Properties

Label 3381.1.o.c.2138.2
Level $3381$
Weight $1$
Character 3381.2138
Analytic conductor $1.687$
Analytic rank $0$
Dimension $16$
Projective image $D_{8}$
CM discriminant -23
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3381,1,Mod(68,3381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3381.68");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3381.o (of order \(6\), degree \(2\), not 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
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: \(D_{8}\)
Projective field: Galois closure of 8.2.18667348829703.1

Embedding invariants

Embedding label 2138.2
Root \(0.608761 - 0.793353i\) of defining polynomial
Character \(\chi\) \(=\) 3381.2138
Dual form 3381.1.o.c.68.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.22474 + 0.707107i) q^{2} +(-0.130526 + 0.991445i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-0.541196 - 1.30656i) q^{6} +(-0.965926 - 0.258819i) q^{9} +O(q^{10})\) \(q+(-1.22474 + 0.707107i) q^{2} +(-0.130526 + 0.991445i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-0.541196 - 1.30656i) q^{6} +(-0.965926 - 0.258819i) q^{9} +(0.793353 + 0.608761i) q^{12} -0.765367i q^{13} +(0.500000 + 0.866025i) q^{16} +(1.36603 - 0.366025i) q^{18} +(-0.866025 + 0.500000i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(0.541196 + 0.937379i) q^{26} +(0.382683 - 0.923880i) q^{27} +(-1.60021 - 0.923880i) q^{31} +(-1.22474 - 0.707107i) q^{32} +(-0.707107 + 0.707107i) q^{36} +(0.758819 + 0.0999004i) q^{39} -0.765367 q^{41} +(0.707107 - 1.22474i) q^{46} +(-0.382683 - 0.662827i) q^{47} +(-0.923880 + 0.382683i) q^{48} -1.41421i q^{50} +(-0.662827 - 0.382683i) q^{52} +(0.184592 + 1.40211i) q^{54} +(0.923880 - 1.60021i) q^{59} +2.61313 q^{62} +1.00000 q^{64} +(-0.382683 - 0.923880i) q^{69} -2.00000i q^{71} +(-1.60021 - 0.923880i) q^{73} +(-0.793353 - 0.608761i) q^{75} +(-1.00000 + 0.414214i) q^{78} +(0.866025 + 0.500000i) q^{81} +(0.937379 - 0.541196i) q^{82} +1.00000i q^{92} +(1.12484 - 1.46593i) q^{93} +(0.937379 + 0.541196i) q^{94} +(0.860919 - 1.12197i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 8 q^{16} + 8 q^{18} - 8 q^{25} + 8 q^{39} + 16 q^{64} - 16 q^{78}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(346\) \(442\) \(2255\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(3\) −0.130526 + 0.991445i −0.130526 + 0.991445i
\(4\) 0.500000 0.866025i 0.500000 0.866025i
\(5\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) −0.541196 1.30656i −0.541196 1.30656i
\(7\) 0 0
\(8\) 0 0
\(9\) −0.965926 0.258819i −0.965926 0.258819i
\(10\) 0 0
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0.793353 + 0.608761i 0.793353 + 0.608761i
\(13\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(17\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 1.36603 0.366025i 1.36603 0.366025i
\(19\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) 0.541196 + 0.937379i 0.541196 + 0.937379i
\(27\) 0.382683 0.923880i 0.382683 0.923880i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.60021 0.923880i −1.60021 0.923880i −0.991445 0.130526i \(-0.958333\pi\)
−0.608761 0.793353i \(-0.708333\pi\)
\(32\) −1.22474 0.707107i −1.22474 0.707107i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(37\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) 0.758819 + 0.0999004i 0.758819 + 0.0999004i
\(40\) 0 0
\(41\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.707107 1.22474i 0.707107 1.22474i
\(47\) −0.382683 0.662827i −0.382683 0.662827i 0.608761 0.793353i \(-0.291667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(48\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(49\) 0 0
\(50\) 1.41421i 1.41421i
\(51\) 0 0
\(52\) −0.662827 0.382683i −0.662827 0.382683i
\(53\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(54\) 0.184592 + 1.40211i 0.184592 + 1.40211i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 2.61313 2.61313
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) −0.382683 0.923880i −0.382683 0.923880i
\(70\) 0 0
\(71\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(72\) 0 0
\(73\) −1.60021 0.923880i −1.60021 0.923880i −0.991445 0.130526i \(-0.958333\pi\)
−0.608761 0.793353i \(-0.708333\pi\)
\(74\) 0 0
\(75\) −0.793353 0.608761i −0.793353 0.608761i
