Properties

Label 1260.1.bp.b.419.1
Level $1260$
Weight $1$
Character 1260.419
Analytic conductor $0.629$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -20
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1260,1,Mod(419,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 3, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.419");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1260.bp (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: \(0.628821915918\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.13502538000.16

Embedding invariants

Embedding label 419.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1260.419
Dual form 1260.1.bp.b.839.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.866025 + 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} -1.00000 q^{6} +(0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.866025 + 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} -1.00000 q^{6} +(0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +1.00000i q^{10} +(0.866025 - 0.500000i) q^{12} -1.00000 q^{14} +(0.866025 - 0.500000i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(-0.866025 - 0.500000i) q^{18} +(-0.500000 - 0.866025i) q^{20} +(0.500000 + 0.866025i) q^{21} +(-0.866025 - 0.500000i) q^{23} +(-0.500000 + 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} +1.00000i q^{27} +(0.866025 - 0.500000i) q^{28} +(-1.50000 + 0.866025i) q^{29} +(-0.500000 + 0.866025i) q^{30} +(0.866025 + 0.500000i) q^{32} +(0.866025 - 0.500000i) q^{35} +1.00000 q^{36} +(0.866025 + 0.500000i) q^{40} +(0.500000 - 0.866025i) q^{41} +(-0.866025 - 0.500000i) q^{42} +1.00000 q^{45} +1.00000 q^{46} +(-0.866025 - 1.50000i) q^{47} -1.00000i q^{48} +(0.500000 + 0.866025i) q^{49} +(0.866025 + 0.500000i) q^{50} +(-0.500000 - 0.866025i) q^{54} +(-0.500000 + 0.866025i) q^{56} +(0.866025 - 1.50000i) q^{58} -1.00000i q^{60} +(1.50000 - 0.866025i) q^{61} +1.00000i q^{63} -1.00000 q^{64} +(-0.866025 + 1.50000i) q^{67} +(-0.500000 - 0.866025i) q^{69} +(-0.500000 + 0.866025i) q^{70} +(-0.866025 + 0.500000i) q^{72} -1.00000i q^{75} -1.00000 q^{80} +(-0.500000 + 0.866025i) q^{81} +1.00000i q^{82} +(0.866025 + 1.50000i) q^{83} +1.00000 q^{84} -1.73205 q^{87} -1.00000 q^{89} +(-0.866025 + 0.500000i) q^{90} +(-0.866025 + 0.500000i) q^{92} +(1.50000 + 0.866025i) q^{94} +(0.500000 + 0.866025i) q^{96} +(-0.866025 - 0.500000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{5} - 4 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{5} - 4 q^{6} + 2 q^{9} - 4 q^{14} - 2 q^{16} - 2 q^{20} + 2 q^{21} - 2 q^{24} - 2 q^{25} - 6 q^{29} - 2 q^{30} + 4 q^{36} + 2 q^{41} + 4 q^{45} + 4 q^{46} + 2 q^{49} - 2 q^{54} - 2 q^{56} + 6 q^{61} - 4 q^{64} - 2 q^{69} - 2 q^{70} - 4 q^{80} - 2 q^{81} + 4 q^{84} - 4 q^{89} + 6 q^{94} + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(3\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(4\) 0.500000 0.866025i 0.500000 0.866025i
\(5\) 0.500000 0.866025i 0.500000 0.866025i
\(6\) −1.00000 −1.00000
\(7\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(8\) 1.00000i 1.00000i
\(9\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(10\) 1.00000i 1.00000i
\(11\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0.866025 0.500000i 0.866025 0.500000i
\(13\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(14\) −1.00000 −1.00000
\(15\) 0.866025 0.500000i 0.866025 0.500000i
\(16\) −0.500000 0.866025i −0.500000 0.866025i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −0.866025 0.500000i −0.866025 0.500000i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −0.500000 0.866025i −0.500000 0.866025i
\(21\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(22\) 0 0
\(23\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(25\) −0.500000 0.866025i −0.500000 0.866025i
\(26\) 0 0
\(27\) 1.00000i 1.00000i
\(28\) 0.866025 0.500000i 0.866025 0.500000i
\(29\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(30\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(31\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(33\) 0 0
\(34\) 0 0
\(35\) 0.866025 0.500000i 0.866025 0.500000i
