Properties

Label 1296.1.x.a.269.2
Level $1296$
Weight $1$
Character 1296.269
Analytic conductor $0.647$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1296,1,Mod(53,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.53");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1296.x (of order \(12\), degree \(4\), 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: \(0.646788256372\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 432)
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.55296.2

Embedding invariants

Embedding label 269.2
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1296.269
Dual form 1296.1.x.a.53.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.965926 - 0.258819i) q^{2} +(0.866025 - 0.500000i) q^{4} +(-0.258819 + 0.965926i) q^{5} +(0.866025 + 0.500000i) q^{7} +(0.707107 - 0.707107i) q^{8} +O(q^{10})\) \(q+(0.965926 - 0.258819i) q^{2} +(0.866025 - 0.500000i) q^{4} +(-0.258819 + 0.965926i) q^{5} +(0.866025 + 0.500000i) q^{7} +(0.707107 - 0.707107i) q^{8} +1.00000i q^{10} +(-0.965926 + 0.258819i) q^{11} +(-0.366025 + 1.36603i) q^{13} +(0.965926 + 0.258819i) q^{14} +(0.500000 - 0.866025i) q^{16} -1.41421i q^{17} +(0.258819 + 0.965926i) q^{20} +(-0.866025 + 0.500000i) q^{22} +1.41421i q^{26} +1.00000 q^{28} +(-0.500000 - 0.866025i) q^{31} +(0.258819 - 0.965926i) q^{32} +(-0.366025 - 1.36603i) q^{34} +(-0.707107 + 0.707107i) q^{35} +(0.500000 + 0.866025i) q^{40} +(-0.366025 - 1.36603i) q^{43} +(-0.707107 + 0.707107i) q^{44} +(-1.22474 - 0.707107i) q^{47} +(0.366025 + 1.36603i) q^{52} +(0.707107 + 0.707107i) q^{53} -1.00000i q^{55} +(0.965926 - 0.258819i) q^{56} +(-0.707107 - 0.707107i) q^{62} -1.00000i q^{64} +(-1.22474 - 0.707107i) q^{65} +(0.366025 - 1.36603i) q^{67} +(-0.707107 - 1.22474i) q^{68} +(-0.500000 + 0.866025i) q^{70} +1.41421 q^{71} +1.00000i q^{73} +(-0.965926 - 0.258819i) q^{77} +(0.707107 + 0.707107i) q^{80} +(0.258819 + 0.965926i) q^{83} +(1.36603 + 0.366025i) q^{85} +(-0.707107 - 1.22474i) q^{86} +(-0.500000 + 0.866025i) q^{88} -1.41421 q^{89} +(-1.00000 + 1.00000i) q^{91} +(-1.36603 - 0.366025i) q^{94} +(-0.500000 + 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{13} + 4 q^{16} + 8 q^{28} - 4 q^{31} + 4 q^{34} + 4 q^{40} + 4 q^{43} - 4 q^{52} - 4 q^{67} - 4 q^{70} + 4 q^{85} - 4 q^{88} - 8 q^{91} - 4 q^{94} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.965926 0.258819i 0.965926 0.258819i
\(3\) 0 0
\(4\) 0.866025 0.500000i 0.866025 0.500000i
\(5\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(6\) 0 0
\(7\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(8\) 0.707107 0.707107i 0.707107 0.707107i
\(9\) 0 0
\(10\) 1.00000i 1.00000i
\(11\) −0.965926 + 0.258819i −0.965926 + 0.258819i −0.707107 0.707107i \(-0.750000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(12\) 0 0
\(13\) −0.366025 + 1.36603i −0.366025 + 1.36603i 0.500000 + 0.866025i \(0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(14\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(15\) 0 0
\(16\) 0.500000 0.866025i 0.500000 0.866025i
\(17\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(18\) 0 0
\(19\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(20\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(21\) 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 0
\(25\) 0 0
\(26\) 1.41421i 1.41421i
\(27\) 0 0
\(28\) 1.00000 1.00000
\(29\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(30\) 0 0
\(31\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(32\) 0.258819 0.965926i 0.258819 0.965926i
\(33\) 0 0
\(34\) −0.366025 1.36603i −0.366025 1.36603i
\(35\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(36\) 0 0
\(37\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(41\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 0 0
\(43\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(44\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(45\) 0 0
\(46\) 0 0
\(47\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(53\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(54\) 0 0
\(55\) 1.00000i 1.00000i
\(56\) 0.965926 0.258819i 0.965926 0.258819i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(60\) 0 0
\(61\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(62\) −0.707107 0.707107i −0.707107 0.707107i
\(63\) 0 0
\(64\) 1.00000i 1.00000i
\(65\) −1.22474 0.707107i −1.22474 0.707107i
\(66\) 0 0
\(67\) 0.366025 1.36603i 0.366025 1.36603i −0.500000 0.866025i \(-0.666667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(68\) −0.707107 1.22474i −0.707107 1.22474i
\(69\) 0 0
\(70\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(71\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(72\) 0 0
\(73\) 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 0
