Properties

Label 1380.1.bn.b.659.1
Level $1380$
Weight $1$
Character 1380.659
Analytic conductor $0.689$
Analytic rank $0$
Dimension $20$
Projective image $D_{22}$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1380,1,Mod(359,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 11, 11, 21]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.359");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1380.bn (of order \(22\), degree \(10\), 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: \(0.688709717434\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{44})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{22}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{22} + \cdots)\)

Embedding invariants

Embedding label 659.1
Root \(0.989821 - 0.142315i\) of defining polynomial
Character \(\chi\) \(=\) 1380.659
Dual form 1380.1.bn.b.779.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.281733 - 0.959493i) q^{2} +(0.909632 - 0.415415i) q^{3} +(-0.841254 + 0.540641i) q^{4} +(-0.755750 + 0.654861i) q^{5} +(-0.654861 - 0.755750i) q^{6} +(0.755750 + 0.654861i) q^{8} +(0.654861 - 0.755750i) q^{9} +O(q^{10})\) \(q+(-0.281733 - 0.959493i) q^{2} +(0.909632 - 0.415415i) q^{3} +(-0.841254 + 0.540641i) q^{4} +(-0.755750 + 0.654861i) q^{5} +(-0.654861 - 0.755750i) q^{6} +(0.755750 + 0.654861i) q^{8} +(0.654861 - 0.755750i) q^{9} +(0.841254 + 0.540641i) q^{10} +(-0.540641 + 0.841254i) q^{12} +(-0.415415 + 0.909632i) q^{15} +(0.415415 - 0.909632i) q^{16} +(-0.540641 - 1.84125i) q^{17} +(-0.909632 - 0.415415i) q^{18} +(1.61435 + 0.474017i) q^{19} +(0.281733 - 0.959493i) q^{20} +(0.989821 - 0.142315i) q^{23} +(0.959493 + 0.281733i) q^{24} +(0.142315 - 0.989821i) q^{25} +(0.281733 - 0.959493i) q^{27} +(0.989821 + 0.142315i) q^{30} +(0.983568 + 0.449181i) q^{31} +(-0.989821 - 0.142315i) q^{32} +(-1.61435 + 1.03748i) q^{34} +(-0.142315 + 0.989821i) q^{36} -1.68251i q^{38} -1.00000 q^{40} +1.00000i q^{45} +(-0.415415 - 0.909632i) q^{46} +0.284630i q^{47} -1.00000i q^{48} +(0.415415 + 0.909632i) q^{49} +(-0.989821 + 0.142315i) q^{50} +(-1.25667 - 1.45027i) q^{51} +(-0.153882 + 0.239446i) q^{53} -1.00000 q^{54} +(1.66538 - 0.239446i) q^{57} +(-0.142315 - 0.989821i) q^{60} +(-1.80075 - 0.822373i) q^{61} +(0.153882 - 1.07028i) q^{62} +(0.142315 + 0.989821i) q^{64} +(1.45027 + 1.25667i) q^{68} +(0.841254 - 0.540641i) q^{69} +(0.989821 - 0.142315i) q^{72} +(-0.281733 - 0.959493i) q^{75} +(-1.61435 + 0.474017i) q^{76} +(-1.10181 + 0.708089i) q^{79} +(0.281733 + 0.959493i) q^{80} +(-0.142315 - 0.989821i) q^{81} +(-1.19136 + 1.37491i) q^{83} +(1.61435 + 1.03748i) q^{85} +(0.959493 - 0.281733i) q^{90} +(-0.755750 + 0.654861i) q^{92} +1.08128 q^{93} +(0.273100 - 0.0801894i) q^{94} +(-1.53046 + 0.698939i) q^{95} +(-0.959493 + 0.281733i) q^{96} +(0.755750 - 0.654861i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{4} - 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{4} - 2 q^{6} + 2 q^{9} - 2 q^{10} + 2 q^{15} - 2 q^{16} + 4 q^{19} + 2 q^{24} + 2 q^{25} - 4 q^{34} - 2 q^{36} - 20 q^{40} + 2 q^{46} - 2 q^{49} + 4 q^{51} - 20 q^{54} - 2 q^{60} + 2 q^{64} - 2 q^{69} - 4 q^{76} - 4 q^{79} - 2 q^{81} + 4 q^{85} + 2 q^{90} - 4 q^{94} - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(691\) \(1201\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{17}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.281733 0.959493i −0.281733 0.959493i
\(3\) 0.909632 0.415415i 0.909632 0.415415i
\(4\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(5\) −0.755750 + 0.654861i −0.755750 + 0.654861i
\(6\) −0.654861 0.755750i −0.654861 0.755750i
\(7\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(8\) 0.755750 + 0.654861i 0.755750 + 0.654861i
\(9\) 0.654861 0.755750i 0.654861 0.755750i
\(10\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(11\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(12\) −0.540641 + 0.841254i −0.540641 + 0.841254i
\(13\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(14\) 0 0
\(15\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(16\) 0.415415 0.909632i 0.415415 0.909632i
\(17\) −0.540641 1.84125i −0.540641 1.84125i −0.540641 0.841254i \(-0.681818\pi\)
1.00000i \(-0.5\pi\)
\(18\) −0.909632 0.415415i −0.909632 0.415415i
\(19\) 1.61435 + 0.474017i 1.61435 + 0.474017i 0.959493 0.281733i \(-0.0909091\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(20\) 0.281733 0.959493i 0.281733 0.959493i
\(21\) 0 0
\(22\) 0 0
\(23\) 0.989821 0.142315i 0.989821 0.142315i
\(24\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(25\) 0.142315 0.989821i 0.142315 0.989821i
\(26\) 0 0
\(27\) 0.281733 0.959493i 0.281733 0.959493i
\(28\) 0 0
\(29\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(30\) 0.989821 + 0.142315i 0.989821 + 0.142315i
\(31\) 0.983568 + 0.449181i 0.983568 + 0.449181i 0.841254 0.540641i \(-0.181818\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(32\) −0.989821 0.142315i −0.989821 0.142315i
