Properties

Label 1617.1.k.e.362.8
Level $1617$
Weight $1$
Character 1617.362
Analytic conductor $0.807$
Analytic rank $0$
Dimension $16$
Projective image $D_{8}$
CM discriminant -11
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1617,1,Mod(362,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, 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("1617.362");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1617.k (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: \(0.806988125428\)
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 8.0.88786994373.1

Embedding invariants

Embedding label 362.8
Root \(-0.793353 + 0.608761i\) of defining polynomial
Character \(\chi\) \(=\) 1617.362
Dual form 1617.1.k.e.1550.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.991445 - 0.130526i) q^{3} +(0.500000 + 0.866025i) q^{4} +(-0.923880 + 1.60021i) q^{5} +(0.965926 - 0.258819i) q^{9} +O(q^{10})\) \(q+(0.991445 - 0.130526i) q^{3} +(0.500000 + 0.866025i) q^{4} +(-0.923880 + 1.60021i) q^{5} +(0.965926 - 0.258819i) q^{9} +(0.866025 - 0.500000i) q^{11} +(0.608761 + 0.793353i) q^{12} +(-0.707107 + 1.70711i) q^{15} +(-0.500000 + 0.866025i) q^{16} -1.84776 q^{20} +(-1.22474 - 0.707107i) q^{23} +(-1.20711 - 2.09077i) q^{25} +(0.923880 - 0.382683i) q^{27} +(-0.662827 + 0.382683i) q^{31} +(0.793353 - 0.608761i) q^{33} +(0.707107 + 0.707107i) q^{36} +(0.866025 + 0.500000i) q^{44} +(-0.478235 + 1.78480i) q^{45} +(0.382683 - 0.662827i) q^{47} +(-0.382683 + 0.923880i) q^{48} +(1.22474 - 0.707107i) q^{53} +1.84776i q^{55} +(-0.382683 - 0.662827i) q^{59} +(-1.83195 + 0.241181i) q^{60} -1.00000 q^{64} +(0.707107 + 1.22474i) q^{67} +(-1.30656 - 0.541196i) q^{69} +1.41421i q^{71} +(-1.46968 - 1.91532i) q^{75} +(-0.923880 - 1.60021i) q^{80} +(0.866025 - 0.500000i) q^{81} +(-0.382683 + 0.662827i) q^{89} -1.41421i q^{92} +(-0.607206 + 0.465926i) q^{93} -1.84776i q^{97} +(0.707107 - 0.707107i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 8 q^{16} - 8 q^{25} - 16 q^{64}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1617\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(442\) \(1079\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(3\) 0.991445 0.130526i 0.991445 0.130526i
\(4\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(5\) −0.923880 + 1.60021i −0.923880 + 1.60021i −0.130526 + 0.991445i \(0.541667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0.965926 0.258819i 0.965926 0.258819i
\(10\) 0 0
\(11\) 0.866025 0.500000i 0.866025 0.500000i
\(12\) 0.608761 + 0.793353i 0.608761 + 0.793353i
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) −1.84776 −1.84776
\(21\) 0 0
\(22\) 0 0
\(23\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(24\) 0 0
\(25\) −1.20711 2.09077i −1.20711 2.09077i
\(26\) 0 0
\(27\) 0.923880 0.382683i 0.923880 0.382683i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −0.662827 + 0.382683i −0.662827 + 0.382683i −0.793353 0.608761i \(-0.791667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(32\) 0 0
\(33\) 0.793353 0.608761i 0.793353 0.608761i
\(34\) 0 0
\(35\) 0 0
\(36\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(37\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(45\) −0.478235 + 1.78480i −0.478235 + 1.78480i
\(46\) 0 0
\(47\) 0.382683 0.662827i 0.382683 0.662827i −0.608761 0.793353i \(-0.708333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(48\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(54\) 0 0
\(55\) 1.84776i 1.84776i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.382683 0.662827i −0.382683 0.662827i 0.608761 0.793353i \(-0.291667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(60\) −1.83195 + 0.241181i −1.83195 + 0.241181i
\(61\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(68\) 0 0
\(69\) −1.30656 0.541196i −1.30656 0.541196i
\(70\) 0 0
\(71\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(72\) 0 0
\(73\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(74\) 0 0
\(75\) −1.46968 1.91532i −1.46968 1.91532i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) −0.923880 1.60021i −0.923880 1.60021i
\(81\) 0.866025 0.500000i 0.866025 0.500000i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.382683 + 0.662827i −0.382683 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.41421i 1.41421i
\(93\) −0.607206 + 0.465926i −0.607206 + 0.465926i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(98\) 0 0
