Properties

Label 1456.1.fx.a.1413.2
Level $1456$
Weight $1$
Character 1456.1413
Analytic conductor $0.727$
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] = [1456,1,Mod(237,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 6, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.237");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1456.fx (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: \(0.726638658394\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.2422784.3

Embedding invariants

Embedding label 1413.2
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1456.1413
Dual form 1456.1.fx.a.237.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(0.965926 - 0.258819i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(-0.707107 + 0.707107i) q^{5} +(-0.258819 + 0.965926i) q^{6} +(-0.866025 + 0.500000i) q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+(-0.500000 + 0.866025i) q^{2} +(0.965926 - 0.258819i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(-0.707107 + 0.707107i) q^{5} +(-0.258819 + 0.965926i) q^{6} +(-0.866025 + 0.500000i) q^{7} +1.00000 q^{8} +(-0.258819 - 0.965926i) q^{10} +(-0.707107 - 0.707107i) q^{12} +(-0.258819 + 0.965926i) q^{13} -1.00000i q^{14} +(-0.500000 + 0.866025i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(0.965926 + 0.258819i) q^{20} +(-0.707107 + 0.707107i) q^{21} +(-0.866025 - 0.500000i) q^{23} +(0.965926 - 0.258819i) q^{24} +(-0.707107 - 0.707107i) q^{26} +(-0.707107 + 0.707107i) q^{27} +(0.866025 + 0.500000i) q^{28} +(-0.500000 - 0.866025i) q^{30} +1.41421i q^{31} +(-0.500000 - 0.866025i) q^{32} +(0.258819 - 0.965926i) q^{35} +1.00000i q^{39} +(-0.707107 + 0.707107i) q^{40} +(-0.707107 + 1.22474i) q^{41} +(-0.258819 - 0.965926i) q^{42} +(0.366025 - 1.36603i) q^{43} +(0.866025 - 0.500000i) q^{46} +(-0.258819 + 0.965926i) q^{48} +(0.500000 - 0.866025i) q^{49} +(0.965926 - 0.258819i) q^{52} +(-0.258819 - 0.965926i) q^{54} +(-0.866025 + 0.500000i) q^{56} +(-0.965926 - 0.258819i) q^{59} +1.00000 q^{60} +(-0.258819 + 0.965926i) q^{61} +(-1.22474 - 0.707107i) q^{62} +1.00000 q^{64} +(-0.500000 - 0.866025i) q^{65} +(0.366025 + 1.36603i) q^{67} +(-0.965926 - 0.258819i) q^{69} +(0.707107 + 0.707107i) q^{70} +(0.866025 - 0.500000i) q^{71} +1.41421 q^{73} +(-0.866025 - 0.500000i) q^{78} +(-0.258819 - 0.965926i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-0.707107 - 1.22474i) q^{82} +(0.965926 + 0.258819i) q^{84} +(1.00000 + 1.00000i) q^{86} +(-0.707107 + 1.22474i) q^{89} +(-0.258819 - 0.965926i) q^{91} +1.00000i q^{92} +(0.366025 + 1.36603i) q^{93} +(-0.707107 - 0.707107i) q^{96} +(1.22474 - 0.707107i) q^{97} +(0.500000 + 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{4} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 4 q^{4} + 8 q^{8} - 4 q^{15} - 4 q^{16} - 4 q^{30} - 4 q^{32} - 4 q^{43} + 4 q^{49} + 8 q^{60} + 8 q^{64} - 4 q^{65} - 4 q^{67} - 4 q^{81} + 8 q^{86} - 4 q^{93} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(3\) 0.965926 0.258819i 0.965926 0.258819i 0.258819 0.965926i \(-0.416667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) −0.500000 0.866025i −0.500000 0.866025i
\(5\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(6\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(7\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(8\) 1.00000 1.00000
\(9\) 0 0
\(10\) −0.258819 0.965926i −0.258819 0.965926i
\(11\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(12\) −0.707107 0.707107i −0.707107 0.707107i
\(13\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(14\) 1.00000i 1.00000i
\(15\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(20\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(21\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(22\) 0 0
\(23\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0.965926 0.258819i 0.965926 0.258819i
\(25\) 0 0
\(26\) −0.707107 0.707107i −0.707107 0.707107i
\(27\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(28\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(29\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(30\) −0.500000 0.866025i −0.500000 0.866025i
\(31\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(32\) −0.500000 0.866025i −0.500000 0.866025i
\(33\) 0 0
\(34\) 0 0
\(35\) 0.258819 0.965926i 0.258819 0.965926i
\(36\) 0 0
\(37\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(38\) 0 0
\(39\) 1.00000i 1.00000i
\(40\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(41\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(42\) −0.258819 0.965926i −0.258819 0.965926i
\(43\) 0.366025 1.36603i 0.366025 1.36603i −0.500000 0.866025i \(-0.666667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.866025 0.500000i 0.866025 0.500000i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(49\) 0.500000 0.866025i 0.500000 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.965926 0.258819i 0.965926 0.258819i
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) −0.258819 0.965926i −0.258819 0.965926i
\(55\) 0 0
