Properties

Label 1800.1.ba.b.1699.1
Level $1800$
Weight $1$
Character 1800.1699
Analytic conductor $0.898$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -8
Inner twists $8$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1800,1,Mod(499,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.499");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1800.ba (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: \(0.898317022739\)
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: no (minimal twist has level 72)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.648.1
Artin image: $S_3\times C_{12}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

Embedding invariants

Embedding label 1699.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1800.1699
Dual form 1800.1.ba.b.499.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{6} -1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.866025 + 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{6} -1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{11} +1.00000i q^{12} +(-0.500000 + 0.866025i) q^{16} -1.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +(-0.866025 + 0.500000i) q^{22} +(0.500000 - 0.866025i) q^{24} +1.00000i q^{27} +(0.866025 - 0.500000i) q^{32} +(0.866025 - 0.500000i) q^{33} +(-0.500000 + 0.866025i) q^{34} +(-0.500000 + 0.866025i) q^{36} +(-0.866025 - 0.500000i) q^{38} +(0.500000 + 0.866025i) q^{41} +(-0.866025 - 0.500000i) q^{43} +1.00000 q^{44} +(-0.866025 + 0.500000i) q^{48} +(0.500000 + 0.866025i) q^{49} +(0.500000 - 0.866025i) q^{51} +(0.500000 - 0.866025i) q^{54} +(0.866025 + 0.500000i) q^{57} +(-0.500000 - 0.866025i) q^{59} -1.00000 q^{64} -1.00000 q^{66} +(-0.866025 + 0.500000i) q^{67} +(0.866025 - 0.500000i) q^{68} +(0.866025 - 0.500000i) q^{72} +1.00000i q^{73} +(0.500000 + 0.866025i) q^{76} +(-0.500000 + 0.866025i) q^{81} -1.00000i q^{82} +(1.73205 + 1.00000i) q^{83} +(0.500000 + 0.866025i) q^{86} +(-0.866025 - 0.500000i) q^{88} -2.00000 q^{89} +1.00000 q^{96} +(0.866025 + 0.500000i) q^{97} -1.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 2 q^{6} + 2 q^{9} + 2 q^{11} - 2 q^{16} + 4 q^{19} + 2 q^{24} - 2 q^{34} - 2 q^{36} + 2 q^{41} + 4 q^{44} + 2 q^{49} + 2 q^{51} + 2 q^{54} - 2 q^{59} - 4 q^{64} - 4 q^{66} + 2 q^{76} - 2 q^{81} + 2 q^{86} - 8 q^{89} + 4 q^{96} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 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 0
\(6\) −0.500000 0.866025i −0.500000 0.866025i
\(7\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(8\) 1.00000i 1.00000i
\(9\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(12\) 1.00000i 1.00000i
\(13\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(18\) 1.00000i 1.00000i
\(19\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0.500000 0.866025i 0.500000 0.866025i
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 1.00000i
\(28\) 0 0
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 0.866025 0.500000i 0.866025 0.500000i
\(33\) 0.866025 0.500000i 0.866025 0.500000i
\(34\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(35\) 0 0
\(36\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) −0.866025 0.500000i −0.866025 0.500000i
\(39\) 0 0
\(40\) 0 0
\(41\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(44\) 1.00000 1.00000
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(49\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(50\) 0 0
\(51\) 0.500000 0.866025i 0.500000 0.866025i
\(52\) 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 0
\(57\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(58\) 0 0
\(59\) −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
\(66\) −1.00000 −1.00000
\(67\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(68\) 0.866025 0.500000i 0.866025 0.500000i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0.866025 0.500000i 0.866025 0.500000i
\(73\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(82\) 1.00000i 1.00000i
\(83\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(87\) 0 0
\(88\) −0.866025 0.500000i −0.866025 0.500000i
\(89\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 1.00000