\(76\) 0 0
\(77\) 0 0
\(78\) −1.00000 + 0.414214i −1.00000 + 0.414214i
\(79\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(82\) 0.937379 0.541196i 0.937379 0.541196i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000i 1.00000i
\(93\) 1.12484 1.46593i 1.12484 1.46593i
\(94\) 0.937379 + 0.541196i 0.937379 + 0.541196i
\(95\) 0 0
\(96\) 0.860919 1.12197i 0.860919 1.12197i
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(101\) −0.382683 + 0.662827i −0.382683 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(102\) 0 0
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(108\) −0.608761 0.793353i −0.608761 0.793353i
\(109\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(110\) 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
\(116\) 0 0
\(117\) −0.198092 + 0.739288i −0.198092 + 0.739288i
\(118\) 2.61313i 2.61313i
\(119\) 0 0
\(120\) 0 0
\(121\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0.0999004 0.758819i 0.0999004 0.758819i
\(124\) −1.60021 + 0.923880i −1.60021 + 0.923880i
\(125\) 0 0
\(126\) 0 0
\(127\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.923880 1.60021i −0.923880 1.60021i −0.793353 0.608761i \(-0.791667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(138\) 1.12197 + 0.860919i 1.12197 + 0.860919i
\(139\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(140\) 0 0
\(141\) 0.707107 0.292893i 0.707107 0.292893i
\(142\) 1.41421 + 2.44949i 1.41421 + 2.44949i
\(143\) 0 0
\(144\) −0.258819 0.965926i −0.258819 0.965926i
\(145\) 0 0
\(146\) 2.61313 2.61313
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 1.40211 + 0.184592i 1.40211 + 0.184592i
\(151\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0.465926 0.607206i 0.465926 0.607206i
\(157\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −1.41421 −1.41421
\(163\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(164\) −0.382683 + 0.662827i −0.382683 + 0.662827i
\(165\) 0 0
\(166\) 0 0
\(167\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(168\) 0 0
\(169\) 0.414214 0.414214
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.923880 1.60021i −0.923880 1.60021i −0.793353 0.608761i \(-0.791667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.46593 + 1.12484i 1.46593 + 1.12484i
\(178\) 0 0
\(179\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
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 0
\(185\) 0 0
\(186\) −0.341081 + 2.59077i −0.341081 + 2.59077i
\(187\) 0 0
\(188\) −0.765367 −0.765367
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) −0.130526 + 0.991445i −0.130526 + 0.991445i
\(193\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1.08239i 1.08239i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.965926 0.258819i 0.965926 0.258819i
\(208\) 0.662827 0.382683i 0.662827 0.382683i
\(209\) 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
\(213\) 1.98289 + 0.261052i 1.98289 + 0.261052i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1.12484 1.46593i 1.12484 1.46593i
\(220\) 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
\(225\) 0.707107 0.707107i 0.707107 0.707107i
\(226\) 0 0
\(227\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(228\) 0 0
\(229\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.73205 1.00000i 1.73205 1.00000i 0.866025 0.500000i \(-0.166667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(234\) −0.280144 1.04551i −0.280144 1.04551i
\(235\) 0 0
\(236\) −0.923880 1.60021i −0.923880 1.60021i
\(237\) 0 0
\(238\) 0 0
\(239\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(240\) 0 0
\(241\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(242\) −1.22474 0.707107i −1.22474 0.707107i
\(243\) −0.608761 + 0.793353i −0.608761 + 0.793353i
\(244\) 0 0
\(245\) 0 0
\(246\) 0.414214 + 1.00000i 0.414214 + 1.00000i
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 1.73205 1.00000i 1.73205 1.00000i
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(257\) 0.382683 + 0.662827i 0.382683 + 0.662827i 0.991445 0.130526i \(-0.0416667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 2.26303 + 1.30656i 2.26303 + 1.30656i
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.382683 0.662827i 0.382683 0.662827i −0.608761 0.793353i \(-0.708333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(270\) 0 0
