Properties

Label 1008.1.ed.a.485.1
Level $1008$
Weight $1$
Character 1008.485
Analytic conductor $0.503$
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] = [1008,1,Mod(53,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 6, 8]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.53");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1008.ed (of order \(12\), degree \(4\), 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: \(0.503057532734\)
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: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.2709504.6

Embedding invariants

Embedding label 485.1
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1008.485
Dual form 1008.1.ed.a.557.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.965926 + 0.258819i) q^{2} +(0.866025 - 0.500000i) q^{4} +(-0.258819 - 0.965926i) q^{5} +(-0.866025 + 0.500000i) q^{7} +(-0.707107 + 0.707107i) q^{8} +O(q^{10})\) \(q+(-0.965926 + 0.258819i) q^{2} +(0.866025 - 0.500000i) q^{4} +(-0.258819 - 0.965926i) q^{5} +(-0.866025 + 0.500000i) q^{7} +(-0.707107 + 0.707107i) q^{8} +(0.500000 + 0.866025i) q^{10} +(-0.965926 - 0.258819i) q^{11} +(0.707107 - 0.707107i) q^{14} +(0.500000 - 0.866025i) q^{16} +(1.22474 - 0.707107i) q^{17} +(-0.366025 - 1.36603i) q^{19} +(-0.707107 - 0.707107i) q^{20} +1.00000 q^{22} +(-0.500000 + 0.866025i) q^{28} +(-0.707107 - 0.707107i) q^{29} +(-0.500000 - 0.866025i) q^{31} +(-0.258819 + 0.965926i) q^{32} +(-1.00000 + 1.00000i) q^{34} +(0.707107 + 0.707107i) q^{35} +(-1.36603 + 0.366025i) q^{37} +(0.707107 + 1.22474i) q^{38} +(0.866025 + 0.500000i) q^{40} -1.41421 q^{41} +(-1.00000 - 1.00000i) q^{43} +(-0.965926 + 0.258819i) q^{44} +(0.500000 - 0.866025i) q^{49} +(0.965926 + 0.258819i) q^{53} +1.00000i q^{55} +(0.258819 - 0.965926i) q^{56} +(0.866025 + 0.500000i) q^{58} +(0.965926 + 0.258819i) q^{59} +(0.707107 + 0.707107i) q^{62} -1.00000i q^{64} +(0.707107 - 1.22474i) q^{68} +(-0.866025 - 0.500000i) q^{70} +(1.22474 - 0.707107i) q^{74} +(-1.00000 - 1.00000i) q^{76} +(0.965926 - 0.258819i) q^{77} +(-0.500000 + 0.866025i) q^{79} +(-0.965926 - 0.258819i) q^{80} +(1.36603 - 0.366025i) q^{82} +(0.707107 + 0.707107i) q^{83} +(-1.00000 - 1.00000i) q^{85} +(1.22474 + 0.707107i) q^{86} +(0.866025 - 0.500000i) q^{88} +(0.707107 - 1.22474i) q^{89} +(-1.22474 + 0.707107i) q^{95} +1.00000 q^{97} +(-0.258819 + 0.965926i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{10} + 4 q^{16} + 4 q^{19} + 8 q^{22} - 4 q^{28} - 4 q^{31} - 8 q^{34} - 4 q^{37} - 8 q^{43} + 4 q^{49} - 8 q^{76} - 4 q^{79} + 4 q^{82} - 8 q^{85} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.965926 + 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 −0.965926 0.258819i \(-0.916667\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(6\) 0 0
\(7\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(8\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(9\) 0 0
\(10\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(11\) −0.965926 0.258819i −0.965926 0.258819i −0.258819 0.965926i \(-0.583333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) 0 0
\(13\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0.707107 0.707107i 0.707107 0.707107i
\(15\) 0 0
\(16\) 0.500000 0.866025i 0.500000 0.866025i
\(17\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(18\) 0 0
\(19\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(20\) −0.707107 0.707107i −0.707107 0.707107i
\(21\) 0 0
\(22\) 1.00000 1.00000
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(29\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−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
\(34\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(35\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(36\) 0 0
\(37\) −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(39\) 0 0
\(40\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(41\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(42\) 0 0
\(43\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(44\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) 0.500000 0.866025i 0.500000 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(54\) 0 0
\(55\) 1.00000i 1.00000i
\(56\) 0.258819 0.965926i 0.258819 0.965926i
\(57\) 0 0
\(58\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(59\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(60\) 0 0
\(61\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(62\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(63\) 0 0
\(64\) 1.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(68\) 0.707107 1.22474i 0.707107 1.22474i
\(69\) 0 0
\(70\) −0.866025 0.500000i −0.866025 0.500000i
