Properties

Label 1089.1.s.a.481.1
Level $1089$
Weight $1$
Character 1089.481
Analytic conductor $0.543$
Analytic rank $0$
Dimension $8$
Projective image $D_{3}$
CM discriminant -11
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,1,Mod(40,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([10, 21]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.40");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1089.s (of order \(30\), degree \(8\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.543481798757\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.891.1
Artin image: $S_3\times C_{15}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{30} + \cdots)\)

Embedding invariants

Embedding label 481.1
Root \(0.913545 - 0.406737i\) of defining polynomial
Character \(\chi\) \(=\) 1089.481
Dual form 1089.1.s.a.403.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.978148 - 0.207912i) q^{3} +(0.669131 + 0.743145i) q^{4} +(-0.913545 + 0.406737i) q^{5} +(0.913545 + 0.406737i) q^{9} +O(q^{10})\) \(q+(-0.978148 - 0.207912i) q^{3} +(0.669131 + 0.743145i) q^{4} +(-0.913545 + 0.406737i) q^{5} +(0.913545 + 0.406737i) q^{9} +(-0.500000 - 0.866025i) q^{12} +(0.978148 - 0.207912i) q^{15} +(-0.104528 + 0.994522i) q^{16} +(-0.913545 - 0.406737i) q^{20} +(-1.00000 + 1.73205i) q^{23} +(-0.809017 - 0.587785i) q^{27} +(0.104528 + 0.994522i) q^{31} +(0.309017 + 0.951057i) q^{36} +(-0.309017 + 0.951057i) q^{37} -1.00000 q^{45} +(-0.669131 + 0.743145i) q^{47} +(0.309017 - 0.951057i) q^{48} +(0.913545 - 0.406737i) q^{49} +(0.809017 - 0.587785i) q^{53} +(-0.669131 - 0.743145i) q^{59} +(0.809017 + 0.587785i) q^{60} +(-0.809017 + 0.587785i) q^{64} +(0.500000 - 0.866025i) q^{67} +(1.33826 - 1.48629i) q^{69} +(0.809017 + 0.587785i) q^{71} +(-0.309017 - 0.951057i) q^{80} +(0.669131 + 0.743145i) q^{81} +2.00000 q^{89} +(-1.95630 + 0.415823i) q^{92} +(0.104528 - 0.994522i) q^{93} +(-0.913545 - 0.406737i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} + q^{4} - q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} + q^{4} - q^{5} + q^{9} - 4 q^{12} - q^{15} + q^{16} - q^{20} - 8 q^{23} - 2 q^{27} - q^{31} - 2 q^{36} + 2 q^{37} - 8 q^{45} - q^{47} - 2 q^{48} + q^{49} + 2 q^{53} - q^{59} + 2 q^{60} - 2 q^{64} + 4 q^{67} + 2 q^{69} + 2 q^{71} + 2 q^{80} + q^{81} + 16 q^{89} + 2 q^{92} - q^{93} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(3\) −0.978148 0.207912i −0.978148 0.207912i
\(4\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(5\) −0.913545 + 0.406737i −0.913545 + 0.406737i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(6\) 0 0
\(7\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(8\) 0 0
\(9\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(10\) 0 0
\(11\) 0 0
\(12\) −0.500000 0.866025i −0.500000 0.866025i
\(13\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(14\) 0 0
\(15\) 0.978148 0.207912i 0.978148 0.207912i
\(16\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(17\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(20\) −0.913545 0.406737i −0.913545 0.406737i
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.809017 0.587785i −0.809017 0.587785i
\(28\) 0 0
\(29\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(30\) 0 0
\(31\) 0.104528 + 0.994522i 0.104528 + 0.994522i 0.913545 + 0.406737i \(0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(37\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(42\) 0 0
\(43\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) −1.00000 −1.00000
\(46\) 0 0
\(47\) −0.669131 + 0.743145i −0.669131 + 0.743145i −0.978148 0.207912i \(-0.933333\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(48\) 0.309017 0.951057i 0.309017 0.951057i
\(49\) 0.913545 0.406737i 0.913545 0.406737i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.669131 0.743145i −0.669131 0.743145i 0.309017 0.951057i \(-0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(60\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(61\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(68\) 0 0
\(69\) 1.33826 1.48629i 1.33826 1.48629i
\(70\) 0 0
\(71\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(72\) 0 0
\(73\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(80\) −0.309017 0.951057i −0.309017 0.951057i
\(81\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(82\) 0 0
