
\(929\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(930\) 0 0
\(931\) −0.459493 + 1.14776i −0.459493 + 1.14776i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.77767 1.77767 0.888835 0.458227i \(-0.151515\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(938\) 0 0
\(939\) −1.44747 −1.44747
\(940\) 0 0
\(941\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(948\) −0.702443 + 1.75462i −0.702443 + 1.75462i
\(949\) −0.331854 + 0.151553i −0.331854 + 0.151553i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.0440780 0.306569i 0.0440780 0.306569i
\(962\) 0 0
\(963\) 0 0
\(964\) 0.173440 0.336426i 0.173440 0.336426i
\(965\) 0 0
\(966\) 0 0
\(967\) −0.786053 + 1.36148i −0.786053 + 1.36148i 0.142315 + 0.989821i \(0.454545\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.690079 0.723734i \(-0.257576\pi\)
−0.690079 + 0.723734i \(0.742424\pi\)
\(972\) −0.235759 + 0.971812i −0.235759 + 0.971812i
\(973\) −0.283341 + 0.0270558i −0.283341 + 0.0270558i
\(974\) 0 0
\(975\) −1.91899 −1.91899
\(976\) 1.57211 1.57211
\(977\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0.983568 + 1.53046i 0.983568 + 1.53046i
\(982\) 0 0
\(983\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 1.08713 2.10874i 1.08713 2.10874i
\(989\) 0 0
\(990\) 0 0
\(991\) 1.62731 + 0.0775182i 1.62731 + 0.0775182i 0.841254 0.540641i \(-0.181818\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(992\) 0 0
\(993\) −1.87076 + 0.647478i −1.87076 + 0.647478i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.702443 + 0.243118i 0.702443 + 0.243118i 0.654861 0.755750i \(-0.272727\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(998\) 0 0
\(999\) −0.271738 + 0.595023i −0.271738 + 0.595023i
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 1407.1.cb.a.446.1 20
3.2 odd 2 CM 1407.1.cb.a.446.1 20
7.3 odd 6 1407.1.ct.a.647.1 yes 20
21.17 even 6 1407.1.ct.a.647.1 yes 20
67.32 odd 66 1407.1.ct.a.635.1 yes 20
201.32 even 66 1407.1.ct.a.635.1 yes 20
469.367 even 66 inner 1407.1.cb.a.836.1 yes 20
1407.836 odd 66 inner 1407.1.cb.a.836.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1407.1.cb.a.446.1 20 1.1 even 1 trivial
1407.1.cb.a.446.1 20 3.2 odd 2 CM
1407.1.cb.a.836.1 yes 20 469.367 even 66 inner
1407.1.cb.a.836.1 yes 20 1407.836 odd 66 inner
1407.1.ct.a.635.1 yes 20 67.32 odd 66
1407.1.ct.a.635.1 yes 20 201.32 even 66
1407.1.ct.a.647.1 yes 20 7.3 odd 6
1407.1.ct.a.647.1 yes 20 21.17 even 6