Properties

Label 3375.1.g.a.568.8
Level $3375$
Weight $1$
Character 3375.568
Analytic conductor $1.684$
Analytic rank $0$
Dimension $16$
Projective image $D_{30}$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3375,1,Mod(568,3375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3375, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3375.568");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3375 = 3^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3375.g (of order \(4\), degree \(2\), 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: \(1.68434441764\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.6879707136000000000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 21x^{12} + 86x^{8} + 36x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{30}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{30} - \cdots)\)

Embedding invariants

Embedding label 568.8
Root \(0.575212 + 0.575212i\) of defining polynomial
Character \(\chi\) \(=\) 3375.568
Dual form 3375.1.g.a.1432.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.40647 - 1.40647i) q^{2} -2.95630i q^{4} +(-2.75146 - 2.75146i) q^{8} +O(q^{10})\) \(q+(1.40647 - 1.40647i) q^{2} -2.95630i q^{4} +(-2.75146 - 2.75146i) q^{8} -4.78339 q^{16} +(1.05097 - 1.05097i) q^{17} +0.209057i q^{19} +(-0.294032 - 0.294032i) q^{23} -1.33826 q^{31} +(-3.97621 + 3.97621i) q^{32} -2.95630i q^{34} +(0.294032 + 0.294032i) q^{38} -0.827091 q^{46} +(0.831254 - 0.831254i) q^{47} +1.00000i q^{49} +(0.575212 + 0.575212i) q^{53} +1.82709 q^{61} +(-1.88222 + 1.88222i) q^{62} +6.40142i q^{64} +(-3.10696 - 3.10696i) q^{68} +0.618034 q^{76} +1.82709i q^{79} +(-0.575212 - 0.575212i) q^{83} +(-0.869244 + 0.869244i) q^{92} -2.33826i q^{94} +(1.40647 + 1.40647i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{16} - 4 q^{31} + 12 q^{46} + 4 q^{61} - 8 q^{76}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3375\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(2377\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.40647 1.40647i 1.40647 1.40647i 0.629320 0.777146i \(-0.283333\pi\)
0.777146 0.629320i \(-0.216667\pi\)
\(3\) 0 0
\(4\) 2.95630i 2.95630i
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) −2.75146 2.75146i −2.75146 2.75146i
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.78339 −4.78339
\(17\) 1.05097 1.05097i 1.05097 1.05097i 0.0523360 0.998630i \(-0.483333\pi\)
0.998630 0.0523360i \(-0.0166667\pi\)
\(18\) 0 0
\(19\) 0.209057i 0.209057i 0.994522 + 0.104528i \(0.0333333\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.294032 0.294032i −0.294032 0.294032i 0.544639 0.838671i \(-0.316667\pi\)
−0.838671 + 0.544639i \(0.816667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(32\) −3.97621 + 3.97621i −3.97621 + 3.97621i
\(33\) 0 0
\(34\) 2.95630i 2.95630i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0.294032 + 0.294032i 0.294032 + 0.294032i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.827091 −0.827091
\(47\) 0.831254 0.831254i 0.831254 0.831254i −0.156434 0.987688i \(-0.550000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(48\) 0 0
\(49\) 1.00000i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.575212 + 0.575212i 0.575212 + 0.575212i 0.933580 0.358368i \(-0.116667\pi\)
−0.358368 + 0.933580i \(0.616667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 1.82709 1.82709 0.913545 0.406737i \(-0.133333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(62\) −1.88222 + 1.88222i −1.88222 + 1.88222i
\(63\) 0 0
\(64\) 6.40142i 6.40142i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) −3.10696 3.10696i −3.10696 3.10696i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0.618034 0.618034
\(77\) 0 0
\(78\) 0 0
\(79\) 1.82709i 1.82709i 0.406737 + 0.913545i \(0.366667\pi\)
−0.406737 + 0.913545i \(0.633333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.575212 0.575212i −0.575212 0.575212i 0.358368 0.933580i \(-0.383333\pi\)
−0.933580 + 0.358368i \(0.883333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.869244 + 0.869244i −0.869244 + 0.869244i
\(93\) 0 0
\(94\) 2.33826i 2.33826i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) 1.40647 + 1.40647i 1.40647 + 1.40647i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1.61803 1.61803
\(107\) 1.34500 1.34500i 1.34500 1.34500i 0.453990 0.891007i \(-0.350000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(108\) 0 0
\(109\) 1.33826i 1.33826i 0.743145 + 0.669131i \(0.233333\pi\)
−0.743145 + 0.669131i \(0.766667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.831254 + 0.831254i 0.831254 + 0.831254i 0.987688 0.156434i \(-0.0500000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.00000 −1.00000
\(122\) 2.56974 2.56974i 2.56974 2.56974i
\(123\) 0 0
\(124\) 3.95630i 3.95630i
