Properties

Label 1024.2.e.k.257.2
Level $1024$
Weight $2$
Character 1024.257
Analytic conductor $8.177$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1024,2,Mod(257,1024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1024.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.e (of order \(4\), 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: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 257.2
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1024.257
Dual form 1024.2.e.k.769.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.82843 - 2.82843i) q^{5} -3.00000i q^{9} +O(q^{10})\) \(q+(2.82843 - 2.82843i) q^{5} -3.00000i q^{9} +(2.82843 + 2.82843i) q^{13} +2.00000 q^{17} -11.0000i q^{25} +(-2.82843 - 2.82843i) q^{29} +(-8.48528 + 8.48528i) q^{37} -10.0000i q^{41} +(-8.48528 - 8.48528i) q^{45} +7.00000 q^{49} +(-2.82843 + 2.82843i) q^{53} +(8.48528 + 8.48528i) q^{61} +16.0000 q^{65} -6.00000i q^{73} -9.00000 q^{81} +(5.65685 - 5.65685i) q^{85} -10.0000i q^{89} -18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{17} + 28 q^{49} + 64 q^{65} - 36 q^{81} - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) 2.82843 2.82843i 1.26491 1.26491i 0.316228 0.948683i \(-0.397584\pi\)
0.948683 0.316228i \(-0.102416\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(12\) 0 0
\(13\) 2.82843 + 2.82843i 0.784465 + 0.784465i 0.980581 0.196116i \(-0.0628330\pi\)
−0.196116 + 0.980581i \(0.562833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 11.0000i 2.20000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.82843 2.82843i −0.525226 0.525226i 0.393919 0.919145i \(-0.371119\pi\)
−0.919145 + 0.393919i \(0.871119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.48528 + 8.48528i −1.39497 + 1.39497i −0.581238 + 0.813733i \(0.697432\pi\)
−0.813733 + 0.581238i \(0.802568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000i 1.56174i −0.624695 0.780869i \(-0.714777\pi\)
0.624695 0.780869i \(-0.285223\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) −8.48528 8.48528i −1.26491 1.26491i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.82843 + 2.82843i −0.388514 + 0.388514i −0.874157 0.485643i \(-0.838586\pi\)
0.485643 + 0.874157i \(0.338586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(60\) 0 0
\(61\) 8.48528 + 8.48528i 1.08643 + 1.08643i 0.995893 + 0.0905357i \(0.0288579\pi\)
0.0905357 + 0.995893i \(0.471142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 16.0000 1.98456
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 5.65685 5.65685i 0.613572 0.613572i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000i 1.06000i −0.847998 0.529999i \(-0.822192\pi\)
0.847998 0.529999i \(-0.177808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.1421 14.1421i 1.40720 1.40720i 0.633238 0.773957i \(-0.281726\pi\)
0.773957 0.633238i \(-0.218274\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) 0 0
\(109\) 14.1421 + 14.1421i 1.35457 + 1.35457i 0.880471 + 0.474100i \(0.157226\pi\)
0.474100 + 0.880471i \(0.342774\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 8.48528 8.48528i 0.784465 0.784465i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000i 1.00000i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −16.9706 16.9706i −1.51789 1.51789i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 22.0000i 1.87959i 0.341743 + 0.939793i \(0.388983\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 0 0
\(139\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −16.0000 −1.32873
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −14.1421 + 14.1421i −1.15857 + 1.15857i −0.173785 + 0.984784i \(0.555600\pi\)
−0.984784 + 0.173785i \(0.944400\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.48528 + 8.48528i 0.677199 + 0.677199i 0.959366 0.282166i \(-0.0910530\pi\)
−0.282166 + 0.959366i \(0.591053\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 3.00000i 0.230769i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.82843 + 2.82843i 0.215041 + 0.215041i 0.806405 0.591364i \(-0.201410\pi\)
