Properties

Label 880.1.x.a.109.2
Level $880$
Weight $1$
Character 880.109
Analytic conductor $0.439$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -55
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [880,1,Mod(109,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, 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("880.109");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 880.x (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: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.7676100608000.1

Embedding invariants

Embedding label 109.2
Root \(0.923880 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 880.109
Dual form 880.1.x.a.549.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.382683 + 0.923880i) q^{2} +(-0.707107 - 0.707107i) q^{4} +(-0.707107 + 0.707107i) q^{5} -1.84776i q^{7} +(0.923880 - 0.382683i) q^{8} +1.00000i q^{9} +O(q^{10})\) \(q+(-0.382683 + 0.923880i) q^{2} +(-0.707107 - 0.707107i) q^{4} +(-0.707107 + 0.707107i) q^{5} -1.84776i q^{7} +(0.923880 - 0.382683i) q^{8} +1.00000i q^{9} +(-0.382683 - 0.923880i) q^{10} +(0.707107 - 0.707107i) q^{11} +(-0.541196 - 0.541196i) q^{13} +(1.70711 + 0.707107i) q^{14} +1.00000i q^{16} +1.84776 q^{17} +(-0.923880 - 0.382683i) q^{18} +1.00000 q^{20} +(0.382683 + 0.923880i) q^{22} -1.00000i q^{25} +(0.707107 - 0.292893i) q^{26} +(-1.30656 + 1.30656i) q^{28} +1.41421 q^{31} +(-0.923880 - 0.382683i) q^{32} +(-0.707107 + 1.70711i) q^{34} +(1.30656 + 1.30656i) q^{35} +(0.707107 - 0.707107i) q^{36} +(-0.382683 + 0.923880i) q^{40} +(-0.541196 + 0.541196i) q^{43} -1.00000 q^{44} +(-0.707107 - 0.707107i) q^{45} -2.41421 q^{49} +(0.923880 + 0.382683i) q^{50} +0.765367i q^{52} +1.00000i q^{55} +(-0.707107 - 1.70711i) q^{56} +(1.00000 - 1.00000i) q^{59} +(-0.541196 + 1.30656i) q^{62} +1.84776 q^{63} +(0.707107 - 0.707107i) q^{64} +0.765367 q^{65} +(-1.30656 - 1.30656i) q^{68} +(-1.70711 + 0.707107i) q^{70} +(0.382683 + 0.923880i) q^{72} -0.765367i q^{73} +(-1.30656 - 1.30656i) q^{77} +(-0.707107 - 0.707107i) q^{80} -1.00000 q^{81} +(-0.541196 - 0.541196i) q^{83} +(-1.30656 + 1.30656i) q^{85} +(-0.292893 - 0.707107i) q^{86} +(0.382683 - 0.923880i) q^{88} +1.41421i q^{89} +(0.923880 - 0.382683i) q^{90} +(-1.00000 + 1.00000i) q^{91} +(0.923880 - 2.23044i) q^{98} +(0.707107 + 0.707107i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{14} + 8 q^{20} - 8 q^{44} - 8 q^{49} + 8 q^{59} - 8 q^{70} - 8 q^{81} - 8 q^{86} - 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(3\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) −0.707107 0.707107i −0.707107 0.707107i
\(5\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(6\) 0 0
\(7\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(8\) 0.923880 0.382683i 0.923880 0.382683i
\(9\) 1.00000i 1.00000i
\(10\) −0.382683 0.923880i −0.382683 0.923880i
\(11\) 0.707107 0.707107i 0.707107 0.707107i
\(12\) 0 0
\(13\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(14\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(15\) 0 0
\(16\) 1.00000i 1.00000i
\(17\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(18\) −0.923880 0.382683i −0.923880 0.382683i
\(19\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(20\) 1.00000 1.00000
\(21\) 0 0
\(22\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000i 1.00000i
\(26\) 0.707107 0.292893i 0.707107 0.292893i
\(27\) 0 0
\(28\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(29\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(30\) 0 0
\(31\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(32\) −0.923880 0.382683i −0.923880 0.382683i
\(33\) 0 0
\(34\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(35\) 1.30656 + 1.30656i 1.30656 + 1.30656i
\(36\) 0.707107 0.707107i 0.707107 0.707107i
\(37\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(44\) −1.00000 −1.00000
\(45\) −0.707107 0.707107i −0.707107 0.707107i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −2.41421 −2.41421
\(50\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(51\) 0 0
\(52\) 0.765367i 0.765367i
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 1.00000i 1.00000i
\(56\) −0.707107 1.70711i −0.707107 1.70711i
