Properties

Label 2888.1.l.b.69.1
Level $2888$
Weight $1$
Character 2888.69
Analytic conductor $1.441$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -152
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2888,1,Mod(69,2888)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2888, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2888.69");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2888 = 2^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2888.l (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: \(1.44129975648\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.152.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of 12.0.1607222233084985344.6

Embedding invariants

Embedding label 69.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2888.69
Dual form 2888.1.l.b.293.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{6} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{12} +(-0.500000 + 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(0.500000 + 0.866025i) q^{17} +(0.500000 + 0.866025i) q^{21} +(0.500000 - 0.866025i) q^{23} +(0.500000 + 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{26} -1.00000 q^{27} +(0.500000 - 0.866025i) q^{28} +(-0.500000 + 0.866025i) q^{29} +(0.500000 - 0.866025i) q^{32} +(-0.500000 + 0.866025i) q^{34} -2.00000 q^{37} +1.00000 q^{39} +(-0.500000 + 0.866025i) q^{42} +1.00000 q^{46} +(-1.00000 + 1.73205i) q^{47} +(-0.500000 + 0.866025i) q^{48} -1.00000 q^{50} +(0.500000 - 0.866025i) q^{51} +(-0.500000 - 0.866025i) q^{52} +(-0.500000 + 0.866025i) q^{53} +(-0.500000 - 0.866025i) q^{54} +1.00000 q^{56} -1.00000 q^{58} +(-0.500000 - 0.866025i) q^{59} +1.00000 q^{64} +(-0.500000 + 0.866025i) q^{67} -1.00000 q^{68} -1.00000 q^{69} +(0.500000 + 0.866025i) q^{73} +(-1.00000 - 1.73205i) q^{74} +1.00000 q^{75} +(0.500000 + 0.866025i) q^{78} +(0.500000 + 0.866025i) q^{81} -1.00000 q^{84} +1.00000 q^{87} +(0.500000 - 0.866025i) q^{91} +(0.500000 + 0.866025i) q^{92} -2.00000 q^{94} -1.00000 q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{12} - q^{13} - q^{14} - q^{16} + q^{17} + q^{21} + q^{23} + q^{24} - q^{25} - 2 q^{26} - 2 q^{27} + q^{28} - q^{29} + q^{32} - q^{34} - 4 q^{37} + 2 q^{39} - q^{42} + 2 q^{46} - 2 q^{47} - q^{48} - 2 q^{50} + q^{51} - q^{52} - q^{53} - q^{54} + 2 q^{56} - 2 q^{58} - q^{59} + 2 q^{64} - q^{67} - 2 q^{68} - 2 q^{69} + q^{73} - 2 q^{74} + 2 q^{75} + q^{78} + q^{81} - 2 q^{84} + 2 q^{87} + q^{91} + q^{92} - 4 q^{94} - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2888\mathbb{Z}\right)^\times\).

\(n\) \(1445\) \(2167\) \(2529\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(3\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(4\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(5\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0.500000 0.866025i 0.500000 0.866025i
\(7\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) −1.00000 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.00000 1.00000
\(13\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(14\) −0.500000 0.866025i −0.500000 0.866025i
\(15\) 0 0
\(16\) −0.500000 0.866025i −0.500000 0.866025i
\(17\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(22\) 0 0
\(23\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(24\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) −1.00000 −1.00000
\(27\) −1.00000 −1.00000
\(28\) 0.500000 0.866025i 0.500000 0.866025i
\(29\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0.500000 0.866025i 0.500000 0.866025i
\(33\) 0 0
\(34\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 1.00000 1.00000
\(40\) 0 0
\(41\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(43\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.00000 1.00000
\(47\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(49\) 0 0
\(50\) −1.00000 −1.00000
\(51\) 0.500000 0.866025i 0.500000 0.866025i
\(52\) −0.500000 0.866025i −0.500000 0.866025i
\(53\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(54\) −0.500000 0.866025i −0.500000 0.866025i
\(55\) 0 0
\(56\) 1.00000 1.00000
\(57\) 0 0
\(58\) −1.00000 −1.00000
\(59\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(68\) −1.00000 −1.00000
\(69\) −1.00000 −1.00000
\(70\) 0 0