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 1.22474 0.707107i 1.22474 0.707107i
\(75\) 0 0
\(76\) −1.00000 1.00000i −1.00000 1.00000i
\(77\) 0.965926 0.258819i 0.965926 0.258819i
\(78\) 0 0
\(79\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(80\) −0.965926 0.258819i −0.965926 0.258819i
\(81\) 0 0
\(82\) 1.36603 0.366025i 1.36603 0.366025i
\(83\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(84\) 0 0
\(85\) −1.00000 1.00000i −1.00000 1.00000i
\(86\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(87\) 0 0
\(88\) 0.866025 0.500000i 0.866025 0.500000i
\(89\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(96\) 0 0
\(97\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(102\) 0 0
\(103\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1.00000 −1.00000
\(107\) −0.258819 0.965926i −0.258819 0.965926i −0.965926 0.258819i \(-0.916667\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(108\) 0 0
\(109\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(110\) −0.258819 0.965926i −0.258819 0.965926i
\(111\) 0 0
\(112\) 1.00000i 1.00000i
\(113\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.965926 0.258819i −0.965926 0.258819i
\(117\) 0 0
\(118\) −1.00000 −1.00000
\(119\) −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
\(130\) 0 0
\(131\) 0.965926 0.258819i 0.965926 0.258819i 0.258819 0.965926i \(-0.416667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(132\) 0 0
\(133\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(134\) 0 0
\(135\) 0 0
\(136\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(137\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(138\) 0 0
\(139\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(140\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(146\) 0 0
\(147\) 0 0
\(148\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(149\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(150\) 0 0
\(151\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(152\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(153\) 0 0
\(154\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(155\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(156\) 0 0
\(157\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(158\) 0.258819 0.965926i 0.258819 0.965926i
\(159\) 0 0
\(160\) 1.00000 1.00000
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(164\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(165\) 0 0
\(166\) −0.866025 0.500000i −0.866025 0.500000i
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000i 1.00000i
\(170\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(171\) 0 0
\(172\) −1.36603 0.366025i −1.36603 0.366025i
\(173\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(177\) 0 0
\(178\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(179\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(180\) 0 0
\(181\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(186\) 0 0
\(187\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(188\) 0 0
\(189\) 0 0
\(190\) 1.00000 1.00000i 1.00000 1.00000i
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 0 0
\(193\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(195\) 0 0
\(196\) 1.00000i 1.00000i
\(197\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(198\) 0 0
\(199\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(204\) 0 0
\(205\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(206\) −1.93185 0.517638i −1.93185 0.517638i
\(207\) 0 0
\(208\) 0 0
\(209\) 1.41421i 1.41421i
\(210\) 0 0
\(211\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(212\) 0.965926 0.258819i 0.965926 0.258819i
\(213\) 0 0
\(214\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(215\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(216\) 0 0
\(217\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(218\) 0 0
\(219\) 0 0
\(220\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(221\) 0 0
\(222\) 0 0
\(223\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(224\) −0.258819 0.965926i −0.258819 0.965926i
\(225\) 0 0
\(226\) −0.366025 1.36603i −0.366025 1.36603i
\(227\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(228\) 0 0
\(229\) 1.36603 0.366025i 1.36603 0.366025i 0.500000 0.866025i \(-0.333333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.00000 1.00000
\(233\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0.965926 0.258819i 0.965926 0.258819i
\(237\) 0 0
\(238\) 0.366025 1.36603i 0.366025 1.36603i
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.965926 0.258819i −0.965926 0.258819i
\(246\) 0 0
\(247\) 0 0
\(248\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(249\) 0 0
