Properties

Label 848.1.x.a.271.1
Level $848$
Weight $1$
Character 848.271
Analytic conductor $0.423$
Analytic rank $0$
Dimension $12$
Projective image $D_{26}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [848,1,Mod(143,848)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(848, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([13, 0, 15]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("848.143");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 848 = 2^{4} \cdot 53 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 848.x (of order \(26\), degree \(12\), 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: \(0.423207130713\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{26})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{26}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{26} - \cdots)\)

Embedding invariants

Embedding label 271.1
Root \(-0.120537 - 0.992709i\) of defining polynomial
Character \(\chi\) \(=\) 848.271
Dual form 848.1.x.a.751.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.09148 + 1.23202i) q^{5} +(0.748511 - 0.663123i) q^{9} +O(q^{10})\) \(q+(1.09148 + 1.23202i) q^{5} +(0.748511 - 0.663123i) q^{9} +(-1.10312 - 1.59814i) q^{13} +(0.213460 + 1.75800i) q^{17} +(-0.206022 + 1.69675i) q^{25} +(-1.45352 + 0.358261i) q^{29} +(0.0854858 + 0.225408i) q^{37} +(0.475142 - 1.92773i) q^{41} +(1.63397 + 0.198399i) q^{45} +(0.568065 - 0.822984i) q^{49} +(0.354605 + 0.935016i) q^{53} +(-1.31658 - 0.159861i) q^{61} +(0.764919 - 3.10340i) q^{65} +(-1.85640 + 0.225408i) q^{73} +(0.120537 - 0.992709i) q^{81} +(-1.93291 + 2.18181i) q^{85} +(-0.0290582 - 0.239316i) q^{89} +(-1.12054 + 0.992709i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + q^{9} - 2 q^{13} - 2 q^{17} - q^{25} + 2 q^{29} + 2 q^{37} - q^{49} + q^{53} - q^{81} - 11 q^{89} - 11 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/848\mathbb{Z}\right)^\times\).

\(n\) \(161\) \(213\) \(319\)
\(\chi(n)\) \(e\left(\frac{9}{26}\right)\) \(1\) \(-1\)

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
\(3\) 0 0 0.935016 0.354605i \(-0.115385\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(4\) 0 0
\(5\) 1.09148 + 1.23202i 1.09148 + 1.23202i 0.970942 + 0.239316i \(0.0769231\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(6\) 0 0
\(7\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(8\) 0 0
\(9\) 0.748511 0.663123i 0.748511 0.663123i
\(10\) 0 0
\(11\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(12\) 0 0
\(13\) −1.10312 1.59814i −1.10312 1.59814i −0.748511 0.663123i \(-0.769231\pi\)
−0.354605 0.935016i \(-0.615385\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.213460 + 1.75800i 0.213460 + 1.75800i 0.568065 + 0.822984i \(0.307692\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(18\) 0 0
\(19\) 0 0 0.822984 0.568065i \(-0.192308\pi\)
−0.822984 + 0.568065i \(0.807692\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −0.206022 + 1.69675i −0.206022 + 1.69675i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.45352 + 0.358261i −1.45352 + 0.358261i −0.885456 0.464723i \(-0.846154\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(30\) 0 0
\(31\) 0 0 −0.239316 0.970942i \(-0.576923\pi\)
0.239316 + 0.970942i \(0.423077\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.0854858 + 0.225408i 0.0854858 + 0.225408i 0.970942 0.239316i \(-0.0769231\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.475142 1.92773i 0.475142 1.92773i 0.120537 0.992709i \(-0.461538\pi\)
0.354605 0.935016i \(-0.384615\pi\)
\(42\) 0 0
\(43\) 0 0 0.354605 0.935016i \(-0.384615\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(44\) 0 0
\(45\) 1.63397 + 0.198399i 1.63397 + 0.198399i
\(46\) 0 0
\(47\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(48\) 0 0
\(49\) 0.568065 0.822984i 0.568065 0.822984i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.354605 + 0.935016i 0.354605 + 0.935016i
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(60\) 0 0
\(61\) −1.31658 0.159861i −1.31658 0.159861i −0.568065 0.822984i \(-0.692308\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.764919 3.10340i 0.764919 3.10340i
\(66\) 0 0
\(67\) 0 0 −0.822984 0.568065i \(-0.807692\pi\)
0.822984 + 0.568065i \(0.192308\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.935016 0.354605i \(-0.884615\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(72\) 0 0
