Properties

Label 1740.1.v.d.347.3
Level $1740$
Weight $1$
Character 1740.347
Analytic conductor $0.868$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -116
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1740,1,Mod(347,1740)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1740.347"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1740, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1740 = 2^{2} \cdot 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1740.v (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.868373121981\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content 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: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

Embedding invariants

Embedding label 347.3
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1740.347
Dual form 1740.1.v.d.1043.3

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 - 0.707107i) q^{2} +(-0.258819 - 0.965926i) q^{3} -1.00000i q^{4} +(0.500000 - 0.866025i) q^{5} +(-0.866025 - 0.500000i) q^{6} +(-0.707107 - 0.707107i) q^{8} +(-0.866025 + 0.500000i) q^{9} +(-0.258819 - 0.965926i) q^{10} +0.517638i q^{11} +(-0.965926 + 0.258819i) q^{12} +(1.36603 - 1.36603i) q^{13} +(-0.965926 - 0.258819i) q^{15} -1.00000 q^{16} +(-0.258819 + 0.965926i) q^{18} +1.41421i q^{19} +(-0.866025 - 0.500000i) q^{20} +(0.366025 + 0.366025i) q^{22} +(-0.500000 + 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} -1.93185i q^{26} +(0.707107 + 0.707107i) q^{27} -1.00000 q^{29} +(-0.866025 + 0.500000i) q^{30} +0.517638 q^{31} +(-0.707107 + 0.707107i) q^{32} +(0.500000 - 0.133975i) q^{33} +(0.500000 + 0.866025i) q^{36} +(1.00000 + 1.00000i) q^{38} +(-1.67303 - 0.965926i) q^{39} +(-0.965926 + 0.258819i) q^{40} +(-1.22474 + 1.22474i) q^{43} +0.517638 q^{44} +1.00000i q^{45} +(0.707107 - 0.707107i) q^{47} +(0.258819 + 0.965926i) q^{48} +1.00000i q^{49} +(-0.965926 - 0.258819i) q^{50} +(-1.36603 - 1.36603i) q^{52} +(1.36603 + 1.36603i) q^{53} +1.00000 q^{54} +(0.448288 + 0.258819i) q^{55} +(1.36603 - 0.366025i) q^{57} +(-0.707107 + 0.707107i) q^{58} +(-0.258819 + 0.965926i) q^{60} +(0.366025 - 0.366025i) q^{62} +1.00000i q^{64} +(-0.500000 - 1.86603i) q^{65} +(0.258819 - 0.448288i) q^{66} +(0.965926 + 0.258819i) q^{72} +(-0.707107 + 0.707107i) q^{75} +1.41421 q^{76} +(-1.86603 + 0.500000i) q^{78} -1.93185i q^{79} +(-0.500000 + 0.866025i) q^{80} +(0.500000 - 0.866025i) q^{81} +1.73205i q^{86} +(0.258819 + 0.965926i) q^{87} +(0.366025 - 0.366025i) q^{88} +(0.707107 + 0.707107i) q^{90} +(-0.133975 - 0.500000i) q^{93} -1.00000i q^{94} +(1.22474 + 0.707107i) q^{95} +(0.866025 + 0.500000i) q^{96} +(0.707107 + 0.707107i) q^{98} +(-0.258819 - 0.448288i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5} + 4 q^{13} - 8 q^{16} - 4 q^{22} - 4 q^{24} - 4 q^{25} - 8 q^{29} + 4 q^{33} + 4 q^{36} + 8 q^{38} - 4 q^{52} + 4 q^{53} + 8 q^{54} + 4 q^{57} - 4 q^{62} - 4 q^{65} - 8 q^{78} - 4 q^{80} + 4 q^{81}+ \cdots - 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1740\mathbb{Z}\right)^\times\).