\(57\) 0 0
\(58\) 0 0
\(59\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(60\) 0 0
\(61\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(62\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(63\) 1.84776 1.84776
\(64\) 0.707107 0.707107i 0.707107 0.707107i
\(65\) 0.765367 0.765367
\(66\) 0 0
\(67\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) −1.30656 1.30656i −1.30656 1.30656i
\(69\) 0 0
\(70\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(73\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.30656 1.30656i −1.30656 1.30656i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −0.707107 0.707107i −0.707107 0.707107i
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(84\) 0 0
\(85\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(86\) −0.292893 0.707107i −0.292893 0.707107i
\(87\) 0 0
\(88\) 0.382683 0.923880i 0.382683 0.923880i
\(89\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) 0.923880 0.382683i 0.923880 0.382683i
\(91\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0.923880 2.23044i 0.923880 2.23044i
\(99\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(100\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(101\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −0.707107 0.292893i −0.707107 0.292893i
\(105\) 0 0
\(106\) 0 0
\(107\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(108\) 0 0
\(109\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(110\) −0.923880 0.382683i −0.923880 0.382683i
\(111\) 0 0
\(112\) 1.84776 1.84776
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.541196 0.541196i 0.541196 0.541196i
\(118\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(119\) 3.41421i 3.41421i
\(120\) 0 0
\(121\) 1.00000i 1.00000i
\(122\) 0 0
\(123\) 0 0
\(124\) −1.00000 1.00000i −1.00000 1.00000i
\(125\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(126\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(127\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(128\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(129\) 0 0
\(130\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(131\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 1.70711 0.707107i 1.70711 0.707107i
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(140\) 1.84776i 1.84776i
\(141\) 0 0
\(142\) 0 0
\(143\) −0.765367 −0.765367
\(144\) −1.00000 −1.00000
\(145\) 0 0
\(146\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 1.84776i 1.84776i
\(154\) 1.70711 0.707107i 1.70711 0.707107i
\(155\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(156\) 0 0
\(157\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.923880 0.382683i 0.923880 0.382683i
\(161\) 0 0
\(162\) 0.382683 0.923880i 0.382683 0.923880i
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0.707107 0.292893i 0.707107 0.292893i
\(167\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(168\) 0 0
\(169\) 0.414214i 0.414214i
\(170\) −0.707107 1.70711i −0.707107 1.70711i
\(171\) 0 0
\(172\) 0.765367 0.765367
\(173\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(174\) 0 0
\(175\) −1.84776 −1.84776
\(176\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(177\) 0 0
\(178\) −1.30656 0.541196i −1.30656 0.541196i
\(179\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(180\) 1.00000i 1.00000i
\(181\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(182\) −0.541196 1.30656i −0.541196 1.30656i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.30656 1.30656i 1.30656 1.30656i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(192\) 0 0
\(193\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.70711 + 1.70711i 1.70711 + 1.70711i
\(197\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(198\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −0.382683 0.923880i −0.382683 0.923880i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.541196 0.541196i 0.541196 0.541196i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −0.707107 1.70711i −0.707107 1.70711i
\(215\) 0.765367i 0.765367i
\(216\) 0 0
\(217\) 2.61313i 2.61313i
\(218\) 0 0
\(219\) 0 0
\(220\) 0.707107 0.707107i 0.707107 0.707107i
\(221\) −1.00000 1.00000i −1.00000 1.00000i