\(33\) 0 0
\(34\) −1.61435 + 1.03748i −1.61435 + 1.03748i
\(35\) 0 0
\(36\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(37\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(38\) 1.68251i 1.68251i
\(39\) 0 0
\(40\) −1.00000 −1.00000
\(41\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(42\) 0 0
\(43\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(44\) 0 0
\(45\) 1.00000i 1.00000i
\(46\) −0.415415 0.909632i −0.415415 0.909632i
\(47\) 0.284630i 0.284630i 0.989821 + 0.142315i \(0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(48\) 1.00000i 1.00000i
\(49\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(50\) −0.989821 + 0.142315i −0.989821 + 0.142315i
\(51\) −1.25667 1.45027i −1.25667 1.45027i
\(52\) 0 0
\(53\) −0.153882 + 0.239446i −0.153882 + 0.239446i −0.909632 0.415415i \(-0.863636\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(54\) −1.00000 −1.00000
\(55\) 0 0
\(56\) 0 0
\(57\) 1.66538 0.239446i 1.66538 0.239446i
\(58\) 0 0
\(59\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(60\) −0.142315 0.989821i −0.142315 0.989821i
\(61\) −1.80075 0.822373i −1.80075 0.822373i −0.959493 0.281733i \(-0.909091\pi\)
−0.841254 0.540641i \(-0.818182\pi\)
\(62\) 0.153882 1.07028i 0.153882 1.07028i
\(63\) 0 0
\(64\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(68\) 1.45027 + 1.25667i 1.45027 + 1.25667i
\(69\) 0.841254 0.540641i 0.841254 0.540641i
\(70\) 0 0
\(71\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(72\) 0.989821 0.142315i 0.989821 0.142315i
\(73\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(74\) 0 0
\(75\) −0.281733 0.959493i −0.281733 0.959493i
\(76\) −1.61435 + 0.474017i −1.61435 + 0.474017i
\(77\) 0 0
\(78\) 0 0
\(79\) −1.10181 + 0.708089i −1.10181 + 0.708089i −0.959493 0.281733i \(-0.909091\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(80\) 0.281733 + 0.959493i 0.281733 + 0.959493i
\(81\) −0.142315 0.989821i −0.142315 0.989821i
\(82\) 0 0
\(83\) −1.19136 + 1.37491i −1.19136 + 1.37491i −0.281733 + 0.959493i \(0.590909\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(84\) 0 0
\(85\) 1.61435 + 1.03748i 1.61435 + 1.03748i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(90\) 0.959493 0.281733i 0.959493 0.281733i
\(91\) 0 0
\(92\) −0.755750 + 0.654861i −0.755750 + 0.654861i
\(93\) 1.08128 1.08128
\(94\) 0.273100 0.0801894i 0.273100 0.0801894i
\(95\) −1.53046 + 0.698939i −1.53046 + 0.698939i
\(96\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(97\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(98\) 0.755750 0.654861i 0.755750 0.654861i
\(99\) 0 0
\(100\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(101\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(102\) −1.03748 + 1.61435i −1.03748 + 1.61435i
\(103\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.273100 + 0.0801894i 0.273100 + 0.0801894i
\(107\) −0.627899 + 1.37491i −0.627899 + 1.37491i 0.281733 + 0.959493i \(0.409091\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(108\) 0.281733 + 0.959493i 0.281733 + 0.959493i
\(109\) 0.425839 + 1.45027i 0.425839 + 1.45027i 0.841254 + 0.540641i \(0.181818\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.822373 0.118239i −0.822373 0.118239i −0.281733 0.959493i \(-0.590909\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(114\) −0.698939 1.53046i −0.698939 1.53046i
\(115\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) −0.909632 + 0.415415i −0.909632 + 0.415415i
\(121\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(122\) −0.281733 + 1.95949i −0.281733 + 1.95949i
\(123\) 0 0
\(124\) −1.07028 + 0.153882i −1.07028 + 0.153882i
\(125\) 0.540641 + 0.841254i 0.540641 + 0.841254i
\(126\) 0 0
\(127\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(128\) 0.909632 0.415415i 0.909632 0.415415i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(136\) 0.797176 1.74557i 0.797176 1.74557i
\(137\) 1.30972i 1.30972i −0.755750 0.654861i \(-0.772727\pi\)
0.755750 0.654861i \(-0.227273\pi\)
\(138\) −0.755750 0.654861i −0.755750 0.654861i
\(139\) 1.97964i 1.97964i −0.142315 0.989821i \(-0.545455\pi\)
0.142315 0.989821i \(-0.454545\pi\)
\(140\) 0 0
\(141\) 0.118239 + 0.258908i 0.118239 + 0.258908i
\(142\) 0 0
\(143\) 0 0
\(144\) −0.415415 0.909632i −0.415415 0.909632i
\(145\) 0 0
\(146\) 0 0
\(147\) 0.755750 + 0.654861i 0.755750 + 0.654861i
\(148\) 0 0
\(149\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(150\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(151\) −0.304632 0.474017i −0.304632 0.474017i 0.654861 0.755750i \(-0.272727\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(152\) 0.909632 + 1.41542i 0.909632 + 1.41542i
\(153\) −1.74557 0.797176i −1.74557 0.797176i
\(154\) 0 0
\(155\) −1.03748 + 0.304632i −1.03748 + 0.304632i
\(156\) 0 0
\(157\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(158\) 0.989821 + 0.857685i 0.989821 + 0.857685i