\(n\) \(581\) \(697\) \(871\) \(901\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 0.707107i 0.707107 0.707107i
\(3\) −0.258819 0.965926i −0.258819 0.965926i
\(4\) 1.00000i 1.00000i
\(5\) 0.500000 0.866025i 0.500000 0.866025i
\(6\) −0.866025 0.500000i −0.866025 0.500000i
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) −0.707107 0.707107i −0.707107 0.707107i
\(9\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(10\) −0.258819 0.965926i −0.258819 0.965926i
\(11\) 0.517638i 0.517638i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(12\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(13\) 1.36603 1.36603i 1.36603 1.36603i 0.500000 0.866025i \(-0.333333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(14\) 0 0
\(15\) −0.965926 0.258819i −0.965926 0.258819i
\(16\) −1.00000 −1.00000
\(17\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(19\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(20\) −0.866025 0.500000i −0.866025 0.500000i
\(21\) 0 0
\(22\) 0.366025 + 0.366025i 0.366025 + 0.366025i
\(23\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(25\) −0.500000 0.866025i −0.500000 0.866025i
\(26\) 1.93185i 1.93185i
\(27\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(28\) 0 0
\(29\) −1.00000 −1.00000
\(30\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(31\) 0.517638 0.517638 0.258819 0.965926i \(-0.416667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(32\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(33\) 0.500000 0.133975i 0.500000 0.133975i
\(34\) 0 0
\(35\) 0 0
\(36\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(37\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(39\) −1.67303 0.965926i −1.67303 0.965926i
\(40\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i \(0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(44\) 0.517638 0.517638
\(45\) 1.00000i 1.00000i
\(46\) 0 0
\(47\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(48\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(49\) 1.00000i 1.00000i
\(50\) −0.965926 0.258819i −0.965926 0.258819i
\(51\) 0 0
\(52\) −1.36603 1.36603i −1.36603 1.36603i
\(53\) 1.36603 + 1.36603i 1.36603 + 1.36603i 0.866025 + 0.500000i \(0.166667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 1.00000 1.00000
\(55\) 0.448288 + 0.258819i 0.448288 + 0.258819i
\(56\) 0 0
\(57\) 1.36603 0.366025i 1.36603 0.366025i
\(58\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0.366025 0.366025i 0.366025 0.366025i
\(63\) 0 0
\(64\) 1.00000i 1.00000i
\(65\) −0.500000 1.86603i −0.500000 1.86603i
\(66\) 0.258819 0.448288i 0.258819 0.448288i
\(67\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(73\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0 0
\(75\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(76\) 1.41421 1.41421
\(77\) 0 0
\(78\) −1.86603 + 0.500000i −1.86603 + 0.500000i
\(79\) 1.93185i 1.93185i −0.258819 0.965926i \(-0.583333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(80\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(81\) 0.500000 0.866025i 0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.73205i 1.73205i
\(87\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(88\) 0.366025 0.366025i 0.366025 0.366025i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(91\) 0 0
\(92\) 0 0
\(93\) −0.133975 0.500000i −0.133975 0.500000i
\(94\) 1.00000i 1.00000i
\(95\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(96\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(97\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(99\) −0.258819 0.448288i −0.258819 0.448288i
\(100\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(104\) −1.93185 −1.93185
\(105\) 0 0
\(106\) 1.93185 1.93185
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) 0.707107 0.707107i 0.707107 0.707107i
\(109\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(110\) 0.500000 0.133975i 0.500000 0.133975i
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(114\) 0.707107 1.22474i 0.707107 1.22474i
\(115\) 0 0
\(116\) 1.00000i 1.00000i
\(117\) −0.500000 + 1.86603i −0.500000 + 1.86603i
\(118\) 0 0
\(119\) 0 0
\(120\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(121\) 0.732051 0.732051
\(122\) 0 0
\(123\) 0 0
\(124\) 0.517638i 0.517638i
\(125\) −1.00000 −1.00000
\(126\) 0 0