−0.591364 + 0.806405i \(0.701410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(180\) 0 0
\(181\) −14.1421 + 14.1421i −1.05118 + 1.05118i −0.0525588 + 0.998618i \(0.516738\pi\)
−0.998618 + 0.0525588i \(0.983262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 48.0000i 3.52903i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −19.7990 + 19.7990i −1.41062 + 1.41062i −0.654931 + 0.755689i \(0.727302\pi\)
−0.755689 + 0.654931i \(0.772698\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −28.2843 28.2843i −1.97546 1.97546i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.65685 + 5.65685i 0.380521 + 0.380521i
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −33.0000 −2.20000
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 2.82843 2.82843i 0.186908 0.186908i −0.607450 0.794358i \(-0.707808\pi\)
0.794358 + 0.607450i \(0.207808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 26.0000i 1.70332i 0.524097 + 0.851658i \(0.324403\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −30.0000 −1.93247 −0.966235 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 19.7990 19.7990i 1.26491 1.26491i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −8.48528 + 8.48528i −0.525226 + 0.525226i
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 16.0000i 0.982872i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 14.1421 + 14.1421i 0.862261 + 0.862261i 0.991600 0.129339i \(-0.0412856\pi\)
−0.129339 + 0.991600i \(0.541286\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 19.7990 19.7990i 1.18961 1.18961i 0.212430 0.977176i \(-0.431862\pi\)
0.977176 0.212430i \(-0.0681376\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000i 0.596550i −0.954480 0.298275i \(-0.903589\pi\)
0.954480 0.298275i \(-0.0964112\pi\)
\(282\) 0 0
\(283\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.82843 2.82843i 0.165238 0.165238i −0.619644 0.784883i \(-0.712723\pi\)
0.784883 + 0.619644i \(0.212723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 48.0000 2.74847
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 26.0000i 1.46961i 0.678280 + 0.734803i \(0.262726\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 19.7990 + 19.7990i 1.11202 + 1.11202i 0.992877 + 0.119145i \(0.0380154\pi\)
0.119145 + 0.992877i \(0.461985\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 31.1127 31.1127i 1.72582 1.72582i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(332\) 0 0
\(333\) 25.4558 + 25.4558i 1.39497 + 1.39497i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) −25.4558 25.4558i −1.36262 1.36262i −0.870563 0.492057i \(-0.836245\pi\)
−0.492057 0.870563i \(-0.663755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 34.0000 1.80964 0.904819 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 19.0000i 1.00000i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −16.9706 16.9706i −0.888280 0.888280i
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −30.0000 −1.56174
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −25.4558 + 25.4558i −1.31805 + 1.31805i −0.402739 + 0.915315i \(0.631942\pi\)
−0.915315 + 0.402739i \(0.868058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.0000i 0.824042i
\(378\) 0 0
\(379\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 14.1421 14.1421i 0.717035 0.717035i −0.250962 0.967997i \(-0.580747\pi\)
0.967997 + 0.250962i \(0.0807470\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −8.48528 8.48528i −0.425864 0.425864i 0.461353 0.887217i \(-0.347364\pi\)
−0.887217 + 0.461353i \(0.847364\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −25.4558 + 25.4558i −1.26491 + 1.26491i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.00000i 0.296681i −0.988936 0.148340i \(-0.952607\pi\)
0.988936 0.148340i \(-0.0473931\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(420\) 0 0
\(421\) −19.7990 + 19.7990i −0.964944 + 0.964944i −0.999406 0.0344623i \(-0.989028\pi\)
