Properties

Label 3381.1.c.b.3380.12
Level $3381$
Weight $1$
Character 3381.3380
Analytic conductor $1.687$
Analytic rank $0$
Dimension $16$
Projective image $D_{24}$
CM discriminant -23
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3381,1,Mod(3380,3381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3381.3380");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3381.c (of order \(2\), degree \(1\), 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: \(1.68733880771\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{24}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

Embedding invariants

Embedding label 3380.12
Root \(0.991445 - 0.130526i\) of defining polynomial
Character \(\chi\) \(=\) 3381.3380
Dual form 3381.1.c.b.3380.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.517638i q^{2} +(0.793353 - 0.608761i) q^{3} +0.732051 q^{4} +(0.315118 + 0.410670i) q^{6} +0.896575i q^{8} +(0.258819 - 0.965926i) q^{9} +O(q^{10})\) \(q+0.517638i q^{2} +(0.793353 - 0.608761i) q^{3} +0.732051 q^{4} +(0.315118 + 0.410670i) q^{6} +0.896575i q^{8} +(0.258819 - 0.965926i) q^{9} +(0.580775 - 0.445644i) q^{12} +1.98289i q^{13} +0.267949 q^{16} +(0.500000 + 0.133975i) q^{18} +1.00000i q^{23} +(0.545801 + 0.711301i) q^{24} +1.00000 q^{25} -1.02642 q^{26} +(-0.382683 - 0.923880i) q^{27} -1.73205i q^{29} -0.261052i q^{31} +1.03528i q^{32} +(0.189469 - 0.707107i) q^{36} +(1.20711 + 1.57313i) q^{39} -1.98289 q^{41} -0.517638 q^{46} -1.21752 q^{47} +(0.212578 - 0.163117i) q^{48} +0.517638i q^{50} +1.45158i q^{52} +(0.478235 - 0.198092i) q^{54} +0.896575 q^{58} +1.84776 q^{59} +0.135131 q^{62} -0.267949 q^{64} +(0.608761 + 0.793353i) q^{69} -1.00000i q^{71} +(0.866025 + 0.232051i) q^{72} -1.58671i q^{73} +(0.793353 - 0.608761i) q^{75} +(-0.814313 + 0.624844i) q^{78} +(-0.866025 - 0.500000i) q^{81} -1.02642i q^{82} +(-1.05441 - 1.37413i) q^{87} +0.732051i q^{92} +(-0.158919 - 0.207107i) q^{93} -0.630236i q^{94} +(0.630236 + 0.821340i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 32 q^{16} + 8 q^{18} + 16 q^{25} + 8 q^{39} - 32 q^{64} + 8 q^{78}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3381\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(2255\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.517638i 0.517638i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(3\) 0.793353 0.608761i 0.793353 0.608761i
\(4\) 0.732051 0.732051
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0.315118 + 0.410670i 0.315118 + 0.410670i
\(7\) 0 0
\(8\) 0.896575i 0.896575i
\(9\) 0.258819 0.965926i 0.258819 0.965926i
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0.580775 0.445644i 0.580775 0.445644i
\(13\) 1.98289i 1.98289i 0.130526 + 0.991445i \(0.458333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.267949 0.267949
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0.500000 + 0.133975i 0.500000 + 0.133975i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000i 1.00000i
\(24\) 0.545801 + 0.711301i 0.545801 + 0.711301i
\(25\) 1.00000 1.00000
\(26\) −1.02642 −1.02642
\(27\) −0.382683 0.923880i −0.382683 0.923880i
\(28\) 0 0
\(29\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(30\) 0 0
\(31\) 0.261052i 0.261052i −0.991445 0.130526i \(-0.958333\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(32\) 1.03528i 1.03528i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.189469 0.707107i 0.189469 0.707107i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 1.20711 + 1.57313i 1.20711 + 1.57313i
\(40\) 0 0
\(41\) −1.98289 −1.98289 −0.991445 0.130526i \(-0.958333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.517638 −0.517638
\(47\) −1.21752 −1.21752 −0.608761 0.793353i \(-0.708333\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(48\) 0.212578 0.163117i 0.212578 0.163117i
\(49\) 0 0
\(50\) 0.517638i 0.517638i
\(51\) 0 0
\(52\) 1.45158i 1.45158i
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0.478235 0.198092i 0.478235 0.198092i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0.896575 0.896575
\(59\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0.135131 0.135131
\(63\) 0 0
\(64\) −0.267949 −0.267949
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0.608761 + 0.793353i 0.608761 + 0.793353i
\(70\) 0 0