\(83\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.95630 + 0.415823i −1.95630 + 0.415823i
\(93\) 0.104528 0.994522i 0.104528 0.994522i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.913545 0.406737i −0.913545 0.406737i −0.104528 0.994522i \(-0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(102\) 0 0
\(103\) −0.669131 0.743145i −0.669131 0.743145i 0.309017 0.951057i \(-0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(108\) −0.104528 0.994522i −0.104528 0.994522i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0.500000 0.866025i 0.500000 0.866025i
\(112\) 0 0
\(113\) 0.978148 + 0.207912i 0.978148 + 0.207912i 0.669131 0.743145i \(-0.266667\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(114\) 0 0
\(115\) 0.209057 1.98904i 0.209057 1.98904i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(125\) 0.309017 0.951057i 0.309017 0.951057i
\(126\) 0 0
\(127\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(136\) 0 0
\(137\) 0.104528 0.994522i 0.104528 0.994522i −0.809017 0.587785i \(-0.800000\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(138\) 0 0
\(139\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(140\) 0 0
\(141\) 0.809017 0.587785i 0.809017 0.587785i
\(142\) 0 0
\(143\) 0 0
\(144\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(148\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(149\) 0 0 0.913545 0.406737i \(-0.133333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(150\) 0 0
\(151\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.500000 0.866025i −0.500000 0.866025i
\(156\) 0 0
\(157\) 0.978148 + 0.207912i 0.978148 + 0.207912i 0.669131 0.743145i \(-0.266667\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(158\) 0 0
\(159\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(168\) 0 0
\(169\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(178\) 0 0
\(179\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(180\) −0.669131 0.743145i −0.669131 0.743145i
\(181\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.104528 0.994522i −0.104528 0.994522i
\(186\) 0 0
\(187\) 0 0
\(188\) −1.00000 −1.00000
\(189\) 0 0
\(190\) 0 0
\(191\) 0.978148 0.207912i 0.978148 0.207912i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(192\) 0.913545 0.406737i 0.913545 0.406737i
\(193\) 0 0 0.913545 0.406737i \(-0.133333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) 0 0
\(201\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.61803 + 1.17557i −1.61803 + 1.17557i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(212\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(213\) −0.669131 0.743145i −0.669131 0.743145i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.33826 1.48629i 1.33826 1.48629i 0.669131 0.743145i \(-0.266667\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(228\) 0 0
\(229\) −0.209057 1.98904i −0.209057 1.98904i −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 0.994522i \(-0.533333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(234\) 0 0
\(235\) 0.309017 0.951057i 0.309017 0.951057i
\(236\) 0.104528 0.994522i 0.104528 0.994522i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(240\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(241\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) 0 0
\(243\) −0.500000 0.866025i −0.500000 0.866025i
\(244\) 0 0
\(245\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.61803 + 1.17557i −1.61803 + 1.17557i −0.809017 + 0.587785i \(0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.978148 0.207912i −0.978148 0.207912i
\(257\) 1.33826 + 1.48629i 1.33826 + 1.48629i 0.669131 + 0.743145i \(0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(266\) 0 0
\(267\) −1.95630 0.415823i −1.95630 0.415823i
\(268\) 0.978148 0.207912i 0.978148 0.207912i
\(269\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(270\) 0 0
\(271\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 2.00000 2.00000
\(277\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(278\) 0 0
\(279\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(280\) 0 0
\(281\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(282\) 0 0
\(283\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(284\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(290\) 0 0