\(36\) 1.00000 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(41\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(42\) −0.866025 0.500000i −0.866025 0.500000i
\(43\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(44\) 0 0
\(45\) 1.00000 1.00000
\(46\) 1.00000 1.00000
\(47\) −0.866025 1.50000i −0.866025 1.50000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(-0.5\pi\)
\(48\) 1.00000i 1.00000i
\(49\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(50\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −0.500000 0.866025i −0.500000 0.866025i
\(55\) 0 0
\(56\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(57\) 0 0
\(58\) 0.866025 1.50000i 0.866025 1.50000i
\(59\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 1.00000i 1.00000i
\(61\) 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(62\) 0 0
\(63\) 1.00000i 1.00000i
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(68\) 0 0
\(69\) −0.500000 0.866025i −0.500000 0.866025i
\(70\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 1.00000i 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) −1.00000 −1.00000
\(81\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(82\) 1.00000i 1.00000i
\(83\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(84\) 1.00000 1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) −1.73205 −1.73205
\(88\) 0 0
\(89\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(91\) 0 0
\(92\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(93\) 0 0
\(94\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(95\) 0 0
\(96\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(97\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(98\) −0.866025 0.500000i −0.866025 0.500000i
\(99\) 0 0
\(100\) −1.00000 −1.00000
\(101\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) −1.73205 1.00000i −1.73205 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 0.500000i \(-0.833333\pi\)
\(104\) 0 0
\(105\) 1.00000 1.00000
\(106\) 0 0
\(107\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(108\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(109\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000i 1.00000i
\(113\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(114\) 0 0
\(115\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(116\) 1.73205i 1.73205i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(121\) 0.500000 0.866025i 0.500000 0.866025i
\(122\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(123\) 0.866025 0.500000i 0.866025 0.500000i
\(124\) 0 0
\(125\) −1.00000 −1.00000
\(126\) −0.500000 0.866025i −0.500000 0.866025i
\(127\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(128\) 0.866025 0.500000i 0.866025 0.500000i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.73205i 1.73205i
\(135\) 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 \(0.833333\pi\)
\(138\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(139\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(140\) 1.00000i 1.00000i
\(141\) 1.73205i 1.73205i
\(142\) 0 0
\(143\) 0 0
\(144\) 0.500000 0.866025i 0.500000 0.866025i
\(145\) 1.73205i 1.73205i
\(146\) 0 0
\(147\) 1.00000i 1.00000i
\(148\) 0 0
\(149\) 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 \(0\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(151\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.866025 0.500000i 0.866025 0.500000i
\(161\) −0.500000 0.866025i −0.500000 0.866025i
\(162\) 1.00000i 1.00000i
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) −0.500000 0.866025i −0.500000 0.866025i
\(165\) 0 0
\(166\) −1.50000 0.866025i −1.50000 0.866025i
\(167\) 0.866025 1.50000i 0.866025 1.50000i 1.00000i \(-0.5\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(168\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(169\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 1.50000 0.866025i 1.50000 0.866025i
\(175\) 1.00000i 1.00000i
\(176\) 0 0
\(177\) 0 0
\(178\) 0.866025 0.500000i 0.866025 0.500000i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0.500000 0.866025i 0.500000 0.866025i
\(181\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(182\) 0 0
\(183\) 1.73205 1.73205
\(184\) 0.500000 0.866025i 0.500000 0.866025i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −1.73205 −1.73205
