Properties

Label 3744.1.ck.a.2239.2
Level $3744$
Weight $1$
Character 3744.2239
Analytic conductor $1.868$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3744,1,Mod(607,3744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3744, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3744.607");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3744.ck (of order \(6\), degree \(2\), 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: \(1.86849940730\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.876096.1

Embedding invariants

Embedding label 2239.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 3744.2239
Dual form 3744.1.ck.a.607.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{5} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{5} +(0.500000 + 0.866025i) q^{9} +1.00000i q^{11} +(0.500000 - 0.866025i) q^{13} +1.00000i q^{15} +(-0.500000 + 0.866025i) q^{17} +(-0.866025 - 0.500000i) q^{19} +1.00000i q^{27} +1.00000 q^{29} +(0.866025 - 0.500000i) q^{31} +(-0.500000 + 0.866025i) q^{33} +(-0.500000 - 0.866025i) q^{37} +(0.866025 - 0.500000i) q^{39} +(-1.73205 + 1.00000i) q^{43} +(-0.500000 + 0.866025i) q^{45} +(0.866025 + 0.500000i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-0.866025 + 0.500000i) q^{51} +(-0.866025 + 0.500000i) q^{55} +(-0.500000 - 0.866025i) q^{57} -1.00000i q^{59} +1.00000 q^{65} +(-0.866025 - 0.500000i) q^{71} +2.00000 q^{73} +(-0.866025 - 0.500000i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(-0.866025 - 0.500000i) q^{83} -1.00000 q^{85} +(0.866025 + 0.500000i) q^{87} +(-0.500000 - 0.866025i) q^{89} +1.00000 q^{93} -1.00000i q^{95} +(-0.866025 + 0.500000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} + 2 q^{9} + 2 q^{13} - 2 q^{17} + 4 q^{29} - 2 q^{33} - 2 q^{37} - 2 q^{45} - 2 q^{49} - 2 q^{57} + 4 q^{65} + 8 q^{73} - 2 q^{81} - 4 q^{85} - 2 q^{89} + 4 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) 0.500000 0.866025i 0.500000 0.866025i
\(14\) 0 0
\(15\) 1.00000i 1.00000i
\(16\) 0 0
\(17\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 1.00000i
\(28\) 0 0
\(29\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 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
\(35\) 0 0
\(36\) 0 0
\(37\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0.866025 0.500000i 0.866025 0.500000i
\(40\) 0 0
\(41\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(44\) 0 0
\(45\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(46\) 0 0
\(47\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(50\) 0 0
\(51\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(56\) 0 0
\(57\) −0.500000 0.866025i −0.500000 0.866025i
\(58\) 0 0
\(59\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.00000 1.00000
\(66\) 0 0
\(67\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 0 0
\(73\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) −1.00000 −1.00000
\(86\) 0 0
\(87\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(88\) 0 0
\(89\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.00000 1.00000
\(94\) 0 0
\(95\) 1.00000i 1.00000i
\(96\) 0 0
\(97\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(98\) 0 0
\(99\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(100\) 0 0
\(101\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 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
\(105\) 0 0
\(106\) 0 0
\(107\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 1.00000i 1.00000i
\(112\) 0 0
\(113\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.00000 1.00000
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 1.00000
\(126\) 0 0
\(127\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) −2.00000 −2.00000
\(130\) 0 0
\(131\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(136\) 0 0
\(137\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(138\) 0 0
\(139\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(140\) 0 0
\(141\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(142\) 0 0
\(143\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(144\) 0 0
\(145\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(146\) 0 0
\(147\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(148\) 0 0
\(149\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 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
\(153\) −1.00000 −1.00000
\(154\) 0 0
\(155\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(156\) 0 0
\(157\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) −1.00000 −1.00000
\(166\) 0 0
\(167\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(168\) 0 0
