Properties

Label 1176.1.e.a.587.6
Level $1176$
Weight $1$
Character 1176.587
Analytic conductor $0.587$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -8
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1176,1,Mod(587,1176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1176.587");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1176.e (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: \(0.586900454856\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) 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 8.2.273231802368.3

Embedding invariants

Embedding label 587.6
Root \(0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 1176.587
Dual form 1176.1.e.a.587.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +(-0.382683 - 0.923880i) q^{3} -1.00000 q^{4} +(0.923880 - 0.382683i) q^{6} -1.00000i q^{8} +(-0.707107 + 0.707107i) q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +(-0.382683 - 0.923880i) q^{3} -1.00000 q^{4} +(0.923880 - 0.382683i) q^{6} -1.00000i q^{8} +(-0.707107 + 0.707107i) q^{9} +1.41421i q^{11} +(0.382683 + 0.923880i) q^{12} +1.00000 q^{16} +1.84776 q^{17} +(-0.707107 - 0.707107i) q^{18} -0.765367i q^{19} -1.41421 q^{22} +(-0.923880 + 0.382683i) q^{24} +1.00000 q^{25} +(0.923880 + 0.382683i) q^{27} +1.00000i q^{32} +(1.30656 - 0.541196i) q^{33} +1.84776i q^{34} +(0.707107 - 0.707107i) q^{36} +0.765367 q^{38} +0.765367 q^{41} +1.41421 q^{43} -1.41421i q^{44} +(-0.382683 - 0.923880i) q^{48} +1.00000i q^{50} +(-0.707107 - 1.70711i) q^{51} +(-0.382683 + 0.923880i) q^{54} +(-0.707107 + 0.292893i) q^{57} -1.84776 q^{59} -1.00000 q^{64} +(0.541196 + 1.30656i) q^{66} -1.84776 q^{68} +(0.707107 + 0.707107i) q^{72} +1.84776i q^{73} +(-0.382683 - 0.923880i) q^{75} +0.765367i q^{76} -1.00000i q^{81} +0.765367i q^{82} -1.84776 q^{83} +1.41421i q^{86} +1.41421 q^{88} -0.765367 q^{89} +(0.923880 - 0.382683i) q^{96} -0.765367i q^{97} +(-1.00000 - 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 8 q^{16} + 8 q^{25} - 8 q^{64} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times\).

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 1.00000i
\(3\) −0.382683 0.923880i −0.382683 0.923880i
\(4\) −1.00000 −1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0.923880 0.382683i 0.923880 0.382683i
\(7\) 0 0
\(8\) 1.00000i 1.00000i
\(9\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(10\) 0 0
\(11\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(18\) −0.707107 0.707107i −0.707107 0.707107i
\(19\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.41421 −1.41421
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(25\) 1.00000 1.00000
\(26\) 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 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000i 1.00000i
\(33\) 1.30656 0.541196i 1.30656 0.541196i
\(34\) 1.84776i 1.84776i
\(35\) 0 0
\(36\) 0.707107 0.707107i 0.707107 0.707107i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0.765367 0.765367
\(39\) 0 0
\(40\) 0 0
\(41\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(42\) 0 0
\(43\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 1.41421i 1.41421i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −0.382683 0.923880i −0.382683 0.923880i
\(49\) 0 0
\(50\) 1.00000i 1.00000i
\(51\) −0.707107 1.70711i −0.707107 1.70711i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(55\) 0 0
\(56\) 0 0
\(57\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(58\) 0 0
\(59\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −1.84776 −1.84776
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(73\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(74\) 0 0
\(75\) −0.382683 0.923880i −0.382683 0.923880i
\(76\) 0.765367i 0.765367i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000i 1.00000i
\(82\) 0.765367i 0.765367i
\(83\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.41421i 1.41421i
\(87\) 0 0
\(88\) 1.41421 1.41421
\(89\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0.923880 0.382683i 0.923880 0.382683i
\(97\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(98\) 0 0
\(99\) −1.00000 1.00000i −1.00000 1.00000i
\(100\) −1.00000 −1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 1.70711 0.707107i 1.70711 0.707107i
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −0.923880 0.382683i −0.923880 0.382683i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) −0.292893 0.707107i −0.292893 0.707107i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 1.84776i 1.84776i
\(119\) 0 0
\(120\) 0 0
\(121\) −1.00000 −1.00000
\(122\) 0 0
\(123\) −0.292893 0.707107i −0.292893 0.707107i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 1.00000i 1.00000i
\(129\) −0.541196 1.30656i −0.541196 1.30656i
\(130\) 0 0
\(131\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(132\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 1.84776i 1.84776i
\(137\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(138\) 0 0
\(139\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(145\) 0 0