\(159\) −0.0405070 + 0.281733i −0.0405070 + 0.281733i
\(160\) 0.841254 0.540641i 0.841254 0.540641i
\(161\) 0 0
\(162\) −0.909632 + 0.415415i −0.909632 + 0.415415i
\(163\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 1.65486 + 0.755750i 1.65486 + 0.755750i
\(167\) −0.368991 1.25667i −0.368991 1.25667i −0.909632 0.415415i \(-0.863636\pi\)
0.540641 0.841254i \(-0.318182\pi\)
\(168\) 0 0
\(169\) 0.415415 0.909632i 0.415415 0.909632i
\(170\) 0.540641 1.84125i 0.540641 1.84125i
\(171\) 1.41542 0.909632i 1.41542 0.909632i
\(172\) 0 0
\(173\) 0.281733 + 1.95949i 0.281733 + 1.95949i 0.281733 + 0.959493i \(0.409091\pi\)
1.00000i \(0.500000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(180\) −0.540641 0.841254i −0.540641 0.841254i
\(181\) −0.512546 + 0.234072i −0.512546 + 0.234072i −0.654861 0.755750i \(-0.727273\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(182\) 0 0
\(183\) −1.97964 −1.97964
\(184\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(185\) 0 0
\(186\) −0.304632 1.03748i −0.304632 1.03748i
\(187\) 0 0
\(188\) −0.153882 0.239446i −0.153882 0.239446i
\(189\) 0 0
\(190\) 1.10181 + 1.27155i 1.10181 + 1.27155i
\(191\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(192\) 0.540641 + 0.841254i 0.540641 + 0.841254i
\(193\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.841254 0.540641i −0.841254 0.540641i
\(197\) 1.27155 0.817178i 1.27155 0.817178i 0.281733 0.959493i \(-0.409091\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(198\) 0 0
\(199\) 0.345139 0.755750i 0.345139 0.755750i −0.654861 0.755750i \(-0.727273\pi\)
1.00000 \(0\)
\(200\) 0.755750 0.654861i 0.755750 0.654861i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 1.84125 + 0.540641i 1.84125 + 0.540641i
\(205\) 0 0
\(206\) 0 0
\(207\) 0.540641 0.841254i 0.540641 0.841254i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.158746 + 0.540641i −0.158746 + 0.540641i 0.841254 + 0.540641i \(0.181818\pi\)
−1.00000 \(\pi\)
\(212\) 0.284630i 0.284630i
\(213\) 0 0
\(214\) 1.49611 + 0.215109i 1.49611 + 0.215109i
\(215\) 0 0
\(216\) 0.841254 0.540641i 0.841254 0.540641i
\(217\) 0 0
\(218\) 1.27155 0.817178i 1.27155 0.817178i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(224\) 0 0
\(225\) −0.654861 0.755750i −0.654861 0.755750i
\(226\) 0.118239 + 0.822373i 0.118239 + 0.822373i
\(227\) 0.822373 + 1.80075i 0.822373 + 1.80075i 0.540641 + 0.841254i \(0.318182\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(228\) −1.27155 + 1.10181i −1.27155 + 1.10181i
\(229\) 1.81926i 1.81926i −0.415415 0.909632i \(-0.636364\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(230\) 0.909632 + 0.415415i 0.909632 + 0.415415i
\(231\) 0 0
\(232\) 0 0
\(233\) 0.627899 + 1.37491i 0.627899 + 1.37491i 0.909632 + 0.415415i \(0.136364\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(234\) 0 0
\(235\) −0.186393 0.215109i −0.186393 0.215109i
\(236\) 0 0
\(237\) −0.708089 + 1.10181i −0.708089 + 1.10181i
\(238\) 0 0
\(239\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(240\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(241\) −0.557730 + 0.0801894i −0.557730 + 0.0801894i −0.415415 0.909632i \(-0.636364\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(242\) 0.540641 + 0.841254i 0.540641 + 0.841254i
\(243\) −0.540641 0.841254i −0.540641 0.841254i
\(244\) 1.95949 0.281733i 1.95949 0.281733i
\(245\) −0.909632 0.415415i −0.909632 0.415415i
\(246\) 0 0
\(247\) 0 0
\(248\) 0.449181 + 0.983568i 0.449181 + 0.983568i
\(249\) −0.512546 + 1.74557i −0.512546 + 1.74557i
\(250\) 0.654861 0.755750i 0.654861 0.755750i
\(251\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 1.89945 + 0.273100i 1.89945 + 0.273100i
\(256\) −0.654861 0.755750i −0.654861 0.755750i
\(257\) −0.540641 0.158746i −0.540641 0.158746i 1.00000i \(-0.5\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.909632 0.584585i 0.909632 0.584585i 1.00000i \(-0.5\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(264\) 0 0
\(265\) −0.0405070 0.281733i −0.0405070 0.281733i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(270\) 0.755750 0.654861i 0.755750 0.654861i
\(271\) −1.49611 + 1.29639i −1.49611 + 1.29639i −0.654861 + 0.755750i \(0.727273\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(272\) −1.89945 0.273100i −1.89945 0.273100i
\(273\) 0 0
\(274\) −1.25667 + 0.368991i −1.25667 + 0.368991i
\(275\) 0 0
\(276\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −1.89945 + 0.557730i −1.89945 + 0.557730i
\(279\) 0.983568 0.449181i 0.983568 0.449181i
\(280\) 0 0
\(281\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(282\) 0.215109 0.186393i 0.215109 0.186393i
\(283\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(284\) 0 0
\(285\) −1.10181 + 1.27155i −1.10181 + 1.27155i
\(286\) 0 0
\(287\) 0 0
\(288\) −0.755750 + 0.654861i −0.755750 + 0.654861i