\(271\) 1.60021 0.923880i 1.60021 0.923880i 0.608761 0.793353i \(-0.291667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −0.991445 0.130526i −0.991445 0.130526i
\(277\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(278\) −1.30656 2.26303i −1.30656 2.26303i
\(279\) 1.30656 + 1.30656i 1.30656 + 1.30656i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −0.658919 + 0.858719i −0.658919 + 0.858719i
\(283\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(284\) −1.73205 1.00000i −1.73205 1.00000i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(289\) −0.500000 0.866025i −0.500000 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) −1.60021 + 0.923880i −1.60021 + 0.923880i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.382683 + 0.662827i 0.382683 + 0.662827i
\(300\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(301\) 0 0
\(302\) 2.00000i 2.00000i
\(303\) −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
\(311\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(312\) 0 0
\(313\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.866025 0.500000i 0.866025 0.500000i
\(325\) 0.662827 + 0.382683i 0.662827 + 0.382683i
\(326\) 2.44949 + 1.41421i 2.44949 + 1.41421i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0.937379 0.541196i 0.937379 0.541196i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −0.507306 + 0.292893i −0.507306 + 0.292893i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 2.26303 + 1.30656i 2.26303 + 1.30656i
\(347\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(348\) 0 0
\(349\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(350\) 0 0
\(351\) −0.707107 0.292893i −0.707107 0.292893i
\(352\) 0 0
\(353\) −0.923880 + 1.60021i −0.923880 + 1.60021i −0.130526 + 0.991445i \(0.541667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(354\) −2.59077 0.341081i −2.59077 0.341081i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −2.00000 −2.00000
\(359\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(360\) 0 0
\(361\) 0.500000 0.866025i 0.500000 0.866025i
\(362\) 0 0
\(363\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(368\) −0.866025 0.500000i −0.866025 0.500000i
\(369\) 0.739288 + 0.198092i 0.739288 + 0.198092i
\(370\) 0 0
\(371\) 0 0
\(372\) −0.707107 1.70711i −0.707107 1.70711i
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0.184592 1.40211i 0.184592 1.40211i
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.82843i 2.82843i
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 1.70711 0.707107i 1.70711 0.707107i
\(394\) −1.41421 2.44949i −1.41421 2.44949i
\(395\) 0 0
\(396\) 0 0
\(397\) 0.662827 0.382683i 0.662827 0.382683i −0.130526 0.991445i \(-0.541667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.00000 −1.00000
\(401\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) 0 0
\(403\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(404\) 0.382683 + 0.662827i 0.382683 + 0.662827i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.662827 0.382683i −0.662827 0.382683i 0.130526 0.991445i \(-0.458333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(415\) 0 0
\(416\) −0.541196 + 0.937379i −0.541196 + 0.937379i
\(417\) −1.83195 0.241181i −1.83195 0.241181i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(423\) 0.198092 + 0.739288i 0.198092 + 0.739288i
\(424\) 0 0
\(425\) 0 0
\(426\) −2.61313 + 1.08239i −2.61313 + 1.08239i
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(432\) 0.991445 0.130526i 0.991445 0.130526i
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −0.341081 + 2.59077i −0.341081 + 2.59077i
\(439\) −1.60021 + 0.923880i −1.60021 + 0.923880i −0.608761 + 0.793353i \(0.708333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0.541196 + 0.937379i 0.541196 + 0.937379i
\(447\) 0 0
\(448\) 0 0
\(449\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(450\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(451\) 0 0
\(452\) 0 0
\(453\) 1.12197 + 0.860919i 1.12197 + 0.860919i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(462\) 0 0
\(463\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −1.41421 + 2.44949i −1.41421 + 2.44949i