\(250\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(251\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(252\) 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 \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(258\) 0 0
\(259\) 1.00000 1.00000i 1.00000 1.00000i
\(260\) 0 0
\(261\) 0 0
\(262\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(263\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(264\) 0 0
\(265\) 1.00000i 1.00000i
\(266\) −1.22474 0.707107i −1.22474 0.707107i
\(267\) 0 0
\(268\) 0 0
\(269\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(270\) 0 0
\(271\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(272\) 1.41421i 1.41421i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(278\) −1.22474 0.707107i −1.22474 0.707107i
\(279\) 0 0
\(280\) −1.00000 −1.00000
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0.366025 1.36603i 0.366025 1.36603i −0.500000 0.866025i \(-0.666667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.22474 0.707107i 1.22474 0.707107i
\(288\) 0 0
\(289\) 0.500000 0.866025i 0.500000 0.866025i
\(290\) 0.258819 0.965926i 0.258819 0.965926i
\(291\) 0 0
\(292\) 0 0
\(293\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(294\) 0 0
\(295\) 1.00000i 1.00000i
\(296\) 0.707107 1.22474i 0.707107 1.22474i
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(302\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(303\) 0 0
\(304\) −1.36603 0.366025i −1.36603 0.366025i
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) 0.707107 0.707107i 0.707107 0.707107i
\(309\) 0 0
\(310\) 0.500000 0.866025i 0.500000 0.866025i
\(311\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(312\) 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
\(316\) 1.00000i 1.00000i
\(317\) −0.965926 + 0.258819i −0.965926 + 0.258819i −0.707107 0.707107i \(-0.750000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(318\) 0 0
\(319\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(320\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(321\) 0 0
\(322\) 0 0
\(323\) −1.41421 1.41421i −1.41421 1.41421i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 1.00000 1.00000i 1.00000 1.00000i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(332\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) −0.258819 0.965926i −0.258819 0.965926i
\(339\) 0 0
\(340\) −1.36603 0.366025i −1.36603 0.366025i
\(341\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(342\) 0 0
\(343\) 1.00000i 1.00000i
\(344\) 1.41421 1.41421
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(348\) 0 0
\(349\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.500000 0.866025i 0.500000 0.866025i
\(353\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1.41421i 1.41421i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(360\) 0 0
\(361\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(362\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −1.00000 1.00000i −1.00000 1.00000i
\(371\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(372\) 0 0
\(373\) −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 1.22474 0.707107i 1.22474 0.707107i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(380\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(381\) 0 0
\(382\) 0 0
\(383\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(384\) 0 0
\(385\) −0.500000 0.866025i −0.500000 0.866025i
\(386\) −0.707107 0.707107i −0.707107 0.707107i
\(387\) 0 0
\(388\) 0.866025 0.500000i 0.866025 0.500000i
\(389\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(393\) 0 0
\(394\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(395\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(396\) 0 0
\(397\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −1.00000 −1.00000
\(407\) 1.41421 1.41421
\(408\) 0 0
\(409\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(410\) −0.707107 1.22474i −0.707107 1.22474i
\(411\) 0 0
\(412\) 2.00000 2.00000
\(413\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(414\) 0 0
\(415\) 0.500000 0.866025i 0.500000 0.866025i
\(416\) 0 0
\(417\) 0 0
\(418\) −0.366025 1.36603i −0.366025 1.36603i
\(419\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(420\) 0 0
\(421\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −0.707107 0.707107i −0.707107 0.707107i
\(429\) 0 0
\(430\) 0.366025 1.36603i 0.366025 1.36603i
\(431\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) −0.965926 0.258819i −0.965926 0.258819i
\(435\) 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
\(442\) 0 0