\(289\) −2.25667 + 1.45027i −2.25667 + 1.45027i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.234072 + 0.797176i 0.234072 + 0.797176i 0.989821 + 0.142315i \(0.0454545\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(294\) 0.415415 0.909632i 0.415415 0.909632i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0.755750 + 0.654861i 0.755750 + 0.654861i
\(301\) 0 0
\(302\) −0.368991 + 0.425839i −0.368991 + 0.425839i
\(303\) 0 0
\(304\) 1.10181 1.27155i 1.10181 1.27155i
\(305\) 1.89945 0.557730i 1.89945 0.557730i
\(306\) −0.273100 + 1.89945i −0.273100 + 1.89945i
\(307\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0.584585 + 0.909632i 0.584585 + 0.909632i
\(311\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(312\) 0 0
\(313\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0.544078 1.19136i 0.544078 1.19136i
\(317\) −0.368991 0.425839i −0.368991 0.425839i 0.540641 0.841254i \(-0.318182\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(318\) 0.281733 0.0405070i 0.281733 0.0405070i
\(319\) 0 0
\(320\) −0.755750 0.654861i −0.755750 0.654861i
\(321\) 1.51150i 1.51150i
\(322\) 0 0
\(323\) 3.22871i 3.22871i
\(324\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(325\) 0 0
\(326\) 0 0
\(327\) 0.989821 + 1.14231i 0.989821 + 1.14231i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.37491 + 1.19136i 1.37491 + 1.19136i 0.959493 + 0.281733i \(0.0909091\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(332\) 0.258908 1.80075i 0.258908 1.80075i
\(333\) 0 0
\(334\) −1.10181 + 0.708089i −1.10181 + 0.708089i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(338\) −0.989821 0.142315i −0.989821 0.142315i
\(339\) −0.797176 + 0.234072i −0.797176 + 0.234072i
\(340\) −1.91899 −1.91899
\(341\) 0 0
\(342\) −1.27155 1.10181i −1.27155 1.10181i
\(343\) 0 0
\(344\) 0 0
\(345\) −0.281733 + 0.959493i −0.281733 + 0.959493i
\(346\) 1.80075 0.822373i 1.80075 0.822373i
\(347\) 1.66538 + 0.239446i 1.66538 + 0.239446i 0.909632 0.415415i \(-0.136364\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(348\) 0 0
\(349\) −0.797176 0.234072i −0.797176 0.234072i −0.142315 0.989821i \(-0.545455\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.449181 0.983568i 0.449181 0.983568i −0.540641 0.841254i \(-0.681818\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(360\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(361\) 1.54019 + 0.989821i 1.54019 + 0.989821i
\(362\) 0.368991 + 0.425839i 0.368991 + 0.425839i
\(363\) −0.755750 + 0.654861i −0.755750 + 0.654861i
\(364\) 0 0
\(365\) 0 0
\(366\) 0.557730 + 1.89945i 0.557730 + 1.89945i
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0.281733 0.959493i 0.281733 0.959493i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −0.909632 + 0.584585i −0.909632 + 0.584585i
\(373\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(374\) 0 0
\(375\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(376\) −0.186393 + 0.215109i −0.186393 + 0.215109i
\(377\) 0 0
\(378\) 0 0
\(379\) 0.118239 + 0.822373i 0.118239 + 0.822373i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(380\) 0.909632 1.41542i 0.909632 1.41542i
\(381\) 0 0
\(382\) 0 0
\(383\) −0.755750 + 1.65486i −0.755750 + 1.65486i 1.00000i \(0.5\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(384\) 0.654861 0.755750i 0.654861 0.755750i
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(390\) 0 0
\(391\) −0.797176 1.74557i −0.797176 1.74557i
\(392\) −0.281733 + 0.959493i −0.281733 + 0.959493i
\(393\) 0 0
\(394\) −1.14231 0.989821i −1.14231 0.989821i
\(395\) 0.368991 1.25667i 0.368991 1.25667i
\(396\) 0 0
\(397\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(398\) −0.822373 0.118239i −0.822373 0.118239i
\(399\) 0 0
\(400\) −0.841254 0.540641i −0.841254 0.540641i
\(401\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0.755750 + 0.654861i 0.755750 + 0.654861i
\(406\) 0 0
\(407\) 0 0
\(408\) 1.91899i 1.91899i
\(409\) −1.25667 1.45027i −1.25667 1.45027i −0.841254 0.540641i \(-0.818182\pi\)
−0.415415 0.909632i \(-0.636364\pi\)
\(410\) 0 0
\(411\) −0.544078 1.19136i −0.544078 1.19136i
\(412\) 0 0
\(413\) 0 0
\(414\) −0.959493 0.281733i −0.959493 0.281733i
\(415\) 1.81926i 1.81926i
\(416\) 0 0
\(417\) −0.822373 1.80075i −0.822373 1.80075i
\(418\) 0 0
\(419\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(420\) 0 0
\(421\) −0.584585 + 0.909632i −0.584585 + 0.909632i 0.415415 + 0.909632i \(0.363636\pi\)
−1.00000 \(1.00000\pi\)
\(422\) 0.563465 0.563465
\(423\) 0.215109 + 0.186393i 0.215109 + 0.186393i
\(424\) −0.273100 + 0.0801894i −0.273100 + 0.0801894i
\(425\) −1.89945 + 0.273100i −1.89945 + 0.273100i
\(426\) 0 0
\(427\) 0 0
\(428\) −0.215109 1.49611i −0.215109 1.49611i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(432\) −0.755750 0.654861i −0.755750 0.654861i
\(433\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.14231 0.989821i −1.14231 0.989821i
\(437\) 1.66538 + 0.239446i 1.66538 + 0.239446i