\(99\) 0.707107 0.707107i 0.707107 0.707107i
\(100\) 1.20711 2.09077i 1.20711 2.09077i
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) 0.662827 + 0.382683i 0.662827 + 0.382683i 0.793353 0.608761i \(-0.208333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(108\) 0.793353 + 0.608761i 0.793353 + 0.608761i
\(109\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(114\) 0 0
\(115\) 2.26303 1.30656i 2.26303 1.30656i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.500000 0.866025i 0.500000 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) −0.662827 0.382683i −0.662827 0.382683i
\(125\) 2.61313 2.61313
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(133\) 0 0
\(134\) 0 0
\(135\) −0.241181 + 1.83195i −0.241181 + 1.83195i
\(136\) 0 0
\(137\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0.292893 0.707107i 0.292893 0.707107i
\(142\) 0 0
\(143\) 0 0
\(144\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(150\) 0 0
\(151\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.41421i 1.41421i
\(156\) 0 0
\(157\) 0.662827 0.382683i 0.662827 0.382683i −0.130526 0.991445i \(-0.541667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(158\) 0 0
\(159\) 1.12197 0.860919i 1.12197 0.860919i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(164\) 0 0
\(165\) 0.241181 + 1.83195i 0.241181 + 1.83195i
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −1.00000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000i 1.00000i
\(177\) −0.465926 0.607206i −0.465926 0.607206i
\(178\) 0 0
\(179\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) −1.78480 + 0.478235i −1.78480 + 0.478235i
\(181\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0.765367 0.765367
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) −0.991445 + 0.130526i −0.991445 + 0.130526i
\(193\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −1.60021 + 0.923880i −1.60021 + 0.923880i −0.608761 + 0.793353i \(0.708333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(200\) 0 0
\(201\) 0.860919 + 1.12197i 0.860919 + 1.12197i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.36603 0.366025i −1.36603 0.366025i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(213\) 0.184592 + 1.40211i 0.184592 + 1.40211i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −1.60021 + 0.923880i −1.60021 + 0.923880i
\(221\) 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
\(225\) −1.70711 1.70711i −1.70711 1.70711i
\(226\) 0 0
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 0 0
\(229\) 1.60021 + 0.923880i 1.60021 + 0.923880i 0.991445 + 0.130526i \(0.0416667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(234\) 0 0
\(235\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(236\) 0.382683 0.662827i 0.382683 0.662827i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.12484 1.46593i −1.12484 1.46593i
\(241\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(242\) 0 0
\(243\) 0.793353 0.608761i 0.793353 0.608761i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(252\) 0 0
\(253\) −1.41421 −1.41421
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.500000 0.866025i
\(257\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 2.61313i 2.61313i
\(266\) 0 0
\(267\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(268\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(269\) −0.923880 1.60021i −0.923880 1.60021i −0.793353 0.608761i \(-0.791667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(270\) 0 0
\(271\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.09077 1.20711i −2.09077 1.20711i
\(276\) −0.184592 1.40211i −0.184592 1.40211i
\(277\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) 0 0
\(279\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(284\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(290\) 0 0
\(291\) −0.241181 1.83195i −0.241181 1.83195i
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 1.41421 1.41421
\(296\) 0 0
\(297\) 0.608761 0.793353i 0.608761 0.793353i
\(298\) 0 0
\(299\) 0 0
\(300\) 0.923880 2.23044i 0.923880 2.23044i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(310\) 0 0
\(311\) 0.923880 + 1.60021i 0.923880 + 1.60021i 0.793353 + 0.608761i \(0.208333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(312\) 0 0
\(313\) −1.60021 0.923880i −1.60021 0.923880i −0.991445 0.130526i \(-0.958333\pi\)