\(97\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(98\) 1.00000i 1.00000i
\(99\) 1.00000 1.00000
\(100\) 0 0
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 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
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(114\) −0.500000 0.866025i −0.500000 0.866025i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 1.00000i 1.00000i
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 1.00000i 1.00000i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(129\) −0.500000 0.866025i −0.500000 0.866025i
\(130\) 0 0
\(131\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(132\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(133\) 0 0
\(134\) 1.00000 1.00000
\(135\) 0 0
\(136\) −1.00000 −1.00000
\(137\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −1.00000
\(145\) 0 0
\(146\) 0.500000 0.866025i 0.500000 0.866025i
\(147\) 1.00000i 1.00000i
\(148\) 0 0
\(149\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(150\) 0 0
\(151\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(152\) 1.00000i 1.00000i
\(153\) 0.866025 0.500000i 0.866025 0.500000i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0.866025 0.500000i 0.866025 0.500000i
\(163\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(164\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(165\) 0 0
\(166\) −1.00000 1.73205i −1.00000 1.73205i
\(167\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(168\) 0 0
\(169\) 0.500000 0.866025i 0.500000 0.866025i
\(170\) 0 0
\(171\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(172\) 1.00000i 1.00000i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(177\) 1.00000i 1.00000i
\(178\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(179\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −0.866025 0.500000i −0.866025 0.500000i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) −0.866025 0.500000i −0.866025 0.500000i
\(193\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(194\) −0.500000 0.866025i −0.500000 0.866025i
\(195\) 0 0
\(196\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) −0.866025 0.500000i −0.866025 0.500000i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) −1.00000 −1.00000
\(202\) 0 0
\(203\) 0 0
\(204\) 1.00000 1.00000
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.500000 0.866025i 0.500000 0.866025i
\(210\) 0 0
\(211\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−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 0
\(219\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.00000 2.00000
\(227\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(228\) 1.00000i 1.00000i
\(229\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0.500000 0.866025i 0.500000 0.866025i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(240\) 0 0
\(241\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(242\) 0 0
\(243\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0.500000 0.866025i 0.500000 0.866025i
\(247\) 0 0
\(248\) 0 0
\(249\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(250\) 0 0
\(251\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−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
\(257\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(258\) 1.00000i 1.00000i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 2.00000i 2.00000i
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) −0.500000 0.866025i −0.500000 0.866025i
\(265\) 0 0
\(266\) 0 0
\(267\) −1.73205 1.00000i −1.73205 1.00000i
\(268\) −0.866025 0.500000i −0.866025 0.500000i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(273\) 0 0
\(274\) −0.500000 0.866025i −0.500000 0.866025i
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(278\) 1.00000i 1.00000i
\(279\) 0 0
\(280\) 0 0
\(281\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 0 0
\(283\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(289\) 0 0
\(290\) 0 0
\(291\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(292\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(293\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(294\) 0.500000 0.866025i 0.500000 0.866025i
\(295\) 0 0
\(296\) 0 0