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(225\) 1.00000 1.00000
\(226\) 0 0
\(227\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(228\) 0 0
\(229\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(234\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(235\) 0 0
\(236\) −1.41421 −1.41421
\(237\) 0 0
\(238\) 3.15432 + 1.30656i 3.15432 + 1.30656i
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(243\) 0 0
\(244\) 0 0
\(245\) 1.70711 1.70711i 1.70711 1.70711i
\(246\) 0 0
\(247\) 0 0
\(248\) 1.30656 0.541196i 1.30656 0.541196i
\(249\) 0 0
\(250\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(251\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(252\) −1.30656 1.30656i −1.30656 1.30656i
\(253\) 0 0
\(254\) 0.292893 0.707107i 0.292893 0.707107i
\(255\) 0 0
\(256\) −1.00000 −1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −0.541196 0.541196i −0.541196 0.541196i
\(261\) 0 0
\(262\) 0 0
\(263\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 1.84776i 1.84776i
\(273\) 0 0
\(274\) 0 0
\(275\) −0.707107 0.707107i −0.707107 0.707107i
\(276\) 0 0
\(277\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(278\) 0 0
\(279\) 1.41421i 1.41421i
\(280\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0.292893 0.707107i 0.292893 0.707107i
\(287\) 0 0
\(288\) 0.382683 0.923880i 0.382683 0.923880i
\(289\) 2.41421 2.41421
\(290\) 0 0
\(291\) 0 0
\(292\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(293\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(294\) 0 0
\(295\) 1.41421i 1.41421i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) −1.70711 0.707107i −1.70711 0.707107i
\(307\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(308\) 1.84776i 1.84776i
\(309\) 0 0
\(310\) −0.541196 1.30656i −0.541196 1.30656i
\(311\) 2.00000i 2.00000i 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 0
\(315\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(316\) 0 0
\(317\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.00000i 1.00000i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(325\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i \(0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(332\) 0.765367i 0.765367i
\(333\) 0 0
\(334\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(335\) 0 0
\(336\) 0 0
\(337\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(338\) 0.382683 + 0.158513i 0.382683 + 0.158513i
\(339\) 0 0
\(340\) 1.84776 1.84776
\(341\) 1.00000 1.00000i 1.00000 1.00000i
\(342\) 0 0
\(343\) 2.61313i 2.61313i
\(344\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(345\) 0 0
\(346\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(347\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(348\) 0 0
\(349\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(350\) 0.707107 1.70711i 0.707107 1.70711i
\(351\) 0 0
\(352\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1.00000 1.00000i 1.00000 1.00000i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −0.923880 0.382683i −0.923880 0.382683i
\(361\) 1.00000i 1.00000i
\(362\) 0 0
\(363\) 0 0
\(364\) 1.41421 1.41421
\(365\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(374\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0.541196 1.30656i 0.541196 1.30656i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 1.84776 1.84776
\(386\) 0.292893 0.707107i 0.292893 0.707107i
\(387\) −0.541196 0.541196i −0.541196 0.541196i
\(388\) 0 0
\(389\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.23044 + 0.923880i −2.23044 + 0.923880i
\(393\) 0 0
\(394\) −0.707107 1.70711i −0.707107 1.70711i
\(395\) 0 0
\(396\) 1.00000i 1.00000i
\(397\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 1.00000
\(401\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(402\) 0 0
\(403\) −0.765367 0.765367i −0.765367 0.765367i
\(404\) 0 0
\(405\) 0.707107 0.707107i 0.707107 0.707107i
\(406\) 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
\(413\) −1.84776 1.84776i −1.84776 1.84776i
\(414\) 0 0
\(415\) 0.765367 0.765367
\(416\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(417\) 0 0
\(418\) 0 0
\(419\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(420\) 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