\(56\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(57\) 0 0
\(58\) 0 0
\(59\) −0.965926 0.258819i −0.965926 0.258819i −0.258819 0.965926i \(-0.583333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 1.00000 1.00000
\(61\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(62\) −1.22474 0.707107i −1.22474 0.707107i
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) −0.500000 0.866025i −0.500000 0.866025i
\(66\) 0 0
\(67\) 0.366025 + 1.36603i 0.366025 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) −0.965926 0.258819i −0.965926 0.258819i
\(70\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(71\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(72\) 0 0
\(73\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) −0.866025 0.500000i −0.866025 0.500000i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −0.258819 0.965926i −0.258819 0.965926i
\(81\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(82\) −0.707107 1.22474i −0.707107 1.22474i
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(85\) 0 0
\(86\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(87\) 0 0
\(88\) 0 0
\(89\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(90\) 0 0
\(91\) −0.258819 0.965926i −0.258819 0.965926i
\(92\) 1.00000i 1.00000i
\(93\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(94\) 0 0
\(95\) 0 0
\(96\) −0.707107 0.707107i −0.707107 0.707107i
\(97\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(98\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(102\) 0 0
\(103\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(104\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(105\) 1.00000i 1.00000i
\(106\) 0 0
\(107\) 1.36603 0.366025i 1.36603 0.366025i 0.500000 0.866025i \(-0.333333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(108\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(109\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000i 1.00000i
\(113\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(114\) 0 0
\(115\) 0.965926 0.258819i 0.965926 0.258819i
\(116\) 0 0
\(117\) 0 0
\(118\) 0.707107 0.707107i 0.707107 0.707107i
\(119\) 0 0
\(120\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(121\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(122\) −0.707107 0.707107i −0.707107 0.707107i
\(123\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(124\) 1.22474 0.707107i 1.22474 0.707107i
\(125\) −0.707107 0.707107i −0.707107 0.707107i
\(126\) 0 0
\(127\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(128\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(129\) 1.41421i 1.41421i
\(130\) 1.00000 1.00000
\(131\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.36603 0.366025i −1.36603 0.366025i
\(135\) 1.00000i 1.00000i
\(136\) 0 0
\(137\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(138\) 0.707107 0.707107i 0.707107 0.707107i
\(139\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(140\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(141\) 0 0
\(142\) 1.00000i 1.00000i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(147\) 0.258819 0.965926i 0.258819 0.965926i
\(148\) 0 0
\(149\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(150\) 0 0
\(151\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.00000 1.00000i −1.00000 1.00000i
\(156\) 0.866025 0.500000i 0.866025 0.500000i
\(157\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(161\) 1.00000 1.00000
\(162\) −0.500000 0.866025i −0.500000 0.866025i
\(163\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(164\) 1.41421 1.41421
\(165\) 0 0
\(166\) 0 0
\(167\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(168\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(169\) −0.866025 0.500000i −0.866025 0.500000i
\(170\) 0 0
\(171\) 0 0
\(172\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(173\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.00000 −1.00000
\(178\) −0.707107 1.22474i −0.707107 1.22474i
\(179\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(180\) 0 0
\(181\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(182\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(183\) 1.00000i 1.00000i
\(184\) −0.866025 0.500000i −0.866025 0.500000i
\(185\) 0 0
\(186\) −1.36603 0.366025i −1.36603 0.366025i
\(187\) 0 0
\(188\) 0 0
\(189\) 0.258819 0.965926i 0.258819 0.965926i
\(190\) 0 0
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) 0.965926 0.258819i 0.965926 0.258819i
\(193\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(194\) 1.41421i 1.41421i
\(195\) −0.707107 0.707107i −0.707107 0.707107i
\(196\) −1.00000 −1.00000
\(197\) −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(198\) 0 0
\(199\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(200\) 0 0
\(201\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −0.366025 1.36603i −0.366025 1.36603i
\(206\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(207\) 0 0
\(208\) −0.707107 0.707107i −0.707107 0.707107i
\(209\) 0 0
\(210\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(211\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(212\) 0 0