\(77\) −0.965926 0.258819i −0.965926 0.258819i
\(78\) 0 0
\(79\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(80\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(81\) 0 0
\(82\) 0 0
\(83\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(86\) −0.707107 1.22474i −0.707107 1.22474i
\(87\) 0 0
\(88\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(89\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) 0 0
\(91\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(92\) 0 0
\(93\) 0 0
\(94\) −1.36603 0.366025i −1.36603 0.366025i
\(95\) 0 0
\(96\) 0 0
\(97\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.965926 0.258819i 0.965926 0.258819i 0.258819 0.965926i \(-0.416667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(102\) 0 0
\(103\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(104\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(105\) 0 0
\(106\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(107\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(108\) 0 0
\(109\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(110\) −0.258819 0.965926i −0.258819 0.965926i
\(111\) 0 0
\(112\) 0.866025 0.500000i 0.866025 0.500000i
\(113\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.707107 1.22474i 0.707107 1.22474i
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) −0.866025 0.500000i −0.866025 0.500000i
\(125\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(126\) 0 0
\(127\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) −0.258819 0.965926i −0.258819 0.965926i
\(129\) 0 0
\(130\) −1.36603 0.366025i −1.36603 0.366025i
\(131\) −0.965926 0.258819i −0.965926 0.258819i −0.258819 0.965926i \(-0.583333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.41421i 1.41421i
\(135\) 0 0
\(136\) −1.00000 1.00000i −1.00000 1.00000i
\(137\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(138\) 0 0
\(139\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(140\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(141\) 0 0
\(142\) 1.36603 0.366025i 1.36603 0.366025i
\(143\) 1.41421i 1.41421i
\(144\) 0 0
\(145\) 0 0
\(146\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(147\) 0 0
\(148\) 0 0
\(149\) 0.258819 0.965926i 0.258819 0.965926i −0.707107 0.707107i \(-0.750000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(150\) 0 0
\(151\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −1.00000 −1.00000
\(155\) 0.965926 0.258819i 0.965926 0.258819i
\(156\) 0 0
\(157\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(161\) 0 0
\(162\) 0 0
\(163\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(167\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(168\) 0 0
\(169\) −0.866025 0.500000i −0.866025 0.500000i
\(170\) 1.41421 1.41421
\(171\) 0 0
\(172\) −1.00000 1.00000i −1.00000 1.00000i
\(173\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(177\) 0 0
\(178\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(179\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(180\) 0 0
\(181\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(182\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(188\) −1.41421 −1.41421
\(189\) 0 0
\(190\) 0 0
\(191\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(192\) 0 0
\(193\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(195\) 0 0
\(196\) 0 0
\(197\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(198\) 0 0
\(199\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0.866025 0.500000i 0.866025 0.500000i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(212\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(213\) 0 0
\(214\) −0.866025 0.500000i −0.866025 0.500000i
\(215\) 1.41421 1.41421
\(216\) 0 0
\(217\) 1.00000i 1.00000i
\(218\) 0 0
\(219\) 0 0
\(220\) −0.500000 0.866025i −0.500000 0.866025i
\(221\) 1.93185 + 0.517638i 1.93185 + 0.517638i
\(222\) 0 0
\(223\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(224\) 0.707107 0.707107i 0.707107 0.707107i
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(228\) 0 0
\(229\) −1.36603 0.366025i −1.36603 0.366025i −0.500000 0.866025i \(-0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(234\) 0 0
\(235\) 1.00000 1.00000i 1.00000 1.00000i
\(236\) 0 0
\(237\) 0 0
\(238\) 0.366025 1.36603i 0.366025 1.36603i
\(239\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(240\) 0 0
\(241\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) −0.965926 0.258819i −0.965926 0.258819i
\(249\) 0 0
\(250\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(251\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.500000 0.866025i
\(257\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.41421 −1.41421
\(261\) 0 0
\(262\) −1.00000 −1.00000
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 0 0