0.0344623 + 0.999406i \(0.489028\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 22.0000i 1.06716i
\(426\) 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 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 21.0000i 1.00000i
\(442\) 0 0
\(443\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(444\) 0 0
\(445\) −28.2843 28.2843i −1.34080 1.34080i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 42.0000i 1.96468i −0.187112 0.982339i \(-0.559913\pi\)
0.187112 0.982339i \(-0.440087\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.1421 + 14.1421i 0.658665 + 0.658665i 0.955064 0.296399i \(-0.0957859\pi\)
−0.296399 + 0.955064i \(0.595786\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 8.48528 + 8.48528i 0.388514 + 0.388514i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −48.0000 −2.18861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −50.9117 + 50.9117i −2.31178 + 2.31178i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(492\) 0 0
\(493\) −5.65685 5.65685i −0.254772 0.254772i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 80.0000i 3.55995i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 31.1127 + 31.1127i 1.37905 + 1.37905i 0.846233 + 0.532813i \(0.178865\pi\)
0.532813 + 0.846233i \(0.321135\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 22.0000i 0.963837i 0.876216 + 0.481919i \(0.160060\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(522\) 0 0
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 28.2843 28.2843i 1.22513 1.22513i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −14.1421 14.1421i −0.608018 0.608018i 0.334410 0.942428i \(-0.391463\pi\)
−0.942428 + 0.334410i \(0.891463\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 80.0000 3.42682
\(546\) 0 0
\(547\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 0 0
\(549\) 25.4558 25.4558i 1.08643 1.08643i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −19.7990 19.7990i −0.838910 0.838910i 0.149805 0.988716i \(-0.452135\pi\)
−0.988716 + 0.149805i \(0.952135\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(564\) 0 0
\(565\) 39.5980 39.5980i 1.66590 1.66590i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.0000i 1.08998i 0.838444 + 0.544988i \(0.183466\pi\)
−0.838444 + 0.544988i \(0.816534\pi\)
\(570\) 0 0
\(571\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 48.0000i 1.98456i
\(586\) 0 0
\(587\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 46.0000 1.88899 0.944497 0.328521i \(-0.106550\pi\)
0.944497 + 0.328521i \(0.106550\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 10.0000i 0.407909i −0.978980 0.203954i \(-0.934621\pi\)
0.978980 0.203954i \(-0.0653794\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 31.1127 + 31.1127i 1.26491 + 1.26491i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 25.4558 25.4558i 1.02815 1.02815i 0.0285598 0.999592i \(-0.490908\pi\)
0.999592 0.0285598i \(-0.00909209\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 38.0000i 1.52982i −0.644136 0.764911i \(-0.722783\pi\)
0.644136 0.764911i \(-0.277217\pi\)
\(618\) 0 0
\(619\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −41.0000 −1.64000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16.9706 + 16.9706i −0.676661 + 0.676661i
\(630\) 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
\(637\) 19.7990 + 19.7990i 0.784465 + 0.784465i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −50.0000 −1.97488 −0.987441 0.157991i \(-0.949498\pi\)
−0.987441 + 0.157991i \(0.949498\pi\)
\(642\) 0 0
\(643\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −31.1127 31.1127i −1.21753 1.21753i −0.968493 0.249041i \(-0.919885\pi\)
−0.249041 0.968493i \(-0.580115\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −18.0000 −0.702247
\(658\) 0 0
\(659\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) 8.48528 8.48528i 0.330039 0.330039i −0.522562 0.852601i \(-0.675024\pi\)