\(467\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(468\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(469\) 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
\(478\) −1.00000 1.73205i −1.00000 1.73205i
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.00000 1.00000
\(485\) 0 0
\(486\) 0.184592 1.40211i 0.184592 1.40211i
\(487\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(488\) 0 0
\(489\) 1.84776 0.765367i 1.84776 0.765367i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) −0.607206 0.465926i −0.607206 0.465926i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.84776i 1.84776i
\(497\) 0 0
\(498\) 0 0
\(499\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(500\) 0 0
\(501\) 0.0999004 0.758819i 0.0999004 0.758819i
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.0540657 + 0.410670i −0.0540657 + 0.410670i
\(508\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(509\) 0.923880 + 1.60021i 0.923880 + 1.60021i 0.793353 + 0.608761i \(0.208333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.41421i 1.41421i
\(513\) 0 0
\(514\) −0.937379 0.541196i −0.937379 0.541196i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 1.70711 0.707107i 1.70711 0.707107i
\(520\) 0 0
\(521\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(524\) −1.84776 −1.84776
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.500000 0.866025i 0.500000 0.866025i
\(530\) 0 0
\(531\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(532\) 0 0
\(533\) 0.585786i 0.585786i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −0.860919 + 1.12197i −0.860919 + 1.12197i
\(538\) 1.08239i 1.08239i
\(539\) 0 0
\(540\) 0 0
\(541\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(542\) −1.30656 + 2.26303i −1.30656 + 2.26303i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 2.00000i 2.00000i
\(555\) 0 0
\(556\) 1.60021 + 0.923880i 1.60021 + 0.923880i
\(557\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(558\) −2.52409 0.676327i −2.52409 0.676327i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 0.0999004 0.758819i 0.0999004 0.758819i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(570\) 0 0
\(571\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000i 1.00000i
\(576\) −0.965926 0.258819i −0.965926 0.258819i
\(577\) 1.60021 + 0.923880i 1.60021 + 0.923880i 0.991445 + 0.130526i \(0.0416667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(578\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(579\) −1.58671 1.21752i −1.58671 1.21752i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −1.98289 0.261052i −1.98289 0.261052i
\(592\) 0 0
\(593\) 0.382683 + 0.662827i 0.382683 + 0.662827i 0.991445 0.130526i \(-0.0416667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −0.937379 0.541196i −0.937379 0.541196i
\(599\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(600\) 0 0
\(601\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −0.707107 1.22474i −0.707107 1.22474i
\(605\) 0 0
\(606\) 1.07313 + 0.141281i 1.07313 + 0.141281i
\(607\) −1.60021 + 0.923880i −1.60021 + 0.923880i −0.608761 + 0.793353i \(0.708333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.507306 + 0.292893i −0.507306 + 0.292893i
\(612\) 0 0
\(613\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) −0.541196 0.937379i −0.541196 0.937379i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(620\) 0 0
\(621\) 0.130526 + 0.991445i 0.130526 + 0.991445i
\(622\) 2.61313i 2.61313i
\(623\) 0 0
\(624\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(625\) −0.500000 0.866025i −0.500000 0.866025i
\(626\) 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
\(633\) −0.184592 + 1.40211i −0.184592 + 1.40211i
\(634\) 1.00000 1.73205i 1.00000 1.73205i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −0.517638 + 1.93185i −0.517638 + 1.93185i
\(640\) 0 0
\(641\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.382683 + 0.662827i −0.382683 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −1.08239 −1.08239
\(651\) 0 0
\(652\) −2.00000 −2.00000
\(653\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.382683 0.662827i −0.382683 0.662827i
\(657\) 1.30656 + 1.30656i 1.30656 + 1.30656i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −0.382683 + 0.662827i −0.382683 + 0.662827i