\(265\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(266\) 0 0
\(267\) 0 0
\(268\) −0.366025 1.36603i −0.366025 1.36603i
\(269\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(270\) 0 0
\(271\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(272\) −1.22474 0.707107i −1.22474 0.707107i
\(273\) 0 0
\(274\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 1.00000i 1.00000i
\(281\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(282\) 0 0
\(283\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(284\) 1.22474 0.707107i 1.22474 0.707107i
\(285\) 0 0
\(286\) −0.366025 1.36603i −0.366025 1.36603i
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(293\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 1.00000i 1.00000i
\(299\) 0 0
\(300\) 0 0
\(301\) 0.366025 1.36603i 0.366025 1.36603i
\(302\) −0.965926 0.258819i −0.965926 0.258819i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(308\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(309\) 0 0
\(310\) 0.866025 0.500000i 0.866025 0.500000i
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.258819 0.965926i −0.258819 0.965926i −0.965926 0.258819i \(-0.916667\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −1.22474 0.707107i −1.22474 0.707107i
\(327\) 0 0
\(328\) 0 0
\(329\) −0.707107 1.22474i −0.707107 1.22474i
\(330\) 0 0
\(331\) 0.366025 + 1.36603i 0.366025 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(333\) 0 0
\(334\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(335\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(336\) 0 0
\(337\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(338\) −0.965926 0.258819i −0.965926 0.258819i
\(339\) 0 0
\(340\) 1.36603 0.366025i 1.36603 0.366025i
\(341\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(342\) 0 0
\(343\) 1.00000i 1.00000i
\(344\) −1.22474 0.707107i −1.22474 0.707107i
\(345\) 0 0
\(346\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(347\) 0.258819 0.965926i 0.258819 0.965926i −0.707107 0.707107i \(-0.750000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(348\) 0 0
\(349\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000i 1.00000i
\(353\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(356\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(357\) 0 0
\(358\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(359\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(360\) 0 0
\(361\) 1.00000i 1.00000i
\(362\) 0 0
\(363\) 0 0
\(364\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(365\) −0.965926 0.258819i −0.965926 0.258819i
\(366\) 0 0
\(367\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(372\) 0 0
\(373\) −1.36603 0.366025i −1.36603 0.366025i −0.500000 0.866025i \(-0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(374\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(375\) 0 0
\(376\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(383\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(384\) 0 0
\(385\) 0.500000 0.866025i 0.500000 0.866025i
\(386\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(387\) 0 0
\(388\) 1.00000i 1.00000i
\(389\) 0.965926 0.258819i 0.965926 0.258819i 0.258819 0.965926i \(-0.416667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(395\) 0 0
\(396\) 0 0
\(397\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(398\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) 1.36603 0.366025i 1.36603 0.366025i
\(404\) 0.707107 0.707107i 0.707107 0.707107i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1.00000 −1.00000
\(416\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(420\) 0 0
\(421\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 1.00000 1.00000
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −0.965926 0.258819i −0.965926 0.258819i
\(429\) 0 0
\(430\) 1.36603 0.366025i 1.36603 0.366025i
\(431\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(432\) 0 0
\(433\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(434\) −0.258819 0.965926i −0.258819 0.965926i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) −0.707107 0.707107i −0.707107 0.707107i
\(441\) 0 0
\(442\) 2.00000 2.00000
\(443\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(444\) 0 0
\(445\) 0.366025 1.36603i 0.366025 1.36603i
\(446\) 0 0
\(447\) 0 0
\(448\) 0.500000 0.866025i 0.500000 0.866025i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.707107 1.22474i −0.707107 1.22474i
\(456\) 0 0
\(457\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(458\) −1.41421 −1.41421
\(459\) 0 0
\(460\) 0 0
\(461\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) 0 0
\(463\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 1.36603 0.366025i 1.36603 0.366025i
\(467\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(468\) 0 0