\(189\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(190\) 0 0
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) −0.866025 0.500000i −0.866025 0.500000i
\(193\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.00000 1.00000
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0.866025 0.500000i 0.866025 0.500000i
\(201\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(202\) −1.73205 1.00000i −1.73205 1.00000i
\(203\) −1.73205 −1.73205
\(204\) 0 0
\(205\) −0.500000 0.866025i −0.500000 0.866025i
\(206\) 2.00000 2.00000
\(207\) 1.00000i 1.00000i
\(208\) 0 0
\(209\) 0 0
\(210\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(211\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(215\) 0 0
\(216\) −1.00000 −1.00000
\(217\) 0 0
\(218\) 0.866025 0.500000i 0.866025 0.500000i
\(219\) 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 \(0.166667\pi\)
\(224\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(225\) 0.500000 0.866025i 0.500000 0.866025i
\(226\) 0 0
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) 0 0
\(229\) −1.50000 0.866025i −1.50000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(230\) 0.500000 0.866025i 0.500000 0.866025i
\(231\) 0 0
\(232\) −0.866025 1.50000i −0.866025 1.50000i
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) −1.73205 −1.73205
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) −0.866025 0.500000i −0.866025 0.500000i
\(241\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(242\) 1.00000i 1.00000i
\(243\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(244\) 1.73205i 1.73205i
\(245\) 1.00000 1.00000
\(246\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(247\) 0 0
\(248\) 0 0
\(249\) 1.73205i 1.73205i
\(250\) 0.866025 0.500000i 0.866025 0.500000i
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(253\) 0 0
\(254\) 1.50000 0.866025i 1.50000 0.866025i
\(255\) 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
\(259\) 0 0
\(260\) 0 0
\(261\) −1.50000 0.866025i −1.50000 0.866025i
\(262\) 0 0
\(263\) 1.73205 1.00000i 1.73205 1.00000i 0.866025 0.500000i \(-0.166667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −0.866025 0.500000i −0.866025 0.500000i
\(268\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(269\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(270\) −1.00000 −1.00000
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −1.00000 −1.00000
\(277\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(281\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(282\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(283\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.866025 0.500000i 0.866025 0.500000i
\(288\) 1.00000i 1.00000i
\(289\) 1.00000 1.00000
\(290\) −0.866025 1.50000i −0.866025 1.50000i
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(294\) −0.500000 0.866025i −0.500000 0.866025i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) −1.73205 −1.73205
\(299\) 0 0
\(300\) −0.866025 0.500000i −0.866025 0.500000i
\(301\) 0 0
\(302\) 0 0
\(303\) 2.00000i 2.00000i
\(304\) 0 0
\(305\) 1.73205i 1.73205i
\(306\) 0 0
\(307\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(308\) 0 0
\(309\) −1.00000 1.73205i −1.00000 1.73205i
\(310\) 0 0
\(311\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) 0 0
\(313\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(314\) 0 0
\(315\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(316\) 0 0
\(317\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(321\) 0.500000 0.866025i 0.500000 0.866025i
\(322\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(323\) 0 0
\(324\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(325\) 0 0
\(326\) 0 0
\(327\) −0.866025 0.500000i −0.866025 0.500000i
\(328\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(329\) 1.73205i 1.73205i
\(330\) 0 0
\(331\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) 1.73205 1.73205
\(333\) 0 0
\(334\) 1.73205i 1.73205i
\(335\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(336\) 0.500000 0.866025i 0.500000 0.866025i
\(337\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) −0.866025 0.500000i −0.866025 0.500000i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 1.00000i