\(443\) −0.258819 0.965926i −0.258819 0.965926i −0.965926 0.258819i \(-0.916667\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(444\) 0 0
\(445\) −1.36603 0.366025i −1.36603 0.366025i
\(446\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(447\) 0 0
\(448\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(452\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(453\) 0 0
\(454\) 1.00000i 1.00000i
\(455\) 0 0
\(456\) 0 0
\(457\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(458\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(459\) 0 0
\(460\) 0 0
\(461\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(462\) 0 0
\(463\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(464\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(465\) 0 0
\(466\) 0.366025 1.36603i 0.366025 1.36603i
\(467\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(473\) 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 \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(483\) 0 0
\(484\) 0 0
\(485\) −0.258819 0.965926i −0.258819 0.965926i
\(486\) 0 0
\(487\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 1.00000 1.00000
\(491\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(492\) 0 0
\(493\) −1.36603 0.366025i −1.36603 0.366025i
\(494\) 0 0
\(495\) 0 0
\(496\) −1.00000 −1.00000
\(497\) 0 0
\(498\) 0 0
\(499\) 0.366025 + 1.36603i 0.366025 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(500\) −0.965926 0.258819i −0.965926 0.258819i
\(501\) 0 0
\(502\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(503\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 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 \(-0.416667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(513\) 0 0
\(514\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(515\) 0.517638 1.93185i 0.517638 1.93185i
\(516\) 0 0
\(517\) 0 0
\(518\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(519\) 0 0
\(520\) 0 0
\(521\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(522\) 0 0
\(523\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(524\) 0.707107 0.707107i 0.707107 0.707107i
\(525\) 0 0
\(526\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(527\) −1.22474 0.707107i −1.22474 0.707107i
\(528\) 0 0
\(529\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(530\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(531\) 0 0
\(532\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(533\) 0 0
\(534\) 0 0
\(535\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(536\) 0 0
\(537\) 0 0
\(538\) 1.00000i 1.00000i
\(539\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(540\) 0 0
\(541\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(542\) 0.258819 0.965926i 0.258819 0.965926i
\(543\) 0 0
\(544\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(545\) 0 0
\(546\) 0 0
\(547\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(552\) 0 0
\(553\) 1.00000i 1.00000i
\(554\) 0 0
\(555\) 0 0
\(556\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(557\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0.965926 0.258819i 0.965926 0.258819i
\(561\) 0 0
\(562\) 0 0
\(563\) 0.258819 0.965926i 0.258819 0.965926i −0.707107 0.707107i \(-0.750000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(564\) 0 0
\(565\) 1.36603 0.366025i 1.36603 0.366025i
\(566\) 1.41421i 1.41421i
\(567\) 0 0
\(568\) 0 0
\(569\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(570\) 0 0
\(571\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(575\) 0 0
\(576\) 0 0
\(577\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(578\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(579\) 0 0
\(580\) 1.00000i 1.00000i
\(581\) −0.965926 0.258819i −0.965926 0.258819i
\(582\) 0 0
\(583\) −0.866025 0.500000i −0.866025 0.500000i
\(584\) 0 0
\(585\) 0 0
\(586\) 0.500000 0.866025i 0.500000 0.866025i
\(587\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(588\) 0 0
\(589\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(590\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(591\) 0 0
\(592\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(593\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(594\) 0 0
\(595\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(602\) −1.41421 −1.41421
\(603\) 0 0
\(604\) 0.500000 0.866025i 0.500000 0.866025i
\(605\) 0 0
\(606\) 0 0
\(607\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(608\) 1.41421 1.41421
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(620\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(621\) 0 0
\(622\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(623\) 1.41421i 1.41421i
\(624\) 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