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(128\) 5.02717 + 5.02717i 5.02717 + 5.02717i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −5.78339 −5.78339
\(137\) 1.22474 1.22474i 1.22474 1.22474i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(138\) 0 0
\(139\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(152\) 0.575212 0.575212i 0.575212 0.575212i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) 2.56974 + 2.56974i 2.56974 + 2.56974i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −1.61803 −1.61803
\(167\) 0.575212 0.575212i 0.575212 0.575212i −0.358368 0.933580i \(-0.616667\pi\)
0.933580 + 0.358368i \(0.116667\pi\)
\(168\) 0 0
\(169\) 1.00000i 1.00000i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.22474 1.22474i −1.22474 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0.209057 0.209057 0.104528 0.994522i \(-0.466667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.61803i 1.61803i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −2.45743 2.45743i −2.45743 2.45743i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2.95630 2.95630
\(197\) 0.294032 0.294032i 0.294032 0.294032i −0.544639 0.838671i \(-0.683333\pi\)
0.838671 + 0.544639i \(0.183333\pi\)
\(198\) 0 0
\(199\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(212\) 1.70050 1.70050i 1.70050 1.70050i
\(213\) 0 0
\(214\) 3.78339i 3.78339i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 1.88222 + 1.88222i 1.88222 + 1.88222i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.33826 2.33826
\(227\) −1.05097 + 1.05097i −1.05097 + 1.05097i −0.0523360 + 0.998630i \(0.516667\pi\)
−0.998630 + 0.0523360i \(0.983333\pi\)
\(228\) 0 0
\(229\) 1.95630i 1.95630i 0.207912 + 0.978148i \(0.433333\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(242\) −1.40647 + 1.40647i −1.40647 + 1.40647i
\(243\) 0 0
\(244\) 5.40142i 5.40142i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 3.68218 + 3.68218i 3.68218 + 3.68218i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 7.73968 7.73968
\(257\) −0.575212 + 0.575212i −0.575212 + 0.575212i −0.933580 0.358368i \(-0.883333\pi\)
0.358368 + 0.933580i \(0.383333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.34500 + 1.34500i 1.34500 + 1.34500i 0.891007 + 0.453990i \(0.150000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(272\) −5.02717 + 5.02717i −5.02717 + 5.02717i
\(273\) 0 0
\(274\) 3.44512i 3.44512i
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(278\) −0.869244 0.869244i −0.869244 0.869244i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.20906i 1.20906i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.294032 + 0.294032i 0.294032 + 0.294032i 0.838671 0.544639i \(-0.183333\pi\)
−0.544639 + 0.838671i \(0.683333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 2.27571 2.27571i 2.27571 2.27571i
\(303\) 0 0
\(304\) 1.00000i 1.00000i
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 5.40142 5.40142
\(317\) −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i \(0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.219712 + 0.219712i 0.219712 + 0.219712i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(332\) −1.70050 + 1.70050i −1.70050 + 1.70050i
\(333\) 0 0
\(334\) 1.61803i 1.61803i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(338\) −1.40647 1.40647i −1.40647 1.40647i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −3.44512 −3.44512
\(347\) −0.831254 + 0.831254i −0.831254 + 0.831254i −0.987688 0.156434i \(-0.950000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(348\) 0 0
\(349\) 1.95630i 1.95630i 0.207912 + 0.978148i \(0.433333\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.34500 1.34500i −1.34500 1.34500i −0.891007 0.453990i \(-0.850000\pi\)
−0.453990 0.891007i \(-0.650000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 0.956295 0.956295
\(362\) 0.294032 0.294032i 0.294032 0.294032i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(368\) 1.40647 + 1.40647i 1.40647 + 1.40647i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −4.57433 −4.57433
\(377\) 0 0
\(378\) 0 0
\(379\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.05097 1.05097i −1.05097 1.05097i −0.998630 0.0523360i \(-0.983333\pi\)
−0.0523360 0.998630i \(-0.516667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) −0.618034 −0.618034
\(392\) 2.75146 2.75146i 2.75146 2.75146i
\(393\) 0 0
\(394\) 0.827091i 0.827091i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) 0.869244 + 0.869244i 0.869244 + 0.869244i
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.82709i 1.82709i −0.406737 0.913545i \(-0.633333\pi\)
0.406737 0.913545i \(-0.366667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0.209057 0.209057 0.104528 0.994522i \(-0.466667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(422\) −2.75146 + 2.75146i −2.75146 + 2.75146i