\(344\) 0 0
\(345\) −1.00000 −1.00000
\(346\) 0 0
\(347\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(348\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(349\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(350\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 1.00000i 1.00000i
\(361\) −1.00000 −1.00000
\(362\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(363\) 0.866025 0.500000i 0.866025 0.500000i
\(364\) 0 0
\(365\) 0 0
\(366\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(367\) 1.73205 1.00000i 1.73205 1.00000i 0.866025 0.500000i \(-0.166667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(368\) 1.00000i 1.00000i
\(369\) 1.00000 1.00000
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) −0.866025 0.500000i −0.866025 0.500000i
\(376\) 1.50000 0.866025i 1.50000 0.866025i
\(377\) 0 0
\(378\) 1.00000i 1.00000i
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −1.50000 0.866025i −1.50000 0.866025i
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 1.00000 1.00000
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0.866025 1.50000i 0.866025 1.50000i
\(403\) 0 0
\(404\) 2.00000 2.00000
\(405\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(406\) 1.50000 0.866025i 1.50000 0.866025i
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(410\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(411\) 0 0
\(412\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(413\) 0 0
\(414\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(415\) 1.73205 1.73205
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0.500000 0.866025i 0.500000 0.866025i
\(421\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(422\) 0 0
\(423\) 0.866025 1.50000i 0.866025 1.50000i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.73205 1.73205
\(428\) −0.866025 0.500000i −0.866025 0.500000i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0.866025 0.500000i 0.866025 0.500000i
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(436\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) 0 0
\(441\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(442\) 0 0
\(443\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(446\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(447\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(448\) −0.866025 0.500000i −0.866025 0.500000i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 1.00000i 1.00000i
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(458\) 1.73205 1.73205
\(459\) 0 0
\(460\) 1.00000i 1.00000i
\(461\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(462\) 0 0
\(463\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(464\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(470\) 1.50000 0.866025i 1.50000 0.866025i
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 1.00000 1.00000
\(481\) 0 0
\(482\) 0.866025 1.50000i 0.866025 1.50000i
\(483\) 1.00000i 1.00000i
\(484\) −0.500000 0.866025i −0.500000 0.866025i
\(485\) 0 0
\(486\) 0.500000 0.866025i 0.500000 0.866025i
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(489\) 0 0
\(490\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(491\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(492\) 1.00000i 1.00000i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −0.866025 1.50000i −0.866025 1.50000i
\(499\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(500\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(501\) 1.50000 0.866025i 1.50000 0.866025i
\(502\) 0 0
\(503\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(504\) −1.00000 −1.00000
\(505\) 2.00000 2.00000
\(506\) 0 0
\(507\) 1.00000i 1.00000i
\(508\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(509\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 1.00000i
\(513\) 0 0
\(514\) 0 0
\(515\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 1.73205 1.73205
\(523\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(524\) 0 0
\(525\) 0.500000 0.866025i 0.500000 0.866025i
\(526\) −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
\(534\) 1.00000 1.00000
\(535\) −0.866025 0.500000i −0.866025 0.500000i
