Properties

Label 2160.1.bn.b.269.1
Level $2160$
Weight $1$
Character 2160.269
Analytic conductor $1.078$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2160,1,Mod(269,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.269");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2160.bn (of order \(4\), degree \(2\), 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: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.276480.2

Embedding invariants

Embedding label 269.1
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2160.269
Dual form 2160.1.bn.b.1349.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +(-0.707107 - 0.707107i) q^{5} +1.41421 q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +(-0.707107 - 0.707107i) q^{5} +1.41421 q^{7} +1.00000i q^{8} +(-0.707107 + 0.707107i) q^{10} +(0.707107 + 0.707107i) q^{11} +(-0.707107 + 0.707107i) q^{13} -1.41421i q^{14} +1.00000 q^{16} +1.00000 q^{17} +(1.00000 + 1.00000i) q^{19} +(0.707107 + 0.707107i) q^{20} +(0.707107 - 0.707107i) q^{22} -1.00000i q^{23} +1.00000i q^{25} +(0.707107 + 0.707107i) q^{26} -1.41421 q^{28} +(-0.707107 + 0.707107i) q^{29} +1.00000 q^{31} -1.00000i q^{32} -1.00000i q^{34} +(-1.00000 - 1.00000i) q^{35} +(1.00000 - 1.00000i) q^{38} +(0.707107 - 0.707107i) q^{40} +(-0.707107 - 0.707107i) q^{43} +(-0.707107 - 0.707107i) q^{44} -1.00000 q^{46} -1.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +(0.707107 - 0.707107i) q^{52} +(1.00000 - 1.00000i) q^{53} -1.00000i q^{55} +1.41421i q^{56} +(0.707107 + 0.707107i) q^{58} -1.00000i q^{62} -1.00000 q^{64} +1.00000 q^{65} -1.00000 q^{68} +(-1.00000 + 1.00000i) q^{70} -1.41421 q^{71} +(-1.00000 - 1.00000i) q^{76} +(1.00000 + 1.00000i) q^{77} +1.00000 q^{79} +(-0.707107 - 0.707107i) q^{80} +(-0.707107 - 0.707107i) q^{85} +(-0.707107 + 0.707107i) q^{86} +(-0.707107 + 0.707107i) q^{88} +1.41421 q^{89} +(-1.00000 + 1.00000i) q^{91} +1.00000i q^{92} +1.00000i q^{94} -1.41421i q^{95} -1.41421i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{16} + 4 q^{17} + 4 q^{19} + 4 q^{31} - 4 q^{35} + 4 q^{38} - 4 q^{46} - 4 q^{47} + 4 q^{49} + 4 q^{50} + 4 q^{53} - 4 q^{64} + 4 q^{65} - 4 q^{68} - 4 q^{70} - 4 q^{76} + 4 q^{77} + 4 q^{79} - 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 1.00000i
\(3\) 0 0
\(4\) −1.00000 −1.00000
\(5\) −0.707107 0.707107i −0.707107 0.707107i
\(6\) 0 0
\(7\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 1.00000i 1.00000i
\(9\) 0 0
\(10\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(11\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(12\) 0 0
\(13\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(14\) 1.41421i 1.41421i
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(20\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(21\) 0 0
\(22\) 0.707107 0.707107i 0.707107 0.707107i
\(23\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(24\) 0 0
\(25\) 1.00000i 1.00000i
\(26\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(27\) 0 0
\(28\) −1.41421 −1.41421
\(29\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(30\) 0 0
\(31\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 1.00000i 1.00000i
\(33\) 0 0
\(34\) 1.00000i 1.00000i
\(35\) −1.00000 1.00000i −1.00000 1.00000i
\(36\) 0 0
\(37\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 1.00000 1.00000i 1.00000 1.00000i
\(39\) 0 0
\(40\) 0.707107 0.707107i 0.707107 0.707107i
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(44\) −0.707107 0.707107i −0.707107 0.707107i
\(45\) 0 0
\(46\) −1.00000 −1.00000
\(47\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) 1.00000 1.00000
\(50\) 1.00000 1.00000
\(51\) 0 0
\(52\) 0.707107 0.707107i 0.707107 0.707107i
\(53\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(54\) 0 0
\(55\) 1.00000i 1.00000i
\(56\) 1.41421i 1.41421i
\(57\) 0 0
\(58\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(59\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(62\) 1.00000i 1.00000i
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 1.00000 1.00000
\(66\) 0 0
\(67\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) −1.00000 −1.00000
\(69\) 0 0
\(70\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(71\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.00000 1.00000i −1.00000 1.00000i
\(77\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(78\) 0 0
\(79\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) −0.707107 0.707107i −0.707107 0.707107i
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) −0.707107 0.707107i −0.707107 0.707107i