\(423\) 0 0
\(424\) 3.16535i 3.16535i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −3.97621 3.97621i −3.97621 3.97621i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.95630 3.95630
\(437\) 0.0614693 0.0614693i 0.0614693 0.0614693i
\(438\) 0 0
\(439\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.40647 + 1.40647i 1.40647 + 1.40647i 0.777146 + 0.629320i \(0.216667\pi\)
0.629320 + 0.777146i \(0.283333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 2.45743 2.45743i 2.45743 2.45743i
\(453\) 0 0
\(454\) 2.95630i 2.95630i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(458\) 2.75146 + 2.75146i 2.75146 + 2.75146i
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.40647 1.40647i 1.40647 1.40647i 0.629320 0.777146i \(-0.283333\pi\)
0.777146 0.629320i \(-0.216667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1.88222 1.88222i 1.88222 1.88222i
\(483\) 0 0
\(484\) 2.95630i 2.95630i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(488\) −5.02717 5.02717i −5.02717 5.02717i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 6.40142 6.40142
\(497\) 0 0
\(498\) 0 0
\(499\) 1.82709i 1.82709i 0.406737 + 0.913545i \(0.366667\pi\)
−0.406737 + 0.913545i \(0.633333\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 5.85843 5.85843i 5.85843 5.85843i
\(513\) 0 0
\(514\) 1.61803i 1.61803i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 3.78339 3.78339
\(527\) −1.40647 + 1.40647i −1.40647 + 1.40647i
\(528\) 0 0
\(529\) 0.827091i 0.827091i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(542\) −2.75146 + 2.75146i −2.75146 + 2.75146i
\(543\) 0 0
\(544\) 8.35772i 8.35772i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(548\) −3.62071 3.62071i −3.62071 3.62071i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −1.82709 −1.82709
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.34500 1.34500i −1.34500 1.34500i −0.891007 0.453990i \(-0.850000\pi\)
−0.453990 0.891007i \(-0.650000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(578\) −1.70050 1.70050i −1.70050 1.70050i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0.827091 0.827091
\(587\) −0.294032 + 0.294032i −0.294032 + 0.294032i −0.838671 0.544639i \(-0.816667\pi\)
0.544639 + 0.838671i \(0.316667\pi\)
\(588\) 0 0
\(589\) 0.279773i 0.279773i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.40647 1.40647i −1.40647 1.40647i −0.777146 0.629320i \(-0.783333\pi\)
−0.629320 0.777146i \(-0.716667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1.95630 1.95630 0.978148 0.207912i \(-0.0666667\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 4.78339i 4.78339i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(608\) −0.831254 0.831254i −0.831254 0.831254i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.294032 + 0.294032i −0.294032 + 0.294032i −0.838671 0.544639i \(-0.816667\pi\)
0.544639 + 0.838671i \(0.316667\pi\)
\(618\) 0 0
\(619\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(632\) 5.02717 5.02717i 5.02717 5.02717i
\(633\) 0 0
\(634\) 3.44512i 3.44512i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 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 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0.618034 0.618034
\(647\) −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i \(0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.40647 + 1.40647i 1.40647 + 1.40647i 0.777146 + 0.629320i \(0.216667\pi\)
0.629320 + 0.777146i \(0.283333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(662\) −2.27571 + 2.27571i −2.27571 + 2.27571i
\(663\) 0 0
\(664\) 3.16535i 3.16535i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −1.70050 1.70050i −1.70050 1.70050i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −2.95630 −2.95630
\(677\) −0.831254 + 0.831254i −0.831254 + 0.831254i −0.987688 0.156434i \(-0.950000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.294032 + 0.294032i 0.294032 + 0.294032i 0.838671 0.544639i \(-0.183333\pi\)
−0.544639 + 0.838671i \(0.683333\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(692\) −3.62071 + 3.62071i −3.62071 + 3.62071i
\(693\) 0 0
\(694\) 2.33826i 2.33826i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 2.75146 + 2.75146i 2.75146 + 2.75146i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −3.78339 −3.78339
\(707\) 0 0
\(708\) 0 0
\(709\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0.393491 + 0.393491i 0.393491 + 0.393491i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.34500 1.34500i 1.34500 1.34500i
\(723\) 0 0
\(724\) 0.618034i 0.618034i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 2.33826 2.33826
\(737\) 0 0
\(738\) 0 0
\(739\) 1.33826i 1.33826i −0.743145 0.669131i \(-0.766667\pi\)