\(297\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(305\) 0 0
\(306\) −1.00000 −1.00000
\(307\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0.500000 0.866025i 0.500000 0.866025i
\(322\) 0 0
\(323\) 1.00000i 1.00000i
\(324\) −1.00000 −1.00000
\(325\) 0 0
\(326\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(327\) 0 0
\(328\) 0.866025 0.500000i 0.866025 0.500000i
\(329\) 0 0
\(330\) 0 0
\(331\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 2.00000i 2.00000i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(338\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(339\) −2.00000 −2.00000
\(340\) 0 0
\(341\) 0 0
\(342\) 1.00000i 1.00000i
\(343\) 0 0
\(344\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(345\) 0 0
\(346\) 0 0
\(347\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000i 1.00000i
\(353\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(354\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(355\) 0 0
\(356\) −1.00000 1.73205i −1.00000 1.73205i
\(357\) 0 0
\(358\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(368\) 0 0
\(369\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(374\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 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 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(385\) 0 0
\(386\) −1.00000 −1.00000
\(387\) 1.00000i 1.00000i
\(388\) 1.00000i 1.00000i
\(389\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.866025 0.500000i 0.866025 0.500000i
\(393\) 2.00000i 2.00000i
\(394\) 0 0
\(395\) 0 0
\(396\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −0.866025 0.500000i −0.866025 0.500000i
\(409\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.00000i 1.00000i
\(418\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(419\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(420\) 0 0
\(421\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(422\) 2.00000i 2.00000i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0.866025 0.500000i 0.866025 0.500000i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −0.866025 0.500000i −0.866025 0.500000i
\(433\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0.866025 0.500000i 0.866025 0.500000i
\(439\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(442\) 0 0
\(443\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(450\) 0 0
\(451\) 1.00000 1.00000
\(452\) −1.73205 1.00000i −1.73205 1.00000i
\(453\) 0 0
\(454\) −0.500000 0.866025i −0.500000 0.866025i
\(455\) 0 0
\(456\) 0.500000 0.866025i 0.500000 0.866025i
\(457\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 1.00000 1.00000
\(460\) 0 0
\(461\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(462\) 0 0
\(463\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0.500000 0.866025i 0.500000 0.866025i
\(467\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(473\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 1.00000 1.00000
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 1.00000 1.73205i 1.00000 1.73205i
\(490\) 0 0
\(491\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 2.00000i 2.00000i
\(499\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.866025 0.500000i 0.866025 0.500000i
\(508\) 0 0
\(509\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 1.00000i
\(513\) 1.00000i 1.00000i
\(514\) 1.00000 1.00000
\(515\) 0 0
\(516\) 0.500000 0.866025i 0.500000 0.866025i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(524\) 1.00000 1.73205i 1.00000 1.73205i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 1.00000i 1.00000i
\(529\) 0.500000 0.866025i 0.500000 0.866025i
\(530\) 0 0
\(531\) 0.500000 0.866025i 0.500000 0.866025i
\(532\) 0 0
\(533\) 0 0
\(534\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(535\) 0 0
\(536\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(537\) −1.73205 1.00000i −1.73205 1.00000i
\(538\) 0 0
\(539\) 1.00000 1.00000
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.500000 0.866025i −0.500000 0.866025i
\(545\) 0 0
\(546\) 0 0
\(547\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(548\) 1.00000i 1.00000i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0.500000 0.866025i 0.500000 0.866025i