\(469\) 1.00000 1.00000i 1.00000 1.00000i
\(470\) 0.707107 1.22474i 0.707107 1.22474i
\(471\) 0 0
\(472\) 0 0
\(473\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(474\) 0 0
\(475\) 0 0
\(476\) 1.41421i 1.41421i
\(477\) 0 0
\(478\) 0 0
\(479\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.707107 0.707107i −0.707107 0.707107i
\(486\) 0 0
\(487\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.00000 −1.00000
\(497\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(498\) 0 0
\(499\) −0.366025 + 1.36603i −0.366025 + 1.36603i 0.500000 + 0.866025i \(0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(500\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 1.00000i 1.00000i
\(506\) 0 0
\(507\) 0 0
\(508\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(509\) −0.965926 0.258819i −0.965926 0.258819i −0.258819 0.965926i \(-0.583333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(510\) 0 0
\(511\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(512\) −0.707107 0.707107i −0.707107 0.707107i
\(513\) 0 0
\(514\) 1.00000 1.00000i 1.00000 1.00000i
\(515\) 0 0
\(516\) 0 0
\(517\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(518\) 0 0
\(519\) 0 0
\(520\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(521\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(522\) 0 0
\(523\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(524\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(525\) 0 0
\(526\) 0 0
\(527\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(528\) 0 0
\(529\) 0.500000 0.866025i 0.500000 0.866025i
\(530\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0.866025 0.500000i 0.866025 0.500000i
\(536\) −0.707107 1.22474i −0.707107 1.22474i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(542\) 0.965926 0.258819i 0.965926 0.258819i
\(543\) 0 0
\(544\) −1.36603 0.366025i −1.36603 0.366025i
\(545\) 0 0
\(546\) 0 0
\(547\) −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(548\) 1.41421i 1.41421i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(558\) 0 0
\(559\) 2.00000 2.00000
\(560\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(561\) 0 0
\(562\) 0.366025 1.36603i 0.366025 1.36603i
\(563\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 1.00000 1.00000i 1.00000 1.00000i
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) −1.36603 0.366025i −1.36603 0.366025i −0.500000 0.866025i \(-0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(572\) −0.707107 1.22474i −0.707107 1.22474i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(579\) 0 0
\(580\) 0 0
\(581\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(582\) 0 0
\(583\) −0.866025 0.500000i −0.866025 0.500000i
\(584\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(585\) 0 0
\(586\) 0 0
\(587\) 0.965926 0.258819i 0.965926 0.258819i 0.258819 0.965926i \(-0.416667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(594\) 0 0
\(595\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(596\) −0.258819 0.965926i −0.258819 0.965926i
\(597\) 0 0
\(598\) 0 0
\(599\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(600\) 0 0
\(601\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(602\) 1.41421i 1.41421i
\(603\) 0 0
\(604\) −1.00000 −1.00000
\(605\) 0 0
\(606\) 0 0
\(607\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.41421 1.41421i 1.41421 1.41421i
\(612\) 0 0
\(613\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(614\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(615\) 0 0
\(616\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(617\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(618\) 0 0
\(619\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(620\) 0.707107 0.707107i 0.707107 0.707107i
\(621\) 0 0
\(622\) 0 0
\(623\) −1.22474 0.707107i −1.22474 0.707107i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.500000 0.866025i
\(626\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −0.500000 0.866025i −0.500000 0.866025i
\(635\) 0.258819 0.965926i 0.258819 0.965926i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 1.00000
\(641\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 0 0
\(643\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −1.36603 0.366025i −1.36603 0.366025i
\(653\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(654\) 0 0
\(655\) 0.500000 0.866025i 0.500000 0.866025i
\(656\) 0 0
\(657\) 0 0
\(658\) −1.00000 1.00000i −1.00000 1.00000i
\(659\) −0.258819 0.965926i −0.258819 0.965926i −0.965926 0.258819i \(-0.916667\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(660\) 0 0
\(661\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(662\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(663\) 0 0
\(664\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(669\) 0 0