\(73\) −1.85640 + 0.225408i −1.85640 + 0.225408i −0.970942 0.239316i \(-0.923077\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.464723 0.885456i \(-0.346154\pi\)
−0.464723 + 0.885456i \(0.653846\pi\)
\(80\) 0 0
\(81\) 0.120537 0.992709i 0.120537 0.992709i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −1.93291 + 2.18181i −1.93291 + 2.18181i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.0290582 0.239316i −0.0290582 0.239316i 0.970942 0.239316i \(-0.0769231\pi\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.12054 + 0.992709i −1.12054 + 0.992709i −0.120537 + 0.992709i \(0.538462\pi\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.616337 0.695701i −0.616337 0.695701i 0.354605 0.935016i \(-0.384615\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(102\) 0 0
\(103\) 0 0 0.935016 0.354605i \(-0.115385\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0.447528 0.169725i 0.447528 0.169725i −0.120537 0.992709i \(-0.538462\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.627974 0.329586i 0.627974 0.329586i −0.120537 0.992709i \(-0.538462\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.88546 0.464723i −1.88546 0.464723i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.885456 + 0.464723i 0.885456 + 0.464723i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −0.960699 + 0.663123i −0.960699 + 0.663123i
\(126\) 0 0
\(127\) 0 0 0.663123 0.748511i \(-0.269231\pi\)
−0.663123 + 0.748511i \(0.730769\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.222431 0.902438i −0.222431 0.902438i −0.970942 0.239316i \(-0.923077\pi\)
0.748511 0.663123i \(-0.230769\pi\)
\(138\) 0 0
\(139\) 0 0 0.992709 0.120537i \(-0.0384615\pi\)
−0.992709 + 0.120537i \(0.961538\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −2.02787 1.39974i −2.02787 1.39974i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −0.0854858 + 0.225408i −0.0854858 + 0.225408i −0.970942 0.239316i \(-0.923077\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(150\) 0 0
\(151\) 0 0 −0.992709 0.120537i \(-0.961538\pi\)
0.992709 + 0.120537i \(0.0384615\pi\)
\(152\) 0 0
\(153\) 1.32555 + 1.17433i 1.32555 + 1.17433i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.869047 1.65583i −0.869047 1.65583i −0.748511 0.663123i \(-0.769231\pi\)
−0.120537 0.992709i \(-0.538462\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.568065 0.822984i \(-0.307692\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.992709 0.120537i \(-0.961538\pi\)
0.992709 + 0.120537i \(0.0384615\pi\)
\(168\) 0 0
\(169\) −0.982579 + 2.59085i −0.982579 + 2.59085i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.09148 + 0.753393i 1.09148 + 0.753393i 0.970942 0.239316i \(-0.0769231\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.992709 0.120537i \(-0.0384615\pi\)
−0.992709 + 0.120537i \(0.961538\pi\)
\(180\) 0 0
\(181\) −0.447528 1.81569i −0.447528 1.81569i −0.568065 0.822984i \(-0.692308\pi\)
0.120537 0.992709i \(-0.461538\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.184402 + 0.351348i −0.184402 + 0.351348i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.663123 0.748511i \(-0.269231\pi\)
−0.663123 + 0.748511i \(0.730769\pi\)
\(192\) 0 0
\(193\) 0.393906 0.271894i 0.393906 0.271894i −0.354605 0.935016i \(-0.615385\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.627974 + 0.329586i 0.627974 + 0.329586i 0.748511 0.663123i \(-0.230769\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(198\) 0 0
\(199\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.89361 1.51868i 2.89361 1.51868i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 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
\(221\) 2.57406 2.28042i 2.57406 2.28042i
\(222\) 0 0
\(223\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(224\) 0 0
\(225\) 0.970942 + 1.40665i 0.970942 + 1.40665i
\(226\) 0 0
\(227\) 0 0 −0.885456 0.464723i \(-0.846154\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(228\) 0 0
\(229\) 0.0854858 + 0.704039i 0.0854858 + 0.704039i 0.970942 + 0.239316i \(0.0769231\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.317391 + 0.358261i −0.317391 + 0.358261i −0.885456 0.464723i \(-0.846154\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.464723 0.885456i \(-0.346154\pi\)
−0.464723 + 0.885456i \(0.653846\pi\)