\(424\) 0 0
\(425\) 1.84776i 1.84776i
\(426\) 0 0
\(427\) 0 0
\(428\) 1.84776 1.84776
\(429\) 0 0
\(430\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 2.41421 + 1.00000i 2.41421 + 1.00000i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(441\) 2.41421i 2.41421i
\(442\) 1.30656 0.541196i 1.30656 0.541196i
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 0 0
\(445\) −1.00000 1.00000i −1.00000 1.00000i
\(446\) 0 0
\(447\) 0 0
\(448\) −1.30656 1.30656i −1.30656 1.30656i
\(449\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(450\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(455\) 1.41421i 1.41421i
\(456\) 0 0
\(457\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(458\) −0.541196 1.30656i −0.541196 1.30656i
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −1.70711 0.707107i −1.70711 0.707107i
\(467\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(468\) −0.765367 −0.765367
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0.541196 1.30656i 0.541196 1.30656i
\(473\) 0.765367i 0.765367i
\(474\) 0 0
\(475\) 0 0
\(476\) −2.41421 + 2.41421i −2.41421 + 2.41421i
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0.923880 + 2.23044i 0.923880 + 2.23044i
\(491\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −1.00000 −1.00000
\(496\) 1.41421i 1.41421i
\(497\) 0 0
\(498\) 0 0
\(499\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(500\) 1.00000i 1.00000i
\(501\) 0 0
\(502\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(503\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(504\) 1.70711 0.707107i 1.70711 0.707107i
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(509\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −1.41421 −1.41421
\(512\) 0.382683 0.923880i 0.382683 0.923880i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0.707107 0.292893i 0.707107 0.292893i
\(521\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(522\) 0 0
\(523\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −0.707107 0.292893i −0.707107 0.292893i
\(527\) 2.61313 2.61313
\(528\) 0 0
\(529\) −1.00000 −1.00000
\(530\) 0 0
\(531\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.84776i 1.84776i
\(536\) 0 0
\(537\) 0 0
\(538\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(539\) −1.70711 + 1.70711i −1.70711 + 1.70711i
\(540\) 0 0
\(541\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −1.70711 0.707107i −1.70711 0.707107i
\(545\) 0 0
\(546\) 0 0
\(547\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0.923880 0.382683i 0.923880 0.382683i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −0.292893 0.707107i −0.292893 0.707107i
\(555\) 0 0
\(556\) 0 0
\(557\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(558\) −1.30656 0.541196i −1.30656 0.541196i
\(559\) 0.585786 0.585786
\(560\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(561\) 0 0
\(562\) 0 0
\(563\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.292893 0.707107i −0.292893 0.707107i
\(567\) 1.84776i 1.84776i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(572\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −0.923880 + 2.23044i −0.923880 + 2.23044i
\(579\) 0 0
\(580\) 0 0
\(581\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(582\) 0 0
\(583\) 0 0
\(584\) −0.292893 0.707107i −0.292893 0.707107i
\(585\) 0.765367i 0.765367i
\(586\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(587\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −1.30656 0.541196i −1.30656 0.541196i
\(591\) 0 0
\(592\) 0 0
\(593\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(594\) 0 0
\(595\) 2.41421 + 2.41421i 2.41421 + 2.41421i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(603\) 0 0
\(604\) 0 0
\(605\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(606\) 0 0
\(607\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 1.30656 1.30656i 1.30656 1.30656i
\(613\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(614\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(615\) 0 0
\(616\) −1.70711 0.707107i −1.70711 0.707107i
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(620\) 1.41421 1.41421
\(621\) 0 0