\(71\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(72\) 0.866025 + 0.232051i 0.866025 + 0.232051i
\(73\) 1.58671i 1.58671i −0.608761 0.793353i \(-0.708333\pi\)
0.608761 0.793353i \(-0.291667\pi\)
\(74\) 0 0
\(75\) 0.793353 0.608761i 0.793353 0.608761i
\(76\) 0 0
\(77\) 0 0
\(78\) −0.814313 + 0.624844i −0.814313 + 0.624844i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −0.866025 0.500000i −0.866025 0.500000i
\(82\) 1.02642i 1.02642i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.05441 1.37413i −1.05441 1.37413i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.732051i 0.732051i
\(93\) −0.158919 0.207107i −0.158919 0.207107i
\(94\) 0.630236i 0.630236i
\(95\) 0 0
\(96\) 0.630236 + 0.821340i 0.630236 + 0.821340i
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0.732051 0.732051
\(101\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −1.77781 −1.77781
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −0.280144 0.676327i −0.280144 0.676327i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.26795i 1.26795i
\(117\) 1.91532 + 0.513210i 1.91532 + 0.513210i
\(118\) 0.956470i 0.956470i
\(119\) 0 0
\(120\) 0 0
\(121\) −1.00000 −1.00000
\(122\) 0 0
\(123\) −1.57313 + 1.20711i −1.57313 + 1.20711i
\(124\) 0.191104i 0.191104i
\(125\) 0 0
\(126\) 0 0
\(127\) −0.517638 −0.517638 −0.258819 0.965926i \(-0.583333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(128\) 0.896575i 0.896575i
\(129\) 0 0
\(130\) 0 0
\(131\) 0.261052 0.261052 0.130526 0.991445i \(-0.458333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) −0.410670 + 0.315118i −0.410670 + 0.315118i
\(139\) 1.58671i 1.58671i −0.608761 0.793353i \(-0.708333\pi\)
0.608761 0.793353i \(-0.291667\pi\)
\(140\) 0 0
\(141\) −0.965926 + 0.741181i −0.965926 + 0.741181i
\(142\) 0.517638 0.517638
\(143\) 0 0
\(144\) 0.0693504 0.258819i 0.0693504 0.258819i
\(145\) 0 0
\(146\) 0.821340 0.821340
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0.315118 + 0.410670i 0.315118 + 0.410670i
\(151\) 1.93185 1.93185 0.965926 0.258819i \(-0.0833333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0.883663 + 1.15161i 0.883663 + 1.15161i
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0.258819 0.448288i 0.258819 0.448288i
\(163\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(164\) −1.45158 −1.45158
\(165\) 0 0
\(166\) 0 0
\(167\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(168\) 0 0
\(169\) −2.93185 −2.93185
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(174\) 0.711301 0.545801i 0.711301 0.545801i
\(175\) 0 0
\(176\) 0 0
\(177\) 1.46593 1.12484i 1.46593 1.12484i
\(178\) 0 0
\(179\) 1.93185i 1.93185i −0.258819 0.965926i \(-0.583333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −0.896575 −0.896575
\(185\) 0 0
\(186\) 0.107206 0.0822623i 0.107206 0.0822623i
\(187\) 0 0
\(188\) −0.891289 −0.891289
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −0.212578 + 0.163117i −0.212578 + 0.163117i
\(193\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0.896575i 0.896575i
\(201\) 0 0
\(202\) 0.396183i 0.396183i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(208\) 0.531314i 0.531314i
\(209\) 0 0
\(210\) 0 0
\(211\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(212\) 0 0
\(213\) −0.608761 0.793353i −0.608761 0.793353i
\(214\) 0 0
\(215\) 0 0
\(216\) 0.828328 0.343105i 0.828328 0.343105i
\(217\) 0 0
\(218\) 0 0
\(219\) −0.965926 1.25882i −0.965926 1.25882i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(224\) 0 0
\(225\) 0.258819 0.965926i 0.258819 0.965926i
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.55291 1.55291
\(233\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(234\) −0.265657 + 0.991445i −0.265657 + 0.991445i
\(235\) 0 0
\(236\) 1.35265 1.35265
\(237\) 0 0
\(238\) 0 0
\(239\) 1.93185i 1.93185i 0.258819 + 0.965926i \(0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0.517638i 0.517638i
\(243\) −0.991445 + 0.130526i −0.991445 + 0.130526i
\(244\) 0 0
\(245\) 0 0
\(246\) −0.624844 0.814313i −0.624844 0.814313i
\(247\) 0 0
\(248\) 0.234053 0.234053
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0.267949i 0.267949i
\(255\) 0 0
\(256\) −0.732051 −0.732051
\(257\) 1.21752 1.21752 0.608761 0.793353i \(-0.291667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.67303 0.448288i −1.67303 0.448288i