\(240\) 0 0
\(241\) 1.88546 0.464723i 1.88546 0.464723i 0.885456 0.464723i \(-0.153846\pi\)
1.00000 \(0\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.63397 0.198399i 1.63397 0.198399i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.822984 0.568065i \(-0.807692\pi\)
0.822984 + 0.568065i \(0.192308\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −0.922670 0.112032i −0.922670 0.112032i −0.354605 0.935016i \(-0.615385\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −0.850405 + 1.23202i −0.850405 + 1.23202i
\(262\) 0 0
\(263\) 0 0 −0.464723 0.885456i \(-0.653846\pi\)
0.464723 + 0.885456i \(0.346154\pi\)
\(264\) 0 0
\(265\) −0.764919 + 1.45743i −0.764919 + 1.45743i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.00599 + 1.45743i −1.00599 + 1.45743i −0.120537 + 0.992709i \(0.538462\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(270\) 0 0
\(271\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −0.475142 + 1.92773i −0.475142 + 1.92773i −0.120537 + 0.992709i \(0.538462\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.402877 + 1.06230i 0.402877 + 1.06230i 0.970942 + 0.239316i \(0.0769231\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(282\) 0 0
\(283\) 0 0 −0.935016 0.354605i \(-0.884615\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.07406 + 0.511209i −2.07406 + 0.511209i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.136945 1.12785i 0.136945 1.12785i −0.748511 0.663123i \(-0.769231\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(294\) 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
\(305\) −1.24006 1.79654i −1.24006 1.79654i
\(306\) 0 0
\(307\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(312\) 0 0
\(313\) 0.317391 + 0.358261i 0.317391 + 0.358261i 0.885456 0.464723i \(-0.153846\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.13613 1.13613 0.568065 0.822984i \(-0.307692\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 2.93891 1.54246i 2.93891 1.54246i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(332\) 0 0
\(333\) 0.213460 + 0.112032i 0.213460 + 0.112032i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.764919 + 0.527986i −0.764919 + 0.527986i −0.885456 0.464723i \(-0.846154\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(338\) 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 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(348\) 0 0
\(349\) 0.447528 + 1.81569i 0.447528 + 1.81569i 0.568065 + 0.822984i \(0.307692\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.85640 + 0.704039i 1.85640 + 0.704039i 0.970942 + 0.239316i \(0.0769231\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.239316 0.970942i \(-0.423077\pi\)
−0.239316 + 0.970942i \(0.576923\pi\)
\(360\) 0 0
\(361\) 0.354605 0.935016i 0.354605 0.935016i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.30393 2.04110i −2.30393 2.04110i
\(366\) 0 0
\(367\) 0 0 0.568065 0.822984i \(-0.307692\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(368\) 0 0
\(369\) −0.922670 1.75800i −0.922670 1.75800i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −0.222431 0.423807i −0.222431 0.423807i 0.748511 0.663123i \(-0.230769\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.17595 + 1.92773i 2.17595 + 1.92773i
\(378\) 0 0
\(379\) 0 0 −0.992709 0.120537i \(-0.961538\pi\)
0.992709 + 0.120537i \(0.0384615\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.239316 0.970942i \(-0.423077\pi\)
−0.239316 + 0.970942i \(0.576923\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.74851 + 0.663123i 1.74851 + 0.663123i 1.00000 \(0\)
0.748511 + 0.663123i \(0.230769\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −0.922670 + 1.75800i −0.922670 + 1.75800i −0.354605 + 0.935016i \(0.615385\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.32625i 1.32625i 0.748511 + 0.663123i \(0.230769\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.35460 0.935016i 1.35460 0.935016i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.32555 + 0.695701i 1.32555 + 0.695701i 0.970942 0.239316i \(-0.0769231\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(410\) 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 −0.663123 0.748511i \(-0.730769\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(420\) 0 0