\(291\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(292\) 0 0
\(293\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(294\) 0 0
\(295\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(296\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(310\) 0 0
\(311\) 0.978148 + 0.207912i 0.978148 + 0.207912i 0.669131 0.743145i \(-0.266667\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(312\) 0 0
\(313\) −0.209057 + 1.98904i −0.209057 + 1.98904i −0.104528 + 0.994522i \(0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.82709 + 0.813473i 1.82709 + 0.813473i 0.913545 + 0.406737i \(0.133333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.500000 0.866025i 0.500000 0.866025i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 0 0
\(333\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(334\) 0 0
\(335\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(336\) 0 0
\(337\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(338\) 0 0
\(339\) −0.913545 0.406737i −0.913545 0.406737i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −0.618034 + 1.90211i −0.618034 + 1.90211i
\(346\) 0 0
\(347\) 0 0 0.913545 0.406737i \(-0.133333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(348\) 0 0
\(349\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(354\) 0 0
\(355\) −0.978148 0.207912i −0.978148 0.207912i
\(356\) 1.33826 + 1.48629i 1.33826 + 1.48629i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(360\) 0 0
\(361\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0.978148 0.207912i 0.978148 0.207912i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(368\) −1.61803 1.17557i −1.61803 1.17557i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0.809017 0.587785i 0.809017 0.587785i
\(373\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.61803 1.17557i −1.61803 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.104528 + 0.994522i 0.104528 + 0.994522i 0.913545 + 0.406737i \(0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −0.309017 0.951057i −0.309017 0.951057i
\(389\) 0.978148 0.207912i 0.978148 0.207912i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.913545 + 0.406737i −0.913545 + 0.406737i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.913545 0.406737i −0.913545 0.406737i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(410\) 0 0
\(411\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(412\) 0.104528 0.994522i 0.104528 0.994522i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(420\) 0 0
\(421\) −0.669131 + 0.743145i −0.669131 + 0.743145i −0.978148 0.207912i \(-0.933333\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(422\) 0 0
\(423\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(432\) 0.669131 0.743145i 0.669131 0.743145i
\(433\) 0.618034 1.90211i 0.618034 1.90211i 0.309017 0.951057i \(-0.400000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) 1.00000 1.00000
\(442\) 0 0
\(443\) −0.669131 + 0.743145i −0.669131 + 0.743145i −0.978148 0.207912i \(-0.933333\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(444\) 0.978148 0.207912i 0.978148 0.207912i
\(445\) −1.82709 + 0.813473i −1.82709 + 0.813473i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 1.61803 1.17557i 1.61803 1.17557i
\(461\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) 0 0
\(463\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 0 0
\(465\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(466\) 0 0
\(467\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −0.913545 0.406737i −0.913545 0.406737i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.978148 0.207912i 0.978148 0.207912i
\(478\) 0 0
\(479\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.00000 1.00000
\(486\) 0 0
\(487\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(488\) 0 0
\(489\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(490\) 0 0
\(491\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.00000 −1.00000
\(497\) 0 0
\(498\) 0 0
\(499\) −0.669131 0.743145i −0.669131 0.743145i 0.309017 0.951057i \(-0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(500\) 0.913545 0.406737i 0.913545 0.406737i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 1.00000
\(508\) 0 0
\(509\) −1.95630 0.415823i −1.95630 0.415823i −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 0.207912i \(-0.933333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(522\) 0 0