−0.608761 0.793353i \(-0.708333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.923880 1.60021i 0.923880 1.60021i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.61313 −2.61313
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) −0.261052 1.98289i −0.261052 1.98289i
\(340\) 0 0
\(341\) −0.382683 + 0.662827i −0.382683 + 0.662827i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2.07313 1.59077i 2.07313 1.59077i
\(346\) 0 0
\(347\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.923880 + 1.60021i 0.923880 + 1.60021i 0.793353 + 0.608761i \(0.208333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(354\) 0 0
\(355\) −2.26303 1.30656i −2.26303 1.30656i
\(356\) −0.765367 −0.765367
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(360\) 0 0
\(361\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(362\) 0 0
\(363\) 0.382683 0.923880i 0.382683 0.923880i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.662827 + 0.382683i −0.662827 + 0.382683i −0.793353 0.608761i \(-0.791667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(368\) 1.22474 0.707107i 1.22474 0.707107i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −0.707107 0.292893i −0.707107 0.292893i
\(373\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) 2.59077 0.341081i 2.59077 0.341081i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.923880 + 1.60021i −0.923880 + 1.60021i −0.130526 + 0.991445i \(0.541667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 1.60021 0.923880i 1.60021 0.923880i
\(389\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(397\) −1.60021 0.923880i −1.60021 0.923880i −0.991445 0.130526i \(-0.958333\pi\)
−0.608761 0.793353i \(-0.708333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 2.41421 2.41421
\(401\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.84776i 1.84776i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.765367i 0.765367i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(420\) 0 0
\(421\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(422\) 0 0
\(423\) 0.198092 0.739288i 0.198092 0.739288i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(432\) −0.130526 + 0.991445i −0.130526 + 0.991445i
\(433\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.73205 1.00000i −1.73205 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 0.500000i \(-0.833333\pi\)
\(444\) 0 0
\(445\) −0.707107 1.22474i −0.707107 1.22474i
\(446\) 0 0
\(447\) 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 0
\(451\) 0 0
\(452\) 1.73205 1.00000i 1.73205 1.00000i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 2.26303 + 1.30656i 2.26303 + 1.30656i
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(464\) 0 0
\(465\) −0.184592 1.40211i −0.184592 1.40211i
\(466\) 0 0
\(467\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0.607206 0.465926i 0.607206 0.465926i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.00000 1.00000i 1.00000 1.00000i
\(478\) 0 0
\(479\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.00000 1.00000
\(485\) 2.95680 + 1.70711i 2.95680 + 1.70711i
\(486\) 0 0
\(487\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(488\) 0 0
\(489\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0.478235 + 1.78480i 0.478235 + 1.78480i
\(496\) 0.765367i 0.765367i
\(497\) 0 0
\(498\) 0 0
\(499\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(500\) 1.30656 + 2.26303i 1.30656 + 2.26303i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.991445 + 0.130526i −0.991445 + 0.130526i
\(508\) 0 0
\(509\) −0.923880 + 1.60021i −0.923880 + 1.60021i −0.130526 + 0.991445i \(0.541667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(516\) 0 0
\(517\) 0.765367i 0.765367i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.382683 + 0.662827i 0.382683 + 0.662827i 0.991445 0.130526i \(-0.0416667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(522\) 0 0
\(523\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0.130526 + 0.991445i 0.130526 + 0.991445i
\(529\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) −0.541196 0.541196i −0.541196 0.541196i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.58671 + 1.21752i −1.58671 + 1.21752i
\(538\) 0 0
\(539\) 0 0
\(540\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(541\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(542\) 0 0