\(146\) −1.84776 −1.84776
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0.923880 0.382683i 0.923880 0.382683i
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) −0.765367 −0.765367
\(153\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 1.00000
\(163\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) −0.765367 −0.765367
\(165\) 0 0
\(166\) 1.84776i 1.84776i
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −1.00000 −1.00000
\(170\) 0 0
\(171\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(172\) −1.41421 −1.41421
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.41421i 1.41421i
\(177\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(178\) 0.765367i 0.765367i
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 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 0
\(187\) 2.61313i 2.61313i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(193\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(194\) 0.765367 0.765367
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 1.00000 1.00000i 1.00000 1.00000i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000i 1.00000i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.08239 1.08239
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0.382683 0.923880i 0.382683 0.923880i
\(217\) 0 0
\(218\) 0 0
\(219\) 1.70711 0.707107i 1.70711 0.707107i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(226\) −1.41421 −1.41421
\(227\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(228\) 0.707107 0.292893i 0.707107 0.292893i
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.84776 1.84776
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(242\) 1.00000i 1.00000i
\(243\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(244\) 0 0
\(245\) 0 0
\(246\) 0.707107 0.292893i 0.707107 0.292893i
\(247\) 0 0
\(248\) 0 0
\(249\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(250\) 0 0
\(251\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(258\) 1.30656 0.541196i 1.30656 0.541196i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0.765367i 0.765367i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −0.541196 1.30656i −0.541196 1.30656i
\(265\) 0 0
\(266\) 0 0
\(267\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 1.84776 1.84776
\(273\) 0 0
\(274\) 1.41421 1.41421
\(275\) 1.41421i 1.41421i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 1.84776 1.84776
\(279\) 0 0
\(280\) 0 0
\(281\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(282\) 0 0
\(283\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.707107 0.707107i −0.707107 0.707107i
\(289\) 2.41421 2.41421
\(290\) 0 0
\(291\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(292\) 1.84776i 1.84776i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(298\) 0 0
\(299\) 0 0
\(300\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0.765367i 0.765367i
\(305\) 0 0
\(306\) −1.30656 1.30656i −1.30656 1.30656i
\(307\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.41421i 1.41421i
\(324\) 1.00000i 1.00000i
\(325\) 0 0
\(326\) 1.41421i 1.41421i
\(327\) 0 0
\(328\) 0.765367i 0.765367i
\(329\) 0 0
\(330\) 0 0
\(331\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(332\) 1.84776 1.84776
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 1.00000i 1.00000i
\(339\) 1.30656 0.541196i 1.30656 0.541196i
\(340\) 0 0
\(341\) 0 0
\(342\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(343\) 0 0
\(344\) 1.41421i 1.41421i
\(345\) 0 0
\(346\) 0 0
\(347\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.41421 −1.41421
\(353\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(354\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(355\) 0 0
\(356\) 0.765367 0.765367
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 0.414214 0.414214
\(362\) 0 0
\(363\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) −2.61313 −2.61313
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(385\) 0 0
\(386\) 1.41421i 1.41421i
\(387\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(388\) 0.765367i 0.765367i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −0.292893 0.707107i −0.292893 0.707107i
\(394\) 0 0
\(395\) 0 0
\(396\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(409\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(410\) 0 0
\(411\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(418\) 1.08239i 1.08239i
\(419\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.84776 1.84776
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(433\) 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
\(438\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(450\) −0.707107 0.707107i −0.707107 0.707107i
\(451\) 1.08239i 1.08239i
\(452\) 1.41421i 1.41421i
\(453\) 0 0
\(454\) 0.765367i 0.765367i
\(455\) 0 0
\(456\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(457\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) 0 0
\(459\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 1.41421 1.41421
\(467\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 1.84776i 1.84776i