\(86\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(87\) 0 0
\(88\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(89\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(90\) 0 0
\(91\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(92\) 1.00000i 1.00000i
\(93\) 0 0
\(94\) 1.00000i 1.00000i
\(95\) 1.41421i 1.41421i
\(96\) 0 0
\(97\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(98\) 1.00000i 1.00000i
\(99\) 0 0
\(100\) 1.00000i 1.00000i
\(101\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(102\) 0 0
\(103\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(104\) −0.707107 0.707107i −0.707107 0.707107i
\(105\) 0 0
\(106\) −1.00000 1.00000i −1.00000 1.00000i
\(107\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) 0 0
\(109\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(110\) −1.00000 −1.00000
\(111\) 0 0
\(112\) 1.41421 1.41421
\(113\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(114\) 0 0
\(115\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(116\) 0.707107 0.707107i 0.707107 0.707107i
\(117\) 0 0
\(118\) 0 0
\(119\) 1.41421 1.41421
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) −1.00000 −1.00000
\(125\) 0.707107 0.707107i 0.707107 0.707107i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 1.00000i 1.00000i
\(129\) 0 0
\(130\) 1.00000i 1.00000i
\(131\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(132\) 0 0
\(133\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(134\) 0 0
\(135\) 0 0
\(136\) 1.00000i 1.00000i
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(140\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(141\) 0 0
\(142\) 1.41421i 1.41421i
\(143\) −1.00000 −1.00000
\(144\) 0 0
\(145\) 1.00000 1.00000
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(150\) 0 0
\(151\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(152\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(153\) 0 0
\(154\) 1.00000 1.00000i 1.00000 1.00000i
\(155\) −0.707107 0.707107i −0.707107 0.707107i
\(156\) 0 0
\(157\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(158\) 1.00000i 1.00000i
\(159\) 0 0
\(160\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(161\) 1.41421i 1.41421i
\(162\) 0 0
\(163\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(171\) 0 0
\(172\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(173\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 1.41421i 1.41421i
\(176\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(177\) 0 0
\(178\) 1.41421i 1.41421i
\(179\) −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i \(0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(180\) 0 0
\(181\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(182\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(183\) 0 0
\(184\) 1.00000 1.00000
\(185\) 0 0
\(186\) 0 0
\(187\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(188\) 1.00000 1.00000
\(189\) 0 0
\(190\) −1.41421 −1.41421
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(194\) −1.41421 −1.41421
\(195\) 0 0
\(196\) −1.00000 −1.00000
\(197\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(200\) −1.00000 −1.00000
\(201\) 0 0
\(202\) 0.707107 0.707107i 0.707107 0.707107i
\(203\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(204\) 0 0
\(205\) 0 0
\(206\) 1.41421i 1.41421i
\(207\) 0 0
\(208\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(209\) 1.41421i 1.41421i
\(210\) 0 0
\(211\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(212\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(213\) 0 0
\(214\) 0 0
\(215\) 1.00000i 1.00000i
\(216\) 0 0
\(217\) 1.41421 1.41421
\(218\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(219\) 0 0
\(220\) 1.00000i 1.00000i
\(221\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(222\) 0 0
\(223\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(224\) 1.41421i 1.41421i
\(225\) 0 0
\(226\) 1.00000i 1.00000i
\(227\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(230\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(231\) 0 0
\(232\) −0.707107 0.707107i −0.707107 0.707107i
\(233\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(236\) 0 0
\(237\) 0 0
\(238\) 1.41421i 1.41421i
\(239\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(240\) 0 0
\(241\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.707107 0.707107i −0.707107 0.707107i
\(246\) 0 0
\(247\) −1.41421 −1.41421
\(248\) 1.00000i 1.00000i
\(249\) 0 0
\(250\) −0.707107 0.707107i −0.707107 0.707107i
\(251\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(252\) 0 0