\(629\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(630\) 0 0
\(631\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(632\) −0.258819 0.965926i −0.258819 0.965926i
\(633\) 0 0
\(634\) 0.866025 0.500000i 0.866025 0.500000i
\(635\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(636\) 0 0
\(637\) 0 0
\(638\) −0.707107 0.707107i −0.707107 0.707107i
\(639\) 0 0
\(640\) 0.866025 0.500000i 0.866025 0.500000i
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) −0.866025 0.500000i −0.866025 0.500000i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.965926 0.258819i 0.965926 0.258819i 0.258819 0.965926i \(-0.416667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 0 0
\(655\) −0.500000 0.866025i −0.500000 0.866025i
\(656\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) −0.366025 + 1.36603i −0.366025 + 1.36603i 0.500000 + 0.866025i \(0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −1.00000 −1.00000
\(665\) 0.707107 1.22474i 0.707107 1.22474i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(674\) 0.965926 0.258819i 0.965926 0.258819i
\(675\) 0 0
\(676\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(677\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(680\) 1.41421 1.41421
\(681\) 0 0
\(682\) −0.500000 0.866025i −0.500000 0.866025i
\(683\) −0.965926 0.258819i −0.965926 0.258819i −0.258819 0.965926i \(-0.583333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.258819 0.965926i −0.258819 0.965926i
\(687\) 0 0
\(688\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(689\) 0 0
\(690\) 0 0
\(691\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.707107 1.22474i 0.707107 1.22474i
\(696\) 0 0
\(697\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(702\) 0 0
\(703\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(704\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.36603 0.366025i 1.36603 0.366025i 0.500000 0.866025i \(-0.333333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(720\) 0 0
\(721\) −2.00000 −2.00000
\(722\) 0.707107 0.707107i 0.707107 0.707107i
\(723\) 0 0
\(724\) −1.36603 0.366025i −1.36603 0.366025i
\(725\) 0 0
\(726\) 0 0
\(727\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.93185 0.517638i −1.93185 0.517638i
\(732\) 0 0
\(733\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(734\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.36603 0.366025i −1.36603 0.366025i −0.500000 0.866025i \(-0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(740\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(741\) 0 0
\(742\) 0.866025 0.500000i 0.866025 0.500000i
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.22474 0.707107i 1.22474 0.707107i
\(747\) 0 0
\(748\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(749\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(750\) 0 0
\(751\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −0.707107 0.707107i −0.707107 0.707107i
\(756\) 0 0
\(757\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(758\) −1.22474 0.707107i −1.22474 0.707107i
\(759\) 0 0
\(760\) 0.366025 1.36603i 0.366025 1.36603i
\(761\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(767\) 0 0
\(768\) 0 0
\(769\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(771\) 0 0
\(772\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(773\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(777\) 0 0
\(778\) 0 0
\(779\) 0.517638 + 1.93185i 0.517638 + 1.93185i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.500000 0.866025i −0.500000 0.866025i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(788\) 0.517638 1.93185i 0.517638 1.93185i
\(789\) 0 0
\(790\) −1.00000 −1.00000
\(791\) −0.707107 1.22474i −0.707107 1.22474i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(810\) 0 0
\(811\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(812\) 0.965926 0.258819i 0.965926 0.258819i
\(813\) 0 0
\(814\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(815\) 0 0
\(816\) 0 0
\(817\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(818\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(819\) 0 0
\(820\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(821\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(822\) 0 0
\(823\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(824\) −1.93185 + 0.517638i −1.93185 + 0.517638i
\(825\) 0 0
\(826\) 0.866025 0.500000i 0.866025 0.500000i
\(827\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(828\) 0 0
\(829\) 1.36603 + 0.366025i 1.36603 + 0.366025i 0.866025 0.500000i \(-0.166667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(830\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(831\) 0 0