\(438\) 0 0
\(439\) −1.80075 0.258908i −1.80075 0.258908i −0.841254 0.540641i \(-0.818182\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(440\) 0 0
\(441\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(442\) 0 0
\(443\) 0.540641 + 1.84125i 0.540641 + 1.84125i 0.540641 + 0.841254i \(0.318182\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(450\) −0.540641 + 0.841254i −0.540641 + 0.841254i
\(451\) 0 0
\(452\) 0.755750 0.345139i 0.755750 0.345139i
\(453\) −0.474017 0.304632i −0.474017 0.304632i
\(454\) 1.49611 1.29639i 1.49611 1.29639i
\(455\) 0 0
\(456\) 1.41542 + 0.909632i 1.41542 + 0.909632i
\(457\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(458\) −1.74557 + 0.512546i −1.74557 + 0.512546i
\(459\) −1.91899 −1.91899
\(460\) 0.142315 0.989821i 0.142315 0.989821i
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(464\) 0 0
\(465\) −0.817178 + 0.708089i −0.817178 + 0.708089i
\(466\) 1.14231 0.989821i 1.14231 0.989821i
\(467\) −0.474017 0.304632i −0.474017 0.304632i 0.281733 0.959493i \(-0.409091\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −0.153882 + 0.239446i −0.153882 + 0.239446i
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 1.25667 + 0.368991i 1.25667 + 0.368991i
\(475\) 0.698939 1.53046i 0.698939 1.53046i
\(476\) 0 0
\(477\) 0.0801894 + 0.273100i 0.0801894 + 0.273100i
\(478\) 0 0
\(479\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(480\) 0.540641 0.841254i 0.540641 0.841254i
\(481\) 0 0
\(482\) 0.234072 + 0.512546i 0.234072 + 0.512546i
\(483\) 0 0
\(484\) 0.654861 0.755750i 0.654861 0.755750i
\(485\) 0 0
\(486\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(487\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(488\) −0.822373 1.80075i −0.822373 1.80075i
\(489\) 0 0
\(490\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(491\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0.817178 0.708089i 0.817178 0.708089i
\(497\) 0 0
\(498\) 1.81926 1.81926
\(499\) −0.817178 + 1.27155i −0.817178 + 1.27155i 0.142315 + 0.989821i \(0.454545\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(500\) −0.909632 0.415415i −0.909632 0.415415i
\(501\) −0.857685 0.989821i −0.857685 0.989821i
\(502\) 0 0
\(503\) 0.449181 + 0.983568i 0.449181 + 0.983568i 0.989821 + 0.142315i \(0.0454545\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000i 1.00000i
\(508\) 0 0
\(509\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(510\) −0.273100 1.89945i −0.273100 1.89945i
\(511\) 0 0
\(512\) −0.540641 + 0.841254i −0.540641 + 0.841254i
\(513\) 0.909632 1.41542i 0.909632 1.41542i
\(514\) 0.563465i 0.563465i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 1.07028 + 1.66538i 1.07028 + 1.66538i
\(520\) 0 0
\(521\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(522\) 0 0
\(523\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −0.817178 0.708089i −0.817178 0.708089i
\(527\) 0.295298 2.05384i 0.295298 2.05384i
\(528\) 0 0
\(529\) 0.959493 0.281733i 0.959493 0.281733i
\(530\) −0.258908 + 0.118239i −0.258908 + 0.118239i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.425839 1.45027i −0.425839 1.45027i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) −0.841254 0.540641i −0.841254 0.540641i
\(541\) −0.239446 1.66538i −0.239446 1.66538i −0.654861 0.755750i \(-0.727273\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(542\) 1.66538 + 1.07028i 1.66538 + 1.07028i
\(543\) −0.368991 + 0.425839i −0.368991 + 0.425839i
\(544\) 0.273100 + 1.89945i 0.273100 + 1.89945i
\(545\) −1.27155 0.817178i −1.27155 0.817178i
\(546\) 0 0
\(547\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(548\) 0.708089 + 1.10181i 0.708089 + 1.10181i
\(549\) −1.80075 + 0.822373i −1.80075 + 0.822373i
\(550\) 0 0
\(551\) 0 0
\(552\) 0.989821 + 0.142315i 0.989821 + 0.142315i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 1.07028 + 1.66538i 1.07028 + 1.66538i
\(557\) −1.27155 + 1.10181i −1.27155 + 1.10181i −0.281733 + 0.959493i \(0.590909\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(558\) −0.708089 0.817178i −0.708089 0.817178i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −0.281733 1.95949i −0.281733 1.95949i −0.281733 0.959493i \(-0.590909\pi\)
1.00000i \(-0.5\pi\)
\(564\) −0.239446 0.153882i −0.239446 0.153882i
\(565\) 0.698939 0.449181i 0.698939 0.449181i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(570\) 1.53046 + 0.698939i 1.53046 + 0.698939i
\(571\) −0.273100 0.0801894i −0.273100 0.0801894i 0.142315 0.989821i \(-0.454545\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000i 1.00000i
\(576\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(577\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(578\) 2.02730 + 1.75667i 2.02730 + 1.75667i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0.698939 0.449181i 0.698939 0.449181i