\(473\) 2.00000i 2.00000i
\(474\) 0 0
\(475\) 0.765367i 0.765367i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1.84776 1.84776
\(483\) 0 0
\(484\) 1.00000 1.00000
\(485\) 0 0
\(486\) −0.382683 0.923880i −0.382683 0.923880i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) −0.541196 1.30656i −0.541196 1.30656i
\(490\) 0 0
\(491\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0.765367i 0.765367i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 1.00000i
\(513\) 0.292893 0.707107i 0.292893 0.707107i
\(514\) 1.84776i 1.84776i
\(515\) 0 0
\(516\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(522\) 0 0
\(523\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(524\) −0.765367 −0.765367
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 1.30656 0.541196i 1.30656 0.541196i
\(529\) −1.00000 −1.00000
\(530\) 0 0
\(531\) 1.30656 1.30656i 1.30656 1.30656i
\(532\) 0 0
\(533\) 0 0
\(534\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.84776i 1.84776i
\(545\) 0 0
\(546\) 0 0
\(547\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 1.41421i 1.41421i
\(549\) 0 0
\(550\) −1.41421 −1.41421
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 1.84776i 1.84776i
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 2.41421 1.00000i 2.41421 1.00000i
\(562\) 1.41421 1.41421
\(563\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.765367 −0.765367
\(567\) 0 0
\(568\) 0 0
\(569\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(570\) 0 0
\(571\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.707107 0.707107i 0.707107 0.707107i
\(577\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(578\) 2.41421i 2.41421i
\(579\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(580\) 0 0
\(581\) 0 0
\(582\) −0.292893 0.707107i −0.292893 0.707107i
\(583\) 0 0
\(584\) 1.84776 1.84776
\(585\) 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
\(591\) 0 0
\(592\) 0 0
\(593\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(594\) −1.30656 0.541196i −1.30656 0.541196i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(601\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0.765367 0.765367
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 1.30656 1.30656i 1.30656 1.30656i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −1.84776 −1.84776
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 1.00000
\(626\) −1.84776 −1.84776
\(627\) −0.414214 1.00000i −0.414214 1.00000i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(642\) 0 0
\(643\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1.41421 1.41421
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −1.00000 −1.00000
\(649\) 2.61313i 2.61313i
\(650\) 0 0
\(651\) 0 0
\(652\) −1.41421 −1.41421
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.765367 0.765367
\(657\) −1.30656 1.30656i −1.30656 1.30656i
\(658\) 0 0
\(659\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 1.41421i 1.41421i
\(663\) 0 0
\(664\) 1.84776i 1.84776i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(674\) 0 0
\(675\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(676\) 1.00000 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(679\) 0 0
\(680\) 0 0
\(681\) −0.292893 0.707107i −0.292893 0.707107i
\(682\) 0 0
\(683\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(684\) −0.541196 0.541196i −0.541196 0.541196i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 1.41421 1.41421
\(689\) 0 0
\(690\) 0 0
\(691\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 1.41421 1.41421
\(695\) 0 0
\(696\) 0 0
\(697\) 1.41421 1.41421
\(698\) 0 0
\(699\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.41421i 1.41421i
\(705\) 0 0
\(706\) 0.765367i 0.765367i
\(707\) 0 0
\(708\) −0.707107 1.70711i −0.707107 1.70711i
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.765367i 0.765367i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.414214i 0.414214i
\(723\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(724\) 0 0
\(725\) 0 0
\(726\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(727\) 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
\(731\) 2.61313 2.61313
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −0.541196 0.541196i −0.541196 0.541196i
\(739\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(740\) 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
\(747\) 1.30656 1.30656i 1.30656 1.30656i
\(748\) 2.61313i 2.61313i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 1.41421i 1.41421i
\(759\) 0 0
\(760\) 0 0
\(761\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.382683 0.923880i −0.382683 0.923880i
\(769\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(770\) 0 0
\(771\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(772\) 1.41421 1.41421
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −1.00000 1.00000i −1.00000 1.00000i
\(775\) 0 0
\(776\) −0.765367 −0.765367
\(777\) 0 0
\(778\) 0 0
\(779\) 0.585786i 0.585786i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0.707107 0.292893i 0.707107 0.292893i