\(262\) 0.135131i 0.135131i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.21752 1.21752 0.608761 0.793353i \(-0.291667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(270\) 0 0
\(271\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0.445644 + 0.580775i 0.445644 + 0.580775i
\(277\) −1.93185 −1.93185 −0.965926 0.258819i \(-0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(278\) 0.821340 0.821340
\(279\) −0.252157 0.0675653i −0.252157 0.0675653i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −0.383663 0.500000i −0.383663 0.500000i
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0.732051i 0.732051i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 + 0.267949i 1.00000 + 0.267949i
\(289\) 1.00000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 1.16155i 1.16155i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.98289 −1.98289
\(300\) 0.580775 0.445644i 0.580775 0.445644i
\(301\) 0 0
\(302\) 1.00000i 1.00000i
\(303\) −0.607206 + 0.465926i −0.607206 + 0.465926i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −0.261052 −0.261052 −0.130526 0.991445i \(-0.541667\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(312\) −1.41043 + 1.08226i −1.41043 + 1.08226i
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.633975 0.366025i −0.633975 0.366025i
\(325\) 1.98289i 1.98289i
\(326\) 0.517638i 0.517638i
\(327\) 0 0
\(328\) 1.77781i 1.77781i
\(329\) 0 0
\(330\) 0 0
\(331\) −1.93185 −1.93185 −0.965926 0.258819i \(-0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0.396183i 0.396183i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 1.51764i 1.51764i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0.956470i 0.956470i
\(347\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(348\) −0.771879 1.00593i −0.771879 1.00593i
\(349\) 0.261052i 0.261052i 0.991445 + 0.130526i \(0.0416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(350\) 0 0
\(351\) 1.83195 0.758819i 1.83195 0.758819i
\(352\) 0 0
\(353\) 0.261052 0.261052 0.130526 0.991445i \(-0.458333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(354\) 0.582262 + 0.758819i 0.582262 + 0.758819i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 1.00000 1.00000
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −1.00000 −1.00000
\(362\) 0 0
\(363\) −0.793353 + 0.608761i −0.793353 + 0.608761i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0.267949i 0.267949i
\(369\) −0.513210 + 1.91532i −0.513210 + 1.91532i
\(370\) 0 0
\(371\) 0 0
\(372\) −0.116337 0.151613i −0.116337 0.151613i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 1.09160i 1.09160i
\(377\) 3.43447 3.43447
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −0.410670 + 0.315118i −0.410670 + 0.315118i
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0.545801 + 0.711301i 0.545801 + 0.711301i
\(385\) 0 0
\(386\) 0.517638i 0.517638i
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0.207107 0.158919i 0.207107 0.158919i
\(394\) −0.517638 −0.517638
\(395\) 0 0
\(396\) 0 0
\(397\) 1.21752i 1.21752i 0.793353 + 0.608761i \(0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.267949 0.267949
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0.517638 0.517638
\(404\) −0.560287 −0.560287
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.21752i 1.21752i 0.793353 + 0.608761i \(0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −0.133975 + 0.500000i −0.133975 + 0.500000i
\(415\) 0 0
\(416\) −2.05284 −2.05284
\(417\) −0.965926 1.25882i −0.965926 1.25882i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0.732051i 0.732051i
\(423\) −0.315118 + 1.17604i −0.315118 + 1.17604i
\(424\) 0 0
\(425\) 0 0
\(426\) 0.410670 0.315118i 0.410670 0.315118i
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −0.102540 0.247553i −0.102540 0.247553i
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0.651613 0.500000i 0.651613 0.500000i
\(439\) 1.58671i 1.58671i 0.608761 + 0.793353i \(0.291667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0.396183 0.396183
\(447\) 0 0
\(448\) 0 0
\(449\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(450\) 0.500000 + 0.133975i 0.500000 + 0.133975i
\(451\) 0 0
\(452\) 0 0
\(453\) 1.53264 1.17604i 1.53264 1.17604i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.58671 −1.58671 −0.793353 0.608761i \(-0.791667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(462\) 0 0