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −0.500000 0.866025i −0.500000 0.866025i
\(562\) 1.73205 1.00000i 1.73205 1.00000i
\(563\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2.00000 2.00000
\(567\) 0 0
\(568\) 0 0
\(569\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.500000 0.866025i −0.500000 0.866025i
\(577\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(578\) 0 0
\(579\) 1.00000 1.00000
\(580\) 0 0
\(581\) 0 0
\(582\) 1.00000i 1.00000i
\(583\) 0 0
\(584\) 1.00000 1.00000
\(585\) 0 0
\(586\) 0 0
\(587\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
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
\(593\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(594\) −0.500000 0.866025i −0.500000 0.866025i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(602\) 0 0
\(603\) −0.866025 0.500000i −0.866025 0.500000i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(608\) 0.866025 0.500000i 0.866025 0.500000i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(615\) 0 0
\(616\) 0 0
\(617\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(627\) 0.866025 0.500000i 0.866025 0.500000i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 2.00000i 2.00000i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(642\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(643\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(649\) −1.00000 −1.00000
\(650\) 0 0
\(651\) 0 0
\(652\) 1.73205 1.00000i 1.73205 1.00000i
\(653\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.00000 −1.00000
\(657\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(658\) 0 0
\(659\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(660\) 0 0
\(661\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(662\) 1.73205 1.00000i 1.73205 1.00000i
\(663\) 0 0
\(664\) 1.00000 1.73205i 1.00000 1.73205i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(674\) 1.00000 1.00000
\(675\) 0 0
\(676\) 1.00000 1.00000
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(679\) 0 0
\(680\) 0 0
\(681\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(682\) 0 0
\(683\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(684\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0.866025 0.500000i 0.866025 0.500000i
\(689\) 0 0
\(690\) 0 0
\(691\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 1.00000 1.00000
\(695\) 0 0
\(696\) 0 0
\(697\) 0.866025 0.500000i 0.866025 0.500000i
\(698\) 0 0
\(699\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(705\) 0 0
\(706\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(707\) 0 0
\(708\) 0.866025 0.500000i 0.866025 0.500000i
\(709\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.00000i 2.00000i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −1.00000 1.73205i −1.00000 1.73205i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0.866025 0.500000i 0.866025 0.500000i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(728\) 0 0
\(729\) −1.00000 −1.00000
\(730\) 0 0
\(731\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(732\) 0 0
\(733\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.00000i 1.00000i
\(738\) 0.866025 0.500000i 0.866025 0.500000i
\(739\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2.00000i 2.00000i
\(748\) 1.00000i 1.00000i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) 0 0
\(753\) −0.866025 0.500000i −0.866025 0.500000i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) −0.866025 0.500000i −0.866025 0.500000i
\(759\) 0 0
\(760\) 0 0
\(761\) −1.00000 1.73205i −1.00000 1.73205i −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 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.00000i 1.00000i
\(769\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(770\) 0 0
\(771\) −1.00000 −1.00000
\(772\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(775\) 0 0
\(776\) 0.500000 0.866025i 0.500000 0.866025i
\(777\) 0 0
\(778\) 0 0
\(779\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −1.00000 −1.00000
\(785\) 0 0
\(786\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(787\) 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