\(523\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.50000 2.59808i −1.50000 2.59808i
\(530\) 0 0
\(531\) −0.309017 0.951057i −0.309017 0.951057i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(538\) 0 0
\(539\) 0 0
\(540\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(541\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(542\) 0 0
\(543\) −0.669131 0.743145i −0.669131 0.743145i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(548\) 0.809017 0.587785i 0.809017 0.587785i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(556\) 0 0
\(557\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(564\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(565\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(570\) 0 0
\(571\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) −1.00000 −1.00000
\(574\) 0 0
\(575\) 0 0
\(576\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(577\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.978148 0.207912i 0.978148 0.207912i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(588\) −0.809017 0.587785i −0.809017 0.587785i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.913545 0.406737i −0.913545 0.406737i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(598\) 0 0
\(599\) 1.82709 0.813473i 1.82709 0.813473i 0.913545 0.406737i \(-0.133333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(600\) 0 0
\(601\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(602\) 0 0
\(603\) 0.809017 0.587785i 0.809017 0.587785i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(618\) 0 0
\(619\) −0.669131 + 0.743145i −0.669131 + 0.743145i −0.978148 0.207912i \(-0.933333\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(620\) 0.309017 0.951057i 0.309017 0.951057i
\(621\) 1.82709 0.813473i 1.82709 0.813473i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(626\) 0 0
\(627\) 0 0
\(628\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(629\) 0 0
\(630\) 0 0
\(631\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −0.913545 0.406737i −0.913545 0.406737i
\(637\) 0 0
\(638\) 0 0
\(639\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(640\) 0 0
\(641\) 1.33826 1.48629i 1.33826 1.48629i 0.669131 0.743145i \(-0.266667\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(642\) 0 0
\(643\) 1.82709 0.813473i 1.82709 0.813473i 0.913545 0.406737i \(-0.133333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.61803 + 1.17557i −1.61803 + 1.17557i −0.809017 + 0.587785i \(0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(653\) −0.669131 0.743145i −0.669131 0.743145i 0.309017 0.951057i \(-0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1.61803 + 1.17557i −1.61803 + 1.17557i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −0.809017 0.587785i −0.809017 0.587785i
\(677\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) 0 0
\(685\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(686\) 0 0
\(687\) −0.209057 + 1.98904i −0.209057 + 1.98904i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −0.913545 0.406737i −0.913545 0.406737i −0.104528 0.994522i \(-0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(706\) 0 0
\(707\) 0 0
\(708\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(709\) 0.104528 0.994522i 0.104528 0.994522i −0.809017 0.587785i \(-0.800000\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.82709 0.813473i −1.82709 0.813473i
\(714\) 0 0
\(715\) 0 0
\(716\) 0.500000 0.866025i 0.500000 0.866025i
\(717\) 0 0
\(718\) 0 0
\(719\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(720\) 0.104528 0.994522i 0.104528 0.994522i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(725\) 0 0
\(726\) 0 0
\(727\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(728\) 0 0
\(729\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(734\) 0 0
\(735\) 0.809017 0.587785i 0.809017 0.587785i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(740\) 0.669131 0.743145i 0.669131 0.743145i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.913545 0.406737i \(-0.133333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0.978148 + 0.207912i 0.978148 + 0.207912i 0.669131 0.743145i \(-0.266667\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(752\) −0.669131 0.743145i −0.669131 0.743145i