\(213\) 0.707107 0.707107i 0.707107 0.707107i
\(214\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(215\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(216\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(217\) −0.707107 1.22474i −0.707107 1.22474i
\(218\) 0 0
\(219\) 1.36603 0.366025i 1.36603 0.366025i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(224\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(225\) 0 0
\(226\) 0 0
\(227\) 0.258819 0.965926i 0.258819 0.965926i −0.707107 0.707107i \(-0.750000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(228\) 0 0
\(229\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(230\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(231\) 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
\(236\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(237\) 0 0
\(238\) 0 0
\(239\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(240\) −0.500000 0.866025i −0.500000 0.866025i
\(241\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) 1.00000i 1.00000i
\(243\) 0 0
\(244\) 0.965926 0.258819i 0.965926 0.258819i
\(245\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(246\) −1.00000 1.00000i −1.00000 1.00000i
\(247\) 0 0
\(248\) 1.41421i 1.41421i
\(249\) 0 0
\(250\) 0.965926 0.258819i 0.965926 0.258819i
\(251\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.500000 0.866025i
\(257\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(258\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(259\) 0 0
\(260\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(261\) 0 0
\(262\) 0.965926 0.258819i 0.965926 0.258819i
\(263\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(268\) 1.00000 1.00000i 1.00000 1.00000i
\(269\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(270\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(271\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(272\) 0 0
\(273\) −0.500000 0.866025i −0.500000 0.866025i
\(274\) 1.00000i 1.00000i
\(275\) 0 0
\(276\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(277\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0.258819 0.965926i 0.258819 0.965926i
\(281\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) −0.866025 0.500000i −0.866025 0.500000i
\(285\) 0 0
\(286\) 0 0
\(287\) 1.41421i 1.41421i
\(288\) 0 0
\(289\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(290\) 0 0
\(291\) 1.00000 1.00000i 1.00000 1.00000i
\(292\) −0.707107 1.22474i −0.707107 1.22474i
\(293\) −1.93185 0.517638i −1.93185 0.517638i −0.965926 0.258819i \(-0.916667\pi\)
−0.965926 0.258819i \(-0.916667\pi\)
\(294\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(295\) 0.866025 0.500000i 0.866025 0.500000i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.707107 0.707107i 0.707107 0.707107i
\(300\) 0 0
\(301\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(302\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(303\) 0 0
\(304\) 0 0
\(305\) −0.500000 0.866025i −0.500000 0.866025i
\(306\) 0 0
\(307\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(308\) 0 0
\(309\) 1.36603 0.366025i 1.36603 0.366025i
\(310\) 1.36603 0.366025i 1.36603 0.366025i
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 1.00000i 1.00000i
\(313\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(321\) 1.22474 0.707107i 1.22474 0.707107i
\(322\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(323\) 0 0
\(324\) 1.00000 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(329\) 0 0
\(330\) 0 0
\(331\) −0.366025 + 1.36603i −0.366025 + 1.36603i 0.500000 + 0.866025i \(0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(335\) −1.22474 0.707107i −1.22474 0.707107i
\(336\) −0.258819 0.965926i −0.258819 0.965926i
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0.866025 0.500000i 0.866025 0.500000i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 1.00000i
\(344\) 0.366025 1.36603i 0.366025 1.36603i
\(345\) 0.866025 0.500000i 0.866025 0.500000i
\(346\) −0.707107 0.707107i −0.707107 0.707107i
\(347\) −0.366025 + 1.36603i −0.366025 + 1.36603i 0.500000 + 0.866025i \(0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(348\) 0 0
\(349\) −0.965926 + 0.258819i −0.965926 + 0.258819i −0.707107 0.707107i \(-0.750000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(350\) 0 0
\(351\) −0.500000 0.866025i −0.500000 0.866025i
\(352\) 0 0
\(353\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(354\) 0.500000 0.866025i 0.500000 0.866025i
\(355\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(356\) 1.41421 1.41421
\(357\) 0 0
\(358\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(359\) 1.00000i 1.00000i −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
\(362\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(363\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(364\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(365\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(366\) −0.866025 0.500000i −0.866025 0.500000i