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 1.00000i 1.00000i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −1.00000 1.73205i −1.00000 1.73205i
\(802\) 1.00000i 1.00000i
\(803\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(804\) −0.500000 0.866025i −0.500000 0.866025i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(810\) 0 0
\(811\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(817\) −0.866025 0.500000i −0.866025 0.500000i
\(818\) 1.00000i 1.00000i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 1.00000i 1.00000i
\(823\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(824\) 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 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.866025 0.500000i 0.866025 0.500000i
\(834\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(835\) 0 0
\(836\) 1.00000 1.00000
\(837\) 0 0
\(838\) 2.00000i 2.00000i
\(839\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(840\) 0 0
\(841\) −0.500000 0.866025i −0.500000 0.866025i
\(842\) 0 0
\(843\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(844\) 1.00000 1.73205i 1.00000 1.73205i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −2.00000 −2.00000
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.00000 −1.00000
\(857\) −1.73205 1.00000i −1.73205 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 0.500000i \(-0.833333\pi\)
\(858\) 0 0
\(859\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(865\) 0 0
\(866\) 0.500000 0.866025i 0.500000 0.866025i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.00000i 1.00000i
\(874\) 0 0
\(875\) 0 0
\(876\) −1.00000 −1.00000
\(877\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(882\) 0.866025 0.500000i 0.866025 0.500000i
\(883\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(887\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −0.866025 0.500000i −0.866025 0.500000i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −0.866025 0.500000i −0.866025 0.500000i
\(903\) 0 0
\(904\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(905\) 0 0
\(906\) 0 0
\(907\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(908\) 1.00000i 1.00000i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(912\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(913\) 1.73205 1.00000i 1.73205 1.00000i
\(914\) −0.500000 0.866025i −0.500000 0.866025i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −0.866025 0.500000i −0.866025 0.500000i
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0.500000 0.866025i 0.500000 0.866025i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 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
\(931\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(932\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(933\) 0 0
\(934\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) −0.500000 0.866025i −0.500000 0.866025i
\(940\) 0 0
\(941\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.00000 1.00000
\(945\) 0 0
\(946\) 1.00000 1.00000
\(947\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(962\) 0 0
\(963\) 0.866025 0.500000i 0.866025 0.500000i
\(964\) 1.00000 1.00000
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(968\) 0 0
\(969\) 0.500000 0.866025i 0.500000 0.866025i
\(970\) 0 0
\(971\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(972\) −0.866025 0.500000i −0.866025 0.500000i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(978\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(979\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(980\) 0 0
\(981\) 0 0
\(982\) 1.00000i 1.00000i
\(983\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(984\) 1.00000 1.00000
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(994\) 0 0
\(995\) 0 0
\(996\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(997\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(998\) 1.00000i 1.00000i
\(999\) 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 1800.1.ba.b.1699.1 4
5.2 odd 4 1800.1.bk.d.1051.1 2
5.3 odd 4 72.1.p.a.43.1 2
5.4 even 2 inner 1800.1.ba.b.1699.2 4
8.3 odd 2 CM 1800.1.ba.b.1699.1 4
9.4 even 3 inner 1800.1.ba.b.499.2 4
15.8 even 4 216.1.p.a.19.1 2
20.3 even 4 288.1.t.a.79.1 2