\(169\) −0.500000 0.866025i −0.500000 0.866025i
\(170\) 0 0
\(171\) 1.00000i 1.00000i
\(172\) 0 0
\(173\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.500000 0.866025i 0.500000 0.866025i
\(178\) 0 0
\(179\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.500000 0.866025i 0.500000 0.866025i
\(186\) 0 0
\(187\) −0.866025 0.500000i −0.866025 0.500000i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.73205 1.00000i 1.73205 1.00000i 0.866025 0.500000i \(-0.166667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(192\) 0 0
\(193\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(194\) 0 0
\(195\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(196\) 0 0
\(197\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.500000 0.866025i 0.500000 0.866025i
\(210\) 0 0
\(211\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(212\) 0 0
\(213\) −0.500000 0.866025i −0.500000 0.866025i
\(214\) 0 0
\(215\) −1.73205 1.00000i −1.73205 1.00000i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(220\) 0 0
\(221\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(222\) 0 0
\(223\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 0 0
\(229\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 1.00000i 1.00000i
\(236\) 0 0
\(237\) −0.500000 0.866025i −0.500000 0.866025i
\(238\) 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 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(242\) 0 0
\(243\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(244\) 0 0
\(245\) −1.00000 −1.00000
\(246\) 0 0
\(247\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(248\) 0 0
\(249\) −0.500000 0.866025i −0.500000 0.866025i
\(250\) 0 0
\(251\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −0.866025 0.500000i −0.866025 0.500000i
\(256\) 0 0
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(262\) 0 0
\(263\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.00000i 1.00000i
\(268\) 0 0
\(269\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(278\) 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 \(0.333333\pi\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(284\) 0 0
\(285\) 0.500000 0.866025i 0.500000 0.866025i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(294\) 0 0
\(295\) 0.866025 0.500000i 0.866025 0.500000i
\(296\) 0 0
\(297\) −1.00000 −1.00000
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −0.866025 0.500000i −0.866025 0.500000i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 1.00000 1.00000
\(310\) 0 0
\(311\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 1.00000i 1.00000i
\(320\) 0 0
\(321\) 1.00000 1.00000
\(322\) 0 0
\(323\) 0.866025 0.500000i 0.866025 0.500000i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) 0 0
\(333\) 0.500000 0.866025i 0.500000 0.866025i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) 0 0
\(339\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(340\) 0 0
\(341\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(348\) 0 0
\(349\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(350\) 0 0
\(351\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(352\) 0 0
\(353\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(354\) 0 0
\(355\) 1.00000i 1.00000i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(366\) 0 0
\(367\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(376\) 0 0
\(377\) 0.500000 0.866025i 0.500000 0.866025i
\(378\) 0 0
\(379\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(380\) 0 0
\(381\) −1.00000 −1.00000
\(382\) 0 0
\(383\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.73205 1.00000i −1.73205 1.00000i
\(388\) 0 0
\(389\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −1.00000 −1.00000
\(394\) 0 0
\(395\) 1.00000i 1.00000i
\(396\) 0 0
\(397\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(402\) 0 0
\(403\) 1.00000i 1.00000i
\(404\) 0 0
\(405\) −1.00000 −1.00000
\(406\) 0 0
\(407\) 0.866025 0.500000i 0.866025 0.500000i
\(408\) 0 0
\(409\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1.00000i 1.00000i
\(416\) 0 0
\(417\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(418\) 0 0
\(419\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 1.00000i 1.00000i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(430\) 0 0
\(431\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(434\) 0 0
\(435\) 1.00000i 1.00000i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(440\) 0 0
\(441\) −1.00000 −1.00000
\(442\) 0 0
\(443\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0.500000 0.866025i 0.500000 0.866025i
\(446\) 0 0