\(587\) 1.66538 0.239446i 1.66538 0.239446i 0.755750 0.654861i \(-0.227273\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(588\) −0.989821 0.142315i −0.989821 0.142315i
\(589\) 1.37491 + 1.19136i 1.37491 + 1.19136i
\(590\) 0 0
\(591\) 0.817178 1.27155i 0.817178 1.27155i
\(592\) 0 0
\(593\) −0.708089 0.817178i −0.708089 0.817178i 0.281733 0.959493i \(-0.409091\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.830830i 0.830830i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0.415415 0.909632i 0.415415 0.909632i
\(601\) 0.544078 + 1.19136i 0.544078 + 1.19136i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0.512546 + 0.234072i 0.512546 + 0.234072i
\(605\) 0.540641 0.841254i 0.540641 0.841254i
\(606\) 0 0
\(607\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(608\) −1.53046 0.698939i −1.53046 0.698939i
\(609\) 0 0
\(610\) −1.07028 1.66538i −1.07028 1.66538i
\(611\) 0 0
\(612\) 1.89945 0.273100i 1.89945 0.273100i
\(613\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.0801894 + 0.273100i −0.0801894 + 0.273100i −0.989821 0.142315i \(-0.954545\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(618\) 0 0
\(619\) 0.186393 1.29639i 0.186393 1.29639i −0.654861 0.755750i \(-0.727273\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(620\) 0.708089 0.817178i 0.708089 0.817178i
\(621\) 0.142315 0.989821i 0.142315 0.989821i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.959493 0.281733i −0.959493 0.281733i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.61435 1.03748i 1.61435 1.03748i 0.654861 0.755750i \(-0.272727\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(632\) −1.29639 0.186393i −1.29639 0.186393i
\(633\) 0.0801894 + 0.557730i 0.0801894 + 0.557730i
\(634\) −0.304632 + 0.474017i −0.304632 + 0.474017i
\(635\) 0 0
\(636\) −0.118239 0.258908i −0.118239 0.258908i
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(641\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(642\) 1.45027 0.425839i 1.45027 0.425839i
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −3.09792 + 0.909632i −3.09792 + 0.909632i
\(647\) −1.74557 + 0.797176i −1.74557 + 0.797176i −0.755750 + 0.654861i \(0.772727\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(648\) 0.540641 0.841254i 0.540641 0.841254i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −0.989821 + 1.14231i −0.989821 + 1.14231i 1.00000i \(0.5\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(654\) 0.817178 1.27155i 0.817178 1.27155i
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(660\) 0 0
\(661\) −0.304632 1.03748i −0.304632 1.03748i −0.959493 0.281733i \(-0.909091\pi\)
0.654861 0.755750i \(-0.272727\pi\)
\(662\) 0.755750 1.65486i 0.755750 1.65486i
\(663\) 0 0
\(664\) −1.80075 + 0.258908i −1.80075 + 0.258908i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0.989821 + 0.857685i 0.989821 + 0.857685i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(674\) 0 0
\(675\) −0.909632 0.415415i −0.909632 0.415415i
\(676\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(677\) −0.708089 1.10181i −0.708089 1.10181i −0.989821 0.142315i \(-0.954545\pi\)
0.281733 0.959493i \(-0.409091\pi\)
\(678\) 0.449181 + 0.698939i 0.449181 + 0.698939i
\(679\) 0 0
\(680\) 0.540641 + 1.84125i 0.540641 + 1.84125i
\(681\) 1.49611 + 1.29639i 1.49611 + 1.29639i
\(682\) 0 0
\(683\) −0.153882 + 0.239446i −0.153882 + 0.239446i −0.909632 0.415415i \(-0.863636\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(684\) −0.698939 + 1.53046i −0.698939 + 1.53046i
\(685\) 0.857685 + 0.989821i 0.857685 + 0.989821i
\(686\) 0 0
\(687\) −0.755750 1.65486i −0.755750 1.65486i
\(688\) 0 0
\(689\) 0 0
\(690\) 1.00000 1.00000
\(691\) 1.81926i 1.81926i 0.415415 + 0.909632i \(0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(692\) −1.29639 1.49611i −1.29639 1.49611i
\(693\) 0 0
\(694\) −0.239446 1.66538i −0.239446 1.66538i
\(695\) 1.29639 + 1.49611i 1.29639 + 1.49611i
\(696\) 0 0
\(697\) 0 0
\(698\) 0.830830i 0.830830i
\(699\) 1.14231 + 0.989821i 1.14231 + 0.989821i
\(700\) 0 0
\(701\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −0.258908 0.118239i −0.258908 0.118239i
\(706\) −1.07028 0.153882i −1.07028 0.153882i
\(707\) 0 0
\(708\) 0 0
\(709\) 0.158746 0.540641i 0.158746 0.540641i −0.841254 0.540641i \(-0.818182\pi\)
1.00000 \(0\)
\(710\) 0 0
\(711\) −0.186393 + 1.29639i −0.186393 + 1.29639i
\(712\) 0 0
\(713\) 1.03748 + 0.304632i 1.03748 + 0.304632i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(720\) 0.909632 + 0.415415i 0.909632 + 0.415415i
\(721\) 0 0
\(722\) 0.515804 1.75667i 0.515804 1.75667i
\(723\) −0.474017 + 0.304632i −0.474017 + 0.304632i
\(724\) 0.304632 0.474017i 0.304632 0.474017i
\(725\) 0 0
\(726\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(727\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(728\) 0 0
\(729\) −0.841254 0.540641i −0.841254 0.540641i
\(730\) 0 0
\(731\) 0 0
\(732\) 1.66538 1.07028i 1.66538 1.07028i
\(733\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(734\) 0 0