\(536\) −1.50000 0.866025i −1.50000 0.866025i
\(537\) 0 0
\(538\) 0.866025 0.500000i 0.866025 0.500000i
\(539\) 0 0
\(540\) 0.866025 0.500000i 0.866025 0.500000i
\(541\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(542\) 0 0
\(543\) 0.866025 1.50000i 0.866025 1.50000i
\(544\) 0 0
\(545\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(546\) 0 0
\(547\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(550\) 0 0
\(551\) 0 0
\(552\) 0.866025 0.500000i 0.866025 0.500000i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −0.866025 0.500000i −0.866025 0.500000i
\(561\) 0 0
\(562\) 0.866025 1.50000i 0.866025 1.50000i
\(563\) 0.866025 1.50000i 0.866025 1.50000i 1.00000i \(-0.5\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(564\) −1.50000 0.866025i −1.50000 0.866025i
\(565\) 0 0
\(566\) 1.00000 1.00000
\(567\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(568\) 0 0
\(569\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(570\) 0 0
\(571\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(575\) 1.00000i 1.00000i
\(576\) −0.500000 0.866025i −0.500000 0.866025i
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(579\) 0 0
\(580\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(581\) 1.73205i 1.73205i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −0.866025 1.50000i −0.866025 1.50000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(-0.5\pi\)
\(588\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.50000 0.866025i 1.50000 0.866025i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 1.00000 1.00000
\(601\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(602\) 0 0
\(603\) −1.73205 −1.73205
\(604\) 0 0
\(605\) −0.500000 0.866025i −0.500000 0.866025i
\(606\) −1.00000 1.73205i −1.00000 1.73205i
\(607\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) −1.50000 0.866025i −1.50000 0.866025i
\(610\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −0.500000 0.866025i −0.500000 0.866025i
\(615\) 1.00000i 1.00000i
\(616\) 0 0
\(617\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(618\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(619\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(620\) 0 0
\(621\) 0.500000 0.866025i 0.500000 0.866025i
\(622\) 0 0
\(623\) −0.866025 0.500000i −0.866025 0.500000i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) −1.00000 −1.00000
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000i 1.00000i
\(641\) 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(642\) 1.00000i 1.00000i
\(643\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(644\) −1.00000 −1.00000
\(645\) 0 0
\(646\) 0 0
\(647\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) −0.866025 0.500000i −0.866025 0.500000i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(654\) 1.00000 1.00000
\(655\) 0 0
\(656\) −1.00000 −1.00000
\(657\) 0 0
\(658\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(659\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(660\) 0 0
\(661\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(665\) 0 0
\(666\) 0 0
\(667\) 1.73205 1.73205
\(668\) −0.866025 1.50000i −0.866025 1.50000i
\(669\) 1.00000 1.00000
\(670\) −1.50000 0.866025i −1.50000 0.866025i
\(671\) 0 0
\(672\) 1.00000i 1.00000i
\(673\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(674\) 0 0
\(675\) 0.866025 0.500000i 0.866025 0.500000i
\(676\) 1.00000 1.00000
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.500000 0.866025i −0.500000 0.866025i
\(687\) −0.866025 1.50000i −0.866025 1.50000i
\(688\) 0 0
\(689\) 0 0
\(690\) 0.866025 0.500000i 0.866025 0.500000i
\(691\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −2.00000 −2.00000
\(695\) 0 0
\(696\) 1.73205i 1.73205i
\(697\) 0 0
\(698\) 0.866025 1.50000i 0.866025 1.50000i
\(699\) 0 0
\(700\) −0.866025 0.500000i −0.866025 0.500000i
\(701\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −1.50000 0.866025i −1.50000 0.866025i
\(706\) 0 0
\(707\) 2.00000i 2.00000i
\(708\) 0 0
\(709\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.00000i 1.00000i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −0.500000 0.866025i −0.500000 0.866025i
\(721\) −1.00000 1.73205i −1.00000 1.73205i