\(253\) 0.707107 0.707107i 0.707107 0.707107i
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.00000 −1.00000
\(261\) 0 0
\(262\) −0.707107 0.707107i −0.707107 0.707107i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −1.41421 −1.41421
\(266\) 1.41421 1.41421i 1.41421 1.41421i
\(267\) 0 0
\(268\) 0 0
\(269\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 1.00000 1.00000
\(273\) 0 0
\(274\) 0 0
\(275\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(276\) 0 0
\(277\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(278\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(279\) 0 0
\(280\) 1.00000 1.00000i 1.00000 1.00000i
\(281\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(282\) 0 0
\(283\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 1.41421 1.41421
\(285\) 0 0
\(286\) 1.00000i 1.00000i
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) 1.00000i 1.00000i
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0.707107 0.707107i 0.707107 0.707107i
\(299\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(300\) 0 0
\(301\) −1.00000 1.00000i −1.00000 1.00000i
\(302\) −1.00000 −1.00000
\(303\) 0 0
\(304\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(305\) 0 0
\(306\) 0 0
\(307\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(308\) −1.00000 1.00000i −1.00000 1.00000i
\(309\) 0 0
\(310\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) −0.707107 0.707107i −0.707107 0.707107i
\(315\) 0 0
\(316\) −1.00000 −1.00000
\(317\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) −1.00000 −1.00000
\(320\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(321\) 0 0
\(322\) −1.41421 −1.41421
\(323\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(324\) 0 0
\(325\) −0.707107 0.707107i −0.707107 0.707107i
\(326\) −0.707107 0.707107i −0.707107 0.707107i
\(327\) 0 0
\(328\) 0 0
\(329\) −1.41421 −1.41421
\(330\) 0 0
\(331\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(341\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(342\) 0 0
\(343\) 0 0
\(344\) 0.707107 0.707107i 0.707107 0.707107i
\(345\) 0 0
\(346\) 1.00000 1.00000i 1.00000 1.00000i
\(347\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(350\) 1.41421 1.41421
\(351\) 0 0
\(352\) 0.707107 0.707107i 0.707107 0.707107i
\(353\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(354\) 0 0
\(355\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(356\) −1.41421 −1.41421
\(357\) 0 0
\(358\) 1.41421 + 1.41421i 1.41421 + 1.41421i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 1.00000i 1.00000i
\(362\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(363\) 0 0
\(364\) 1.00000 1.00000i 1.00000 1.00000i
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 1.00000i 1.00000i
\(369\) 0 0
\(370\) 0 0
\(371\) 1.41421 1.41421i 1.41421 1.41421i
\(372\) 0 0
\(373\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(374\) 0.707107 0.707107i 0.707107 0.707107i
\(375\) 0 0
\(376\) 1.00000i 1.00000i
\(377\) 1.00000i 1.00000i
\(378\) 0 0
\(379\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(380\) 1.41421i 1.41421i
\(381\) 0 0
\(382\) 0 0
\(383\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(384\) 0 0
\(385\) 1.41421i 1.41421i
\(386\) 1.41421 1.41421
\(387\) 0 0
\(388\) 1.41421i 1.41421i
\(389\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(390\) 0 0
\(391\) 1.00000i 1.00000i
\(392\) 1.00000i 1.00000i
\(393\) 0 0
\(394\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(395\) −0.707107 0.707107i −0.707107 0.707107i
\(396\) 0 0
\(397\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(398\) −1.00000 −1.00000
\(399\) 0 0
\(400\) 1.00000i 1.00000i
\(401\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(402\) 0 0
\(403\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(404\) −0.707107 0.707107i −0.707107 0.707107i
\(405\) 0 0
\(406\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(407\) 0 0
\(408\) 0 0
\(409\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.41421 1.41421
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(417\) 0 0
\(418\) 1.41421 1.41421
\(419\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(420\) 0 0
\(421\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(425\) 1.00000i 1.00000i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 1.00000 1.00000
\(431\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(432\) 0 0
\(433\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) 1.41421i 1.41421i
\(435\) 0 0
\(436\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(437\) 1.00000 1.00000i 1.00000 1.00000i