\(735\) −1.00000 −1.00000
\(736\) −1.00000 −1.00000
\(737\) 0 0
\(738\) 0 0
\(739\) −1.65486 + 0.755750i −1.65486 + 0.755750i −0.654861 + 0.755750i \(0.727273\pi\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.53046 0.983568i −1.53046 0.983568i −0.989821 0.142315i \(-0.954545\pi\)
−0.540641 0.841254i \(-0.681818\pi\)
\(744\) 0.817178 + 0.708089i 0.817178 + 0.708089i
\(745\) 0 0
\(746\) 0 0
\(747\) 0.258908 + 1.80075i 0.258908 + 1.80075i
\(748\) 0 0
\(749\) 0 0
\(750\) 0.281733 0.959493i 0.281733 0.959493i
\(751\) −0.797176 + 1.74557i −0.797176 + 1.74557i −0.142315 + 0.989821i \(0.545455\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(752\) 0.258908 + 0.118239i 0.258908 + 0.118239i
\(753\) 0 0
\(754\) 0 0
\(755\) 0.540641 + 0.158746i 0.540641 + 0.158746i
\(756\) 0 0
\(757\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(758\) 0.755750 0.345139i 0.755750 0.345139i
\(759\) 0 0
\(760\) −1.61435 0.474017i −1.61435 0.474017i
\(761\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1.84125 0.540641i 1.84125 0.540641i
\(766\) 1.80075 + 0.258908i 1.80075 + 0.258908i
\(767\) 0 0
\(768\) −0.909632 0.415415i −0.909632 0.415415i
\(769\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(770\) 0 0
\(771\) −0.557730 + 0.0801894i −0.557730 + 0.0801894i
\(772\) 0 0
\(773\) −1.27155 1.10181i −1.27155 1.10181i −0.989821 0.142315i \(-0.954545\pi\)
−0.281733 0.959493i \(-0.590909\pi\)
\(774\) 0 0
\(775\) 0.584585 0.909632i 0.584585 0.909632i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −1.45027 + 1.25667i −1.45027 + 1.25667i
\(783\) 0 0
\(784\) 1.00000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(788\) −0.627899 + 1.37491i −0.627899 + 1.37491i
\(789\) 0.584585 0.909632i 0.584585 0.909632i
\(790\) −1.30972 −1.30972
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −0.153882 0.239446i −0.153882 0.239446i
\(796\) 0.118239 + 0.822373i 0.118239 + 0.822373i
\(797\) 1.53046 + 0.698939i 1.53046 + 0.698939i 0.989821 0.142315i \(-0.0454545\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(798\) 0 0
\(799\) 0.524075 0.153882i 0.524075 0.153882i
\(800\) −0.281733 + 0.959493i −0.281733 + 0.959493i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(810\) 0.415415 0.909632i 0.415415 0.909632i
\(811\) 0.425839 + 1.45027i 0.425839 + 1.45027i 0.841254 + 0.540641i \(0.181818\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(812\) 0 0
\(813\) −0.822373 + 1.80075i −0.822373 + 1.80075i
\(814\) 0 0
\(815\) 0 0
\(816\) −1.84125 + 0.540641i −1.84125 + 0.540641i
\(817\) 0 0
\(818\) −1.03748 + 1.61435i −1.03748 + 1.61435i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(822\) −0.989821 + 0.857685i −0.989821 + 0.857685i
\(823\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.08128 −1.08128 −0.540641 0.841254i \(-0.681818\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(828\) 1.00000i 1.00000i
\(829\) 1.30972 1.30972 0.654861 0.755750i \(-0.272727\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(830\) −1.74557 + 0.512546i −1.74557 + 0.512546i
\(831\) 0 0
\(832\) 0 0
\(833\) 1.45027 1.25667i 1.45027 1.25667i
\(834\) −1.49611 + 1.29639i −1.49611 + 1.29639i
\(835\) 1.10181 + 0.708089i 1.10181 + 0.708089i
\(836\) 0 0
\(837\) 0.708089 0.817178i 0.708089 0.817178i
\(838\) 0 0
\(839\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(840\) 0 0
\(841\) 0.841254 0.540641i 0.841254 0.540641i
\(842\) 1.03748 + 0.304632i 1.03748 + 0.304632i
\(843\) 0 0
\(844\) −0.158746 0.540641i −0.158746 0.540641i
\(845\) 0.281733 + 0.959493i 0.281733 + 0.959493i
\(846\) 0.118239 0.258908i 0.118239 0.258908i
\(847\) 0 0
\(848\) 0.153882 + 0.239446i 0.153882 + 0.239446i
\(849\) 0 0
\(850\) 0.797176 + 1.74557i 0.797176 + 1.74557i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(854\) 0 0
\(855\) −0.474017 + 1.61435i −0.474017 + 1.61435i
\(856\) −1.37491 + 0.627899i −1.37491 + 0.627899i
\(857\) −1.03748 + 0.304632i −1.03748 + 0.304632i −0.755750 0.654861i \(-0.772727\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(858\) 0 0
\(859\) 0.512546 + 0.234072i 0.512546 + 0.234072i 0.654861 0.755750i \(-0.272727\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.29639 0.186393i 1.29639 0.186393i 0.540641 0.841254i \(-0.318182\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(864\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(865\) −1.49611 1.29639i −1.49611 1.29639i
\(866\) 0 0
\(867\) −1.45027 + 2.25667i −1.45027 + 2.25667i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −0.627899 + 1.37491i −0.627899 + 1.37491i
\(873\) 0 0
\(874\) −0.239446 1.66538i −0.239446 1.66538i
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(878\) 0.258908 + 1.80075i 0.258908 + 1.80075i
\(879\) 0.544078 + 0.627899i 0.544078 + 0.627899i
\(880\) 0 0
\(881\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(882\) 1.00000i 1.00000i