\(421\) 1.24006 0.470293i 1.24006 0.470293i 0.354605 0.935016i \(-0.384615\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.02686 −3.02686
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(432\) 0 0
\(433\) −0.850405 + 0.753393i −0.850405 + 0.753393i −0.970942 0.239316i \(-0.923077\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.885456 0.464723i \(-0.846154\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(440\) 0 0
\(441\) −0.120537 0.992709i −0.120537 0.992709i
\(442\) 0 0
\(443\) 0 0 0.822984 0.568065i \(-0.192308\pi\)
−0.822984 + 0.568065i \(0.807692\pi\)
\(444\) 0 0
\(445\) 0.263126 0.297008i 0.263126 0.297008i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.136945 1.12785i 0.136945 1.12785i −0.748511 0.663123i \(-0.769231\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.31658 0.159861i 1.31658 0.159861i 0.568065 0.822984i \(-0.307692\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.251489 + 0.663123i 0.251489 + 0.663123i 1.00000 \(0\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(462\) 0 0
\(463\) 0 0 −0.822984 0.568065i \(-0.807692\pi\)
0.822984 + 0.568065i \(0.192308\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.354605 0.935016i \(-0.384615\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(468\) 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
\(477\) 0.885456 + 0.464723i 0.885456 + 0.464723i
\(478\) 0 0
\(479\) 0 0 −0.464723 0.885456i \(-0.653846\pi\)
0.464723 + 0.885456i \(0.346154\pi\)
\(480\) 0 0
\(481\) 0.265932 0.385269i 0.265932 0.385269i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.44608 0.297008i −2.44608 0.297008i
\(486\) 0 0
\(487\) 0 0 0.354605 0.935016i \(-0.384615\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.822984 0.568065i \(-0.807692\pi\)
0.822984 + 0.568065i \(0.192308\pi\)
\(492\) 0 0
\(493\) −0.940091 2.47882i −0.940091 2.47882i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.239316 0.970942i \(-0.576923\pi\)
0.239316 + 0.970942i \(0.423077\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.464723 0.885456i \(-0.346154\pi\)
−0.464723 + 0.885456i \(0.653846\pi\)
\(504\) 0 0
\(505\) 0.184402 1.51868i 0.184402 1.51868i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.24006 1.39974i 1.24006 1.39974i 0.354605 0.935016i \(-0.384615\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(510\) 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
\(521\) −0.850405 + 0.753393i −0.850405 + 0.753393i −0.970942 0.239316i \(-0.923077\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(522\) 0 0
\(523\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.00000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −3.60491 + 1.36716i −3.60491 + 1.36716i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.10312 + 0.271894i 1.10312 + 0.271894i 0.748511 0.663123i \(-0.230769\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.697573 + 0.366114i 0.697573 + 0.366114i
\(546\) 0 0
\(547\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(548\) 0 0
\(549\) −1.09148 + 0.753393i −1.09148 + 0.753393i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −0.431935 + 0.822984i −0.431935 + 0.822984i 0.568065 + 0.822984i \(0.307692\pi\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.992709 0.120537i \(-0.0384615\pi\)
−0.992709 + 0.120537i \(0.961538\pi\)
\(564\) 0 0
\(565\) 1.09148 + 0.413943i 1.09148 + 0.413943i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.35460 0.935016i −1.35460 0.935016i −0.354605 0.935016i \(-0.615385\pi\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 0.239316 0.970942i \(-0.423077\pi\)
−0.239316 + 0.970942i \(0.576923\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.12054 0.992709i −1.12054 0.992709i −0.120537 0.992709i \(-0.538462\pi\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −1.48538 2.83016i −1.48538 2.83016i
\(586\) 0 0
\(587\) 0 0 0.568065 0.822984i \(-0.307692\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −0.688601 + 1.81569i −0.688601 + 1.81569i −0.120537 + 0.992709i \(0.538462\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(600\) 0 0
\(601\) −1.53901 0.583668i −1.53901 0.583668i −0.568065 0.822984i \(-0.692308\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.393906 + 1.59814i 0.393906 + 1.59814i
\(606\) 0 0
\(607\) 0 0 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.64597i 1.64597i −0.568065 0.822984i \(-0.692308\pi\)