0.743145 0.669131i \(-0.233333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.82709 −1.82709 −0.913545 0.406737i \(-0.866667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(752\) −3.97621 + 3.97621i −3.97621 + 3.97621i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) 1.40647 + 1.40647i 1.40647 + 1.40647i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −2.95630 −2.95630
\(767\) 0 0
\(768\) 0 0
\(769\) 0.209057i 0.209057i −0.994522 0.104528i \(-0.966667\pi\)
0.994522 0.104528i \(-0.0333333\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.40647 1.40647i −1.40647 1.40647i −0.777146 0.629320i \(-0.783333\pi\)
−0.629320 0.777146i \(-0.716667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −0.869244 + 0.869244i −0.869244 + 0.869244i
\(783\) 0 0
\(784\) 4.78339i 4.78339i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(788\) −0.869244 0.869244i −0.869244 0.869244i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 1.82709 1.82709
\(797\) −1.05097 + 1.05097i −1.05097 + 1.05097i −0.0523360 + 0.998630i \(0.516667\pi\)
−0.998630 + 0.0523360i \(0.983333\pi\)
\(798\) 0 0
\(799\) 1.74724i 1.74724i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −2.56974 2.56974i −2.56974 2.56974i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −0.575212 + 0.575212i −0.575212 + 0.575212i −0.933580 0.358368i \(-0.883333\pi\)
0.358368 + 0.933580i \(0.383333\pi\)
\(828\) 0 0
\(829\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.05097 + 1.05097i 1.05097 + 1.05097i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 0.294032 0.294032i 0.294032 0.294032i
\(843\) 0 0
\(844\) 5.78339i 5.78339i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −2.75146 2.75146i −2.75146 2.75146i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −7.40142 −7.40142
\(857\) −1.40647 + 1.40647i −1.40647 + 1.40647i −0.629320 + 0.777146i \(0.716667\pi\)
−0.777146 + 0.629320i \(0.783333\pi\)
\(858\) 0 0
\(859\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.22474 1.22474i −1.22474 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 3.68218 3.68218i 3.68218 3.68218i
\(873\) 0 0
\(874\) 0.172909i 0.172909i
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(878\) −1.40647 1.40647i −1.40647 1.40647i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 3.95630 3.95630
\(887\) −1.40647 + 1.40647i −1.40647 + 1.40647i −0.629320 + 0.777146i \(0.716667\pi\)
−0.777146 + 0.629320i \(0.783333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0.173779 + 0.173779i 0.173779 + 0.173779i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 1.20906 1.20906
\(902\) 0 0
\(903\) 0 0
\(904\) 4.57433i 4.57433i
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(908\) 3.10696 + 3.10696i 3.10696 + 3.10696i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 5.78339 5.78339
\(917\) 0 0
\(918\) 0 0
\(919\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −0.209057 −0.209057
\(932\) 0 0
\(933\) 0 0
\(934\) 3.95630i 3.95630i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −0.294032 + 0.294032i −0.294032 + 0.294032i −0.838671 0.544639i \(-0.816667\pi\)
0.544639 + 0.838671i \(0.316667\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.34500 + 1.34500i 1.34500 + 1.34500i 0.891007 + 0.453990i \(0.150000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.790943 0.790943
\(962\) 0 0
\(963\) 0 0
\(964\) 3.95630i 3.95630i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(968\) 2.75146 + 2.75146i 2.75146 + 2.75146i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −8.73968 −8.73968
\(977\) −1.34500 + 1.34500i −1.34500 + 1.34500i −0.453990 + 0.891007i \(0.650000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.40647 1.40647i −1.40647 1.40647i −0.777146 0.629320i \(-0.783333\pi\)
−0.629320 0.777146i \(-0.716667\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −0.209057 −0.209057 −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(992\) 5.32121 5.32121i 5.32121 5.32121i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(998\) 2.56974 + 2.56974i 2.56974 + 2.56974i
\(999\) 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 3375.1.g.a.568.8 yes 16
3.2 odd 2 inner 3375.1.g.a.568.1 16
5.2 odd 4 inner 3375.1.g.a.1432.8 yes 16
5.3 odd 4 inner 3375.1.g.a.1432.1 yes 16
5.4 even 2 inner 3375.1.g.a.568.1 16
15.2 even 4 inner 3375.1.g.a.1432.1 yes 16
15.8 even 4 inner 3375.1.g.a.1432.8 yes 16
15.14 odd 2 CM 3375.1.g.a.568.8 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3375.1.g.a.568.1 16 3.2 odd 2 inner
3375.1.g.a.568.1 16 5.4 even 2 inner
3375.1.g.a.568.8 yes 16 1.1 even 1 trivial
3375.1.g.a.568.8 yes 16 15.14 odd 2 CM
3375.1.g.a.1432.1 yes 16 5.3 odd 4 inner
3375.1.g.a.1432.1 yes 16 15.2 even 4 inner
3375.1.g.a.1432.8 yes 16 5.2 odd 4 inner
3375.1.g.a.1432.8 yes 16 15.8 even 4 inner