Properties

Label 2304.1.t.a.1663.2
Level $2304$
Weight $1$
Character 2304.1663
Analytic conductor $1.150$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,1,Mod(895,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.895");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2304.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.5184.1

Embedding invariants

Embedding label 1663.2
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1663
Dual form 2304.1.t.a.895.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +(0.866025 - 0.500000i) q^{5} +(0.866025 + 0.500000i) q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +(0.866025 - 0.500000i) q^{5} +(0.866025 + 0.500000i) q^{7} +1.00000 q^{9} +(0.500000 - 0.866025i) q^{11} +(0.866025 - 0.500000i) q^{13} +(-0.866025 + 0.500000i) q^{15} +(-0.866025 - 0.500000i) q^{21} +(-0.866025 + 0.500000i) q^{23} -1.00000 q^{27} +(-0.866025 - 0.500000i) q^{29} +(0.866025 - 0.500000i) q^{31} +(-0.500000 + 0.866025i) q^{33} +1.00000 q^{35} +(-0.866025 + 0.500000i) q^{39} +(0.500000 + 0.866025i) q^{41} +(-0.500000 + 0.866025i) q^{43} +(0.866025 - 0.500000i) q^{45} +(-0.866025 - 0.500000i) q^{47} -1.00000i q^{55} +(0.500000 + 0.866025i) q^{59} +(-0.866025 - 0.500000i) q^{61} +(0.866025 + 0.500000i) q^{63} +(0.500000 - 0.866025i) q^{65} +(0.500000 + 0.866025i) q^{67} +(0.866025 - 0.500000i) q^{69} -2.00000i q^{71} +(0.866025 - 0.500000i) q^{77} +(0.866025 + 0.500000i) q^{79} +1.00000 q^{81} +(0.500000 - 0.866025i) q^{83} +(0.866025 + 0.500000i) q^{87} +1.00000 q^{91} +(-0.866025 + 0.500000i) q^{93} +(-0.500000 + 0.866025i) q^{97} +(0.500000 - 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 4 q^{9} + 2 q^{11} - 4 q^{27} - 2 q^{33} + 4 q^{35} + 2 q^{41} - 2 q^{43} + 2 q^{59} + 2 q^{65} + 2 q^{67} + 4 q^{81} + 2 q^{83} + 4 q^{91} - 2 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-1\)

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
\(3\) −1.00000 −1.00000
\(4\) 0 0
\(5\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(12\) 0 0
\(13\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(14\) 0 0
\(15\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −0.866025 0.500000i −0.866025 0.500000i
\(22\) 0 0
\(23\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −1.00000
\(28\) 0 0
\(29\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(30\) 0 0
\(31\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0 0
\(33\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(34\) 0 0
\(35\) 1.00000 1.00000
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(40\) 0 0
\(41\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0.866025 0.500000i 0.866025 0.500000i
\(46\) 0 0
\(47\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 1.00000i 1.00000i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 0 0
\(63\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(64\) 0 0
\(65\) 0.500000 0.866025i 0.500000 0.866025i
\(66\) 0 0
\(67\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 0.866025 0.500000i 0.866025 0.500000i
\(70\) 0 0
\(71\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.866025 0.500000i 0.866025 0.500000i
\(78\) 0 0
\(79\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 1.00000 1.00000
\(92\) 0 0
\(93\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0.500000 0.866025i 0.500000 0.866025i
\(100\) 0 0
\(101\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(104\) 0 0
\(105\) −1.00000 −1.00000
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(114\) 0 0
\(115\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(116\) 0 0
\(117\) 0.866025 0.500000i 0.866025 0.500000i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −0.500000 0.866025i −0.500000 0.866025i
\(124\) 0 0
\(125\) 1.00000i 1.00000i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0.500000 0.866025i 0.500000 0.866025i
\(130\) 0 0
\(131\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(136\) 0 0
\(137\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(138\) 0 0
\(139\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(140\) 0 0
\(141\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(142\) 0 0
\(143\) 1.00000i 1.00000i
\(144\) 0 0
\(145\) −1.00000 −1.00000
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.500000 0.866025i 0.500000 0.866025i
\(156\) 0 0
\(157\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.00000 −1.00000
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 1.00000i 1.00000i
\(166\) 0 0
\(167\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −0.500000 0.866025i −0.500000 0.866025i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −0.866025 0.500000i −0.866025 0.500000i
\(190\) 0 0
\(191\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) 0 0