\(367\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(368\) 0.866025 0.500000i 0.866025 0.500000i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 1.00000 1.00000i 1.00000 1.00000i
\(373\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(374\) 0 0
\(375\) −0.866025 0.500000i −0.866025 0.500000i
\(376\) 0 0
\(377\) 0 0
\(378\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(379\) −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(380\) 0 0
\(381\) 0.258819 0.965926i 0.258819 0.965926i
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(385\) 0 0
\(386\) −0.500000 0.866025i −0.500000 0.866025i
\(387\) 0 0
\(388\) −1.22474 0.707107i −1.22474 0.707107i
\(389\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(390\) 0.965926 0.258819i 0.965926 0.258819i
\(391\) 0 0
\(392\) 0.500000 0.866025i 0.500000 0.866025i
\(393\) −0.866025 0.500000i −0.866025 0.500000i
\(394\) 0.366025 1.36603i 0.366025 1.36603i
\(395\) 0 0
\(396\) 0 0
\(397\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(398\) −1.41421 −1.41421
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(402\) −1.41421 −1.41421
\(403\) −1.36603 0.366025i −1.36603 0.366025i
\(404\) 0 0
\(405\) −0.258819 0.965926i −0.258819 0.965926i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(410\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(411\) 0.707107 0.707107i 0.707107 0.707107i
\(412\) −0.707107 1.22474i −0.707107 1.22474i
\(413\) 0.965926 0.258819i 0.965926 0.258819i
\(414\) 0 0
\(415\) 0 0
\(416\) 0.965926 0.258819i 0.965926 0.258819i
\(417\) 0 0
\(418\) 0 0
\(419\) −1.93185 + 0.517638i −1.93185 + 0.517638i −0.965926 + 0.258819i \(0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(420\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(421\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(427\) −0.258819 0.965926i −0.258819 0.965926i
\(428\) −1.00000 1.00000i −1.00000 1.00000i
\(429\) 0 0
\(430\) −1.41421 −1.41421
\(431\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(432\) −0.258819 0.965926i −0.258819 0.965926i
\(433\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(434\) 1.41421 1.41421
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 0 0
\(445\) −0.366025 1.36603i −0.366025 1.36603i
\(446\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(447\) 0 0
\(448\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(449\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −0.258819 0.965926i −0.258819 0.965926i
\(454\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(455\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(456\) 0 0
\(457\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −0.707107 0.707107i −0.707107 0.707107i
\(461\) 0.258819 0.965926i 0.258819 0.965926i −0.707107 0.707107i \(-0.750000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(462\) 0 0
\(463\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 0 0
\(465\) −1.22474 0.707107i −1.22474 0.707107i
\(466\) −0.866025 0.500000i −0.866025 0.500000i
\(467\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(468\) 0 0
\(469\) −1.00000 1.00000i −1.00000 1.00000i
\(470\) 0 0
\(471\) 0 0
\(472\) −0.965926 0.258819i −0.965926 0.258819i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(479\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(480\) 1.00000 1.00000
\(481\) 0 0
\(482\) 0 0
\(483\) 0.965926 0.258819i 0.965926 0.258819i
\(484\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(485\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(486\) 0 0
\(487\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(488\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(489\) 0 0
\(490\) −0.965926 0.258819i −0.965926 0.258819i
\(491\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(492\) 1.36603 0.366025i 1.36603 0.366025i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.22474 0.707107i −1.22474 0.707107i
\(497\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(498\) 0 0
\(499\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(500\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(501\) 0.366025 1.36603i 0.366025 1.36603i
\(502\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(503\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.965926 0.258819i −0.965926 0.258819i
\(508\) −1.00000 −1.00000
\(509\) 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
\(511\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(512\) 1.00000 1.00000
\(513\) 0 0
\(514\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(515\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(516\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(517\) 0 0
\(518\) 0 0
\(519\) 1.00000i 1.00000i
\(520\) −0.500000 0.866025i −0.500000 0.866025i
\(521\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(522\) 0 0
\(523\) −0.258819 0.965926i −0.258819 0.965926i −0.965926 0.258819i \(-0.916667\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(524\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(525\) 0 0
\(526\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(527\) 0 0