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 1.00000 1.00000
\(441\) 0 0
\(442\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(443\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(444\) 0 0
\(445\) −1.00000 1.00000i −1.00000 1.00000i
\(446\) −1.41421 −1.41421
\(447\) 0 0
\(448\) −1.41421 −1.41421
\(449\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 1.00000 1.00000
\(453\) 0 0
\(454\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(455\) 1.41421 1.41421
\(456\) 0 0
\(457\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0.707107 0.707107i 0.707107 0.707107i
\(461\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(462\) 0 0
\(463\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(464\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(465\) 0 0
\(466\) 2.00000 2.00000
\(467\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0.707107 0.707107i 0.707107 0.707107i
\(471\) 0 0
\(472\) 0 0
\(473\) 1.00000i 1.00000i
\(474\) 0 0
\(475\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(476\) −1.41421 −1.41421
\(477\) 0 0
\(478\) 1.41421 1.41421
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1.00000i 1.00000i
\(483\) 0 0
\(484\) 0 0
\(485\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(491\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(492\) 0 0
\(493\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(494\) 1.41421i 1.41421i
\(495\) 0 0
\(496\) 1.00000 1.00000
\(497\) −2.00000 −2.00000
\(498\) 0 0
\(499\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(500\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(501\) 0 0
\(502\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(503\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(504\) 0 0
\(505\) 1.00000i 1.00000i
\(506\) −0.707107 0.707107i −0.707107 0.707107i
\(507\) 0 0
\(508\) 0 0
\(509\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 1.00000i
\(513\) 0 0
\(514\) 1.00000i 1.00000i
\(515\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(516\) 0 0
\(517\) −0.707107 0.707107i −0.707107 0.707107i
\(518\) 0 0
\(519\) 0 0
\(520\) 1.00000i 1.00000i
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(524\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(525\) 0 0
\(526\) 0 0
\(527\) 1.00000 1.00000
\(528\) 0 0
\(529\) 0 0
\(530\) 1.41421i 1.41421i
\(531\) 0 0
\(532\) −1.41421 1.41421i −1.41421 1.41421i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −0.707107 0.707107i −0.707107 0.707107i
\(539\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(540\) 0 0
\(541\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.00000i 1.00000i
\(545\) 1.41421i 1.41421i
\(546\) 0 0
\(547\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(551\) −1.41421 −1.41421
\(552\) 0 0
\(553\) 1.41421 1.41421
\(554\) 0 0
\(555\) 0 0
\(556\) 1.00000 1.00000i 1.00000 1.00000i
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 1.00000 1.00000
\(560\) −1.00000 1.00000i −1.00000 1.00000i
\(561\) 0 0
\(562\) 1.41421i 1.41421i
\(563\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(566\) 0 0
\(567\) 0 0
\(568\) 1.41421i 1.41421i
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(572\) 1.00000 1.00000
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 1.00000
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −1.00000 −1.00000
\(581\) 0 0
\(582\) 0 0
\(583\) 1.41421 1.41421
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 0 0
\(589\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(594\) 0 0
\(595\) −1.00000 1.00000i −1.00000 1.00000i
\(596\) −0.707107 0.707107i −0.707107 0.707107i
\(597\) 0 0
\(598\) 0.707107 0.707107i 0.707107 0.707107i
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(602\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(603\) 0 0
\(604\) 1.00000i 1.00000i
\(605\) 0 0
\(606\) 0 0
\(607\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 1.00000 1.00000i 1.00000 1.00000i
\(609\) 0 0
\(610\) 0 0
\(611\) 0.707107 0.707107i 0.707107 0.707107i
\(612\) 0 0
\(613\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(614\) −0.707107 0.707107i −0.707107 0.707107i
\(615\) 0 0
\(616\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(617\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(618\) 0 0
\(619\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(620\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(621\) 0 0
\(622\) 0 0
\(623\) 2.00000 2.00000
\(624\) 0 0
\(625\) −1.00000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 1.00000i 1.00000i
\(633\) 0 0
\(634\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(635\) 0 0