\(127\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(128\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(129\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(130\) −1.67303 0.965926i −1.67303 0.965926i
\(131\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(132\) −0.133975 0.500000i −0.133975 0.500000i
\(133\) 0 0
\(134\) 0 0
\(135\) 0.965926 0.258819i 0.965926 0.258819i
\(136\) 0 0
\(137\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −0.866025 0.500000i −0.866025 0.500000i
\(142\) 0 0
\(143\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(144\) 0.866025 0.500000i 0.866025 0.500000i
\(145\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(146\) 0 0
\(147\) 0.965926 0.258819i 0.965926 0.258819i
\(148\) 0 0
\(149\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 1.00000i 1.00000i
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 1.00000 1.00000i 1.00000 1.00000i
\(153\) 0 0
\(154\) 0 0
\(155\) 0.258819 0.448288i 0.258819 0.448288i
\(156\) −0.965926 + 1.67303i −0.965926 + 1.67303i
\(157\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) −1.36603 1.36603i −1.36603 1.36603i
\(159\) 0.965926 1.67303i 0.965926 1.67303i
\(160\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(161\) 0 0
\(162\) −0.258819 0.965926i −0.258819 0.965926i
\(163\) 1.22474 1.22474i 1.22474 1.22474i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(164\) 0 0
\(165\) 0.133975 0.500000i 0.133975 0.500000i
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 2.73205i 2.73205i
\(170\) 0 0
\(171\) −0.707107 1.22474i −0.707107 1.22474i
\(172\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(173\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(174\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(175\) 0 0
\(176\) 0.517638i 0.517638i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 1.00000 1.00000
\(181\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) −0.448288 0.258819i −0.448288 0.258819i
\(187\) 0 0
\(188\) −0.707107 0.707107i −0.707107 0.707107i
\(189\) 0 0
\(190\) 1.36603 0.366025i 1.36603 0.366025i
\(191\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(192\) 0.965926 0.258819i 0.965926 0.258819i
\(193\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(194\) 0 0
\(195\) −1.67303 + 0.965926i −1.67303 + 0.965926i
\(196\) 1.00000 1.00000
\(197\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(198\) −0.500000 0.133975i −0.500000 0.133975i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.36603 + 1.36603i −1.36603 + 1.36603i
\(209\) −0.732051 −0.732051
\(210\) 0 0
\(211\) −0.517638 −0.517638 −0.258819 0.965926i \(-0.583333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(212\) 1.36603 1.36603i 1.36603 1.36603i
\(213\) 0 0
\(214\) 0 0
\(215\) 0.448288 + 1.67303i 0.448288 + 1.67303i
\(216\) 1.00000i 1.00000i
\(217\) 0 0
\(218\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(219\) 0 0
\(220\) 0.258819 0.448288i 0.258819 0.448288i
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(224\) 0 0
\(225\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) −0.366025 1.36603i −0.366025 1.36603i
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(233\) 0.366025 + 0.366025i 0.366025 + 0.366025i 0.866025 0.500000i \(-0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0.965926 + 1.67303i 0.965926 + 1.67303i
\(235\) −0.258819 0.965926i −0.258819 0.965926i
\(236\) 0 0
\(237\) −1.86603 + 0.500000i −1.86603 + 0.500000i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(241\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) 0.517638 0.517638i 0.517638 0.517638i
\(243\) −0.965926 0.258819i −0.965926 0.258819i
\(244\) 0 0
\(245\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(246\) 0 0
\(247\) 1.93185 + 1.93185i 1.93185 + 1.93185i
\(248\) −0.366025 0.366025i −0.366025 0.366025i
\(249\) 0 0
\(250\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(251\) 1.93185i 1.93185i 0.258819 + 0.965926i \(0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −2.00000 −2.00000
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) −0.366025 + 0.366025i −0.366025 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(258\) 1.67303 0.448288i 1.67303 0.448288i
\(259\) 0 0
\(260\) −1.86603 + 0.500000i −1.86603 + 0.500000i
\(261\) 0.866025 0.500000i 0.866025 0.500000i
\(262\) −1.00000 1.00000i −1.00000 1.00000i