\(753\) 1.82709 0.813473i 1.82709 0.813473i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(769\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 0 0
\(771\) −1.00000 1.73205i −1.00000 1.73205i
\(772\) 0 0
\(773\) 0.618034 + 1.90211i 0.618034 + 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(774\) 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
\(784\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(785\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(786\) 0 0
\(787\) 0 0 0.913545 0.406737i \(-0.133333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0.669131 0.743145i 0.669131 0.743145i
\(796\) −0.669131 0.743145i −0.669131 0.743145i
\(797\) −0.913545 + 0.406737i −0.913545 + 0.406737i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 1.82709 + 0.813473i 1.82709 + 0.813473i
\(802\) 0 0
\(803\) 0 0
\(804\) −1.00000 −1.00000
\(805\) 0 0
\(806\) 0 0
\(807\) −0.669131 0.743145i −0.669131 0.743145i
\(808\) 0 0
\(809\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(810\) 0 0
\(811\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(822\) 0 0
\(823\) −0.209057 1.98904i −0.209057 1.98904i −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 0.994522i \(-0.533333\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(828\) −1.95630 0.415823i −1.95630 0.415823i
\(829\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.500000 0.866025i 0.500000 0.866025i
\(838\) 0 0
\(839\) 1.33826 1.48629i 1.33826 1.48629i 0.669131 0.743145i \(-0.266667\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(840\) 0 0
\(841\) 0.913545 0.406737i 0.913545 0.406737i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.809017 0.587785i 0.809017 0.587785i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(849\) 0 0
\(850\) 0 0
\(851\) −1.33826 1.48629i −1.33826 1.48629i
\(852\) 0.104528 0.994522i 0.104528 0.994522i
\(853\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(858\) 0 0
\(859\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.61803 1.17557i −1.61803 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.104528 0.994522i −0.104528 0.994522i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −0.669131 0.743145i −0.669131 0.743145i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(882\) 0 0
\(883\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(884\) 0 0
\(885\) −0.809017 0.587785i −0.809017 0.587785i
\(886\) 0 0
\(887\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 2.00000 2.00000
\(893\) 0 0
\(894\) 0 0
\(895\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −0.978148 0.207912i −0.978148 0.207912i
\(906\) 0 0
\(907\) −0.209057 + 1.98904i −0.209057 + 1.98904i −0.104528 + 0.994522i \(0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −0.913545 0.406737i −0.913545 0.406737i −0.104528 0.994522i \(-0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.33826 1.48629i 1.33826 1.48629i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −0.309017 0.951057i −0.309017 0.951057i
\(928\) 0 0
\(929\) 0.104528 0.994522i 0.104528 0.994522i −0.809017 0.587785i \(-0.800000\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −0.913545 0.406737i −0.913545 0.406737i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(938\) 0 0
\(939\) 0.618034 1.90211i 0.618034 1.90211i
\(940\) 0.913545 0.406737i 0.913545 0.406737i
\(941\) 0 0 0.913545 0.406737i \(-0.133333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0.809017 0.587785i 0.809017 0.587785i
\(945\) 0 0
\(946\) 0 0
\(947\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −1.61803 1.17557i −1.61803 1.17557i
\(952\) 0 0
\(953\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(954\) 0 0
\(955\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0.618034 + 1.90211i 0.618034 + 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(972\) 0.309017 0.951057i 0.309017 0.951057i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.209057 1.98904i −0.209057 1.98904i −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 0.994522i \(-0.533333\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.00000 −1.00000
\(981\) 0 0
\(982\) 0 0
\(983\) 0.978148 0.207912i 0.978148 0.207912i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(992\) 0 0
\(993\) −0.309017 0.951057i −0.309017 0.951057i
\(994\) 0 0