\(636\) 0 0
\(637\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(638\) 1.00000i 1.00000i
\(639\) 0 0
\(640\) 0.707107 0.707107i 0.707107 0.707107i
\(641\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(642\) 0 0
\(643\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(644\) 1.41421i 1.41421i
\(645\) 0 0
\(646\) 1.00000 1.00000i 1.00000 1.00000i
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(651\) 0 0
\(652\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(653\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(654\) 0 0
\(655\) −1.00000 −1.00000
\(656\) 0 0
\(657\) 0 0
\(658\) 1.41421i 1.41421i
\(659\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(660\) 0 0
\(661\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.00000i 2.00000i
\(666\) 0 0
\(667\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 1.41421 1.41421
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(678\) 0 0
\(679\) 2.00000i 2.00000i
\(680\) 0.707107 0.707107i 0.707107 0.707107i
\(681\) 0 0
\(682\) 0.707107 0.707107i 0.707107 0.707107i
\(683\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −0.707107 0.707107i −0.707107 0.707107i
\(689\) 1.41421i 1.41421i
\(690\) 0 0
\(691\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(692\) −1.00000 1.00000i −1.00000 1.00000i
\(693\) 0 0
\(694\) 0 0
\(695\) 1.41421 1.41421
\(696\) 0 0
\(697\) 0 0
\(698\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(699\) 0 0
\(700\) 1.41421i 1.41421i
\(701\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.707107 0.707107i −0.707107 0.707107i
\(705\) 0 0
\(706\) 1.00000i 1.00000i
\(707\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(708\) 0 0
\(709\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(710\) 1.00000 1.00000i 1.00000 1.00000i
\(711\) 0 0
\(712\) 1.41421i 1.41421i
\(713\) 1.00000i 1.00000i
\(714\) 0 0
\(715\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(716\) 1.41421 1.41421i 1.41421 1.41421i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) −2.00000 −2.00000
\(722\) 1.00000 1.00000
\(723\) 0 0
\(724\) 1.00000 1.00000i 1.00000 1.00000i
\(725\) −0.707107 0.707107i −0.707107 0.707107i
\(726\) 0 0
\(727\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) −1.00000 1.00000i −1.00000 1.00000i
\(729\) 0 0
\(730\) 0 0
\(731\) −0.707107 0.707107i −0.707107 0.707107i
\(732\) 0 0
\(733\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −1.00000 −1.00000
\(737\) 0 0
\(738\) 0 0
\(739\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.41421 1.41421i −1.41421 1.41421i
\(743\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(744\) 0 0
\(745\) 1.00000i 1.00000i
\(746\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(747\) 0 0
\(748\) −0.707107 0.707107i −0.707107 0.707107i
\(749\) 0 0
\(750\) 0 0
\(751\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) −1.00000 −1.00000
\(753\) 0 0
\(754\) −1.00000 −1.00000
\(755\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(756\) 0 0
\(757\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 1.41421 1.41421
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −1.41421 1.41421i −1.41421 1.41421i
\(764\) 0 0
\(765\) 0 0
\(766\) 1.00000i 1.00000i
\(767\) 0 0
\(768\) 0 0
\(769\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(770\) −1.41421 −1.41421
\(771\) 0 0
\(772\) 1.41421i 1.41421i
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0 0
\(775\) 1.00000i 1.00000i
\(776\) 1.41421 1.41421
\(777\) 0 0
\(778\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(779\) 0 0
\(780\) 0 0
\(781\) −1.00000 1.00000i −1.00000 1.00000i
\(782\) −1.00000 −1.00000
\(783\) 0 0
\(784\) 1.00000 1.00000
\(785\) −1.00000 −1.00000
\(786\) 0 0
\(787\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(788\) 1.00000 1.00000i 1.00000 1.00000i
\(789\) 0 0
\(790\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(791\) −1.41421 −1.41421
\(792\) 0 0
\(793\) 0 0
\(794\) −0.707107 0.707107i −0.707107 0.707107i
\(795\) 0 0
\(796\) 1.00000i 1.00000i
\(797\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) −1.00000 −1.00000
\(800\) 1.00000 1.00000
\(801\) 0 0
\(802\) −1.41421 −1.41421
\(803\) 0 0
\(804\) 0 0
\(805\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(806\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(807\) 0 0
\(808\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(809\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(810\) 0 0
\(811\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(812\) 1.00000 1.00000i 1.00000 1.00000i
\(813\) 0 0
\(814\) 0 0
\(815\) −1.00000 −1.00000