\(263\) 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(264\) −0.448288 0.258819i −0.448288 0.258819i
\(265\) 1.86603 0.500000i 1.86603 0.500000i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0.500000 0.866025i 0.500000 0.866025i
\(271\) 1.93185 1.93185 0.965926 0.258819i \(-0.0833333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.448288 0.258819i 0.448288 0.258819i
\(276\) 0 0
\(277\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) −0.448288 + 0.258819i −0.448288 + 0.258819i
\(280\) 0 0
\(281\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(282\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(283\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 0 0
\(285\) 0.366025 1.36603i 0.366025 1.36603i
\(286\) 1.00000 1.00000
\(287\) 0 0
\(288\) 0.258819 0.965926i 0.258819 0.965926i
\(289\) 1.00000i 1.00000i
\(290\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0.500000 0.866025i 0.500000 0.866025i
\(295\) 0 0
\(296\) 0 0
\(297\) −0.366025 + 0.366025i −0.366025 + 0.366025i
\(298\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(299\) 0 0
\(300\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 1.41421i 1.41421i
\(305\) 0 0
\(306\) 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
\(309\) 0 0
\(310\) −0.133975 0.500000i −0.133975 0.500000i
\(311\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(312\) 0.500000 + 1.86603i 0.500000 + 1.86603i
\(313\) −0.366025 + 0.366025i −0.366025 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −1.93185 −1.93185
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) −0.500000 1.86603i −0.500000 1.86603i
\(319\) 0.517638i 0.517638i
\(320\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.866025 0.500000i −0.866025 0.500000i
\(325\) −1.86603 0.500000i −1.86603 0.500000i
\(326\) 1.73205i 1.73205i
\(327\) 0.965926 0.258819i 0.965926 0.258819i
\(328\) 0 0
\(329\) 0 0
\(330\) −0.258819 0.448288i −0.258819 0.448288i
\(331\) −1.93185 −1.93185 −0.965926 0.258819i \(-0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(338\) −1.93185 1.93185i −1.93185 1.93185i
\(339\) 0 0
\(340\) 0 0
\(341\) 0.267949i 0.267949i
\(342\) −1.36603 0.366025i −1.36603 0.366025i
\(343\) 0 0
\(344\) 1.73205 1.73205
\(345\) 0 0
\(346\) 1.41421 1.41421
\(347\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0.965926 0.258819i 0.965926 0.258819i
\(349\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(350\) 0 0
\(351\) 1.93185 1.93185
\(352\) −0.366025 0.366025i −0.366025 0.366025i
\(353\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.93185 −1.93185 −0.965926 0.258819i \(-0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(360\) 0.707107 0.707107i 0.707107 0.707107i
\(361\) −1.00000 −1.00000
\(362\) −1.22474 + 1.22474i −1.22474 + 1.22474i
\(363\) −0.189469 0.707107i −0.189469 0.707107i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −0.500000 + 0.133975i −0.500000 + 0.133975i
\(373\) −0.366025 + 0.366025i −0.366025 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(376\) −1.00000 −1.00000
\(377\) −1.36603 + 1.36603i −1.36603 + 1.36603i
\(378\) 0 0
\(379\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(380\) 0.707107 1.22474i 0.707107 1.22474i
\(381\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(382\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0.500000 0.866025i 0.500000 0.866025i
\(385\) 0 0
\(386\) 0 0
\(387\) 0.448288 1.67303i 0.448288 1.67303i
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) −0.500000 + 1.86603i −0.500000 + 1.86603i
\(391\) 0 0
\(392\) 0.707107 0.707107i 0.707107 0.707107i
\(393\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(394\) 1.41421i 1.41421i
\(395\) −1.67303 0.965926i −1.67303 0.965926i
\(396\) −0.448288 + 0.258819i −0.448288 + 0.258819i
\(397\) −0.366025 0.366025i −0.366025 0.366025i 0.500000 0.866025i \(-0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(401\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) 0 0
\(403\) 0.707107 0.707107i 0.707107 0.707107i
\(404\) 0 0
\(405\) −0.500000 0.866025i −0.500000 0.866025i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.93185i 1.93185i
\(417\) 0 0
\(418\) −0.517638 + 0.517638i −0.517638 + 0.517638i
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −0.366025 + 0.366025i −0.366025 + 0.366025i