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.00000 1.00000i −1.00000 1.00000i
\(534\) −1.00000 1.00000i −1.00000 1.00000i
\(535\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(536\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(537\) −0.707107 1.22474i −0.707107 1.22474i
\(538\) −0.707107 0.707107i −0.707107 0.707107i
\(539\) 0 0
\(540\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(541\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(542\) 1.22474 0.707107i 1.22474 0.707107i
\(543\) 0.500000 0.866025i 0.500000 0.866025i
\(544\) 0 0
\(545\) 0 0
\(546\) 1.00000 1.00000
\(547\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(548\) −0.866025 0.500000i −0.866025 0.500000i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) −0.965926 0.258819i −0.965926 0.258819i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(558\) 0 0
\(559\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(560\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(561\) 0 0
\(562\) −1.73205 1.00000i −1.73205 1.00000i
\(563\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.965926 0.258819i −0.965926 0.258819i
\(567\) 1.00000i 1.00000i
\(568\) 0.866025 0.500000i 0.866025 0.500000i
\(569\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(570\) 0 0
\(571\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(575\) 0 0
\(576\) 0 0
\(577\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(578\) −0.500000 0.866025i −0.500000 0.866025i
\(579\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(580\) 0 0
\(581\) 0 0
\(582\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(583\) 0 0
\(584\) 1.41421 1.41421
\(585\) 0 0
\(586\) 1.41421 1.41421i 1.41421 1.41421i
\(587\) −0.258819 0.965926i −0.258819 0.965926i −0.965926 0.258819i \(-0.916667\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(588\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(589\) 0 0
\(590\) 1.00000i 1.00000i
\(591\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(592\) 0 0
\(593\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(598\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(599\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(602\) −1.36603 0.366025i −1.36603 0.366025i
\(603\) 0 0
\(604\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(605\) 0.258819 0.965926i 0.258819 0.965926i
\(606\) 0 0
\(607\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 1.00000 1.00000
\(611\) 0 0
\(612\) 0 0
\(613\) 1.36603 0.366025i 1.36603 0.366025i 0.500000 0.866025i \(-0.333333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(614\) 0.965926 0.258819i 0.965926 0.258819i
\(615\) −0.707107 1.22474i −0.707107 1.22474i
\(616\) 0 0
\(617\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(618\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(619\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(620\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(621\) 0.965926 0.258819i 0.965926 0.258819i
\(622\) 0 0
\(623\) 1.41421i 1.41421i
\(624\) −0.866025 0.500000i −0.866025 0.500000i
\(625\) 1.00000 1.00000
\(626\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(635\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(636\) 0 0
\(637\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(638\) 0 0
\(639\) 0 0
\(640\) −0.258819 0.965926i −0.258819 0.965926i
\(641\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(642\) 1.41421i 1.41421i
\(643\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(644\) −0.500000 0.866025i −0.500000 0.866025i
\(645\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(646\) 0 0
\(647\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(648\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(649\) 0 0
\(650\) 0 0
\(651\) −1.00000 1.00000i −1.00000 1.00000i
\(652\) 0 0
\(653\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(654\) 0 0
\(655\) 1.00000 1.00000
\(656\) −0.707107 1.22474i −0.707107 1.22474i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(660\) 0 0
\(661\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(662\) −1.00000 1.00000i −1.00000 1.00000i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −1.41421 −1.41421
\(669\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(670\) 1.22474 0.707107i 1.22474 0.707107i
\(671\) 0 0
\(672\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(673\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 1.00000i 1.00000i
\(677\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(678\) 0 0
\(679\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(680\) 0 0
\(681\) 1.00000i 1.00000i
\(682\) 0 0
\(683\) 0.366025 1.36603i 0.366025 1.36603i −0.500000 0.866025i \(-0.666667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(684\) 0 0
\(685\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(686\) −0.866025 0.500000i −0.866025 0.500000i
\(687\) 0 0
\(688\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(689\) 0 0
\(690\) 1.00000i 1.00000i