\(622\) −1.84776 0.765367i −1.84776 0.765367i
\(623\) 2.61313 2.61313
\(624\) 0 0
\(625\) −1.00000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) −0.707107 1.70711i −0.707107 1.70711i
\(631\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.541196 0.541196i 0.541196 0.541196i
\(636\) 0 0
\(637\) 1.30656 + 1.30656i 1.30656 + 1.30656i
\(638\) 0 0
\(639\) 0 0
\(640\) −0.923880 0.382683i −0.923880 0.382683i
\(641\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(642\) 0 0
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(649\) 1.41421i 1.41421i
\(650\) −0.292893 0.707107i −0.292893 0.707107i
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.765367 0.765367
\(658\) 0 0
\(659\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(660\) 0 0
\(661\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(662\) −0.765367 1.84776i −0.765367 1.84776i
\(663\) 0 0
\(664\) −0.707107 0.292893i −0.707107 0.292893i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(674\) 0.707107 1.70711i 0.707107 1.70711i
\(675\) 0 0
\(676\) −0.292893 + 0.292893i −0.292893 + 0.292893i
\(677\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(681\) 0 0
\(682\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(683\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.41421 1.00000i −2.41421 1.00000i
\(687\) 0 0
\(688\) −0.541196 0.541196i −0.541196 0.541196i
\(689\) 0 0
\(690\) 0 0
\(691\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(692\) 1.84776i 1.84776i
\(693\) 1.30656 1.30656i 1.30656 1.30656i
\(694\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.30656 + 1.30656i 1.30656 + 1.30656i
\(701\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.00000i 1.00000i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(713\) 0 0
\(714\) 0 0
\(715\) 0.541196 0.541196i 0.541196 0.541196i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(720\) 0.707107 0.707107i 0.707107 0.707107i
\(721\) 0 0
\(722\) −0.923880 0.382683i −0.923880 0.382683i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(729\) 1.00000i 1.00000i
\(730\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(731\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(732\) 0 0
\(733\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.707107 + 1.70711i 0.707107 + 1.70711i
\(747\) 0.541196 0.541196i 0.541196 0.541196i
\(748\) −1.84776 −1.84776
\(749\) 2.41421 + 2.41421i 2.41421 + 2.41421i
\(750\) 0 0
\(751\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(758\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(765\) −1.30656 1.30656i −1.30656 1.30656i
\(766\) 0 0
\(767\) −1.08239 −1.08239
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(771\) 0 0
\(772\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(773\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 0.707107 0.292893i 0.707107 0.292893i
\(775\) 1.41421i 1.41421i
\(776\) 0 0
\(777\) 0 0
\(778\) −0.541196 1.30656i −0.541196 1.30656i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 2.41421i 2.41421i
\(785\) 0 0
\(786\) 0 0
\(787\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(788\) 1.84776 1.84776
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(801\) −1.41421 −1.41421
\(802\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(803\) −0.541196 0.541196i −0.541196 0.541196i
\(804\) 0 0
\(805\) 0 0
\(806\) 1.00000 0.414214i 1.00000 0.414214i
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(811\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −1.00000 1.00000i −1.00000 1.00000i
\(820\) 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 \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 2.41421 1.00000i 2.41421 1.00000i
\(827\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(828\) 0 0
\(829\) −1.41421 1.41421i −1.41421 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(830\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(831\) 0 0
\(832\) −0.765367 −0.765367
\(833\) −4.46088 −4.46088
\(834\) 0 0
\(835\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(836\) 0 0
\(837\) 0 0
\(838\) 1.84776 0.765367i 1.84776 0.765367i
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.00000i 1.00000i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.292893 + 0.292893i 0.292893 + 0.292893i