\(447\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(448\) 0 0
\(449\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −0.500000 0.866025i −0.500000 0.866025i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 0 0
\(459\) −0.866025 0.500000i −0.866025 0.500000i
\(460\) 0 0
\(461\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(462\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 1.00000i 1.00000i
\(472\) 0 0
\(473\) −1.00000 1.73205i −1.00000 1.73205i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(480\) 0 0
\(481\) −1.00000 −1.00000
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(490\) 0 0
\(491\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) 0 0
\(493\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(494\) 0 0
\(495\) −0.866025 0.500000i −0.866025 0.500000i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −0.500000 0.866025i −0.500000 0.866025i
\(506\) 0 0
\(507\) 1.00000i 1.00000i
\(508\) 0 0
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0.500000 0.866025i 0.500000 0.866025i
\(514\) 0 0
\(515\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(516\) 0 0
\(517\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(518\) 0 0
\(519\) 2.00000i 2.00000i
\(520\) 0 0
\(521\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(522\) 0 0
\(523\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.00000i 1.00000i
\(528\) 0 0
\(529\) −0.500000 0.866025i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0.866025 0.500000i 0.866025 0.500000i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(536\) 0 0
\(537\) −1.00000 −1.00000
\(538\) 0 0
\(539\) −0.866025 0.500000i −0.866025 0.500000i
\(540\) 0 0
\(541\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.866025 0.500000i −0.866025 0.500000i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0.866025 0.500000i 0.866025 0.500000i
\(556\) 0 0
\(557\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 2.00000i 2.00000i
\(560\) 0 0
\(561\) −0.500000 0.866025i −0.500000 0.866025i
\(562\) 0 0
\(563\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(564\) 0 0
\(565\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(572\) 0 0
\(573\) 2.00000 2.00000
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) 1.73205 1.00000i 1.73205 1.00000i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(586\) 0 0
\(587\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(588\) 0 0
\(589\) −1.00000 −1.00000
\(590\) 0 0
\(591\) 1.00000i 1.00000i
\(592\) 0 0
\(593\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.500000 0.866025i −0.500000 0.866025i
\(598\) 0 0
\(599\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.866025 0.500000i 0.866025 0.500000i
\(612\) 0 0
\(613\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(618\) 0 0
\(619\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0.866025 0.500000i 0.866025 0.500000i
\(628\) 0 0
\(629\) 1.00000 1.00000
\(630\) 0 0
\(631\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −0.866025 0.500000i −0.866025 0.500000i
\(636\) 0 0
\(637\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(638\) 0 0
\(639\) 1.00000i 1.00000i
\(640\) 0 0
\(641\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(642\) 0 0
\(643\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(644\) 0 0
\(645\) −1.00000 1.73205i −1.00000 1.73205i
\(646\) 0 0
\(647\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 1.00000 1.00000
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(654\) 0 0
\(655\) −0.866025 0.500000i −0.866025 0.500000i
\(656\) 0 0
\(657\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(658\) 0 0
\(659\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(660\) 0 0
\(661\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) 0 0
\(663\) 1.00000i 1.00000i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.00000i 1.00000i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 1.00000i 1.00000i
\(704\) 0 0
\(705\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(710\) 0 0
\(711\) 1.00000i 1.00000i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 1.00000i 1.00000i
\(716\) 0 0
\(717\) 1.00000 1.00000
\(718\) 0 0
\(719\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −1.00000 −1.00000
\(730\) 0 0
\(731\) 2.00000i 2.00000i
\(732\) 0 0
\(733\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −0.866025 0.500000i −0.866025 0.500000i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) −1.00000 −1.00000
\(742\) 0 0
\(743\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(744\) 0 0
\(745\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(746\) 0 0