\(193\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(196\) 0 0
\(197\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −0.500000 0.866025i −0.500000 0.866025i
\(202\) 0 0
\(203\) −0.500000 0.866025i −0.500000 0.866025i
\(204\) 0 0
\(205\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(206\) 0 0
\(207\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(212\) 0 0
\(213\) 2.00000i 2.00000i
\(214\) 0 0
\(215\) 1.00000i 1.00000i
\(216\) 0 0
\(217\) 1.00000 1.00000
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(230\) 0 0
\(231\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(232\) 0 0
\(233\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −1.00000 −1.00000
\(236\) 0 0
\(237\) −0.866025 0.500000i −0.866025 0.500000i
\(238\) 0 0
\(239\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) 0 0
\(241\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(242\) 0 0
\(243\) −1.00000 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 1.00000i 1.00000i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −0.866025 0.500000i −0.866025 0.500000i
\(262\) 0 0
\(263\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) −1.00000 −1.00000
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(278\) 0 0
\(279\) 0.866025 0.500000i 0.866025 0.500000i
\(280\) 0 0
\(281\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(282\) 0 0
\(283\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.00000i 1.00000i
\(288\) 0 0
\(289\) −1.00000 −1.00000
\(290\) 0 0
\(291\) 0.500000 0.866025i 0.500000 0.866025i
\(292\) 0 0
\(293\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(294\) 0 0
\(295\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(296\) 0 0
\(297\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(298\) 0 0
\(299\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(300\) 0 0
\(301\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(302\) 0 0
\(303\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(304\) 0 0
\(305\) −1.00000 −1.00000
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(310\) 0 0
\(311\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 1.00000 1.00000
\(316\) 0 0
\(317\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(318\) 0 0
\(319\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −0.500000 0.866025i −0.500000 0.866025i
\(330\) 0 0
\(331\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(336\) 0 0
\(337\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) 0 0
\(339\) −0.500000 0.866025i −0.500000 0.866025i
\(340\) 0 0
\(341\) 1.00000i 1.00000i
\(342\) 0 0
\(343\) 1.00000i 1.00000i
\(344\) 0 0
\(345\) 0.500000 0.866025i 0.500000 0.866025i
\(346\) 0 0
\(347\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(352\) 0 0
\(353\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −1.00000 1.73205i −1.00000 1.73205i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −1.00000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 1.00000i 1.00000i
\(376\) 0 0
\(377\) −1.00000 −1.00000
\(378\) 0 0
\(379\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0.500000 0.866025i 0.500000 0.866025i
\(386\) 0 0
\(387\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(388\) 0 0
\(389\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(394\) 0 0
\(395\) 1.00000 1.00000
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0.500000 0.866025i 0.500000 0.866025i
\(404\) 0 0
\(405\) 0.866025 0.500000i 0.866025 0.500000i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(412\) 0 0
\(413\) 1.00000i 1.00000i
\(414\) 0 0
\(415\) 1.00000i 1.00000i
\(416\) 0 0
\(417\) −0.500000 0.866025i −0.500000 0.866025i
\(418\) 0 0
\(419\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) −0.866025 0.500000i −0.866025 0.500000i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.500000 0.866025i −0.500000 0.866025i
\(428\) 0 0
\(429\) 1.00000i 1.00000i
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 1.00000 1.00000
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0.866025 0.500000i 0.866025 0.500000i
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 1.00000 1.00000
\(452\) 0 0
\(453\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(454\) 0 0
\(455\) 0.866025 0.500000i 0.866025 0.500000i
\(456\) 0 0
\(457\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(462\) 0 0
\(463\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(464\) 0 0
\(465\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 1.00000i 1.00000i
\(470\) 0 0
\(471\) 0.866025 0.500000i 0.866025 0.500000i
\(472\) 0 0
\(473\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 1.00000 1.00000
\(484\) 0 0
\(485\) 1.00000i 1.00000i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 1.00000i 1.00000i
\(496\) 0 0
\(497\) 1.00000 1.73205i 1.00000 1.73205i
\(498\) 0 0