\(787\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000i 1.00000i
\(801\) 0.541196 0.541196i 0.541196 0.541196i
\(802\) 0 0
\(803\) −2.61313 −2.61313
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(810\) 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 0
\(814\) 0 0
\(815\) 0 0
\(816\) −0.707107 1.70711i −0.707107 1.70711i
\(817\) 1.08239i 1.08239i
\(818\) −0.765367 −0.765367
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) −0.541196 1.30656i −0.541196 1.30656i
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 1.30656 0.541196i 1.30656 0.541196i
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) −0.707107 1.70711i −0.707107 1.70711i
\(835\) 0 0
\(836\) −1.08239 −1.08239
\(837\) 0 0
\(838\) 1.84776i 1.84776i
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −1.00000 −1.00000
\(842\) 0 0
\(843\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0.707107 0.292893i 0.707107 0.292893i
\(850\) 1.84776i 1.84776i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(858\) 0 0
\(859\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(865\) 0 0
\(866\) −0.765367 −0.765367
\(867\) −0.923880 2.23044i −0.923880 2.23044i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(874\) 0 0
\(875\) 0 0
\(876\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(882\) 0 0
\(883\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(884\) 0 0
\(885\) 0 0
\(886\) −2.00000 −2.00000
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.41421 1.41421
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 2.00000 2.00000
\(899\) 0 0
\(900\) 0.707107 0.707107i 0.707107 0.707107i
\(901\) 0 0
\(902\) −1.08239 −1.08239
\(903\) 0 0
\(904\) 1.41421 1.41421
\(905\) 0 0
\(906\) 0 0
\(907\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(908\) −0.765367 −0.765367
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(913\) 2.61313i 2.61313i
\(914\) 1.41421i 1.41421i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 1.70711 0.707107i 1.70711 0.707107i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.41421i 1.41421i
\(933\) 0 0
\(934\) 1.84776i 1.84776i
\(935\) 0 0
\(936\) 0 0
\(937\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(938\) 0 0
\(939\) 1.70711 0.707107i 1.70711 0.707107i
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.84776 −1.84776
\(945\) 0 0
\(946\) −2.00000 −2.00000
\(947\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0.765367 0.765367
\(951\) 0 0
\(952\) 0 0
\(953\) 2.00000i 2.00000i 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 0
\(961\) −1.00000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 1.84776i 1.84776i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 1.00000i 1.00000i
\(969\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(970\) 0 0
\(971\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(972\) 0.923880 0.382683i 0.923880 0.382683i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(978\) 1.30656 0.541196i 1.30656 0.541196i
\(979\) 1.08239i 1.08239i
\(980\) 0 0
\(981\) 0 0
\(982\) −2.00000 −2.00000
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(994\) 0 0
\(995\) 0 0
\(996\) −0.707107 1.70711i −0.707107 1.70711i
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 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 1176.1.e.a.587.6 yes 8
3.2 odd 2 inner 1176.1.e.a.587.3 yes 8
7.2 even 3 1176.1.be.a.227.8 16
7.3 odd 6 1176.1.be.a.803.3 16
7.4 even 3 1176.1.be.a.803.2 16
7.5 odd 6 1176.1.be.a.227.5 16
7.6 odd 2 inner 1176.1.e.a.587.7 yes 8
8.3 odd 2 CM 1176.1.e.a.587.6 yes 8
21.2 odd 6 1176.1.be.a.227.3 16
21.5 even 6 1176.1.be.a.227.2 16
21.11 odd 6 1176.1.be.a.803.5 16
21.17 even 6 1176.1.be.a.803.8 16
21.20 even 2 inner 1176.1.e.a.587.2 8
24.11 even 2 inner 1176.1.e.a.587.3 yes 8
56.3 even 6 1176.1.be.a.803.3 16
56.11 odd 6 1176.1.be.a.803.2 16
56.19 even 6 1176.1.be.a.227.5 16
56.27 even 2 inner 1176.1.e.a.587.7 yes 8
56.51 odd 6 1176.1.be.a.227.8 16
168.11 even 6 1176.1.be.a.803.5 16
168.59 odd 6 1176.1.be.a.803.8 16
168.83 odd 2 inner 1176.1.e.a.587.2 8
168.107 even 6 1176.1.be.a.227.3 16
168.131 odd 6 1176.1.be.a.227.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.1.e.a.587.2 8 21.20 even 2 inner
1176.1.e.a.587.2 8 168.83 odd 2 inner
1176.1.e.a.587.3 yes 8 3.2 odd 2 inner
1176.1.e.a.587.3 yes 8 24.11 even 2 inner
1176.1.e.a.587.6 yes 8 1.1 even 1 trivial
1176.1.e.a.587.6 yes 8 8.3 odd 2 CM
1176.1.e.a.587.7 yes 8 7.6 odd 2 inner
1176.1.e.a.587.7 yes 8 56.27 even 2 inner
1176.1.be.a.227.2 16 21.5 even 6
1176.1.be.a.227.2 16 168.131 odd 6
1176.1.be.a.227.3 16 21.2 odd 6
1176.1.be.a.227.3 16 168.107 even 6
1176.1.be.a.227.5 16 7.5 odd 6
1176.1.be.a.227.5 16 56.19 even 6
1176.1.be.a.227.8 16 7.2 even 3
1176.1.be.a.227.8 16 56.51 odd 6
1176.1.be.a.803.2 16 7.4 even 3
1176.1.be.a.803.2 16 56.11 odd 6
1176.1.be.a.803.3 16 7.3 odd 6
1176.1.be.a.803.3 16 56.3 even 6
1176.1.be.a.803.5 16 21.11 odd 6
1176.1.be.a.803.5 16 168.11 even 6
1176.1.be.a.803.8 16 21.17 even 6
1176.1.be.a.803.8 16 168.59 odd 6