\(423\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(424\) 1.93185i 1.93185i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.500000 0.866025i 0.500000 0.866025i
\(430\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −0.707107 0.707107i −0.707107 0.707107i
\(433\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) 0 0
\(435\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(436\) 1.00000 1.00000
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −0.133975 0.500000i −0.133975 0.500000i
\(441\) −0.500000 0.866025i −0.500000 0.866025i
\(442\) 0 0
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0.965926 0.258819i 0.965926 0.258819i
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) −1.22474 0.707107i −1.22474 0.707107i
\(457\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(464\) 1.00000 1.00000
\(465\) −0.500000 0.133975i −0.500000 0.133975i
\(466\) 0.517638 0.517638
\(467\) −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i \(0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(468\) 1.86603 + 0.500000i 1.86603 + 0.500000i
\(469\) 0 0
\(470\) −0.866025 0.500000i −0.866025 0.500000i
\(471\) 0 0
\(472\) 0 0
\(473\) −0.633975 0.633975i −0.633975 0.633975i
\(474\) −0.965926 + 1.67303i −0.965926 + 1.67303i
\(475\) 1.22474 0.707107i 1.22474 0.707107i
\(476\) 0 0
\(477\) −1.86603 0.500000i −1.86603 0.500000i
\(478\) 0 0
\(479\) 1.93185 1.93185 0.965926 0.258819i \(-0.0833333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(480\) 0.866025 0.500000i 0.866025 0.500000i
\(481\) 0 0
\(482\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(483\) 0 0
\(484\) 0.732051i 0.732051i
\(485\) 0 0
\(486\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(487\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(488\) 0 0
\(489\) −1.50000 0.866025i −1.50000 0.866025i
\(490\) 0.965926 0.258819i 0.965926 0.258819i
\(491\) 1.93185i 1.93185i −0.258819 0.965926i \(-0.583333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 2.73205 2.73205
\(495\) −0.517638 −0.517638
\(496\) −0.517638 −0.517638
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 1.00000i 1.00000i
\(501\) 0 0
\(502\) 1.36603 + 1.36603i 1.36603 + 1.36603i
\(503\) −1.22474 1.22474i −1.22474 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2.63896 + 0.707107i −2.63896 + 0.707107i
\(508\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(509\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.707107 0.707107i 0.707107 0.707107i
\(513\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(514\) 0.517638i 0.517638i
\(515\) 0 0
\(516\) 0.866025 1.50000i 0.866025 1.50000i
\(517\) 0.366025 + 0.366025i 0.366025 + 0.366025i
\(518\) 0 0
\(519\) 0.707107 1.22474i 0.707107 1.22474i
\(520\) −0.965926 + 1.67303i −0.965926 + 1.67303i
\(521\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(522\) 0.258819 0.965926i 0.258819 0.965926i
\(523\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(524\) −1.41421 −1.41421
\(525\) 0 0
\(526\) 1.73205 1.73205
\(527\) 0 0
\(528\) −0.500000 + 0.133975i −0.500000 + 0.133975i
\(529\) 1.00000i 1.00000i
\(530\) 0.965926 1.67303i 0.965926 1.67303i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.517638 −0.517638
\(540\) −0.258819 0.965926i −0.258819 0.965926i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 1.36603 1.36603i 1.36603 1.36603i
\(543\) 0.448288 + 1.67303i 0.448288 + 1.67303i
\(544\) 0 0
\(545\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(546\) 0 0
\(547\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0.133975 0.500000i 0.133975 0.500000i
\(551\) 1.41421i 1.41421i
\(552\) 0 0
\(553\) 0 0
\(554\) 1.41421 1.41421
\(555\) 0 0
\(556\) 0 0
\(557\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(558\) −0.133975 + 0.500000i −0.133975 + 0.500000i
\(559\) 3.34607i 3.34607i
\(560\) 0 0
\(561\) 0 0
\(562\) −0.707107 0.707107i −0.707107 0.707107i
\(563\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(564\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) −0.707107 1.22474i −0.707107 1.22474i
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0.707107 0.707107i 0.707107 0.707107i
\(573\) 1.36603 0.366025i 1.36603 0.366025i
\(574\) 0 0
\(575\) 0 0
\(576\) −0.500000 0.866025i −0.500000 0.866025i
\(577\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(578\) −0.707107 0.707107i −0.707107 0.707107i
\(579\) 0 0
\(580\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(581\) 0 0