0.852601 + 0.522562i \(0.175024\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 46.0000 1.77317 0.886585 0.462566i \(-0.153071\pi\)
0.886585 + 0.462566i \(0.153071\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 36.7696 36.7696i 1.41317 1.41317i 0.679408 0.733761i \(-0.262237\pi\)
0.733761 0.679408i \(-0.237763\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 0 0
\(685\) 62.2254 + 62.2254i 2.37751 + 2.37751i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −16.0000 −0.609551
\(690\) 0 0
\(691\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 20.0000i 0.757554i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −36.7696 36.7696i −1.38877 1.38877i −0.827919 0.560848i \(-0.810475\pi\)
−0.560848 0.827919i \(-0.689525\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −31.1127 + 31.1127i −1.16846 + 1.16846i −0.185892 + 0.982570i \(0.559517\pi\)
−0.982570 + 0.185892i \(0.940483\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −31.1127 + 31.1127i −1.15550 + 1.15550i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −2.82843 2.82843i −0.104470 0.104470i 0.652940 0.757410i \(-0.273536\pi\)
−0.757410 + 0.652940i \(0.773536\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 80.0000i 2.93097i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −36.7696 + 36.7696i −1.33641 + 1.33641i −0.436904 + 0.899508i \(0.643925\pi\)
−0.899508 + 0.436904i \(0.856075\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 38.0000i 1.37750i −0.724999 0.688749i \(-0.758160\pi\)
0.724999 0.688749i \(-0.241840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −16.9706 16.9706i −0.613572 0.613572i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −31.1127 + 31.1127i −1.11905 + 1.11905i −0.127164 + 0.991882i \(0.540588\pi\)
−0.991882 + 0.127164i \(0.959412\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 48.0000 1.71319
\(786\) 0 0
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 48.0000i 1.70453i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −36.7696 36.7696i −1.30244 1.30244i −0.926739 0.375705i \(-0.877401\pi\)
−0.375705 0.926739i \(-0.622599\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 10.0000i 0.351581i −0.984428 0.175791i \(-0.943752\pi\)
0.984428 0.175791i \(-0.0562482\pi\)
\(810\) 0 0
\(811\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19.7990 19.7990i 0.690990 0.690990i −0.271460 0.962450i \(-0.587507\pi\)
0.962450 + 0.271460i \(0.0875065\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(828\) 0 0
\(829\) −14.1421 14.1421i −0.491177 0.491177i 0.417500 0.908677i \(-0.362906\pi\)
−0.908677 + 0.417500i \(0.862906\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 14.0000 0.485071
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 13.0000i 0.448276i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 8.48528 + 8.48528i 0.291903 + 0.291903i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −25.4558 + 25.4558i −0.871592 + 0.871592i −0.992646 0.121054i \(-0.961372\pi\)
0.121054 + 0.992646i \(0.461372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 58.0000i 1.98124i 0.136637 + 0.990621i \(0.456370\pi\)
−0.136637 + 0.990621i \(0.543630\pi\)
\(858\) 0 0
\(859\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 16.0000 0.544016
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 54.0000i 1.82762i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −8.48528 8.48528i −0.286528 0.286528i 0.549178 0.835705i \(-0.314941\pi\)
−0.835705 + 0.549178i \(0.814941\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −50.0000 −1.68454 −0.842271 0.539054i \(-0.818782\pi\)
−0.842271 + 0.539054i \(0.818782\pi\)
\(882\) 0 0
\(883\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −5.65685 + 5.65685i −0.188457 + 0.188457i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 80.0000i 2.65929i
\(906\) 0 0
\(907\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(908\) 0 0
\(909\) −42.4264 42.4264i −1.40720 1.40720i