\(995\) 0.913545 0.406737i 0.913545 0.406737i
\(996\) 0 0
\(997\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(998\) 0 0
\(999\) 0.809017 0.587785i 0.809017 0.587785i
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 1089.1.s.a.481.1 8
3.2 odd 2 3267.1.w.a.118.1 8
9.2 odd 6 3267.1.w.a.1207.1 8
9.7 even 3 inner 1089.1.s.a.844.1 8
11.2 odd 10 99.1.h.a.76.1 yes 2
11.3 even 5 inner 1089.1.s.a.94.1 8
11.4 even 5 inner 1089.1.s.a.40.1 8
11.5 even 5 inner 1089.1.s.a.112.1 8
11.6 odd 10 inner 1089.1.s.a.112.1 8
11.7 odd 10 inner 1089.1.s.a.40.1 8
11.8 odd 10 inner 1089.1.s.a.94.1 8
11.9 even 5 99.1.h.a.76.1 yes 2
11.10 odd 2 CM 1089.1.s.a.481.1 8
33.2 even 10 297.1.h.a.10.1 2
33.5 odd 10 3267.1.w.a.1927.1 8
33.8 even 10 3267.1.w.a.820.1 8
33.14 odd 10 3267.1.w.a.820.1 8
33.17 even 10 3267.1.w.a.1927.1 8
33.20 odd 10 297.1.h.a.10.1 2
33.26 odd 10 3267.1.w.a.766.1 8
33.29 even 10 3267.1.w.a.766.1 8
33.32 even 2 3267.1.w.a.118.1 8
44.31 odd 10 1584.1.bf.b.769.1 2
44.35 even 10 1584.1.bf.b.769.1 2
55.2 even 20 2475.1.t.a.274.2 4
55.9 even 10 2475.1.y.a.76.1 2
55.13 even 20 2475.1.t.a.274.1 4
55.24 odd 10 2475.1.y.a.76.1 2
55.42 odd 20 2475.1.t.a.274.2 4
55.53 odd 20 2475.1.t.a.274.1 4
99.2 even 30 297.1.h.a.208.1 2
99.7 odd 30 inner 1089.1.s.a.403.1 8
99.13 odd 30 891.1.c.a.406.1 1
99.16 even 15 inner 1089.1.s.a.475.1 8
99.20 odd 30 297.1.h.a.208.1 2
99.25 even 15 inner 1089.1.s.a.457.1 8
99.29 even 30 3267.1.w.a.1855.1 8
99.31 even 15 891.1.c.a.406.1 1
99.38 odd 30 3267.1.w.a.3016.1 8
99.43 odd 6 inner 1089.1.s.a.844.1 8
99.47 odd 30 3267.1.w.a.1909.1 8
99.52 odd 30 inner 1089.1.s.a.457.1 8
99.61 odd 30 inner 1089.1.s.a.475.1 8
99.65 even 6 3267.1.w.a.1207.1 8
99.68 even 30 891.1.c.b.406.1 1
99.70 even 15 inner 1089.1.s.a.403.1 8
99.74 even 30 3267.1.w.a.1909.1 8
99.79 odd 30 99.1.h.a.43.1 2
99.83 even 30 3267.1.w.a.3016.1 8
99.86 odd 30 891.1.c.b.406.1 1
99.92 odd 30 3267.1.w.a.1855.1 8
99.97 even 15 99.1.h.a.43.1 2
396.79 even 30 1584.1.bf.b.241.1 2
396.295 odd 30 1584.1.bf.b.241.1 2
495.79 odd 30 2475.1.y.a.1726.1 2
495.97 odd 60 2475.1.t.a.1924.1 4
495.178 even 60 2475.1.t.a.1924.2 4
495.277 even 60 2475.1.t.a.1924.1 4
495.394 even 30 2475.1.y.a.1726.1 2
495.493 odd 60 2475.1.t.a.1924.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.1.h.a.43.1 2 99.79 odd 30
99.1.h.a.43.1 2 99.97 even 15
99.1.h.a.76.1 yes 2 11.2 odd 10
99.1.h.a.76.1 yes 2 11.9 even 5
297.1.h.a.10.1 2 33.2 even 10
297.1.h.a.10.1 2 33.20 odd 10
297.1.h.a.208.1 2 99.2 even 30
297.1.h.a.208.1 2 99.20 odd 30
891.1.c.a.406.1 1 99.13 odd 30
891.1.c.a.406.1 1 99.31 even 15
891.1.c.b.406.1 1 99.68 even 30
891.1.c.b.406.1 1 99.86 odd 30
1089.1.s.a.40.1 8 11.4 even 5 inner
1089.1.s.a.40.1 8 11.7 odd 10 inner
1089.1.s.a.94.1 8 11.3 even 5 inner
1089.1.s.a.94.1 8 11.8 odd 10 inner
1089.1.s.a.112.1 8 11.5 even 5 inner
1089.1.s.a.112.1 8 11.6 odd 10 inner
1089.1.s.a.403.1 8 99.7 odd 30 inner
1089.1.s.a.403.1 8 99.70 even 15 inner
1089.1.s.a.457.1 8 99.25 even 15 inner
1089.1.s.a.457.1 8 99.52 odd 30 inner
1089.1.s.a.475.1 8 99.16 even 15 inner
1089.1.s.a.475.1 8 99.61 odd 30 inner
1089.1.s.a.481.1 8 1.1 even 1 trivial
1089.1.s.a.481.1 8 11.10 odd 2 CM
1089.1.s.a.844.1 8 9.7 even 3 inner
1089.1.s.a.844.1 8 99.43 odd 6 inner
1584.1.bf.b.241.1 2 396.79 even 30
1584.1.bf.b.241.1 2 396.295 odd 30
1584.1.bf.b.769.1 2 44.31 odd 10
1584.1.bf.b.769.1 2 44.35 even 10
2475.1.t.a.274.1 4 55.13 even 20
2475.1.t.a.274.1 4 55.53 odd 20
2475.1.t.a.274.2 4 55.2 even 20
2475.1.t.a.274.2 4 55.42 odd 20
2475.1.t.a.1924.1 4 495.97 odd 60
2475.1.t.a.1924.1 4 495.277 even 60
2475.1.t.a.1924.2 4 495.178 even 60
2475.1.t.a.1924.2 4 495.493 odd 60
2475.1.y.a.76.1 2 55.9 even 10
2475.1.y.a.76.1 2 55.24 odd 10
2475.1.y.a.1726.1 2 495.79 odd 30
2475.1.y.a.1726.1 2 495.394 even 30
3267.1.w.a.118.1 8 3.2 odd 2
3267.1.w.a.118.1 8 33.32 even 2
3267.1.w.a.766.1 8 33.26 odd 10
3267.1.w.a.766.1 8 33.29 even 10
3267.1.w.a.820.1 8 33.8 even 10
3267.1.w.a.820.1 8 33.14 odd 10
3267.1.w.a.1207.1 8 9.2 odd 6
3267.1.w.a.1207.1 8 99.65 even 6
3267.1.w.a.1855.1 8 99.29 even 30
3267.1.w.a.1855.1 8 99.92 odd 30
3267.1.w.a.1909.1 8 99.47 odd 30
3267.1.w.a.1909.1 8 99.74 even 30
3267.1.w.a.1927.1 8 33.5 odd 10
3267.1.w.a.1927.1 8 33.17 even 10
3267.1.w.a.3016.1 8 99.38 odd 30
3267.1.w.a.3016.1 8 99.83 even 30