\(71\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 0 0
\(73\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) −1.00000 1.73205i −1.00000 1.73205i
\(75\) 1.00000 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(79\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −1.00000 −1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000 1.00000
\(88\) 0 0
\(89\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) 0.500000 0.866025i 0.500000 0.866025i
\(92\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(93\) 0 0
\(94\) −2.00000 −2.00000
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −0.500000 0.866025i −0.500000 0.866025i
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 1.00000 1.00000
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0.500000 0.866025i 0.500000 0.866025i
\(105\) 0 0
\(106\) −1.00000 −1.00000
\(107\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) 0.500000 0.866025i 0.500000 0.866025i
\(109\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(112\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.500000 0.866025i −0.500000 0.866025i
\(117\) 0 0
\(118\) 0.500000 0.866025i 0.500000 0.866025i
\(119\) −0.500000 0.866025i −0.500000 0.866025i
\(120\) 0 0
\(121\) 1.00000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.00000 −1.00000
\(135\) 0 0
\(136\) −0.500000 0.866025i −0.500000 0.866025i
\(137\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(138\) −0.500000 0.866025i −0.500000 0.866025i
\(139\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(140\) 0 0
\(141\) 2.00000 2.00000
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(147\) 0 0
\(148\) 1.00000 1.73205i 1.00000 1.73205i
\(149\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(157\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(158\) 0 0
\(159\) 1.00000 1.00000
\(160\) 0 0
\(161\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(162\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(168\) −0.500000 0.866025i −0.500000 0.866025i
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(175\) 0.500000 0.866025i 0.500000 0.866025i
\(176\) 0 0
\(177\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(178\) 0 0
\(179\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(182\) 1.00000 1.00000
\(183\) 0 0
\(184\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −1.00000 1.73205i −1.00000 1.73205i
\(189\) 1.00000 1.00000
\(190\) 0 0
\(191\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) −0.500000 0.866025i −0.500000 0.866025i
\(193\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(200\) 0.500000 0.866025i 0.500000 0.866025i
\(201\) 1.00000 1.00000
\(202\) 0 0
\(203\) 0.500000 0.866025i 0.500000 0.866025i
\(204\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1.00000 1.00000
\(209\) 0 0
\(210\) 0 0
\(211\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(212\) −0.500000 0.866025i −0.500000 0.866025i
\(213\) 0 0
\(214\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(215\) 0 0
\(216\) 1.00000 1.00000
\(217\) 0 0
\(218\) 0.500000 0.866025i 0.500000 0.866025i
\(219\) 0.500000 0.866025i 0.500000 0.866025i
\(220\) 0 0
\(221\) −1.00000 −1.00000
\(222\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(223\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(224\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(225\) 0 0
\(226\) 0 0
\(227\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.500000 0.866025i 0.500000 0.866025i
\(233\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.00000 1.00000
\(237\) 0 0
\(238\) 0.500000 0.866025i 0.500000 0.866025i
\(239\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(240\) 0 0
\(241\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(257\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0 0
\(259\) 2.00000 2.00000
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −0.500000 0.866025i −0.500000 0.866025i
\(269\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(272\) 0.500000 0.866025i 0.500000 0.866025i
\(273\) −1.00000 −1.00000
\(274\) 1.00000 1.00000
\(275\) 0 0
\(276\) 0.500000 0.866025i 0.500000 0.866025i
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(283\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 0 0
\(292\) −1.00000 −1.00000