\(670\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(671\) 0 0
\(672\) 0 0
\(673\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(674\) 0 0
\(675\) 0 0
\(676\) −1.00000 −1.00000
\(677\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(678\) 0 0
\(679\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(680\) 1.22474 0.707107i 1.22474 0.707107i
\(681\) 0 0
\(682\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(683\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 0 0
\(685\) −1.00000 1.00000i −1.00000 1.00000i
\(686\) −0.258819 0.965926i −0.258819 0.965926i
\(687\) 0 0
\(688\) −1.36603 0.366025i −1.36603 0.366025i
\(689\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(690\) 0 0
\(691\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(692\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(693\) 0 0
\(694\) 1.00000i 1.00000i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(705\) 0 0
\(706\) 0 0
\(707\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(708\) 0 0
\(709\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(710\) 1.41421i 1.41421i
\(711\) 0 0
\(712\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(713\) 0 0
\(714\) 0 0
\(715\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(716\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(717\) 0 0
\(718\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −0.258819 0.965926i −0.258819 0.965926i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(728\) 1.41421i 1.41421i
\(729\) 0 0
\(730\) −1.00000 −1.00000
\(731\) −1.93185 + 0.517638i −1.93185 + 0.517638i
\(732\) 0 0
\(733\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(734\) 0.258819 0.965926i 0.258819 0.965926i
\(735\) 0 0
\(736\) 0 0
\(737\) 1.41421i 1.41421i
\(738\) 0 0
\(739\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(743\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(744\) 0 0
\(745\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(746\) −1.41421 −1.41421
\(747\) 0 0
\(748\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(749\) −0.258819 0.965926i −0.258819 0.965926i
\(750\) 0 0
\(751\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(753\) 0 0
\(754\) 0 0
\(755\) 0.707107 0.707107i 0.707107 0.707107i
\(756\) 0 0
\(757\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 1.41421 1.41421
\(765\) 0 0
\(766\) 1.00000 1.00000i 1.00000 1.00000i
\(767\) 0 0
\(768\) 0 0
\(769\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(770\) 0.258819 0.965926i 0.258819 0.965926i
\(771\) 0 0
\(772\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(773\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(777\) 0 0
\(778\) 0.866025 0.500000i 0.866025 0.500000i
\(779\) 0 0
\(780\) 0 0
\(781\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(788\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(795\) 0 0
\(796\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(797\) −0.965926 0.258819i −0.965926 0.258819i −0.258819 0.965926i \(-0.583333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.258819 0.965926i −0.258819 0.965926i
\(804\) 0 0
\(805\) 0 0
\(806\) 1.22474 0.707107i 1.22474 0.707107i
\(807\) 0 0
\(808\) 0.500000 0.866025i 0.500000 0.866025i
\(809\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(810\) 0 0
\(811\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.22474 0.707107i 1.22474 0.707107i
\(816\) 0 0
\(817\) 0 0
\(818\) 0.707107 0.707107i 0.707107 0.707107i
\(819\) 0 0
\(820\) 0 0
\(821\) −1.93185 + 0.517638i −1.93185 + 0.517638i −0.965926 + 0.258819i \(0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(822\) 0 0
\(823\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(828\) 0 0
\(829\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(830\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(831\) 0 0
\(832\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(833\) 0 0
\(834\) 0 0
\(835\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(840\) 0 0
\(841\) 0.866025 0.500000i 0.866025 0.500000i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.707107 0.707107i 0.707107 0.707107i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.965926 0.258819i 0.965926 0.258819i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0.366025 + 1.36603i 0.366025 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.00000 −1.00000
\(857\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(858\) 0 0
\(859\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(860\) 1.22474 0.707107i 1.22474 0.707107i
\(861\) 0 0
\(862\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −1.00000 −1.00000
\(866\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(867\) 0 0
\(868\) −0.500000 0.866025i −0.500000 0.866025i
\(869\) 0 0