\(816\) 0 0
\(817\) 1.41421i 1.41421i
\(818\) 1.00000 1.00000
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 1.41421i 1.41421i
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(828\) 0 0
\(829\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.707107 0.707107i 0.707107 0.707107i
\(833\) 1.00000 1.00000
\(834\) 0 0
\(835\) 0 0
\(836\) 1.41421i 1.41421i
\(837\) 0 0
\(838\) −0.707107 0.707107i −0.707107 0.707107i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 1.00000 1.00000i 1.00000 1.00000i
\(849\) 0 0
\(850\) 1.00000 1.00000
\(851\) 0 0
\(852\) 0 0
\(853\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(860\) 1.00000i 1.00000i
\(861\) 0 0
\(862\) 1.41421 1.41421
\(863\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(864\) 0 0
\(865\) 1.41421i 1.41421i
\(866\) 1.41421 1.41421
\(867\) 0 0
\(868\) −1.41421 −1.41421
\(869\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(870\) 0 0
\(871\) 0 0
\(872\) 1.00000 1.00000i 1.00000 1.00000i
\(873\) 0 0
\(874\) −1.00000 1.00000i −1.00000 1.00000i
\(875\) 1.00000 1.00000i 1.00000 1.00000i
\(876\) 0 0
\(877\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 1.00000i 1.00000i
\(881\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(882\) 0 0
\(883\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(884\) 0.707107 0.707107i 0.707107 0.707107i
\(885\) 0 0
\(886\) 0 0
\(887\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(891\) 0 0
\(892\) 1.41421i 1.41421i
\(893\) −1.00000 1.00000i −1.00000 1.00000i
\(894\) 0 0
\(895\) 2.00000 2.00000
\(896\) 1.41421i 1.41421i
\(897\) 0 0
\(898\) 1.41421 1.41421
\(899\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(900\) 0 0
\(901\) 1.00000 1.00000i 1.00000 1.00000i
\(902\) 0 0
\(903\) 0 0
\(904\) 1.00000i 1.00000i
\(905\) 1.41421 1.41421
\(906\) 0 0
\(907\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(908\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(909\) 0 0
\(910\) 1.41421i 1.41421i
\(911\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1.41421i 1.41421i
\(915\) 0 0
\(916\) 0 0
\(917\) 1.00000 1.00000i 1.00000 1.00000i
\(918\) 0 0
\(919\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(920\) −0.707107 0.707107i −0.707107 0.707107i
\(921\) 0 0
\(922\) 0 0
\(923\) 1.00000 1.00000i 1.00000 1.00000i
\(924\) 0 0
\(925\) 0 0
\(926\) −1.41421 −1.41421
\(927\) 0 0
\(928\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(932\) 2.00000i 2.00000i
\(933\) 0 0
\(934\) 0 0
\(935\) 1.00000i 1.00000i
\(936\) 0 0
\(937\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −0.707107 0.707107i −0.707107 0.707107i
\(941\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −1.00000 −1.00000
\(947\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(951\) 0 0
\(952\) 1.41421i 1.41421i
\(953\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.41421i 1.41421i
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) −1.00000 −1.00000
\(965\) 1.00000 1.00000i 1.00000 1.00000i
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(971\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(972\) 0 0
\(973\) −1.41421 + 1.41421i −1.41421 + 1.41421i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(978\) 0 0
\(979\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(980\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(981\) 0 0
\(982\) 0 0
\(983\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(984\) 0 0
\(985\) 1.41421 1.41421
\(986\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(987\) 0 0
\(988\) 1.41421 1.41421
\(989\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(990\) 0 0
\(991\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 1.00000i 1.00000i
\(993\) 0 0
\(994\) 2.00000i 2.00000i
\(995\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(996\) 0 0
\(997\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(998\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2160.1.bn.b.269.1 yes 4
3.2 odd 2 2160.1.bn.a.269.2 yes 4
5.4 even 2 2160.1.bn.a.269.1 4
15.14 odd 2 inner 2160.1.bn.b.269.2 yes 4
16.5 even 4 inner 2160.1.bn.b.1349.2 yes 4
48.5 odd 4 2160.1.bn.a.1349.1 yes 4
80.69 even 4 2160.1.bn.a.1349.2 yes 4
240.149 odd 4 inner 2160.1.bn.b.1349.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2160.1.bn.a.269.1 4 5.4 even 2
2160.1.bn.a.269.2 yes 4 3.2 odd 2
2160.1.bn.a.1349.1 yes 4 48.5 odd 4
2160.1.bn.a.1349.2 yes 4 80.69 even 4
2160.1.bn.b.269.1 yes 4 1.1 even 1 trivial
2160.1.bn.b.269.2 yes 4 15.14 odd 2 inner
2160.1.bn.b.1349.1 yes 4 240.149 odd 4 inner
2160.1.bn.b.1349.2 yes 4 16.5 even 4 inner