\(293\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 2.00000
\(297\) 0 0
\(298\) 0 0
\(299\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(300\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) −1.00000 −1.00000
\(313\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(314\) 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 \(\pi\)
\(318\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(319\) 0 0
\(320\) 0 0
\(321\) −0.500000 0.866025i −0.500000 0.866025i
\(322\) −1.00000 −1.00000
\(323\) 0 0
\(324\) −1.00000 −1.00000
\(325\) −0.500000 0.866025i −0.500000 0.866025i
\(326\) 0 0
\(327\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(328\) 0 0
\(329\) 1.00000 1.73205i 1.00000 1.73205i
\(330\) 0 0
\(331\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0.500000 0.866025i 0.500000 0.866025i
\(337\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 1.00000 1.00000
\(351\) 0.500000 0.866025i 0.500000 0.866025i
\(352\) 0 0
\(353\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) −1.00000 −1.00000
\(355\) 0 0
\(356\) 0 0
\(357\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(358\) −1.00000 1.73205i −1.00000 1.73205i
\(359\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 2.00000 2.00000
\(363\) −0.500000 0.866025i −0.500000 0.866025i
\(364\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(365\) 0 0
\(366\) 0 0
\(367\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(368\) −1.00000 −1.00000
\(369\) 0 0
\(370\) 0 0
\(371\) 0.500000 0.866025i 0.500000 0.866025i
\(372\) 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 0
\(376\) 1.00000 1.73205i 1.00000 1.73205i
\(377\) −0.500000 0.866025i −0.500000 0.866025i
\(378\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(379\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −0.500000 0.866025i −0.500000 0.866025i
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 0.500000 0.866025i 0.500000 0.866025i
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) 1.00000 1.00000
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(398\) 1.00000 1.00000
\(399\) 0 0
\(400\) 1.00000 1.00000
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 1.00000 1.00000
\(407\) 0 0
\(408\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(409\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(410\) 0 0
\(411\) −1.00000 −1.00000
\(412\) 0 0
\(413\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(414\) 0 0
\(415\) 0 0
\(416\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(422\) 0.500000 0.866025i 0.500000 0.866025i
\(423\) 0 0
\(424\) 0.500000 0.866025i 0.500000 0.866025i
\(425\) −1.00000 −1.00000
\(426\) 0 0
\(427\) 0 0
\(428\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(433\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.00000 1.00000
\(437\) 0 0
\(438\) 1.00000 1.00000
\(439\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −0.500000 0.866025i −0.500000 0.866025i
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) −2.00000 −2.00000
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.00000 −1.00000
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(455\) 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
\(459\) −0.500000 0.866025i −0.500000 0.866025i
\(460\) 0 0
\(461\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) 0 0
\(463\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(464\) 1.00000 1.00000
\(465\) 0 0
\(466\) 1.00000 1.73205i 1.00000 1.73205i
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0.500000 0.866025i 0.500000 0.866025i
\(470\) 0 0
\(471\) 0 0
\(472\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 1.00000 1.00000
\(477\) 0 0
\(478\) −0.500000 0.866025i −0.500000 0.866025i
\(479\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 1.00000 1.73205i 1.00000 1.73205i
\(482\) 0 0
\(483\) 1.00000 1.00000
\(484\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(492\) 0 0
\(493\) −1.00000 −1.00000
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(510\) 0 0
\(511\) −0.500000 0.866025i −0.500000 0.866025i
\(512\) −1.00000 −1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(519\) 1.00000 1.73205i 1.00000 1.73205i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) −1.00000 −1.00000