\(722\) 0.866025 0.500000i 0.866025 0.500000i
\(723\) −1.73205 −1.73205
\(724\) −1.50000 0.866025i −1.50000 0.866025i
\(725\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(726\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(727\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −1.00000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0.866025 1.50000i 0.866025 1.50000i
\(733\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(734\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(735\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(736\) −0.500000 0.866025i −0.500000 0.866025i
\(737\) 0 0
\(738\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 1.50000 0.866025i 1.50000 0.866025i
\(746\) 0 0
\(747\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(748\) 0 0
\(749\) 0.500000 0.866025i 0.500000 0.866025i
\(750\) 1.00000 1.00000
\(751\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(762\) 1.73205 1.73205
\(763\) −0.866025 0.500000i −0.866025 0.500000i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(769\) 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 \(0\)
0.500000 + 0.866025i \(0.333333\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −0.866025 1.50000i −0.866025 1.50000i
\(784\) 0.500000 0.866025i 0.500000 0.866025i
\(785\) 0 0
\(786\) 0 0
\(787\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(788\) 0 0
\(789\) 2.00000 2.00000
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000i 1.00000i
\(801\) −0.500000 0.866025i −0.500000 0.866025i
\(802\) 0 0
\(803\) 0 0
\(804\) 1.73205i 1.73205i
\(805\) −1.00000 −1.00000
\(806\) 0 0
\(807\) −0.866025 0.500000i −0.866025 0.500000i
\(808\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) −0.866025 0.500000i −0.866025 0.500000i
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) −1.00000 −1.00000
\(821\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0.866025 1.50000i 0.866025 1.50000i 1.00000i \(-0.5\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(824\) 1.00000 1.73205i 1.00000 1.73205i
\(825\) 0 0
\(826\) 0 0
\(827\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(828\) −0.866025 0.500000i −0.866025 0.500000i
\(829\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(830\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −0.866025 1.50000i −0.866025 1.50000i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(840\) 1.00000i 1.00000i
\(841\) 1.00000 1.73205i 1.00000 1.73205i
\(842\) −1.73205 1.00000i −1.73205 1.00000i
\(843\) −1.73205 −1.73205
\(844\) 0 0
\(845\) 1.00000 1.00000
\(846\) 1.73205i 1.73205i
\(847\) 0.866025 0.500000i 0.866025 0.500000i
\(848\) 0 0
\(849\) −0.500000 0.866025i −0.500000 0.866025i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(854\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(855\) 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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(860\) 0 0
\(861\) 1.00000 1.00000
\(862\) 0 0
\(863\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(864\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(865\) 0 0
\(866\) 0 0
\(867\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(868\) 0 0
\(869\) 0 0
\(870\) 1.73205i 1.73205i
\(871\) 0 0
\(872\) 1.00000i 1.00000i
\(873\) 0 0
\(874\) 0 0
\(875\) −0.866025 0.500000i −0.866025 0.500000i
\(876\) 0 0
\(877\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(882\) 1.00000i 1.00000i
\(883\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.500000 0.866025i 0.500000 0.866025i
\(887\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(888\) 0 0
\(889\) −1.50000 0.866025i −1.50000 0.866025i
\(890\) 1.00000i 1.00000i
\(891\) 0 0
\(892\) 1.00000i 1.00000i
\(893\) 0 0
\(894\) −1.50000 0.866025i −1.50000 0.866025i
\(895\) 0 0
\(896\) 1.00000 1.00000
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.500000 0.866025i −0.500000 0.866025i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.50000 0.866025i −1.50000 0.866025i
\(906\) 0 0
\(907\) −0.866025 1.50000i −0.866025 1.50000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(-0.5\pi\)
\(908\) 0 0
\(909\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(910\) 0 0