\(846\) 0 0
\(847\) −1.84776 −1.84776
\(848\) 0 0
\(849\) 0 0
\(850\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(851\) 0 0
\(852\) 0 0
\(853\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(857\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(858\) 0 0
\(859\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(860\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −1.84776 −1.84776
\(866\) 0 0
\(867\) 0 0
\(868\) −1.84776 + 1.84776i −1.84776 + 1.84776i
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.30656 1.30656i 1.30656 1.30656i
\(876\) 0 0
\(877\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −1.00000 −1.00000
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 2.23044 + 0.923880i 2.23044 + 0.923880i
\(883\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(884\) 1.41421i 1.41421i
\(885\) 0 0
\(886\) 0 0
\(887\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(888\) 0 0
\(889\) 1.41421i 1.41421i
\(890\) 1.30656 0.541196i 1.30656 0.541196i
\(891\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.70711 0.707107i 1.70711 0.707107i
\(897\) 0 0
\(898\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(899\) 0 0
\(900\) −0.707107 0.707107i −0.707107 0.707107i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(908\) 0.765367i 0.765367i
\(909\) 0 0
\(910\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(911\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(912\) 0 0
\(913\) −0.765367 −0.765367
\(914\) −0.707107 0.292893i −0.707107 0.292893i
\(915\) 0 0
\(916\) 1.41421 1.41421
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.30656 1.30656i 1.30656 1.30656i
\(933\) 0 0
\(934\) 0 0
\(935\) 1.84776i 1.84776i
\(936\) 0.292893 0.707107i 0.292893 0.707107i
\(937\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(945\) 0 0
\(946\) −0.707107 0.292893i −0.707107 0.292893i
\(947\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(948\) 0 0
\(949\) −0.414214 + 0.414214i −0.414214 + 0.414214i
\(950\) 0 0
\(951\) 0 0
\(952\) −1.30656 3.15432i −1.30656 3.15432i
\(953\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(954\) 0 0
\(955\) 1.00000 1.00000i 1.00000 1.00000i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) −1.30656 1.30656i −1.30656 1.30656i
\(964\) 0 0
\(965\) 0.541196 0.541196i 0.541196 0.541196i
\(966\) 0 0
\(967\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(968\) −0.382683 0.923880i −0.382683 0.923880i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(980\) −2.41421 −2.41421
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 1.84776i 1.84776i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0.382683 0.923880i 0.382683 0.923880i
\(991\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(992\) −1.30656 0.541196i −1.30656 0.541196i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(998\) 1.30656 0.541196i 1.30656 0.541196i
\(999\) 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 880.1.x.a.109.2 8
4.3 odd 2 3520.1.x.a.3409.2 8
5.4 even 2 inner 880.1.x.a.109.3 yes 8
11.10 odd 2 inner 880.1.x.a.109.3 yes 8
16.5 even 4 inner 880.1.x.a.549.2 yes 8
16.11 odd 4 3520.1.x.a.1649.1 8
20.19 odd 2 3520.1.x.a.3409.1 8
44.43 even 2 3520.1.x.a.3409.1 8
55.54 odd 2 CM 880.1.x.a.109.2 8
80.59 odd 4 3520.1.x.a.1649.2 8
80.69 even 4 inner 880.1.x.a.549.3 yes 8
176.21 odd 4 inner 880.1.x.a.549.3 yes 8
176.43 even 4 3520.1.x.a.1649.2 8
220.219 even 2 3520.1.x.a.3409.2 8
880.219 even 4 3520.1.x.a.1649.1 8
880.549 odd 4 inner 880.1.x.a.549.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
880.1.x.a.109.2 8 1.1 even 1 trivial
880.1.x.a.109.2 8 55.54 odd 2 CM
880.1.x.a.109.3 yes 8 5.4 even 2 inner
880.1.x.a.109.3 yes 8 11.10 odd 2 inner
880.1.x.a.549.2 yes 8 16.5 even 4 inner
880.1.x.a.549.2 yes 8 880.549 odd 4 inner
880.1.x.a.549.3 yes 8 80.69 even 4 inner
880.1.x.a.549.3 yes 8 176.21 odd 4 inner
3520.1.x.a.1649.1 8 16.11 odd 4
3520.1.x.a.1649.1 8 880.219 even 4
3520.1.x.a.1649.2 8 80.59 odd 4
3520.1.x.a.1649.2 8 176.43 even 4
3520.1.x.a.3409.1 8 20.19 odd 2
3520.1.x.a.3409.1 8 44.43 even 2
3520.1.x.a.3409.2 8 4.3 odd 2
3520.1.x.a.3409.2 8 220.219 even 2