\(669\) 0.758819 + 0.0999004i 0.758819 + 0.0999004i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0.608761 + 0.793353i 0.608761 + 0.793353i
\(676\) 0.207107 0.358719i 0.207107 0.358719i
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.73205 1.00000i −1.73205 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
−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
\(691\) −1.60021 + 0.923880i −1.60021 + 0.923880i −0.608761 + 0.793353i \(0.708333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(692\) −1.84776 −1.84776
\(693\) 0 0
\(694\) 2.00000 2.00000
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 1.30656 + 2.26303i 1.30656 + 2.26303i
\(699\) 0.765367 + 1.84776i 0.765367 + 1.84776i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 1.07313 0.141281i 1.07313 0.141281i
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 2.61313i 2.61313i
\(707\) 0 0
\(708\) 1.70711 0.707107i 1.70711 0.707107i
\(709\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.84776 1.84776
\(714\) 0 0
\(715\) 0 0
\(716\) 1.22474 0.707107i 1.22474 0.707107i
\(717\) −1.40211 0.184592i −1.40211 0.184592i
\(718\) 0 0
\(719\) 0.923880 + 1.60021i 0.923880 + 1.60021i 0.793353 + 0.608761i \(0.208333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.41421i 1.41421i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0.860919 1.12197i 0.860919 1.12197i
\(727\) 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 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 1.41421 1.41421
\(737\) 0 0
\(738\) −1.04551 + 0.280144i −1.04551 + 0.280144i
\(739\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) 0.382683 0.662827i 0.382683 0.662827i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −0.923880 1.60021i −0.923880 1.60021i −0.793353 0.608761i \(-0.791667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(762\) 0.765367 + 1.84776i 0.765367 + 1.84776i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.22474 0.707107i −1.22474 0.707107i
\(768\) −0.793353 0.608761i −0.793353 0.608761i
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(772\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(773\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(774\) 0 0
\(775\) 1.60021 0.923880i 1.60021 0.923880i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) −1.59077 + 2.07313i −1.59077 + 2.07313i
\(787\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(788\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −0.541196 + 0.937379i −0.541196 + 0.937379i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.22474 0.707107i 1.22474 0.707107i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 2.00000i 2.00000i
\(807\) 0.607206 + 0.465926i 0.607206 + 0.465926i
\(808\) 0 0
\(809\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(810\) 0 0
\(811\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(812\) 0 0
\(813\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 1.08239 1.08239
\(819\) 0 0
\(820\) 0 0
\(821\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(822\) 0 0
\(823\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
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 \(0.5\pi\)
\(828\) 0.258819 0.965926i 0.258819 0.965926i
\(829\) 0.662827 + 0.382683i 0.662827 + 0.382683i 0.793353 0.608761i \(-0.208333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(830\) 0 0
\(831\) −1.12197 0.860919i −1.12197 0.860919i
\(832\) 0.765367i 0.765367i
\(833\) 0 0
\(834\) 2.41421 1.00000i 2.41421 1.00000i
\(835\) 0 0
\(836\) 0 0
\(837\) −1.46593 + 1.12484i −1.46593 + 1.12484i
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0.707107 1.22474i 0.707107 1.22474i
\(845\) 0 0
\(846\) −0.765367 0.765367i −0.765367 0.765367i
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 1.21752 1.58671i 1.21752 1.58671i
\(853\) 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
\(857\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(858\) 0 0
\(859\) −0.662827 + 0.382683i −0.662827 + 0.382683i −0.793353 0.608761i \(-0.791667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(864\) −1.12197 + 0.860919i −1.12197 + 0.860919i
\(865\) 0 0
\(866\) 0 0
\(867\) 0.923880 0.382683i 0.923880 0.382683i
\(868\) 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 1.70711i −0.707107 1.70711i