\(911\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0.866025 1.50000i 0.866025 1.50000i
\(916\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) −0.500000 0.866025i −0.500000 0.866025i
\(921\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(922\) −0.866025 0.500000i −0.866025 0.500000i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 2.00000i 2.00000i
\(928\) −1.73205 −1.73205
\(929\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0.866025 1.50000i 0.866025 1.50000i
\(939\) 0 0
\(940\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(941\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(942\) 0 0
\(943\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(944\) 0 0
\(945\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(946\) 0 0
\(947\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(961\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(962\) 0 0
\(963\) 0.866025 0.500000i 0.866025 0.500000i
\(964\) 1.73205i 1.73205i
\(965\) 0 0
\(966\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(967\) 0.866025 1.50000i 0.866025 1.50000i 1.00000i \(-0.5\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(968\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 1.00000i 1.00000i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −1.50000 0.866025i −1.50000 0.866025i
\(977\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.500000 0.866025i 0.500000 0.866025i
\(981\) −0.500000 0.866025i −0.500000 0.866025i
\(982\) 0 0
\(983\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(984\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(985\) 0 0
\(986\) 0 0
\(987\) 0.866025 1.50000i 0.866025 1.50000i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(997\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−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 1260.1.bp.b.419.1 yes 4
3.2 odd 2 3780.1.bp.a.1259.2 4
4.3 odd 2 inner 1260.1.bp.b.419.2 yes 4
5.4 even 2 inner 1260.1.bp.b.419.2 yes 4
7.6 odd 2 1260.1.bp.a.419.1 4
9.2 odd 6 1260.1.bp.a.839.1 yes 4
9.7 even 3 3780.1.bp.b.2519.2 4
12.11 even 2 3780.1.bp.a.1259.1 4
15.14 odd 2 3780.1.bp.a.1259.1 4
20.19 odd 2 CM 1260.1.bp.b.419.1 yes 4
21.20 even 2 3780.1.bp.b.1259.2 4
28.27 even 2 1260.1.bp.a.419.2 yes 4
35.34 odd 2 1260.1.bp.a.419.2 yes 4
36.7 odd 6 3780.1.bp.b.2519.1 4
36.11 even 6 1260.1.bp.a.839.2 yes 4
45.29 odd 6 1260.1.bp.a.839.2 yes 4
45.34 even 6 3780.1.bp.b.2519.1 4
60.59 even 2 3780.1.bp.a.1259.2 4
63.20 even 6 inner 1260.1.bp.b.839.1 yes 4
63.34 odd 6 3780.1.bp.a.2519.2 4
84.83 odd 2 3780.1.bp.b.1259.1 4
105.104 even 2 3780.1.bp.b.1259.1 4
140.139 even 2 1260.1.bp.a.419.1 4
180.79 odd 6 3780.1.bp.b.2519.2 4
180.119 even 6 1260.1.bp.a.839.1 yes 4
252.83 odd 6 inner 1260.1.bp.b.839.2 yes 4
252.223 even 6 3780.1.bp.a.2519.1 4
315.34 odd 6 3780.1.bp.a.2519.1 4
315.209 even 6 inner 1260.1.bp.b.839.2 yes 4
420.419 odd 2 3780.1.bp.b.1259.2 4
1260.839 odd 6 inner 1260.1.bp.b.839.1 yes 4
1260.979 even 6 3780.1.bp.a.2519.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1260.1.bp.a.419.1 4 7.6 odd 2
1260.1.bp.a.419.1 4 140.139 even 2
1260.1.bp.a.419.2 yes 4 28.27 even 2
1260.1.bp.a.419.2 yes 4 35.34 odd 2
1260.1.bp.a.839.1 yes 4 9.2 odd 6
1260.1.bp.a.839.1 yes 4 180.119 even 6
1260.1.bp.a.839.2 yes 4 36.11 even 6
1260.1.bp.a.839.2 yes 4 45.29 odd 6
1260.1.bp.b.419.1 yes 4 1.1 even 1 trivial
1260.1.bp.b.419.1 yes 4 20.19 odd 2 CM
1260.1.bp.b.419.2 yes 4 4.3 odd 2 inner
1260.1.bp.b.419.2 yes 4 5.4 even 2 inner
1260.1.bp.b.839.1 yes 4 63.20 even 6 inner
1260.1.bp.b.839.1 yes 4 1260.839 odd 6 inner
1260.1.bp.b.839.2 yes 4 252.83 odd 6 inner
1260.1.bp.b.839.2 yes 4 315.209 even 6 inner
3780.1.bp.a.1259.1 4 12.11 even 2
3780.1.bp.a.1259.1 4 15.14 odd 2
3780.1.bp.a.1259.2 4 3.2 odd 2
3780.1.bp.a.1259.2 4 60.59 even 2
3780.1.bp.a.2519.1 4 252.223 even 6
3780.1.bp.a.2519.1 4 315.34 odd 6
3780.1.bp.a.2519.2 4 63.34 odd 6
3780.1.bp.a.2519.2 4 1260.979 even 6
3780.1.bp.b.1259.1 4 84.83 odd 2
3780.1.bp.b.1259.1 4 105.104 even 2
3780.1.bp.b.1259.2 4 21.20 even 2
3780.1.bp.b.1259.2 4 420.419 odd 2
3780.1.bp.b.2519.1 4 36.7 odd 6
3780.1.bp.b.2519.1 4 45.34 even 6
3780.1.bp.b.2519.2 4 9.7 even 3
3780.1.bp.b.2519.2 4 180.79 odd 6