\(526\) 1.00000 1.73205i 1.00000 1.73205i
\(527\) 0 0
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0.500000 0.866025i 0.500000 0.866025i
\(537\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(538\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(542\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(543\) −2.00000 −2.00000
\(544\) 1.00000 1.00000
\(545\) 0 0
\(546\) −0.500000 0.866025i −0.500000 0.866025i
\(547\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(548\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 1.00000 1.00000
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(564\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(565\) 0 0
\(566\) 0 0
\(567\) −0.500000 0.866025i −0.500000 0.866025i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(574\) 0 0
\(575\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(576\) 0 0
\(577\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.500000 0.866025i −0.500000 0.866025i
\(585\) 0 0
\(586\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(587\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(593\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.00000 −1.00000
\(598\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) −1.00000 −1.00000
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) −1.00000 −1.00000
\(610\) 0 0
\(611\) −1.00000 1.73205i −1.00000 1.73205i
\(612\) 0 0
\(613\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(615\) 0 0
\(616\) 0 0
\(617\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(622\) −0.500000 0.866025i −0.500000 0.866025i
\(623\) 0 0
\(624\) −0.500000 0.866025i −0.500000 0.866025i
\(625\) −0.500000 0.866025i −0.500000 0.866025i
\(626\) 1.00000 1.00000
\(627\) 0 0
\(628\) 0 0
\(629\) −1.00000 1.73205i −1.00000 1.73205i
\(630\) 0 0
\(631\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(632\) 0 0
\(633\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(634\) −1.00000 −1.00000
\(635\) 0 0
\(636\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0.500000 0.866025i 0.500000 0.866025i
\(643\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(644\) −0.500000 0.866025i −0.500000 0.866025i
\(645\) 0 0
\(646\) 0 0
\(647\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(648\) −0.500000 0.866025i −0.500000 0.866025i
\(649\) 0 0
\(650\) 0.500000 0.866025i 0.500000 0.866025i
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) −1.00000 −1.00000
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 2.00000 2.00000
\(659\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(662\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(663\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 1.00000 1.00000
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0.500000 0.866025i 0.500000 0.866025i
\(676\) 0 0
\(677\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −0.500000 0.866025i −0.500000 0.866025i
\(682\) 0 0
\(683\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(687\) 0 0
\(688\) 0 0
\(689\) −0.500000 0.866025i −0.500000 0.866025i
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) −2.00000 −2.00000
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) −1.00000 −1.00000
\(697\) 0 0
\(698\) 0 0
\(699\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(700\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(701\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(702\) 1.00000 1.00000
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −0.500000 0.866025i −0.500000 0.866025i
\(707\) 0 0
\(708\) −0.500000 0.866025i −0.500000 0.866025i
\(709\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) −1.00000 −1.00000
\(715\) 0 0
\(716\) 1.00000 1.73205i 1.00000 1.73205i
\(717\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(718\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(719\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(725\) −0.500000 0.866025i −0.500000 0.866025i
\(726\) 0.500000 0.866025i 0.500000 0.866025i
\(727\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(728\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) −2.00000 −2.00000
\(735\) 0 0
\(736\) −0.500000 0.866025i −0.500000 0.866025i
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.00000 1.00000