0.568065 0.822984i \(-0.307692\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.63397 + 1.12785i −1.63397 + 1.12785i −0.748511 + 0.663123i \(0.769231\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(618\) 0 0
\(619\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.206022 0.0507800i −0.206022 0.0507800i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.378019 + 0.198399i −0.378019 + 0.198399i
\(630\) 0 0
\(631\) 0 0 −0.663123 0.748511i \(-0.730769\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.94188 −1.94188
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.24006 + 1.39974i 1.24006 + 1.39974i 0.885456 + 0.464723i \(0.153846\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(642\) 0 0
\(643\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.136945 + 1.12785i 0.136945 + 1.12785i 0.885456 + 0.464723i \(0.153846\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.24006 + 1.39974i −1.24006 + 1.39974i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0.213460 1.75800i 0.213460 1.75800i −0.354605 0.935016i \(-0.615385\pi\)
0.568065 0.822984i \(-0.307692\pi\)
\(662\) 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
\(673\) −0.530851 1.39974i −0.530851 1.39974i −0.885456 0.464723i \(-0.846154\pi\)
0.354605 0.935016i \(-0.384615\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −0.393906 + 1.59814i −0.393906 + 1.59814i 0.354605 + 0.935016i \(0.384615\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(684\) 0 0
\(685\) 0.869047 1.25903i 0.869047 1.25903i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.10312 1.59814i 1.10312 1.59814i
\(690\) 0 0
\(691\) 0 0 −0.464723 0.885456i \(-0.653846\pi\)
0.464723 + 0.885456i \(0.346154\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3.49036 + 0.423807i 3.49036 + 0.423807i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0.222431 0.902438i 0.222431 0.902438i −0.748511 0.663123i \(-0.769231\pi\)
0.970942 0.239316i \(-0.0769231\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.475142 0.0576926i 0.475142 0.0576926i 0.120537 0.992709i \(-0.461538\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(710\) 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
\(725\) −0.308420 2.54007i −0.308420 2.54007i
\(726\) 0 0
\(727\) 0 0 −0.885456 0.464723i \(-0.846154\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(728\) 0 0
\(729\) −0.568065 0.822984i −0.568065 0.822984i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.32555 1.17433i 1.32555 1.17433i 0.354605 0.935016i \(-0.384615\pi\)
0.970942 0.239316i \(-0.0769231\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.935016 0.354605i \(-0.115385\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −0.371013 + 0.140707i −0.371013 + 0.140707i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.748511 0.663123i \(-0.230769\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.00599 + 0.527986i 1.00599 + 0.527986i 0.885456 0.464723i \(-0.153846\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.53901 + 1.06230i −1.53901 + 1.06230i −0.568065 + 0.822984i \(0.692308\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.91486i 2.91486i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −0.616337 + 1.17433i −0.616337 + 1.17433i 0.354605 + 0.935016i \(0.384615\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −0.317391 1.28771i −0.317391 1.28771i −0.885456 0.464723i \(-0.846154\pi\)
0.568065 0.822984i \(-0.307692\pi\)
\(774\) 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
\(785\) 1.09148 2.87799i 1.09148 2.87799i
\(786\) 0 0
\(787\) 0 0 −0.992709 0.120537i \(-0.961538\pi\)
0.992709 + 0.120537i \(0.0384615\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.19685 + 2.28042i 1.19685 + 2.28042i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −0.922670 1.75800i −0.922670 1.75800i −0.568065 0.822984i \(-0.692308\pi\)
−0.354605 0.935016i \(-0.615385\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −0.180446 0.159861i −0.180446 0.159861i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.393906 + 0.271894i 0.393906 + 0.271894i 0.748511 0.663123i \(-0.230769\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(810\) 0 0
\(811\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(812\) 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