35.3 even 12 3528.1.ba.a.1843.1 2
35.13 even 4 3528.1.cg.a.2059.1 2
35.18 odd 12 3528.1.ba.b.1843.1 2
35.23 odd 12 3528.1.ce.a.2419.1 2
35.33 even 12 3528.1.ce.b.2419.1 2
40.3 even 4 72.1.p.a.43.1 2
40.13 odd 4 288.1.t.a.79.1 2
40.19 odd 2 inner 1800.1.ba.b.1699.2 4
40.27 even 4 1800.1.bk.d.1051.1 2
45.4 even 6 inner 1800.1.ba.b.499.1 4
45.13 odd 12 72.1.p.a.67.1 yes 2
45.22 odd 12 1800.1.bk.d.1651.1 2
45.23 even 12 216.1.p.a.91.1 2
45.38 even 12 648.1.b.a.163.1 1
45.43 odd 12 648.1.b.b.163.1 1
60.23 odd 4 864.1.t.a.559.1 2
72.67 odd 6 inner 1800.1.ba.b.499.2 4
80.3 even 4 2304.1.o.c.511.2 4
80.13 odd 4 2304.1.o.c.511.1 4
80.43 even 4 2304.1.o.c.511.1 4
80.53 odd 4 2304.1.o.c.511.2 4
120.53 even 4 864.1.t.a.559.1 2
120.83 odd 4 216.1.p.a.19.1 2
180.23 odd 12 864.1.t.a.847.1 2
180.43 even 12 2592.1.b.b.1135.1 1
180.83 odd 12 2592.1.b.a.1135.1 1
180.103 even 12 288.1.t.a.175.1 2
280.3 odd 12 3528.1.ba.a.1843.1 2
280.83 odd 4 3528.1.cg.a.2059.1 2
280.123 even 12 3528.1.ba.b.1843.1 2
280.163 even 12 3528.1.ce.a.2419.1 2
280.243 odd 12 3528.1.ce.b.2419.1 2
315.13 even 12 3528.1.cg.a.3235.1 2
315.58 odd 12 3528.1.ba.b.67.1 2
315.103 even 12 3528.1.ba.a.67.1 2
315.193 odd 12 3528.1.ce.a.3019.1 2
315.283 even 12 3528.1.ce.b.3019.1 2
360.13 odd 12 288.1.t.a.175.1 2
360.43 even 12 648.1.b.b.163.1 1
360.67 even 12 1800.1.bk.d.1651.1 2
360.83 odd 12 648.1.b.a.163.1 1
360.133 odd 12 2592.1.b.b.1135.1 1
360.139 odd 6 inner 1800.1.ba.b.499.1 4
360.173 even 12 2592.1.b.a.1135.1 1
360.203 odd 12 216.1.p.a.91.1 2
360.283 even 12 72.1.p.a.67.1 yes 2
360.293 even 12 864.1.t.a.847.1 2
720.13 odd 12 2304.1.o.c.2047.2 4
720.283 even 12 2304.1.o.c.2047.2 4
720.373 odd 12 2304.1.o.c.2047.1 4
720.643 even 12 2304.1.o.c.2047.1 4
2520.283 odd 12 3528.1.ce.b.3019.1 2
2520.643 odd 12 3528.1.cg.a.3235.1 2
2520.1003 even 12 3528.1.ba.b.67.1 2
2520.1363 odd 12 3528.1.ba.a.67.1 2
2520.2083 even 12 3528.1.ce.a.3019.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.1.p.a.43.1 2 5.3 odd 4
72.1.p.a.43.1 2 40.3 even 4
72.1.p.a.67.1 yes 2 45.13 odd 12
72.1.p.a.67.1 yes 2 360.283 even 12
216.1.p.a.19.1 2 15.8 even 4
216.1.p.a.19.1 2 120.83 odd 4
216.1.p.a.91.1 2 45.23 even 12
216.1.p.a.91.1 2 360.203 odd 12
288.1.t.a.79.1 2 20.3 even 4
288.1.t.a.79.1 2 40.13 odd 4
288.1.t.a.175.1 2 180.103 even 12
288.1.t.a.175.1 2 360.13 odd 12
648.1.b.a.163.1 1 45.38 even 12
648.1.b.a.163.1 1 360.83 odd 12
648.1.b.b.163.1 1 45.43 odd 12
648.1.b.b.163.1 1 360.43 even 12
864.1.t.a.559.1 2 60.23 odd 4
864.1.t.a.559.1 2 120.53 even 4
864.1.t.a.847.1 2 180.23 odd 12
864.1.t.a.847.1 2 360.293 even 12
1800.1.ba.b.499.1 4 45.4 even 6 inner
1800.1.ba.b.499.1 4 360.139 odd 6 inner
1800.1.ba.b.499.2 4 9.4 even 3 inner
1800.1.ba.b.499.2 4 72.67 odd 6 inner
1800.1.ba.b.1699.1 4 1.1 even 1 trivial
1800.1.ba.b.1699.1 4 8.3 odd 2 CM
1800.1.ba.b.1699.2 4 5.4 even 2 inner
1800.1.ba.b.1699.2 4 40.19 odd 2 inner
1800.1.bk.d.1051.1 2 5.2 odd 4
1800.1.bk.d.1051.1 2 40.27 even 4
1800.1.bk.d.1651.1 2 45.22 odd 12
1800.1.bk.d.1651.1 2 360.67 even 12
2304.1.o.c.511.1 4 80.13 odd 4
2304.1.o.c.511.1 4 80.43 even 4
2304.1.o.c.511.2 4 80.3 even 4
2304.1.o.c.511.2 4 80.53 odd 4
2304.1.o.c.2047.1 4 720.373 odd 12
2304.1.o.c.2047.1 4 720.643 even 12
2304.1.o.c.2047.2 4 720.13 odd 12
2304.1.o.c.2047.2 4 720.283 even 12
2592.1.b.a.1135.1 1 180.83 odd 12
2592.1.b.a.1135.1 1 360.173 even 12
2592.1.b.b.1135.1 1 180.43 even 12
2592.1.b.b.1135.1 1 360.133 odd 12
3528.1.ba.a.67.1 2 315.103 even 12
3528.1.ba.a.67.1 2 2520.1363 odd 12
3528.1.ba.a.1843.1 2 35.3 even 12
3528.1.ba.a.1843.1 2 280.3 odd 12
3528.1.ba.b.67.1 2 315.58 odd 12
3528.1.ba.b.67.1 2 2520.1003 even 12
3528.1.ba.b.1843.1 2 35.18 odd 12
3528.1.ba.b.1843.1 2 280.123 even 12
3528.1.ce.a.2419.1 2 35.23 odd 12
3528.1.ce.a.2419.1 2 280.163 even 12
3528.1.ce.a.3019.1 2 315.193 odd 12
3528.1.ce.a.3019.1 2 2520.2083 even 12
3528.1.ce.b.2419.1 2 35.33 even 12
3528.1.ce.b.2419.1 2 280.243 odd 12
3528.1.ce.b.3019.1 2 315.283 even 12
3528.1.ce.b.3019.1 2 2520.283 odd 12
3528.1.cg.a.2059.1 2 35.13 even 4
3528.1.cg.a.2059.1 2 280.83 odd 4
3528.1.cg.a.3235.1 2 315.13 even 12
3528.1.cg.a.3235.1 2 2520.643 odd 12