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 93.3381 + 93.3381i 3.06894 + 3.06894i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 46.0000 1.50921 0.754606 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 38.0000i 1.24141i −0.784046 0.620703i \(-0.786847\pi\)
0.784046 0.620703i \(-0.213153\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 14.1421 + 14.1421i 0.461020 + 0.461020i 0.898990 0.437969i \(-0.144302\pi\)
−0.437969 + 0.898990i \(0.644302\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(948\) 0 0
\(949\) 16.9706 16.9706i 0.550888 0.550888i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 26.0000i 0.842223i 0.907009 + 0.421111i \(0.138360\pi\)
−0.907009 + 0.421111i \(0.861640\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 39.5980 39.5980i 1.27470 1.27470i
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −62.0000 −1.98356 −0.991778 0.127971i \(-0.959153\pi\)
−0.991778 + 0.127971i \(0.959153\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 42.4264 42.4264i 1.35457 1.35457i
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 112.000i 3.56862i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −8.48528 + 8.48528i −0.268732 + 0.268732i −0.828589 0.559857i \(-0.810856\pi\)
0.559857 + 0.828589i \(0.310856\pi\)
\(998\) 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 1024.2.e.k.257.2 4
4.3 odd 2 CM 1024.2.e.k.257.2 4
8.3 odd 2 inner 1024.2.e.k.257.1 4
8.5 even 2 inner 1024.2.e.k.257.1 4
16.3 odd 4 inner 1024.2.e.k.769.1 4
16.5 even 4 inner 1024.2.e.k.769.2 4
16.11 odd 4 inner 1024.2.e.k.769.2 4
16.13 even 4 inner 1024.2.e.k.769.1 4
32.3 odd 8 128.2.b.b.65.2 yes 2
32.5 even 8 256.2.a.c.1.1 1
32.11 odd 8 256.2.a.b.1.1 1
32.13 even 8 128.2.b.b.65.1 2
32.19 odd 8 128.2.b.b.65.1 2
32.21 even 8 256.2.a.b.1.1 1
32.27 odd 8 256.2.a.c.1.1 1
32.29 even 8 128.2.b.b.65.2 yes 2
96.5 odd 8 2304.2.a.a.1.1 1
96.11 even 8 2304.2.a.p.1.1 1
96.29 odd 8 1152.2.d.d.577.1 2
96.35 even 8 1152.2.d.d.577.1 2
96.53 odd 8 2304.2.a.p.1.1 1
96.59 even 8 2304.2.a.a.1.1 1
96.77 odd 8 1152.2.d.d.577.2 2
96.83 even 8 1152.2.d.d.577.2 2
160.3 even 8 3200.2.f.d.449.2 2
160.13 odd 8 3200.2.f.c.449.2 2
160.19 odd 8 3200.2.d.e.1601.1 2
160.29 even 8 3200.2.d.e.1601.2 2
160.59 odd 8 6400.2.a.l.1.1 1
160.67 even 8 3200.2.f.c.449.1 2
160.69 even 8 6400.2.a.l.1.1 1
160.77 odd 8 3200.2.f.d.449.1 2
160.83 even 8 3200.2.f.c.449.2 2
160.93 odd 8 3200.2.f.d.449.2 2
160.99 odd 8 3200.2.d.e.1601.2 2
160.109 even 8 3200.2.d.e.1601.1 2
160.139 odd 8 6400.2.a.m.1.1 1
160.147 even 8 3200.2.f.d.449.1 2
160.149 even 8 6400.2.a.m.1.1 1
160.157 odd 8 3200.2.f.c.449.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.b.b.65.1 2 32.13 even 8
128.2.b.b.65.1 2 32.19 odd 8
128.2.b.b.65.2 yes 2 32.3 odd 8
128.2.b.b.65.2 yes 2 32.29 even 8
256.2.a.b.1.1 1 32.11 odd 8
256.2.a.b.1.1 1 32.21 even 8
256.2.a.c.1.1 1 32.5 even 8
256.2.a.c.1.1 1 32.27 odd 8
1024.2.e.k.257.1 4 8.3 odd 2 inner
1024.2.e.k.257.1 4 8.5 even 2 inner
1024.2.e.k.257.2 4 1.1 even 1 trivial
1024.2.e.k.257.2 4 4.3 odd 2 CM
1024.2.e.k.769.1 4 16.3 odd 4 inner
1024.2.e.k.769.1 4 16.13 even 4 inner
1024.2.e.k.769.2 4 16.5 even 4 inner
1024.2.e.k.769.2 4 16.11 odd 4 inner
1152.2.d.d.577.1 2 96.29 odd 8
1152.2.d.d.577.1 2 96.35 even 8
1152.2.d.d.577.2 2 96.77 odd 8
1152.2.d.d.577.2 2 96.83 even 8
2304.2.a.a.1.1 1 96.5 odd 8
2304.2.a.a.1.1 1 96.59 even 8
2304.2.a.p.1.1 1 96.11 even 8
2304.2.a.p.1.1 1 96.53 odd 8
3200.2.d.e.1601.1 2 160.19 odd 8
3200.2.d.e.1601.1 2 160.109 even 8
3200.2.d.e.1601.2 2 160.29 even 8
3200.2.d.e.1601.2 2 160.99 odd 8
3200.2.f.c.449.1 2 160.67 even 8
3200.2.f.c.449.1 2 160.157 odd 8
3200.2.f.c.449.2 2 160.13 odd 8
3200.2.f.c.449.2 2 160.83 even 8
3200.2.f.d.449.1 2 160.77 odd 8
3200.2.f.d.449.1 2 160.147 even 8
3200.2.f.d.449.2 2 160.3 even 8
3200.2.f.d.449.2 2 160.93 odd 8
6400.2.a.l.1.1 1 160.59 odd 8
6400.2.a.l.1.1 1 160.69 even 8
6400.2.a.m.1.1 1 160.139 odd 8
6400.2.a.m.1.1 1 160.149 even 8