\(463\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(464\) 0.464102i 0.464102i
\(465\) 0 0
\(466\) −0.517638 −0.517638
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 1.40211 + 0.375696i 1.40211 + 0.375696i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 1.65666i 1.65666i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −1.00000 −1.00000
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −0.732051 −0.732051
\(485\) 0 0
\(486\) −0.0675653 0.513210i −0.0675653 0.513210i
\(487\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(488\) 0 0
\(489\) −0.793353 + 0.608761i −0.793353 + 0.608761i
\(490\) 0 0
\(491\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) −1.15161 + 0.883663i −1.15161 + 0.883663i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0.0699488i 0.0699488i
\(497\) 0 0
\(498\) 0 0
\(499\) −0.517638 −0.517638 −0.258819 0.965926i \(-0.583333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(500\) 0 0
\(501\) 0.607206 0.465926i 0.607206 0.465926i
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2.32599 + 1.78480i −2.32599 + 1.78480i
\(508\) −0.378937 −0.378937
\(509\) −1.58671 −1.58671 −0.793353 0.608761i \(-0.791667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.517638i 0.517638i
\(513\) 0 0
\(514\) 0.630236i 0.630236i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −1.46593 + 1.12484i −1.46593 + 1.12484i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0.232051 0.866025i 0.232051 0.866025i
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0.191104 0.191104
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.00000 −1.00000
\(530\) 0 0
\(531\) 0.478235 1.78480i 0.478235 1.78480i
\(532\) 0 0
\(533\) 3.93185i 3.93185i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.17604 1.53264i −1.17604 1.53264i
\(538\) 0.630236i 0.630236i
\(539\) 0 0
\(540\) 0 0
\(541\) 0.517638 0.517638 0.258819 0.965926i \(-0.416667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(542\) −0.956470 −0.956470
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.93185 1.93185 0.965926 0.258819i \(-0.0833333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) −0.711301 + 0.545801i −0.711301 + 0.545801i
\(553\) 0 0
\(554\) 1.00000i 1.00000i
\(555\) 0 0
\(556\) 1.16155i 1.16155i
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0.0349744 0.130526i 0.0349744 0.130526i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) −0.707107 + 0.542582i −0.707107 + 0.542582i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0.896575 0.896575
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000i 1.00000i
\(576\) −0.0693504 + 0.258819i −0.0693504 + 0.258819i
\(577\) 0.261052i 0.261052i 0.991445 + 0.130526i \(0.0416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(578\) 0.517638i 0.517638i
\(579\) −0.793353 + 0.608761i −0.793353 + 0.608761i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 1.42260 1.42260
\(585\) 0 0
\(586\) 0 0
\(587\) 1.98289 1.98289 0.991445 0.130526i \(-0.0416667\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0.608761 + 0.793353i 0.608761 + 0.793353i
\(592\) 0 0
\(593\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 1.02642i 1.02642i
\(599\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(600\) 0.545801 + 0.711301i 0.545801 + 0.711301i
\(601\) 1.21752i 1.21752i −0.793353 0.608761i \(-0.791667\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.41421 1.41421
\(605\) 0 0
\(606\) −0.241181 0.314313i −0.241181 0.314313i
\(607\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.41421i 2.41421i
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −0.396183 −0.396183
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0.923880 0.382683i 0.923880 0.382683i
\(622\) 0.135131i 0.135131i
\(623\) 0 0
\(624\) 0.323443 + 0.421519i 0.323443 + 0.421519i
\(625\) 1.00000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 1.12197 0.860919i 1.12197 0.860919i
\(634\) −0.732051 −0.732051
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −0.965926 0.258819i −0.965926 0.258819i
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.98289 1.98289 0.991445 0.130526i \(-0.0416667\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(648\) 0.448288 0.776457i 0.448288 0.776457i
\(649\) 0 0