\(747\) 1.00000i 1.00000i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 0 0
\(753\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(754\) 0 0
\(755\) 1.00000i 1.00000i
\(756\) 0 0
\(757\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −0.500000 0.866025i −0.500000 0.866025i
\(766\) 0 0
\(767\) −0.866025 0.500000i −0.866025 0.500000i
\(768\) 0 0
\(769\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0.500000 0.866025i 0.500000 0.866025i
\(782\) 0 0
\(783\) 1.00000i 1.00000i
\(784\) 0 0
\(785\) 0.500000 0.866025i 0.500000 0.866025i
\(786\) 0 0
\(787\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(788\) 0 0
\(789\) 0.500000 0.866025i 0.500000 0.866025i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(798\) 0 0
\(799\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(800\) 0 0
\(801\) 0.500000 0.866025i 0.500000 0.866025i
\(802\) 0 0
\(803\) 2.00000i 2.00000i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(808\) 0 0
\(809\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 1.00000 1.00000
\(814\) 0 0
\(815\) 1.00000i 1.00000i
\(816\) 0 0
\(817\) 2.00000 2.00000
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 0 0
\(831\) 2.00000i 2.00000i
\(832\) 0 0
\(833\) −0.500000 0.866025i −0.500000 0.866025i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(838\) 0 0
\(839\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 0 0
\(843\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(844\) 0 0
\(845\) 0.500000 0.866025i 0.500000 0.866025i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0.866025 0.500000i 0.866025 0.500000i
\(856\) 0 0
\(857\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(858\) 0 0
\(859\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0.500000 0.866025i 0.500000 0.866025i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(878\) 0 0
\(879\) −0.866025 0.500000i −0.866025 0.500000i
\(880\) 0 0
\(881\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(882\) 0 0
\(883\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(884\) 0 0
\(885\) 1.00000 1.00000
\(886\) 0 0
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.866025 0.500000i −0.866025 0.500000i
\(892\) 0 0
\(893\) −0.500000 0.866025i −0.500000 0.866025i
\(894\) 0 0
\(895\) −0.866025 0.500000i −0.866025 0.500000i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.866025 0.500000i 0.866025 0.500000i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(908\) 0 0
\(909\) −0.500000 0.866025i −0.500000 0.866025i
\(910\) 0 0
\(911\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(912\) 0 0
\(913\) 0.500000 0.866025i 0.500000 0.866025i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(928\) 0 0
\(929\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(930\) 0 0
\(931\) 0.866025 0.500000i 0.866025 0.500000i
\(932\) 0 0
\(933\) 1.00000 1.00000
\(934\) 0 0
\(935\) 1.00000i 1.00000i
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(940\) 0 0
\(941\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(948\) 0 0
\(949\) 1.00000 1.73205i 1.00000 1.73205i
\(950\) 0 0
\(951\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(952\) 0 0
\(953\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(956\) 0 0
\(957\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(964\) 0 0
\(965\) 2.00000 2.00000
\(966\) 0 0
\(967\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(968\) 0 0
\(969\) 1.00000 1.00000
\(970\) 0 0
\(971\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(978\) 0 0
\(979\) 0.866025 0.500000i 0.866025 0.500000i
\(980\) 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
\(985\) 0.500000 0.866025i 0.500000 0.866025i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.00000i 1.00000i
\(996\) 0 0
\(997\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(998\) 0 0
\(999\) 0.866025 0.500000i 0.866025 0.500000i
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 3744.1.ck.a.2239.2 yes 4
4.3 odd 2 inner 3744.1.ck.a.2239.1 yes 4
9.4 even 3 3744.1.cc.a.3487.1 yes 4
13.9 even 3 3744.1.cc.a.3103.1 4
36.31 odd 6 3744.1.cc.a.3487.2 yes 4
52.35 odd 6 3744.1.cc.a.3103.2 yes 4
117.22 even 3 inner 3744.1.ck.a.607.1 yes 4
468.139 odd 6 inner 3744.1.ck.a.607.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3744.1.cc.a.3103.1 4 13.9 even 3
3744.1.cc.a.3103.2 yes 4 52.35 odd 6
3744.1.cc.a.3487.1 yes 4 9.4 even 3
3744.1.cc.a.3487.2 yes 4 36.31 odd 6
3744.1.ck.a.607.1 yes 4 117.22 even 3 inner
3744.1.ck.a.607.2 yes 4 468.139 odd 6 inner
3744.1.ck.a.2239.1 yes 4 4.3 odd 2 inner
3744.1.ck.a.2239.2 yes 4 1.1 even 1 trivial