\(691\) −0.965926 + 0.258819i −0.965926 + 0.258819i −0.707107 0.707107i \(-0.750000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(692\) 0.965926 0.258819i 0.965926 0.258819i
\(693\) 0 0
\(694\) −1.00000 1.00000i −1.00000 1.00000i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0.258819 0.965926i 0.258819 0.965926i
\(699\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(700\) 0 0
\(701\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(702\) 1.00000 1.00000
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 1.22474 0.707107i 1.22474 0.707107i
\(707\) 0 0
\(708\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(709\) 0.366025 1.36603i 0.366025 1.36603i −0.500000 0.866025i \(-0.666667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(710\) −0.707107 0.707107i −0.707107 0.707107i
\(711\) 0 0
\(712\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(713\) 0.707107 1.22474i 0.707107 1.22474i
\(714\) 0 0
\(715\) 0 0
\(716\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(717\) 0.965926 0.258819i 0.965926 0.258819i
\(718\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(719\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) 0 0
\(721\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(722\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(723\) 0 0
\(724\) −0.965926 0.258819i −0.965926 0.258819i
\(725\) 0 0
\(726\) −0.258819 0.965926i −0.258819 0.965926i
\(727\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) −0.258819 0.965926i −0.258819 0.965926i
\(729\) 1.00000i 1.00000i
\(730\) −0.366025 1.36603i −0.366025 1.36603i
\(731\) 0 0
\(732\) 0.866025 0.500000i 0.866025 0.500000i
\(733\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(734\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(735\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(736\) 1.00000i 1.00000i
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(744\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(750\) 0.866025 0.500000i 0.866025 0.500000i
\(751\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 1.00000 1.00000
\(754\) 0 0
\(755\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(756\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(757\) −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(758\) 0.366025 1.36603i 0.366025 1.36603i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.500000 0.866025i 0.500000 0.866025i
\(768\) −0.707107 0.707107i −0.707107 0.707107i
\(769\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(770\) 0 0
\(771\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(772\) 1.00000 1.00000
\(773\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.22474 0.707107i 1.22474 0.707107i
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(785\) 0 0
\(786\) 0.866025 0.500000i 0.866025 0.500000i
\(787\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(788\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(789\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.866025 0.500000i −0.866025 0.500000i
\(794\) −0.707107 0.707107i −0.707107 0.707107i
\(795\) 0 0
\(796\) 0.707107 1.22474i 0.707107 1.22474i
\(797\) 0.258819 0.965926i 0.258819 0.965926i −0.707107 0.707107i \(-0.750000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0.707107 1.22474i 0.707107 1.22474i
\(805\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(806\) 1.00000 1.00000i 1.00000 1.00000i
\(807\) 1.00000i 1.00000i
\(808\) 0 0
\(809\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(811\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(812\) 0 0
\(813\) −1.36603 0.366025i −1.36603 0.366025i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 1.41421 1.41421
\(819\) 0 0
\(820\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(821\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(822\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(823\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(824\) 1.41421 1.41421
\(825\) 0 0
\(826\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(827\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(828\) 0 0
\(829\) −0.965926 + 0.258819i −0.965926 + 0.258819i −0.707107 0.707107i \(-0.750000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(833\) 0 0
\(834\) 0 0
\(835\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(836\) 0 0
\(837\) −1.00000 1.00000i −1.00000 1.00000i
\(838\) 0.517638 1.93185i 0.517638 1.93185i
\(839\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(840\) 1.00000i 1.00000i
\(841\) 0.866025 0.500000i 0.866025 0.500000i
\(842\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(843\) 0.517638 + 1.93185i 0.517638 + 1.93185i
\(844\) 0 0
\(845\) 0.965926 0.258819i 0.965926 0.258819i
\(846\) 0 0
\(847\) 0.500000 0.866025i 0.500000 0.866025i
\(848\) 0 0
\(849\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(850\) 0 0
\(851\) 0 0
\(852\) −0.965926 0.258819i −0.965926 0.258819i
\(853\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(854\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(855\) 0 0
\(856\) 1.36603 0.366025i 1.36603 0.366025i