\(743\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(747\) 0 0
\(748\) 0 0
\(749\) −1.00000 −1.00000
\(750\) 0 0
\(751\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 2.00000 2.00000
\(753\) 0 0
\(754\) 0.500000 0.866025i 0.500000 0.866025i
\(755\) 0 0
\(756\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(757\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(758\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(759\) 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
\(763\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(764\) 0.500000 0.866025i 0.500000 0.866025i
\(765\) 0 0
\(766\) 0 0
\(767\) 1.00000 1.00000
\(768\) 1.00000 1.00000
\(769\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.00000 1.73205i −1.00000 1.73205i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(783\) 0.500000 0.866025i 0.500000 0.866025i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) 0 0
\(789\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(797\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(798\) 0 0
\(799\) −2.00000 −2.00000
\(800\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(805\) 0 0
\(806\) 0 0
\(807\) 1.00000 1.73205i 1.00000 1.73205i
\(808\) 0 0
\(809\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 0 0
\(811\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(812\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(813\) 0.500000 0.866025i 0.500000 0.866025i
\(814\) 0 0
\(815\) 0 0
\(816\) −1.00000 −1.00000
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) −0.500000 0.866025i −0.500000 0.866025i
\(823\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(824\) 0 0
\(825\) 0 0
\(826\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(827\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 0.500000 0.866025i 0.500000 0.866025i
\(843\) 0 0
\(844\) 1.00000 1.00000
\(845\) 0 0
\(846\) 0 0
\(847\) −1.00000 −1.00000
\(848\) 1.00000 1.00000
\(849\) 0 0
\(850\) −0.500000 0.866025i −0.500000 0.866025i
\(851\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(852\) 0 0
\(853\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.00000 −1.00000
\(857\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(858\) 0 0
\(859\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −0.500000 0.866025i −0.500000 0.866025i
\(872\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(877\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) −0.500000 0.866025i −0.500000 0.866025i
\(880\) 0 0
\(881\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(882\) 0 0
\(883\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(884\) 0.500000 0.866025i 0.500000 0.866025i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) −1.00000 1.73205i −1.00000 1.73205i
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −0.500000 0.866025i −0.500000 0.866025i
\(897\) 0.500000 0.866025i 0.500000 0.866025i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −1.00000 −1.00000
\(902\) 0 0
\(903\) 0 0
\(904\) 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 \(\pi\)
\(908\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.500000 0.866025i −0.500000 0.866025i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0.500000 0.866025i 0.500000 0.866025i
\(919\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(920\) 0 0
\(921\) 1.00000 1.73205i 1.00000 1.73205i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.00000 1.73205i 1.00000 1.73205i
\(926\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(927\) 0 0
\(928\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(929\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.00000 2.00000
\(933\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(938\) 1.00000 1.00000
\(939\) −1.00000 −1.00000
\(940\) 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
\(944\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(948\) 0 0
\(949\) −1.00000 −1.00000
\(950\) 0 0
\(951\) 1.00000 1.00000
\(952\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(953\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0.500000 0.866025i 0.500000 0.866025i
\(957\) 0 0
\(958\) −2.00000 −2.00000
\(959\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(960\) 0 0
\(961\) 1.00000 1.00000
\(962\) 2.00000 2.00000