\(650\) −1.02642 −1.02642
\(651\) 0 0
\(652\) −0.732051 −0.732051
\(653\) 1.93185i 1.93185i −0.258819 0.965926i \(-0.583333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.531314 −0.531314
\(657\) −1.53264 0.410670i −1.53264 0.410670i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 1.00000i 1.00000i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.73205 1.73205
\(668\) 0.560287 0.560287
\(669\) −0.465926 0.607206i −0.465926 0.607206i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(674\) 0 0
\(675\) −0.382683 0.923880i −0.382683 0.923880i
\(676\) −2.14626 −2.14626
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(692\) −1.35265 −1.35265
\(693\) 0 0
\(694\) 0.732051 0.732051
\(695\) 0 0
\(696\) 1.23201 0.945354i 1.23201 0.945354i
\(697\) 0 0
\(698\) −0.135131 −0.135131
\(699\) 0.608761 + 0.793353i 0.608761 + 0.793353i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0.392794 + 0.948288i 0.392794 + 0.948288i
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0.135131i 0.135131i
\(707\) 0 0
\(708\) 1.07313 0.823443i 1.07313 0.823443i
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0.261052 0.261052
\(714\) 0 0
\(715\) 0 0
\(716\) 1.41421i 1.41421i
\(717\) 1.17604 + 1.53264i 1.17604 + 1.53264i
\(718\) 0 0
\(719\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.517638i 0.517638i
\(723\) 0 0
\(724\) 0 0
\(725\) 1.73205i 1.73205i
\(726\) −0.315118 0.410670i −0.315118 0.410670i
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −1.03528 −1.03528
\(737\) 0 0
\(738\) −0.991445 0.265657i −0.991445 0.265657i
\(739\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0.185687 0.142483i 0.185687 0.142483i
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) −0.326234 −0.326234
\(753\) 0 0
\(754\) 1.77781i 1.77781i
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.58671 1.58671 0.793353 0.608761i \(-0.208333\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(762\) −0.163117 0.212578i −0.163117 0.212578i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.66390i 3.66390i
\(768\) −0.580775 + 0.445644i −0.580775 + 0.445644i
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0.965926 0.741181i 0.965926 0.741181i
\(772\) −0.732051 −0.732051
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0.261052i 0.261052i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1.60021 + 0.662827i −1.60021 + 0.662827i
\(784\) 0 0
\(785\) 0 0
\(786\) 0.0822623 + 0.107206i 0.0822623 + 0.107206i
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0.732051i 0.732051i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −0.630236 −0.630236
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.03528i 1.03528i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0.267949i 0.267949i
\(807\) 0.965926 0.741181i 0.965926 0.741181i
\(808\) 0.686209i 0.686209i
\(809\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 1.98289i 1.98289i −0.130526 0.991445i \(-0.541667\pi\)
0.130526 0.991445i \(-0.458333\pi\)
\(812\) 0 0
\(813\) 1.12484 + 1.46593i 1.12484 + 1.46593i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −0.630236 −0.630236
\(819\) 0 0
\(820\) 0 0
\(821\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(822\) 0 0
\(823\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0.707107 + 0.189469i 0.707107 + 0.189469i
\(829\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(830\) 0 0
\(831\) −1.53264 + 1.17604i −1.53264 + 1.17604i
\(832\) 0.531314i 0.531314i
\(833\) 0 0
\(834\) 0.651613 0.500000i 0.651613 0.500000i
\(835\) 0 0
\(836\) 0 0
\(837\) −0.241181 + 0.0999004i −0.241181 + 0.0999004i
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −2.00000 −2.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 1.03528 1.03528
\(845\) 0 0
\(846\) −0.608761 0.163117i −0.608761 0.163117i
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) −0.445644 0.580775i −0.445644 0.580775i
\(853\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.58671 −1.58671 −0.793353 0.608761i \(-0.791667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(858\) 0 0
\(859\) 1.98289i 1.98289i 0.130526 + 0.991445i \(0.458333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(864\) 0.956470 0.396183i 0.956470 0.396183i
\(865\) 0 0
\(866\) 0 0