\(821\) 0.764919 1.45743i 0.764919 1.45743i −0.120537 0.992709i \(-0.538462\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(822\) 0 0
\(823\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.663123 0.748511i \(-0.269231\pi\)
−0.663123 + 0.748511i \(0.730769\pi\)
\(828\) 0 0
\(829\) 1.09148 0.753393i 1.09148 0.753393i 0.120537 0.992709i \(-0.461538\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.56806 + 0.822984i 1.56806 + 0.822984i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.748511 0.663123i \(-0.230769\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(840\) 0 0
\(841\) 1.09892 0.576756i 1.09892 0.576756i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −4.26445 + 1.61729i −4.26445 + 1.61729i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0.616337 + 0.695701i 0.616337 + 0.695701i 0.970942 0.239316i \(-0.0769231\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.180446 + 0.159861i −0.180446 + 0.159861i −0.748511 0.663123i \(-0.769231\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(858\) 0 0
\(859\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.885456 0.464723i \(-0.846154\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(864\) 0 0
\(865\) 0.263126 + 2.16704i 0.263126 + 2.16704i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −0.180446 + 1.48611i −0.180446 + 1.48611i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.71945 0.423807i 1.71945 0.423807i 0.748511 0.663123i \(-0.230769\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.85640 + 0.225408i −1.85640 + 0.225408i −0.970942 0.239316i \(-0.923077\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(882\) 0 0
\(883\) 0 0 −0.935016 0.354605i \(-0.884615\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.822984 0.568065i \(-0.807692\pi\)
0.822984 + 0.568065i \(0.192308\pi\)
\(888\) 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
\(901\) −1.56806 + 0.822984i −1.56806 + 0.822984i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.74851 2.53316i 1.74851 2.53316i
\(906\) 0 0
\(907\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(908\) 0 0
\(909\) −0.922670 0.112032i −0.922670 0.112032i
\(910\) 0 0
\(911\) 0 0 0.354605 0.935016i \(-0.384615\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.935016 0.354605i \(-0.884615\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −0.400072 + 0.0986088i −0.400072 + 0.0986088i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −0.234068 + 1.92773i −0.234068 + 1.92773i 0.120537 + 0.992709i \(0.461538\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −0.136945 1.12785i −0.136945 1.12785i −0.885456 0.464723i \(-0.846154\pi\)
0.748511 0.663123i \(-0.230769\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.00599 + 1.45743i 1.00599 + 1.45743i 0.885456 + 0.464723i \(0.153846\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(948\) 0 0
\(949\) 2.40805 + 2.71813i 2.40805 + 2.71813i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −0.241073 −0.241073 −0.120537 0.992709i \(-0.538462\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.885456 + 0.464723i −0.885456 + 0.464723i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0.764919 + 0.188536i 0.764919 + 0.188536i
\(966\) 0 0
\(967\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0.478631i 0.478631i 0.970942 + 0.239316i \(0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0.222431 0.423807i 0.222431 0.423807i
\(982\) 0 0
\(983\) 0 0 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(984\) 0 0
\(985\) 0.279362 + 1.13342i 0.279362 + 1.13342i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.251489 0.663123i 0.251489 0.663123i −0.748511 0.663123i \(-0.769231\pi\)
1.00000 \(0\)
\(998\) 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 848.1.x.a.271.1 12
4.3 odd 2 CM 848.1.x.a.271.1 12
8.3 odd 2 3392.1.bn.a.2815.1 12
8.5 even 2 3392.1.bn.a.2815.1 12
53.9 even 26 inner 848.1.x.a.751.1 yes 12
212.115 odd 26 inner 848.1.x.a.751.1 yes 12
424.115 odd 26 3392.1.bn.a.1599.1 12
424.221 even 26 3392.1.bn.a.1599.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
848.1.x.a.271.1 12 1.1 even 1 trivial
848.1.x.a.271.1 12 4.3 odd 2 CM
848.1.x.a.751.1 yes 12 53.9 even 26 inner
848.1.x.a.751.1 yes 12 212.115 odd 26 inner
3392.1.bn.a.1599.1 12 424.115 odd 26
3392.1.bn.a.1599.1 12 424.221 even 26
3392.1.bn.a.2815.1 12 8.3 odd 2
3392.1.bn.a.2815.1 12 8.5 even 2