\(857\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(858\) 0 0
\(859\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(860\) 0.707107 1.22474i 0.707107 1.22474i
\(861\) −0.366025 1.36603i −0.366025 1.36603i
\(862\) −0.500000 0.866025i −0.500000 0.866025i
\(863\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(864\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(865\) −0.500000 0.866025i −0.500000 0.866025i
\(866\) 0 0
\(867\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(868\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(869\) 0 0
\(870\) 0 0
\(871\) −1.41421 −1.41421
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(876\) −1.00000 1.00000i −1.00000 1.00000i
\(877\) 1.36603 + 0.366025i 1.36603 + 0.366025i 0.866025 0.500000i \(-0.166667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(878\) 0 0
\(879\) −2.00000 −2.00000
\(880\) 0 0
\(881\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(882\) 0 0
\(883\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(884\) 0 0
\(885\) 0.707107 0.707107i 0.707107 0.707107i
\(886\) 0 0
\(887\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(888\) 0 0
\(889\) 1.00000i 1.00000i
\(890\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(891\) 0 0
\(892\) 1.41421i 1.41421i
\(893\) 0 0
\(894\) 0 0
\(895\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(896\) 1.00000i 1.00000i
\(897\) 0.500000 0.866025i 0.500000 0.866025i
\(898\) −1.00000 −1.00000
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(904\) 0 0
\(905\) 1.00000i 1.00000i
\(906\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(907\) 1.36603 0.366025i 1.36603 0.366025i 0.500000 0.866025i \(-0.333333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(908\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(909\) 0 0
\(910\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(911\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(915\) −0.707107 0.707107i −0.707107 0.707107i
\(916\) 0 0
\(917\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(918\) 0 0
\(919\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(920\) 0.965926 0.258819i 0.965926 0.258819i
\(921\) −0.866025 0.500000i −0.866025 0.500000i
\(922\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(923\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(924\) 0 0
\(925\) 0 0
\(926\) 0.500000 0.866025i 0.500000 0.866025i
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(930\) 1.22474 0.707107i 1.22474 0.707107i
\(931\) 0 0
\(932\) 0.866025 0.500000i 0.866025 0.500000i
\(933\) 0 0
\(934\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 1.36603 0.366025i 1.36603 0.366025i
\(939\) 1.36603 0.366025i 1.36603 0.366025i
\(940\) 0 0
\(941\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(942\) 0 0
\(943\) 1.22474 0.707107i 1.22474 0.707107i
\(944\) 0.707107 0.707107i 0.707107 0.707107i
\(945\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(946\) 0 0
\(947\) 0.366025 + 1.36603i 0.366025 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(948\) 0 0
\(949\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(950\) 0 0
\(951\) 0.707107 1.22474i 0.707107 1.22474i
\(952\) 0 0
\(953\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −0.500000 0.866025i −0.500000 0.866025i
\(957\) 0 0
\(958\) 1.22474 0.707107i 1.22474 0.707107i
\(959\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(960\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(961\) −1.00000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.258819 0.965926i −0.258819 0.965926i
\(966\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(969\) 0 0
\(970\) −1.00000 1.00000i −1.00000 1.00000i
\(971\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1.00000i 1.00000i
\(975\) 0 0
\(976\) −0.707107 0.707107i −0.707107 0.707107i
\(977\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.707107 0.707107i 0.707107 0.707107i
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(985\) 0.707107 1.22474i 0.707107 1.22474i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(990\) 0 0
\(991\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(992\) 1.22474 0.707107i 1.22474 0.707107i
\(993\) 1.41421i 1.41421i
\(994\) −0.500000 0.866025i −0.500000 0.866025i
\(995\) −1.36603 0.366025i −1.36603 0.366025i
\(996\) 0 0
\(997\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1456.1.fx.a.1413.2 yes 8
7.6 odd 2 inner 1456.1.fx.a.1413.1 yes 8
13.3 even 3 inner 1456.1.fx.a.965.1 yes 8
16.13 even 4 inner 1456.1.fx.a.685.2 yes 8
91.55 odd 6 inner 1456.1.fx.a.965.2 yes 8
112.13 odd 4 inner 1456.1.fx.a.685.1 yes 8
208.29 even 12 inner 1456.1.fx.a.237.1 8
1456.237 odd 12 inner 1456.1.fx.a.237.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1456.1.fx.a.237.1 8 208.29 even 12 inner
1456.1.fx.a.237.2 yes 8 1456.237 odd 12 inner
1456.1.fx.a.685.1 yes 8 112.13 odd 4 inner
1456.1.fx.a.685.2 yes 8 16.13 even 4 inner
1456.1.fx.a.965.1 yes 8 13.3 even 3 inner
1456.1.fx.a.965.2 yes 8 91.55 odd 6 inner
1456.1.fx.a.1413.1 yes 8 7.6 odd 2 inner
1456.1.fx.a.1413.2 yes 8 1.1 even 1 trivial