\(867\) 0.793353 0.608761i 0.793353 0.608761i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) −0.707107 0.921519i −0.707107 0.921519i
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) −0.821340 −0.821340
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.896575 0.896575
\(887\) −0.261052 −0.261052 −0.130526 0.991445i \(-0.541667\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0.560287i 0.560287i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1.57313 + 1.20711i −1.57313 + 1.20711i
\(898\) 0.732051 0.732051
\(899\) −0.452156 −0.452156
\(900\) 0.189469 0.707107i 0.189469 0.707107i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0.608761 + 0.793353i 0.608761 + 0.793353i
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −0.198092 + 0.739288i −0.198092 + 0.739288i
\(910\) 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
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0.465926 + 0.607206i 0.465926 + 0.607206i
\(922\) 0.821340i 0.821340i
\(923\) 1.98289 1.98289
\(924\) 0 0
\(925\) 0 0
\(926\) 0.732051i 0.732051i
\(927\) 0 0
\(928\) 1.79315 1.79315
\(929\) −0.261052 −0.261052 −0.130526 0.991445i \(-0.541667\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0.732051i 0.732051i
\(933\) −0.207107 + 0.158919i −0.207107 + 0.158919i
\(934\) 0 0
\(935\) 0 0
\(936\) −0.460131 + 1.71723i −0.460131 + 1.71723i
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 1.98289i 1.98289i
\(944\) 0.495106 0.495106
\(945\) 0 0
\(946\) 0 0
\(947\) 1.93185i 1.93185i 0.258819 + 0.965926i \(0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(948\) 0 0
\(949\) 3.14626 3.14626
\(950\) 0 0
\(951\) 0.860919 + 1.12197i 0.860919 + 1.12197i
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.41421i 1.41421i
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.931852 0.931852
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(968\) 0.896575i 0.896575i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −0.725788 + 0.0955518i −0.725788 + 0.0955518i
\(973\) 0 0
\(974\) 0.517638i 0.517638i
\(975\) 1.20711 + 1.57313i 1.20711 + 1.57313i
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) −0.315118 0.410670i −0.315118 0.410670i
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −0.896575 −0.896575
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −1.08226 1.41043i −1.08226 1.41043i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0.270261 0.270261
\(993\) −1.53264 + 1.17604i −1.53264 + 1.17604i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(998\) 0.267949i 0.267949i
\(999\) 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 3381.1.c.b.3380.12 yes 16
3.2 odd 2 inner 3381.1.c.b.3380.5 16
7.2 even 3 3381.1.o.b.2138.6 16
7.3 odd 6 3381.1.o.a.68.4 16
7.4 even 3 3381.1.o.a.68.3 16
7.5 odd 6 3381.1.o.b.2138.5 16
7.6 odd 2 inner 3381.1.c.b.3380.9 yes 16
21.2 odd 6 3381.1.o.a.2138.4 16
21.5 even 6 3381.1.o.a.2138.3 16
21.11 odd 6 3381.1.o.b.68.5 16
21.17 even 6 3381.1.o.b.68.6 16
21.20 even 2 inner 3381.1.c.b.3380.8 yes 16
23.22 odd 2 CM 3381.1.c.b.3380.12 yes 16
69.68 even 2 inner 3381.1.c.b.3380.5 16
161.45 even 6 3381.1.o.a.68.4 16
161.68 even 6 3381.1.o.b.2138.5 16
161.114 odd 6 3381.1.o.b.2138.6 16
161.137 odd 6 3381.1.o.a.68.3 16
161.160 even 2 inner 3381.1.c.b.3380.9 yes 16
483.68 odd 6 3381.1.o.a.2138.3 16
483.137 even 6 3381.1.o.b.68.5 16
483.206 odd 6 3381.1.o.b.68.6 16
483.275 even 6 3381.1.o.a.2138.4 16
483.482 odd 2 inner 3381.1.c.b.3380.8 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3381.1.c.b.3380.5 16 3.2 odd 2 inner
3381.1.c.b.3380.5 16 69.68 even 2 inner
3381.1.c.b.3380.8 yes 16 21.20 even 2 inner
3381.1.c.b.3380.8 yes 16 483.482 odd 2 inner
3381.1.c.b.3380.9 yes 16 7.6 odd 2 inner
3381.1.c.b.3380.9 yes 16 161.160 even 2 inner
3381.1.c.b.3380.12 yes 16 1.1 even 1 trivial
3381.1.c.b.3380.12 yes 16 23.22 odd 2 CM
3381.1.o.a.68.3 16 7.4 even 3
3381.1.o.a.68.3 16 161.137 odd 6
3381.1.o.a.68.4 16 7.3 odd 6
3381.1.o.a.68.4 16 161.45 even 6
3381.1.o.a.2138.3 16 21.5 even 6
3381.1.o.a.2138.3 16 483.68 odd 6
3381.1.o.a.2138.4 16 21.2 odd 6
3381.1.o.a.2138.4 16 483.275 even 6
3381.1.o.b.68.5 16 21.11 odd 6
3381.1.o.b.68.5 16 483.137 even 6
3381.1.o.b.68.6 16 21.17 even 6
3381.1.o.b.68.6 16 483.206 odd 6
3381.1.o.b.2138.5 16 7.5 odd 6
3381.1.o.b.2138.5 16 161.68 even 6
3381.1.o.b.2138.6 16 7.2 even 3
3381.1.o.b.2138.6 16 161.114 odd 6