\(499\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(502\) 0 0
\(503\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −1.00000 −1.00000
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.500000 0.866025i 0.500000 0.866025i
\(516\) 0 0
\(517\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(518\) 0 0
\(519\) −0.866025 0.500000i −0.866025 0.500000i
\(520\) 0 0
\(521\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(522\) 0 0
\(523\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(532\) 0 0
\(533\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) −0.866025 0.500000i −0.866025 0.500000i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 1.00000i 1.00000i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 0 0
\(565\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(566\) 0 0
\(567\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(568\) 0 0
\(569\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(580\) 0 0
\(581\) 0.866025 0.500000i 0.866025 0.500000i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0.500000 0.866025i 0.500000 0.866025i
\(586\) 0 0
\(587\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 2.00000i 2.00000i
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.00000i 2.00000i
\(598\) 0 0
\(599\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 0 0
\(601\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(602\) 0 0
\(603\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(610\) 0 0
\(611\) −1.00000 −1.00000
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) −0.866025 0.500000i −0.866025 0.500000i
\(616\) 0 0
\(617\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(618\) 0 0
\(619\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(620\) 0 0
\(621\) 0.866025 0.500000i 0.866025 0.500000i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −0.500000 0.866025i −0.500000 0.866025i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2.00000i 2.00000i
\(640\) 0 0
\(641\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(642\) 0 0
\(643\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(644\) 0 0
\(645\) 1.00000i 1.00000i
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 1.00000 1.00000
\(650\) 0 0
\(651\) −1.00000 −1.00000
\(652\) 0 0
\(653\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) −0.866025 0.500000i −0.866025 0.500000i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.00000 1.00000
\(668\) 0 0
\(669\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(670\) 0 0
\(671\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(672\) 0 0
\(673\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(678\) 0 0
\(679\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(680\) 0 0
\(681\) 0.500000 0.866025i 0.500000 0.866025i
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 1.00000i 1.00000i
\(686\) 0 0
\(687\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(692\) 0 0
\(693\) 0.866025 0.500000i 0.866025 0.500000i
\(694\) 0 0
\(695\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 2.00000 2.00000
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 1.00000 1.00000
\(706\) 0 0
\(707\) −0.500000 0.866025i −0.500000 0.866025i
\(708\) 0 0
\(709\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(710\) 0 0
\(711\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(712\) 0 0
\(713\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(714\) 0 0
\(715\) −0.500000 0.866025i −0.500000 0.866025i
\(716\) 0 0
\(717\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 1.00000 1.00000
\(722\) 0 0
\(723\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.00000 1.00000
\(738\) 0 0
\(739\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(744\) 0 0
\(745\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(746\) 0 0
\(747\) 0.500000 0.866025i 0.500000 0.866025i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.00000 −1.00000
\(756\) 0 0
\(757\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(758\) 0 0
\(759\) 1.00000i 1.00000i
\(760\) 0 0
\(761\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(768\) 0 0
\(769\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 0 0
\(771\) −0.500000 0.866025i −0.500000 0.866025i
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −1.73205 1.00000i −1.73205 1.00000i
\(782\) 0 0
\(783\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(784\) 0 0
\(785\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(786\) 0 0
\(787\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(790\) 0 0
\(791\) 1.00000i 1.00000i
\(792\) 0 0
\(793\) −1.00000 −1.00000
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 1.00000 1.00000
\(820\) 0 0
\(821\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0.500000 0.866025i 0.500000 0.866025i
\(836\) 0 0