\(870\) 0 0
\(871\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(876\) 0 0
\(877\) 0.366025 1.36603i 0.366025 1.36603i −0.500000 0.866025i \(-0.666667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(878\) −0.965926 0.258819i −0.965926 0.258819i
\(879\) 0 0
\(880\) −0.866025 0.500000i −0.866025 0.500000i
\(881\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(882\) 0 0
\(883\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(884\) 1.93185 0.517638i 1.93185 0.517638i
\(885\) 0 0
\(886\) 0 0
\(887\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(888\) 0 0
\(889\) −0.866025 0.500000i −0.866025 0.500000i
\(890\) 1.41421i 1.41421i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −0.500000 0.866025i −0.500000 0.866025i
\(896\) 0.258819 0.965926i 0.258819 0.965926i
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 1.00000 1.00000i 1.00000 1.00000i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) −1.00000 1.00000i −1.00000 1.00000i
\(911\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(912\) 0 0
\(913\) −0.500000 0.866025i −0.500000 0.866025i
\(914\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(915\) 0 0
\(916\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(917\) −0.707107 0.707107i −0.707107 0.707107i
\(918\) 0 0
\(919\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(923\) −0.517638 + 1.93185i −0.517638 + 1.93185i
\(924\) 0 0
\(925\) 0 0
\(926\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.22474 0.707107i 1.22474 0.707107i
\(933\) 0 0
\(934\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(935\) −1.41421 −1.41421
\(936\) 0 0
\(937\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(938\) 0.707107 1.22474i 0.707107 1.22474i
\(939\) 0 0
\(940\) 0.366025 1.36603i 0.366025 1.36603i
\(941\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(947\) −0.258819 0.965926i −0.258819 0.965926i −0.965926 0.258819i \(-0.916667\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(948\) 0 0
\(949\) −1.36603 0.366025i −1.36603 0.366025i
\(950\) 0 0
\(951\) 0 0
\(952\) −0.366025 1.36603i −0.366025 1.36603i
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(956\) 0 0
\(957\) 0 0
\(958\) −1.36603 0.366025i −1.36603 0.366025i
\(959\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(966\) 0 0
\(967\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −0.866025 0.500000i −0.866025 0.500000i
\(971\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0.517638 + 1.93185i 0.517638 + 1.93185i
\(975\) 0 0
\(976\) 0 0
\(977\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(978\) 0 0
\(979\) 1.36603 0.366025i 1.36603 0.366025i
\(980\) 0 0
\(981\) 0 0
\(982\) 1.00000i 1.00000i
\(983\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(984\) 0 0
\(985\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(993\) 0 0
\(994\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(995\) −0.965926 0.258819i −0.965926 0.258819i
\(996\) 0 0
\(997\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(998\) 1.41421i 1.41421i
\(999\) 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 1296.1.x.a.269.2 8
3.2 odd 2 inner 1296.1.x.a.269.1 8
9.2 odd 6 432.1.j.a.269.2 yes 4
9.4 even 3 inner 1296.1.x.a.701.1 8
9.5 odd 6 inner 1296.1.x.a.701.2 8
9.7 even 3 432.1.j.a.269.1 yes 4
16.5 even 4 inner 1296.1.x.a.917.2 8
36.7 odd 6 1728.1.j.a.593.1 4
36.11 even 6 1728.1.j.a.593.2 4
48.5 odd 4 inner 1296.1.x.a.917.1 8
72.11 even 6 3456.1.j.a.161.1 4
72.29 odd 6 3456.1.j.b.161.1 4
72.43 odd 6 3456.1.j.a.161.2 4
72.61 even 6 3456.1.j.b.161.2 4
144.5 odd 12 inner 1296.1.x.a.53.2 8
144.11 even 12 1728.1.j.a.1457.1 4
144.29 odd 12 3456.1.j.b.1889.2 4
144.43 odd 12 1728.1.j.a.1457.2 4
144.61 even 12 3456.1.j.b.1889.1 4
144.83 even 12 3456.1.j.a.1889.2 4
144.85 even 12 inner 1296.1.x.a.53.1 8
144.101 odd 12 432.1.j.a.53.1 4
144.115 odd 12 3456.1.j.a.1889.1 4
144.133 even 12 432.1.j.a.53.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
432.1.j.a.53.1 4 144.101 odd 12
432.1.j.a.53.2 yes 4 144.133 even 12
432.1.j.a.269.1 yes 4 9.7 even 3
432.1.j.a.269.2 yes 4 9.2 odd 6
1296.1.x.a.53.1 8 144.85 even 12 inner
1296.1.x.a.53.2 8 144.5 odd 12 inner
1296.1.x.a.269.1 8 3.2 odd 2 inner
1296.1.x.a.269.2 8 1.1 even 1 trivial
1296.1.x.a.701.1 8 9.4 even 3 inner
1296.1.x.a.701.2 8 9.5 odd 6 inner
1296.1.x.a.917.1 8 48.5 odd 4 inner
1296.1.x.a.917.2 8 16.5 even 4 inner
1728.1.j.a.593.1 4 36.7 odd 6
1728.1.j.a.593.2 4 36.11 even 6
1728.1.j.a.1457.1 4 144.11 even 12
1728.1.j.a.1457.2 4 144.43 odd 12
3456.1.j.a.161.1 4 72.11 even 6
3456.1.j.a.161.2 4 72.43 odd 6
3456.1.j.a.1889.1 4 144.115 odd 12
3456.1.j.a.1889.2 4 144.83 even 12
3456.1.j.b.161.1 4 72.29 odd 6
3456.1.j.b.161.2 4 72.61 even 6
3456.1.j.b.1889.1 4 144.61 even 12
3456.1.j.b.1889.2 4 144.29 odd 12