\(582\) 0 0
\(583\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(584\) 0 0
\(585\) 1.36603 + 1.36603i 1.36603 + 1.36603i
\(586\) 0 0
\(587\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(588\) −0.258819 0.965926i −0.258819 0.965926i
\(589\) 0.732051i 0.732051i
\(590\) 0 0
\(591\) −1.22474 0.707107i −1.22474 0.707107i
\(592\) 0 0
\(593\) −1.36603 1.36603i −1.36603 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(594\) 0.517638i 0.517638i
\(595\) 0 0
\(596\) 1.00000i 1.00000i
\(597\) 0 0
\(598\) 0 0
\(599\) 0.517638 0.517638 0.258819 0.965926i \(-0.416667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(600\) 1.00000 1.00000
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.366025 0.633975i 0.366025 0.633975i
\(606\) 0 0
\(607\) −1.22474 1.22474i −1.22474 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(608\) −1.00000 1.00000i −1.00000 1.00000i
\(609\) 0 0
\(610\) 0 0
\(611\) 1.93185i 1.93185i
\(612\) 0 0
\(613\) −1.36603 + 1.36603i −1.36603 + 1.36603i −0.500000 + 0.866025i \(0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(614\) −1.00000 −1.00000
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(618\) 0 0
\(619\) 1.93185i 1.93185i 0.258819 + 0.965926i \(0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(620\) −0.448288 0.258819i −0.448288 0.258819i
\(621\) 0 0
\(622\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(623\) 0 0
\(624\) 1.67303 + 0.965926i 1.67303 + 0.965926i
\(625\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(626\) 0.517638i 0.517638i
\(627\) 0.189469 + 0.707107i 0.189469 + 0.707107i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) −1.36603 + 1.36603i −1.36603 + 1.36603i
\(633\) 0.133975 + 0.500000i 0.133975 + 0.500000i
\(634\) 0 0
\(635\) −1.93185 + 0.517638i −1.93185 + 0.517638i
\(636\) −1.67303 0.965926i −1.67303 0.965926i
\(637\) 1.36603 + 1.36603i 1.36603 + 1.36603i
\(638\) −0.366025 0.366025i −0.366025 0.366025i
\(639\) 0 0
\(640\) 0.965926 0.258819i 0.965926 0.258819i
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 0 0
\(645\) 1.50000 0.866025i 1.50000 0.866025i
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(649\) 0 0
\(650\) −1.67303 + 0.965926i −1.67303 + 0.965926i
\(651\) 0 0
\(652\) −1.22474 1.22474i −1.22474 1.22474i
\(653\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 0.500000 0.866025i 0.500000 0.866025i
\(655\) −1.22474 0.707107i −1.22474 0.707107i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.93185 1.93185 0.965926 0.258819i \(-0.0833333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(660\) −0.500000 0.133975i −0.500000 0.133975i
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) −1.36603 + 1.36603i −1.36603 + 1.36603i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.366025 0.366025i 0.366025 0.366025i −0.500000 0.866025i \(-0.666667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(674\) 0 0
\(675\) 0.258819 0.965926i 0.258819 0.965926i
\(676\) −2.73205 −2.73205
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0.189469 + 0.189469i 0.189469 + 0.189469i
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 1.22474 1.22474i 1.22474 1.22474i
\(689\) 3.73205 3.73205
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 1.00000 1.00000i 1.00000 1.00000i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0.500000 0.866025i 0.500000 0.866025i
\(697\) 0 0
\(698\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(699\) 0.258819 0.448288i 0.258819 0.448288i
\(700\) 0 0
\(701\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(702\) 1.36603 1.36603i 1.36603 1.36603i
\(703\) 0 0
\(704\) −0.517638 −0.517638
\(705\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(706\) 1.41421 1.41421
\(707\) 0 0
\(708\) 0 0
\(709\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(710\) 0 0
\(711\) 0.965926 + 1.67303i 0.965926 + 1.67303i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0.965926 0.258819i 0.965926 0.258819i
\(716\) 0 0
\(717\) 0 0
\(718\) −1.36603 + 1.36603i −1.36603 + 1.36603i
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 1.00000i 1.00000i
\(721\) 0 0
\(722\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(723\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(724\) 1.73205i 1.73205i
\(725\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(726\) −0.633975 0.366025i −0.633975 0.366025i