\(837\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(838\) 0 0
\(839\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 0 0
\(843\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −0.500000 0.866025i −0.500000 0.866025i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(858\) 0 0
\(859\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 1.00000i 1.00000i
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 1.00000 1.00000
\(866\) 0 0
\(867\) 1.00000 1.00000
\(868\) 0 0
\(869\) 0.866025 0.500000i 0.866025 0.500000i
\(870\) 0 0
\(871\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(872\) 0 0
\(873\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(874\) 0 0
\(875\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(876\) 0 0
\(877\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(878\) 0 0
\(879\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) −0.866025 0.500000i −0.866025 0.500000i
\(886\) 0 0
\(887\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.500000 0.866025i 0.500000 0.866025i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0.500000 0.866025i 0.500000 0.866025i
\(898\) 0 0
\(899\) −1.00000 −1.00000
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0.866025 0.500000i 0.866025 0.500000i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −0.866025 0.500000i −0.866025 0.500000i
\(910\) 0 0
\(911\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −0.500000 0.866025i −0.500000 0.866025i
\(914\) 0 0
\(915\) 1.00000 1.00000
\(916\) 0 0
\(917\) 1.00000i 1.00000i
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.00000 1.73205i −1.00000 1.73205i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0.866025 0.500000i 0.866025 0.500000i
\(928\) 0 0
\(929\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0.866025 0.500000i 0.866025 0.500000i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0.500000 0.866025i 0.500000 0.866025i
\(940\) 0 0
\(941\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) −0.866025 0.500000i −0.866025 0.500000i
\(944\) 0 0
\(945\) −1.00000 −1.00000
\(946\) 0 0
\(947\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) −1.00000 −1.00000
\(956\) 0 0
\(957\) 0.866025 0.500000i 0.866025 0.500000i
\(958\) 0 0
\(959\) 0.866025 0.500000i 0.866025 0.500000i
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.866025 0.500000i −0.866025 0.500000i
\(966\) 0 0
\(967\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 1.00000i 1.00000i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(984\) 0 0
\(985\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(986\) 0 0
\(987\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(988\) 0 0
\(989\) 1.00000i 1.00000i
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(994\) 0 0
\(995\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(996\) 0 0
\(997\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2304.1.t.a.1663.2 4
4.3 odd 2 2304.1.t.b.1663.2 4
8.3 odd 2 inner 2304.1.t.a.1663.1 4
8.5 even 2 2304.1.t.b.1663.1 4
9.4 even 3 inner 2304.1.t.a.895.1 4
16.3 odd 4 576.1.o.a.511.2 4
16.5 even 4 288.1.o.a.223.2 yes 4
16.11 odd 4 288.1.o.a.223.1 yes 4
16.13 even 4 576.1.o.a.511.1 4
36.31 odd 6 2304.1.t.b.895.1 4
48.5 odd 4 864.1.o.a.127.1 4
48.11 even 4 864.1.o.a.127.2 4
48.29 odd 4 1728.1.o.a.127.1 4
48.35 even 4 1728.1.o.a.127.2 4
72.13 even 6 2304.1.t.b.895.2 4
72.67 odd 6 inner 2304.1.t.a.895.2 4
144.5 odd 12 864.1.o.a.415.2 4
144.11 even 12 2592.1.g.b.2431.1 2
144.13 even 12 576.1.o.a.319.1 4
144.43 odd 12 2592.1.g.a.2431.1 2
144.59 even 12 864.1.o.a.415.1 4
144.67 odd 12 576.1.o.a.319.2 4
144.77 odd 12 1728.1.o.a.1279.2 4
144.85 even 12 288.1.o.a.31.2 yes 4
144.101 odd 12 2592.1.g.b.2431.2 2
144.131 even 12 1728.1.o.a.1279.1 4
144.133 even 12 2592.1.g.a.2431.2 2
144.139 odd 12 288.1.o.a.31.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.1.o.a.31.1 4 144.139 odd 12
288.1.o.a.31.2 yes 4 144.85 even 12
288.1.o.a.223.1 yes 4 16.11 odd 4
288.1.o.a.223.2 yes 4 16.5 even 4
576.1.o.a.319.1 4 144.13 even 12
576.1.o.a.319.2 4 144.67 odd 12
576.1.o.a.511.1 4 16.13 even 4
576.1.o.a.511.2 4 16.3 odd 4
864.1.o.a.127.1 4 48.5 odd 4
864.1.o.a.127.2 4 48.11 even 4
864.1.o.a.415.1 4 144.59 even 12
864.1.o.a.415.2 4 144.5 odd 12
1728.1.o.a.127.1 4 48.29 odd 4
1728.1.o.a.127.2 4 48.35 even 4
1728.1.o.a.1279.1 4 144.131 even 12
1728.1.o.a.1279.2 4 144.77 odd 12
2304.1.t.a.895.1 4 9.4 even 3 inner
2304.1.t.a.895.2 4 72.67 odd 6 inner
2304.1.t.a.1663.1 4 8.3 odd 2 inner
2304.1.t.a.1663.2 4 1.1 even 1 trivial
2304.1.t.b.895.1 4 36.31 odd 6
2304.1.t.b.895.2 4 72.13 even 6
2304.1.t.b.1663.1 4 8.5 even 2
2304.1.t.b.1663.2 4 4.3 odd 2
2592.1.g.a.2431.1 2 144.43 odd 12
2592.1.g.a.2431.2 2 144.133 even 12
2592.1.g.b.2431.1 2 144.11 even 12
2592.1.g.b.2431.2 2 144.101 odd 12