\(832\) 0 0
\(833\) 1.41421i 1.41421i
\(834\) 0 0
\(835\) 0 0
\(836\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(837\) 0 0
\(838\) 0 0
\(839\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.965926 0.258819i 0.965926 0.258819i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.707107 0.707107i 0.707107 0.707107i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(857\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(858\) 0 0
\(859\) −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(860\) 1.41421i 1.41421i
\(861\) 0 0
\(862\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(863\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 1.00000 1.00000
\(869\) 0.707107 0.707107i 0.707107 0.707107i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(876\) 0 0
\(877\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(878\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(879\) 0 0
\(880\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(884\) 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
\(889\) 0.866025 0.500000i 0.866025 0.500000i
\(890\) 1.41421 1.41421
\(891\) 0 0
\(892\) 0.866025 0.500000i 0.866025 0.500000i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −0.707107 0.707107i −0.707107 0.707107i
\(897\) 0 0
\(898\) 0 0
\(899\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(900\) 0 0
\(901\) 1.36603 0.366025i 1.36603 0.366025i
\(902\) −1.41421 −1.41421
\(903\) 0 0
\(904\) −1.00000 1.00000i −1.00000 1.00000i
\(905\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(906\) 0 0
\(907\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(908\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −0.500000 0.866025i −0.500000 0.866025i
\(914\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(915\) 0 0
\(916\) 1.00000 1.00000i 1.00000 1.00000i
\(917\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(918\) 0 0
\(919\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.73205 1.00000i −1.73205 1.00000i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −1.93185 + 0.517638i −1.93185 + 0.517638i
\(927\) 0 0
\(928\) 0.866025 0.500000i 0.866025 0.500000i
\(929\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(930\) 0 0
\(931\) −1.36603 0.366025i −1.36603 0.366025i
\(932\) 1.41421i 1.41421i
\(933\) 0 0
\(934\) 0 0
\(935\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(936\) 0 0
\(937\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.258819 0.965926i 0.258819 0.965926i −0.707107 0.707107i \(-0.750000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0.707107 0.707107i 0.707107 0.707107i
\(945\) 0 0
\(946\) −1.00000 1.00000i −1.00000 1.00000i
\(947\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) −0.366025 1.36603i −0.366025 1.36603i
\(953\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(959\) 0 0
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) −0.866025 0.500000i −0.866025 0.500000i
\(965\) 0.707107 0.707107i 0.707107 0.707107i
\(966\) 0 0
\(967\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(971\) −0.258819 0.965926i −0.258819 0.965926i −0.965926 0.258819i \(-0.916667\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(972\) 0 0
\(973\) −1.36603 0.366025i −1.36603 0.366025i
\(974\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(975\) 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
\(979\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(980\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(981\) 0 0
\(982\) 0.500000 0.866025i 0.500000 0.866025i
\(983\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(984\) 0 0
\(985\) −1.73205 1.00000i −1.73205 1.00000i
\(986\) 1.41421 1.41421
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(992\) 0.965926 0.258819i 0.965926 0.258819i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(998\) −0.707107 1.22474i −0.707107 1.22474i
\(999\) 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 1008.1.ed.a.485.1 yes 8
3.2 odd 2 inner 1008.1.ed.a.485.2 yes 8
7.4 even 3 inner 1008.1.ed.a.53.2 yes 8
16.13 even 4 inner 1008.1.ed.a.989.1 yes 8
21.11 odd 6 inner 1008.1.ed.a.53.1 8
48.29 odd 4 inner 1008.1.ed.a.989.2 yes 8
112.109 even 12 inner 1008.1.ed.a.557.2 yes 8
336.221 odd 12 inner 1008.1.ed.a.557.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.1.ed.a.53.1 8 21.11 odd 6 inner
1008.1.ed.a.53.2 yes 8 7.4 even 3 inner
1008.1.ed.a.485.1 yes 8 1.1 even 1 trivial
1008.1.ed.a.485.2 yes 8 3.2 odd 2 inner
1008.1.ed.a.557.1 yes 8 336.221 odd 12 inner
1008.1.ed.a.557.2 yes 8 112.109 even 12 inner
1008.1.ed.a.989.1 yes 8 16.13 even 4 inner
1008.1.ed.a.989.2 yes 8 48.29 odd 4 inner