\(543\) −0.241181 1.83195i −0.241181 1.83195i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0.758819 0.0999004i 0.758819 0.0999004i
\(565\) 3.20041 + 1.84776i 3.20041 + 1.84776i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.41421i 3.41421i
\(576\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(577\) −0.662827 + 0.382683i −0.662827 + 0.382683i −0.793353 0.608761i \(-0.791667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.707107 1.22474i 0.707107 1.22474i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.46593 + 1.12484i −1.46593 + 1.12484i
\(598\) 0 0
\(599\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(604\) 0 0
\(605\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(606\) 0 0
\(607\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −1.60021 + 0.923880i −1.60021 + 0.923880i −0.608761 + 0.793353i \(0.708333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(620\) 1.22474 0.707107i 1.22474 0.707107i
\(621\) −1.40211 0.184592i −1.40211 0.184592i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.20711 + 2.09077i −1.20711 + 2.09077i
\(626\) 0 0
\(627\) 0 0
\(628\) 0.662827 + 0.382683i 0.662827 + 0.382683i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(637\) 0 0
\(638\) 0 0
\(639\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.382683 + 0.662827i 0.382683 + 0.662827i 0.991445 0.130526i \(-0.0416667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(648\) 0 0
\(649\) −0.662827 0.382683i −0.662827 0.382683i
\(650\) 0 0
\(651\) 0 0
\(652\) −1.41421 −1.41421
\(653\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) −1.46593 + 1.12484i −1.46593 + 1.12484i
\(661\) 1.60021 0.923880i 1.60021 0.923880i 0.608761 0.793353i \(-0.291667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −0.0999004 0.758819i −0.0999004 0.758819i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −1.91532 1.46968i −1.91532 1.46968i
\(676\) −0.500000 0.866025i −0.500000 0.866025i
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0.662827 + 0.382683i 0.662827 + 0.382683i 0.793353 0.608761i \(-0.208333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(705\) 0.860919 + 1.12197i 0.860919 + 1.12197i
\(706\) 0 0
\(707\) 0 0
\(708\) 0.292893 0.707107i 0.292893 0.707107i
\(709\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.08239 1.08239
\(714\) 0 0
\(715\) 0 0
\(716\) −1.73205 1.00000i −1.73205 1.00000i
\(717\) 0 0
\(718\) 0 0
\(719\) −0.382683 + 0.662827i −0.382683 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(720\) −1.30656 1.30656i −1.30656 1.30656i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 1.60021 0.923880i 1.60021 0.923880i
\(725\) 0 0
\(726\) 0 0
\(727\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(728\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(738\) 0 0
\(739\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(740\) 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
\(751\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(752\) 0.382683 + 0.662827i 0.382683 + 0.662827i
\(753\) 1.83195 0.241181i 1.83195 0.241181i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) 0 0
\(759\) −1.40211 + 0.184592i −1.40211 + 0.184592i
\(760\) 0 0
\(761\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.608761 0.793353i −0.608761 0.793353i
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0.707107 1.70711i 0.707107 1.70711i
\(772\) 0 0
\(773\) −0.923880 1.60021i −0.923880 1.60021i −0.793353 0.608761i \(-0.791667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(774\) 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
\(781\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.41421i 1.41421i
\(786\) 0 0
\(787\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0.341081 + 2.59077i 0.341081 + 2.59077i
\(796\) −1.60021 0.923880i −1.60021 0.923880i
\(797\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −0.198092 + 0.739288i −0.198092 + 0.739288i
\(802\) 0 0
\(803\) 0 0
\(804\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(805\) 0 0
\(806\) 0 0
\(807\) −1.12484 1.46593i −1.12484 1.46593i
\(808\) 0 0
\(809\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.30656 2.26303i −1.30656 2.26303i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(822\) 0 0
\(823\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(824\) 0 0
\(825\) −2.23044 0.923880i −2.23044 0.923880i