\(877\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(878\) 1.30656 2.26303i 1.30656 2.26303i
\(879\) 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 0
\(887\) 0.923880 + 1.60021i 0.923880 + 1.60021i 0.793353 + 0.608761i \(0.208333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −0.662827 0.382683i −0.662827 0.382683i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(898\) −1.00000 1.73205i −1.00000 1.73205i
\(899\) 0 0
\(900\) −0.258819 0.965926i −0.258819 0.965926i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) −1.98289 0.261052i −1.98289 0.261052i
\(907\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(908\) 0 0
\(909\) 0.541196 0.541196i 0.541196 0.541196i
\(910\) 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 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(920\) 0 0
\(921\) −0.758819 0.0999004i −0.758819 0.0999004i
\(922\) 2.26303 1.30656i 2.26303 1.30656i
\(923\) −1.53073 −1.53073
\(924\) 0 0
\(925\) 0 0
\(926\) 1.73205 1.00000i 1.73205 1.00000i
\(927\) 0 0
\(928\) 0 0
\(929\) 0.923880 + 1.60021i 0.923880 + 1.60021i 0.793353 + 0.608761i \(0.208333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.00000i 2.00000i
\(933\) 1.46593 + 1.12484i 1.46593 + 1.12484i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(942\) 0 0
\(943\) 0.662827 0.382683i 0.662827 0.382683i
\(944\) 1.84776 1.84776
\(945\) 0 0
\(946\) 0 0
\(947\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(948\) 0 0
\(949\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(950\) 0 0
\(951\) −0.541196 1.30656i −0.541196 1.30656i
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.20711 + 2.09077i 1.20711 + 2.09077i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(972\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(973\) 0 0
\(974\) 2.82843i 2.82843i
\(975\) −0.465926 + 0.607206i −0.465926 + 0.607206i
\(976\) 0 0
\(977\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(978\) −1.72184 + 2.24394i −1.72184 + 2.24394i
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(992\) 1.30656 + 2.26303i 1.30656 + 2.26303i
\(993\) 1.30656 0.541196i 1.30656 0.541196i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −0.662827 0.382683i −0.662827 0.382683i 0.130526 0.991445i \(-0.458333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(998\) −1.73205 1.00000i −1.73205 1.00000i
\(999\) 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.o.c.2138.2 16
3.2 odd 2 inner 3381.1.o.c.2138.7 16
7.2 even 3 inner 3381.1.o.c.68.6 16
7.3 odd 6 3381.1.c.a.3380.1 8
7.4 even 3 3381.1.c.a.3380.4 yes 8
7.5 odd 6 inner 3381.1.o.c.68.7 16
7.6 odd 2 inner 3381.1.o.c.2138.3 16
21.2 odd 6 inner 3381.1.o.c.68.3 16
21.5 even 6 inner 3381.1.o.c.68.2 16
21.11 odd 6 3381.1.c.a.3380.5 yes 8
21.17 even 6 3381.1.c.a.3380.8 yes 8
21.20 even 2 inner 3381.1.o.c.2138.6 16
23.22 odd 2 CM 3381.1.o.c.2138.2 16
69.68 even 2 inner 3381.1.o.c.2138.7 16
161.45 even 6 3381.1.c.a.3380.1 8
161.68 even 6 inner 3381.1.o.c.68.7 16
161.114 odd 6 inner 3381.1.o.c.68.6 16
161.137 odd 6 3381.1.c.a.3380.4 yes 8
161.160 even 2 inner 3381.1.o.c.2138.3 16
483.68 odd 6 inner 3381.1.o.c.68.2 16
483.137 even 6 3381.1.c.a.3380.5 yes 8
483.206 odd 6 3381.1.c.a.3380.8 yes 8
483.275 even 6 inner 3381.1.o.c.68.3 16
483.482 odd 2 inner 3381.1.o.c.2138.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3381.1.c.a.3380.1 8 7.3 odd 6
3381.1.c.a.3380.1 8 161.45 even 6
3381.1.c.a.3380.4 yes 8 7.4 even 3
3381.1.c.a.3380.4 yes 8 161.137 odd 6
3381.1.c.a.3380.5 yes 8 21.11 odd 6
3381.1.c.a.3380.5 yes 8 483.137 even 6
3381.1.c.a.3380.8 yes 8 21.17 even 6
3381.1.c.a.3380.8 yes 8 483.206 odd 6
3381.1.o.c.68.2 16 21.5 even 6 inner
3381.1.o.c.68.2 16 483.68 odd 6 inner
3381.1.o.c.68.3 16 21.2 odd 6 inner
3381.1.o.c.68.3 16 483.275 even 6 inner
3381.1.o.c.68.6 16 7.2 even 3 inner
3381.1.o.c.68.6 16 161.114 odd 6 inner
3381.1.o.c.68.7 16 7.5 odd 6 inner
3381.1.o.c.68.7 16 161.68 even 6 inner
3381.1.o.c.2138.2 16 1.1 even 1 trivial
3381.1.o.c.2138.2 16 23.22 odd 2 CM
3381.1.o.c.2138.3 16 7.6 odd 2 inner
3381.1.o.c.2138.3 16 161.160 even 2 inner
3381.1.o.c.2138.6 16 21.20 even 2 inner
3381.1.o.c.2138.6 16 483.482 odd 2 inner
3381.1.o.c.2138.7 16 3.2 odd 2 inner
3381.1.o.c.2138.7 16 69.68 even 2 inner