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(967\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(968\) −1.00000 −1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −0.500000 0.866025i −0.500000 0.866025i
\(987\) −2.00000 −2.00000
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(992\) 0 0
\(993\) −0.500000 0.866025i −0.500000 0.866025i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(998\) 0 0
\(999\) 2.00000 2.00000
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 2888.1.l.b.69.1 2
8.5 even 2 2888.1.l.a.69.1 2
19.2 odd 18 2888.1.s.a.333.1 6
19.3 odd 18 2888.1.s.a.2293.1 6
19.4 even 9 2888.1.s.b.477.1 6
19.5 even 9 2888.1.s.b.2789.1 6
19.6 even 9 2888.1.s.b.1021.1 6
19.7 even 3 152.1.g.a.37.1 1
19.8 odd 6 2888.1.l.a.293.1 2
19.9 even 9 2888.1.s.b.1029.1 6
19.10 odd 18 2888.1.s.a.1029.1 6
19.11 even 3 inner 2888.1.l.b.293.1 2
19.12 odd 6 152.1.g.b.37.1 yes 1
19.13 odd 18 2888.1.s.a.1021.1 6
19.14 odd 18 2888.1.s.a.2789.1 6
19.15 odd 18 2888.1.s.a.477.1 6
19.16 even 9 2888.1.s.b.2293.1 6
19.17 even 9 2888.1.s.b.333.1 6
19.18 odd 2 2888.1.l.a.69.1 2
57.26 odd 6 1368.1.i.b.37.1 1
57.50 even 6 1368.1.i.a.37.1 1
76.7 odd 6 608.1.g.a.113.1 1
76.31 even 6 608.1.g.b.113.1 1
95.7 odd 12 3800.1.b.a.949.1 2
95.12 even 12 3800.1.b.b.949.2 2
95.64 even 6 3800.1.o.b.1101.1 1
95.69 odd 6 3800.1.o.a.1101.1 1
95.83 odd 12 3800.1.b.a.949.2 2
95.88 even 12 3800.1.b.b.949.1 2
152.5 even 18 2888.1.s.a.2789.1 6
152.13 odd 18 2888.1.s.b.1021.1 6
152.21 odd 18 2888.1.s.b.333.1 6
152.29 odd 18 2888.1.s.b.1029.1 6
152.37 odd 2 CM 2888.1.l.b.69.1 2
152.45 even 6 152.1.g.b.37.1 yes 1
152.53 odd 18 2888.1.s.b.477.1 6
152.61 even 18 2888.1.s.a.477.1 6
152.69 odd 6 152.1.g.a.37.1 1
152.83 odd 6 608.1.g.b.113.1 1
152.85 even 18 2888.1.s.a.1029.1 6
152.93 even 18 2888.1.s.a.333.1 6
152.101 even 18 2888.1.s.a.1021.1 6
152.107 even 6 608.1.g.a.113.1 1
152.109 odd 18 2888.1.s.b.2789.1 6
152.117 odd 18 2888.1.s.b.2293.1 6
152.125 even 6 2888.1.l.a.293.1 2
152.141 odd 6 inner 2888.1.l.b.293.1 2
152.149 even 18 2888.1.s.a.2293.1 6
456.197 odd 6 1368.1.i.a.37.1 1
456.221 even 6 1368.1.i.b.37.1 1
760.69 odd 6 3800.1.o.b.1101.1 1
760.197 odd 12 3800.1.b.b.949.2 2
760.349 even 6 3800.1.o.a.1101.1 1
760.373 even 12 3800.1.b.a.949.2 2
760.653 odd 12 3800.1.b.b.949.1 2
760.677 even 12 3800.1.b.a.949.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.1.g.a.37.1 1 19.7 even 3
152.1.g.a.37.1 1 152.69 odd 6
152.1.g.b.37.1 yes 1 19.12 odd 6
152.1.g.b.37.1 yes 1 152.45 even 6
608.1.g.a.113.1 1 76.7 odd 6
608.1.g.a.113.1 1 152.107 even 6
608.1.g.b.113.1 1 76.31 even 6
608.1.g.b.113.1 1 152.83 odd 6
1368.1.i.a.37.1 1 57.50 even 6
1368.1.i.a.37.1 1 456.197 odd 6
1368.1.i.b.37.1 1 57.26 odd 6
1368.1.i.b.37.1 1 456.221 even 6
2888.1.l.a.69.1 2 8.5 even 2
2888.1.l.a.69.1 2 19.18 odd 2
2888.1.l.a.293.1 2 19.8 odd 6
2888.1.l.a.293.1 2 152.125 even 6
2888.1.l.b.69.1 2 1.1 even 1 trivial
2888.1.l.b.69.1 2 152.37 odd 2 CM
2888.1.l.b.293.1 2 19.11 even 3 inner
2888.1.l.b.293.1 2 152.141 odd 6 inner
2888.1.s.a.333.1 6 19.2 odd 18
2888.1.s.a.333.1 6 152.93 even 18
2888.1.s.a.477.1 6 19.15 odd 18
2888.1.s.a.477.1 6 152.61 even 18
2888.1.s.a.1021.1 6 19.13 odd 18
2888.1.s.a.1021.1 6 152.101 even 18
2888.1.s.a.1029.1 6 19.10 odd 18
2888.1.s.a.1029.1 6 152.85 even 18
2888.1.s.a.2293.1 6 19.3 odd 18
2888.1.s.a.2293.1 6 152.149 even 18
2888.1.s.a.2789.1 6 19.14 odd 18
2888.1.s.a.2789.1 6 152.5 even 18
2888.1.s.b.333.1 6 19.17 even 9
2888.1.s.b.333.1 6 152.21 odd 18
2888.1.s.b.477.1 6 19.4 even 9
2888.1.s.b.477.1 6 152.53 odd 18
2888.1.s.b.1021.1 6 19.6 even 9
2888.1.s.b.1021.1 6 152.13 odd 18
2888.1.s.b.1029.1 6 19.9 even 9
2888.1.s.b.1029.1 6 152.29 odd 18
2888.1.s.b.2293.1 6 19.16 even 9
2888.1.s.b.2293.1 6 152.117 odd 18
2888.1.s.b.2789.1 6 19.5 even 9
2888.1.s.b.2789.1 6 152.109 odd 18
3800.1.b.a.949.1 2 95.7 odd 12
3800.1.b.a.949.1 2 760.677 even 12
3800.1.b.a.949.2 2 95.83 odd 12
3800.1.b.a.949.2 2 760.373 even 12
3800.1.b.b.949.1 2 95.88 even 12
3800.1.b.b.949.1 2 760.653 odd 12
3800.1.b.b.949.2 2 95.12 even 12
3800.1.b.b.949.2 2 760.197 odd 12
3800.1.o.a.1101.1 1 95.69 odd 6
3800.1.o.a.1101.1 1 760.349 even 6
3800.1.o.b.1101.1 1 95.64 even 6
3800.1.o.b.1101.1 1 760.69 odd 6