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) −0.366025 1.36603i −0.366025 1.36603i
\(829\) 1.60021 0.923880i 1.60021 0.923880i 0.608761 0.793353i \(-0.291667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −0.465926 + 0.607206i −0.465926 + 0.607206i
\(838\) 0 0
\(839\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(840\) 0 0
\(841\) −1.00000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.923880 1.60021i 0.923880 1.60021i
\(846\) 0 0
\(847\) 0 0
\(848\) 1.41421i 1.41421i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) −1.12197 + 0.860919i −1.12197 + 0.860919i
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(858\) 0 0
\(859\) −0.662827 0.382683i −0.662827 0.382683i 0.130526 0.991445i \(-0.458333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −0.478235 1.78480i −0.478235 1.78480i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −1.60021 0.923880i −1.60021 0.923880i
\(881\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 1.40211 0.184592i 1.40211 0.184592i
\(886\) 0 0
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.500000 0.866025i 0.500000 0.866025i
\(892\) 0.662827 0.382683i 0.662827 0.382683i
\(893\) 0 0
\(894\) 0 0
\(895\) 3.69552i 3.69552i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0.624844 2.33195i 0.624844 2.33195i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.95680 + 1.70711i 2.95680 + 1.70711i
\(906\) 0 0
\(907\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.84776i 1.84776i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0.739288 + 0.198092i 0.739288 + 0.198092i
\(928\) 0 0
\(929\) −0.382683 + 0.662827i −0.382683 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.12484 + 1.46593i 1.12484 + 1.46593i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) −1.70711 0.707107i −1.70711 0.707107i
\(940\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(941\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0.765367 0.765367
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0.707107 1.70711i 0.707107 1.70711i
\(961\) −0.207107 + 0.358719i −0.207107 + 0.358719i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(972\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(978\) 0 0
\(979\) 0.765367i 0.765367i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0.923880 + 1.60021i 0.923880 + 1.60021i 0.793353 + 0.608761i \(0.208333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3.41421i 3.41421i
\(996\) 0 0
\(997\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(998\) 0 0
\(999\) 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 1617.1.k.e.362.8 16
3.2 odd 2 inner 1617.1.k.e.362.6 16
7.2 even 3 1617.1.h.a.1616.4 yes 8
7.3 odd 6 inner 1617.1.k.e.1550.6 16
7.4 even 3 inner 1617.1.k.e.1550.3 16
7.5 odd 6 1617.1.h.a.1616.5 yes 8
7.6 odd 2 inner 1617.1.k.e.362.1 16
11.10 odd 2 CM 1617.1.k.e.362.8 16
21.2 odd 6 1617.1.h.a.1616.6 yes 8
21.5 even 6 1617.1.h.a.1616.3 8
21.11 odd 6 inner 1617.1.k.e.1550.1 16
21.17 even 6 inner 1617.1.k.e.1550.8 16
21.20 even 2 inner 1617.1.k.e.362.3 16
33.32 even 2 inner 1617.1.k.e.362.6 16
77.10 even 6 inner 1617.1.k.e.1550.6 16
77.32 odd 6 inner 1617.1.k.e.1550.3 16
77.54 even 6 1617.1.h.a.1616.5 yes 8
77.65 odd 6 1617.1.h.a.1616.4 yes 8
77.76 even 2 inner 1617.1.k.e.362.1 16
231.32 even 6 inner 1617.1.k.e.1550.1 16
231.65 even 6 1617.1.h.a.1616.6 yes 8
231.131 odd 6 1617.1.h.a.1616.3 8
231.164 odd 6 inner 1617.1.k.e.1550.8 16
231.230 odd 2 inner 1617.1.k.e.362.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.1.h.a.1616.3 8 21.5 even 6
1617.1.h.a.1616.3 8 231.131 odd 6
1617.1.h.a.1616.4 yes 8 7.2 even 3
1617.1.h.a.1616.4 yes 8 77.65 odd 6
1617.1.h.a.1616.5 yes 8 7.5 odd 6
1617.1.h.a.1616.5 yes 8 77.54 even 6
1617.1.h.a.1616.6 yes 8 21.2 odd 6
1617.1.h.a.1616.6 yes 8 231.65 even 6
1617.1.k.e.362.1 16 7.6 odd 2 inner
1617.1.k.e.362.1 16 77.76 even 2 inner
1617.1.k.e.362.3 16 21.20 even 2 inner
1617.1.k.e.362.3 16 231.230 odd 2 inner
1617.1.k.e.362.6 16 3.2 odd 2 inner
1617.1.k.e.362.6 16 33.32 even 2 inner
1617.1.k.e.362.8 16 1.1 even 1 trivial
1617.1.k.e.362.8 16 11.10 odd 2 CM
1617.1.k.e.1550.1 16 21.11 odd 6 inner
1617.1.k.e.1550.1 16 231.32 even 6 inner
1617.1.k.e.1550.3 16 7.4 even 3 inner
1617.1.k.e.1550.3 16 77.32 odd 6 inner
1617.1.k.e.1550.6 16 7.3 odd 6 inner
1617.1.k.e.1550.6 16 77.10 even 6 inner
1617.1.k.e.1550.8 16 21.17 even 6 inner
1617.1.k.e.1550.8 16 231.164 odd 6 inner