\(883\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.61435 1.03748i 1.61435 1.03748i
\(887\) −1.03748 1.61435i −1.03748 1.61435i −0.755750 0.654861i \(-0.772727\pi\)
−0.281733 0.959493i \(-0.590909\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −0.134919 + 0.459493i −0.134919 + 0.459493i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(901\) 0.524075 + 0.153882i 0.524075 + 0.153882i
\(902\) 0 0
\(903\) 0 0
\(904\) −0.544078 0.627899i −0.544078 0.627899i
\(905\) 0.234072 0.512546i 0.234072 0.512546i
\(906\) −0.158746 + 0.540641i −0.158746 + 0.540641i
\(907\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(908\) −1.66538 1.07028i −1.66538 1.07028i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(912\) 0.474017 1.61435i 0.474017 1.61435i
\(913\) 0 0
\(914\) 0 0
\(915\) 1.49611 1.29639i 1.49611 1.29639i
\(916\) 0.983568 + 1.53046i 0.983568 + 1.53046i
\(917\) 0 0
\(918\) 0.540641 + 1.84125i 0.540641 + 1.84125i
\(919\) 1.91899 1.91899 0.959493 0.281733i \(-0.0909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(920\) −0.989821 + 0.142315i −0.989821 + 0.142315i
\(921\) 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 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(930\) 0.909632 + 0.584585i 0.909632 + 0.584585i
\(931\) 0.239446 + 1.66538i 0.239446 + 1.66538i
\(932\) −1.27155 0.817178i −1.27155 0.817178i
\(933\) 0 0
\(934\) −0.158746 + 0.540641i −0.158746 + 0.540641i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0.273100 + 0.0801894i 0.273100 + 0.0801894i
\(941\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −0.0801894 + 0.273100i −0.0801894 + 0.273100i −0.989821 0.142315i \(-0.954545\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(948\) 1.30972i 1.30972i
\(949\) 0 0
\(950\) −1.66538 0.239446i −1.66538 0.239446i
\(951\) −0.512546 0.234072i −0.512546 0.234072i
\(952\) 0 0
\(953\) −0.153882 0.239446i −0.153882 0.239446i 0.755750 0.654861i \(-0.227273\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(954\) 0.239446 0.153882i 0.239446 0.153882i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −0.959493 0.281733i −0.959493 0.281733i
\(961\) 0.110783 + 0.127850i 0.110783 + 0.127850i
\(962\) 0 0
\(963\) 0.627899 + 1.37491i 0.627899 + 1.37491i
\(964\) 0.425839 0.368991i 0.425839 0.368991i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −0.909632 0.415415i −0.909632 0.415415i
\(969\) −1.34125 2.93694i −1.34125 2.93694i
\(970\) 0 0
\(971\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(972\) 0.909632 + 0.415415i 0.909632 + 0.415415i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −1.49611 + 1.29639i −1.49611 + 1.29639i
\(977\) 1.29639 0.186393i 1.29639 0.186393i 0.540641 0.841254i \(-0.318182\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.989821 0.142315i 0.989821 0.142315i
\(981\) 1.37491 + 0.627899i 1.37491 + 0.627899i
\(982\) 0 0
\(983\) 1.45027 0.425839i 1.45027 0.425839i 0.540641 0.841254i \(-0.318182\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(984\) 0 0
\(985\) −0.425839 + 1.45027i −0.425839 + 1.45027i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.49611 + 0.215109i 1.49611 + 0.215109i 0.841254 0.540641i \(-0.181818\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(992\) −0.909632 0.584585i −0.909632 0.584585i
\(993\) 1.74557 + 0.512546i 1.74557 + 0.512546i
\(994\) 0 0
\(995\) 0.234072 + 0.797176i 0.234072 + 0.797176i
\(996\) −0.512546 1.74557i −0.512546 1.74557i
\(997\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(998\) 1.45027 + 0.425839i 1.45027 + 0.425839i
\(999\) 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 1380.1.bn.b.659.1 20
3.2 odd 2 inner 1380.1.bn.b.659.2 yes 20
4.3 odd 2 1380.1.bn.c.659.2 yes 20
5.4 even 2 inner 1380.1.bn.b.659.2 yes 20
12.11 even 2 1380.1.bn.c.659.1 yes 20
15.14 odd 2 CM 1380.1.bn.b.659.1 20
20.19 odd 2 1380.1.bn.c.659.1 yes 20
23.20 odd 22 1380.1.bn.c.779.2 yes 20
60.59 even 2 1380.1.bn.c.659.2 yes 20
69.20 even 22 1380.1.bn.c.779.1 yes 20
92.43 even 22 inner 1380.1.bn.b.779.1 yes 20
115.89 odd 22 1380.1.bn.c.779.1 yes 20
276.227 odd 22 inner 1380.1.bn.b.779.2 yes 20
345.89 even 22 1380.1.bn.c.779.2 yes 20
460.319 even 22 inner 1380.1.bn.b.779.2 yes 20
1380.779 odd 22 inner 1380.1.bn.b.779.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.1.bn.b.659.1 20 1.1 even 1 trivial
1380.1.bn.b.659.1 20 15.14 odd 2 CM
1380.1.bn.b.659.2 yes 20 3.2 odd 2 inner
1380.1.bn.b.659.2 yes 20 5.4 even 2 inner
1380.1.bn.b.779.1 yes 20 92.43 even 22 inner
1380.1.bn.b.779.1 yes 20 1380.779 odd 22 inner
1380.1.bn.b.779.2 yes 20 276.227 odd 22 inner
1380.1.bn.b.779.2 yes 20 460.319 even 22 inner
1380.1.bn.c.659.1 yes 20 12.11 even 2
1380.1.bn.c.659.1 yes 20 20.19 odd 2
1380.1.bn.c.659.2 yes 20 4.3 odd 2
1380.1.bn.c.659.2 yes 20 60.59 even 2
1380.1.bn.c.779.1 yes 20 69.20 even 22
1380.1.bn.c.779.1 yes 20 115.89 odd 22
1380.1.bn.c.779.2 yes 20 23.20 odd 22
1380.1.bn.c.779.2 yes 20 345.89 even 22