\(727\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) 0 0
\(729\) 1.00000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) 0 0
\(735\) 0.258819 0.965926i 0.258819 0.965926i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0.517638i 0.517638i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(740\) 0 0
\(741\) 1.36603 2.36603i 1.36603 2.36603i
\(742\) 0 0
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) −0.258819 + 0.448288i −0.258819 + 0.448288i
\(745\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(746\) 0.517638i 0.517638i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(751\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(752\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(753\) 1.86603 0.500000i 1.86603 0.500000i
\(754\) 1.93185i 1.93185i
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(759\) 0 0
\(760\) −0.366025 1.36603i −0.366025 1.36603i
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0.517638 + 1.93185i 0.517638 + 1.93185i
\(763\) 0 0
\(764\) 1.41421 1.41421
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.258819 0.965926i −0.258819 0.965926i
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0.448288 + 0.258819i 0.448288 + 0.258819i
\(772\) 0 0
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) −0.866025 1.50000i −0.866025 1.50000i
\(775\) −0.258819 0.448288i −0.258819 0.448288i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0.965926 + 1.67303i 0.965926 + 1.67303i
\(781\) 0 0
\(782\) 0 0
\(783\) −0.707107 0.707107i −0.707107 0.707107i
\(784\) 1.00000i 1.00000i
\(785\) 0 0
\(786\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(787\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(788\) −1.00000 1.00000i −1.00000 1.00000i
\(789\) 0.866025 1.50000i 0.866025 1.50000i
\(790\) −1.86603 + 0.500000i −1.86603 + 0.500000i
\(791\) 0 0
\(792\) −0.133975 + 0.500000i −0.133975 + 0.500000i
\(793\) 0 0
\(794\) −0.517638 −0.517638
\(795\) −0.965926 1.67303i −0.965926 1.67303i
\(796\) 0 0
\(797\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(801\) 0 0
\(802\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 1.00000i 1.00000i
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) −0.965926 0.258819i −0.965926 0.258819i
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) −0.500000 1.86603i −0.500000 1.86603i
\(814\) 0 0
\(815\) −0.448288 1.67303i −0.448288 1.67303i
\(816\) 0 0
\(817\) −1.73205 1.73205i −1.73205 1.73205i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(824\) 0 0
\(825\) −0.366025 0.366025i −0.366025 0.366025i
\(826\) 0 0
\(827\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0.707107 1.22474i 0.707107 1.22474i
\(832\) 1.36603 + 1.36603i 1.36603 + 1.36603i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0.732051i 0.732051i
\(837\) 0.366025 + 0.366025i 0.366025 + 0.366025i
\(838\) 0 0
\(839\) 0.517638 0.517638 0.258819 0.965926i \(-0.416667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(844\) 0.517638i 0.517638i
\(845\) −2.36603 1.36603i −2.36603 1.36603i
\(846\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(847\) 0 0
\(848\) −1.36603 1.36603i −1.36603 1.36603i
\(849\) 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
\(855\) −1.41421 −1.41421
\(856\) 0 0
\(857\) −1.36603 + 1.36603i −1.36603 + 1.36603i −0.500000 + 0.866025i \(0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(858\) −0.258819 0.965926i −0.258819 0.965926i
\(859\) 0.517638i 0.517638i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(860\) 1.67303 0.448288i 1.67303 0.448288i
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(864\) −1.00000 −1.00000
\(865\) 1.36603 0.366025i 1.36603 0.366025i
\(866\) 0 0
\(867\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(868\) 0 0
\(869\) 1.00000 1.00000
\(870\) 0.866025 0.500000i 0.866025 0.500000i
\(871\) 0 0
\(872\) 0.707107 0.707107i 0.707107 0.707107i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.366025 + 0.366025i 0.366025 + 0.366025i 0.866025 0.500000i \(-0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −0.448288 0.258819i −0.448288 0.258819i
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) −0.965926 0.258819i −0.965926 0.258819i
\(883\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.22474 1.22474i 1.22474 1.22474i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.448288 + 0.258819i 0.448288 + 0.258819i
\(892\) 0 0
\(893\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(894\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −0.517638 −0.517638
\(900\) 0.500000 0.866025i 0.500000 0.866025i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(906\) 0 0
\(907\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.93185i 1.93185i −0.258819 0.965926i \(-0.583333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(912\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(913\) 0 0
\(914\) −1.41421 −1.41421
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0.707107 0.707107i 0.707107 0.707107i
\(929\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(930\) −0.448288 + 0.258819i −0.448288 + 0.258819i
\(931\) −1.41421 −1.41421
\(932\) 0.366025 0.366025i 0.366025 0.366025i
\(933\) 1.36603 0.366025i 1.36603 0.366025i
\(934\) 1.73205i 1.73205i
\(935\) 0 0
\(936\) 1.67303 0.965926i 1.67303 0.965926i
\(937\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0.448288 + 0.258819i 0.448288 + 0.258819i
\(940\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(941\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −0.896575 −0.896575
\(947\) 1.22474 1.22474i 1.22474 1.22474i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(948\) 0.500000 + 1.86603i 0.500000 + 1.86603i
\(949\) 0 0
\(950\) 0.366025 1.36603i 0.366025 1.36603i
\(951\) 0 0
\(952\) 0 0
\(953\) −0.366025 0.366025i −0.366025 0.366025i 0.500000 0.866025i \(-0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(954\) −1.67303 + 0.965926i −1.67303 + 0.965926i
\(955\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(956\) 0 0
\(957\) −0.500000 + 0.133975i −0.500000 + 0.133975i
\(958\) 1.36603 1.36603i 1.36603 1.36603i
\(959\) 0 0
\(960\) 0.258819 0.965926i 0.258819 0.965926i
\(961\) −0.732051 −0.732051
\(962\) 0 0
\(963\) 0 0
\(964\) 1.00000i 1.00000i
\(965\) 0 0
\(966\) 0 0
\(967\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(968\) −0.517638 0.517638i −0.517638 0.517638i
\(969\) 0 0
\(970\) 0 0
\(971\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(972\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(973\) 0 0
\(974\) 0 0
\(975\) 1.93185i 1.93185i
\(976\) 0 0
\(977\) 0.366025 0.366025i 0.366025 0.366025i −0.500000 0.866025i \(-0.666667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(978\) −1.67303 + 0.448288i −1.67303 + 0.448288i
\(979\) 0 0
\(980\) 0.500000 0.866025i 0.500000 0.866025i
\(981\) −0.500000 0.866025i −0.500000 0.866025i
\(982\) −1.36603 1.36603i −1.36603 1.36603i
\(983\) 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(984\) 0 0
\(985\) −0.366025 1.36603i −0.366025 1.36603i
\(986\) 0 0
\(987\) 0 0
\(988\) 1.93185 1.93185i 1.93185 1.93185i
\(989\) 0 0
\(990\) −0.366025 + 0.366025i −0.366025 + 0.366025i
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) −0.366025 + 0.366025i −0.366025 + 0.366025i
\(993\) 0.500000 + 1.86603i 0.500000 + 1.86603i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(998\) 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 1740.1.v.d.347.3 yes 8
3.2 odd 2 1740.1.v.c.347.2 8
4.3 odd 2 inner 1740.1.v.d.347.2 yes 8
5.3 odd 4 1740.1.v.c.1043.2 yes 8
12.11 even 2 1740.1.v.c.347.3 yes 8
15.8 even 4 inner 1740.1.v.d.1043.3 yes 8
20.3 even 4 1740.1.v.c.1043.3 yes 8
29.28 even 2 inner 1740.1.v.d.347.2 yes 8
60.23 odd 4 inner 1740.1.v.d.1043.2 yes 8
87.86 odd 2 1740.1.v.c.347.3 yes 8
116.115 odd 2 CM 1740.1.v.d.347.3 yes 8
145.28 odd 4 1740.1.v.c.1043.3 yes 8
348.347 even 2 1740.1.v.c.347.2 8
435.173 even 4 inner 1740.1.v.d.1043.2 yes 8
580.463 even 4 1740.1.v.c.1043.2 yes 8
1740.1043 odd 4 inner 1740.1.v.d.1043.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1740.1.v.c.347.2 8 3.2 odd 2
1740.1.v.c.347.2 8 348.347 even 2
1740.1.v.c.347.3 yes 8 12.11 even 2
1740.1.v.c.347.3 yes 8 87.86 odd 2
1740.1.v.c.1043.2 yes 8 5.3 odd 4
1740.1.v.c.1043.2 yes 8 580.463 even 4
1740.1.v.c.1043.3 yes 8 20.3 even 4
1740.1.v.c.1043.3 yes 8 145.28 odd 4
1740.1.v.d.347.2 yes 8 4.3 odd 2 inner
1740.1.v.d.347.2 yes 8 29.28 even 2 inner
1740.1.v.d.347.3 yes 8 1.1 even 1 trivial
1740.1.v.d.347.3 yes 8 116.115 odd 2 CM
1740.1.v.d.1043.2 yes 8 60.23 odd 4 inner
1740.1.v.d.1043.2 yes 8 435.173 even 4 inner
1740.1.v.d.1043.3 yes 8 15.8 even 4 inner
1740.1.v.d.1043.3 yes 8 1740.1043 odd 4 inner