Properties

Label 161.1.l.a.48.1
Level $161$
Weight $1$
Character 161.48
Analytic conductor $0.080$
Analytic rank $0$
Dimension $10$
Projective image $D_{11}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [161,1,Mod(6,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 18]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.6");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 161.l (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: \(0.0803494670339\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{11}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{11} - \cdots)\)

Embedding invariants

Embedding label 48.1
Root \(-0.841254 + 0.540641i\) of defining polynomial
Character \(\chi\) \(=\) 161.48
Dual form 161.1.l.a.104.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.698939 + 0.449181i) q^{2} +(-0.128663 - 0.281733i) q^{4} +(-0.654861 + 0.755750i) q^{7} +(0.154861 - 1.07708i) q^{8} +(-0.959493 + 0.281733i) q^{9} +(-0.239446 + 0.153882i) q^{11} +(-0.797176 + 0.234072i) q^{14} +(0.389217 - 0.449181i) q^{16} +(-0.797176 - 0.234072i) q^{18} -0.236479 q^{22} +(-0.959493 - 0.281733i) q^{23} +(0.841254 + 0.540641i) q^{25} +(0.297176 + 0.0872586i) q^{28} +(0.698939 - 1.53046i) q^{29} +(-0.570276 + 0.167448i) q^{32} +(0.202824 + 0.234072i) q^{36} +(1.25667 - 0.368991i) q^{37} +(0.273100 + 1.89945i) q^{43} +(0.0741615 + 0.0476607i) q^{44} +(-0.544078 - 0.627899i) q^{46} +(-0.142315 - 0.989821i) q^{49} +(0.345139 + 0.755750i) q^{50} +(-1.10181 + 1.27155i) q^{53} +(0.712591 + 0.822373i) q^{56} +(1.17597 - 0.755750i) q^{58} +(0.415415 - 0.909632i) q^{63} +(-1.04408 - 0.306569i) q^{64} +(0.698939 + 0.449181i) q^{67} +(-0.239446 - 0.153882i) q^{71} +(0.154861 + 1.07708i) q^{72} +(1.04408 + 0.306569i) q^{74} +(0.0405070 - 0.281733i) q^{77} +(-1.10181 - 1.27155i) q^{79} +(0.841254 - 0.540641i) q^{81} +(-0.662317 + 1.45027i) q^{86} +(0.128663 + 0.281733i) q^{88} +(0.0440780 + 0.306569i) q^{92} +(0.345139 - 0.755750i) q^{98} +(0.186393 - 0.215109i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} - 3 q^{4} - q^{7} - 4 q^{8} - q^{9} - 2 q^{11} - 2 q^{14} + 6 q^{16} - 2 q^{18} - 4 q^{22} - q^{23} - q^{25} - 3 q^{28} - 2 q^{29} + 5 q^{32} + 8 q^{36} - 2 q^{37} - 2 q^{43} + 5 q^{44}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/161\mathbb{Z}\right)^\times\).

\(n\) \(24\) \(120\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.698939 + 0.449181i 0.698939 + 0.449181i 0.841254 0.540641i \(-0.181818\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(3\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(4\) −0.128663 0.281733i −0.128663 0.281733i
\(5\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(6\) 0 0
\(7\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(8\) 0.154861 1.07708i 0.154861 1.07708i
\(9\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(10\) 0 0
\(11\) −0.239446 + 0.153882i −0.239446 + 0.153882i −0.654861 0.755750i \(-0.727273\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(12\) 0 0
\(13\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(14\) −0.797176 + 0.234072i −0.797176 + 0.234072i
\(15\) 0 0
\(16\) 0.389217 0.449181i 0.389217 0.449181i
\(17\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(18\) −0.797176 0.234072i −0.797176 0.234072i
\(19\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.236479 −0.236479
\(23\) −0.959493 0.281733i −0.959493 0.281733i
\(24\) 0 0
\(25\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(26\) 0 0
\(27\) 0 0
\(28\) 0.297176 + 0.0872586i 0.297176 + 0.0872586i
\(29\) 0.698939 1.53046i 0.698939 1.53046i −0.142315 0.989821i \(-0.545455\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(30\) 0 0
\(31\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(32\) −0.570276 + 0.167448i −0.570276 + 0.167448i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.202824 + 0.234072i 0.202824 + 0.234072i
\(37\) 1.25667 0.368991i 1.25667 0.368991i 0.415415 0.909632i \(-0.363636\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(42\) 0 0
\(43\) 0.273100 + 1.89945i 0.273100 + 1.89945i 0.415415 + 0.909632i \(0.363636\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(44\) 0.0741615 + 0.0476607i 0.0741615 + 0.0476607i
\(45\) 0 0
\(46\) −0.544078 0.627899i −0.544078 0.627899i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −0.142315 0.989821i −0.142315 0.989821i
\(50\) 0.345139 + 0.755750i 0.345139 + 0.755750i
\(51\) 0 0
\(52\) 0 0
\(53\) −1.10181 + 1.27155i −1.10181 + 1.27155i −0.142315 + 0.989821i \(0.545455\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.712591 + 0.822373i 0.712591 + 0.822373i
\(57\) 0 0
\(58\) 1.17597 0.755750i 1.17597 0.755750i
\(59\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(60\) 0 0
\(61\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(62\) 0 0
\(63\) 0.415415 0.909632i 0.415415 0.909632i
\(64\) −1.04408 0.306569i −1.04408 0.306569i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.698939 + 0.449181i 0.698939 + 0.449181i 0.841254 0.540641i \(-0.181818\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.239446 0.153882i −0.239446 0.153882i 0.415415 0.909632i \(-0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(72\) 0.154861 + 1.07708i 0.154861 + 1.07708i
\(73\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(74\) 1.04408 + 0.306569i 1.04408 + 0.306569i
\(75\) 0 0
\(76\) 0 0
\(77\) 0.0405070 0.281733i 0.0405070 0.281733i
\(78\) 0 0
\(79\) −1.10181 1.27155i −1.10181 1.27155i −0.959493 0.281733i \(-0.909091\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(80\) 0 0
\(81\) 0.841254 0.540641i 0.841254 0.540641i
\(82\) 0 0
\(83\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.662317 + 1.45027i −0.662317 + 1.45027i
\(87\) 0 0
\(88\) 0.128663 + 0.281733i 0.128663 + 0.281733i
\(89\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.0440780 + 0.306569i 0.0440780 + 0.306569i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(98\) 0.345139 0.755750i 0.345139 0.755750i
\(99\) 0.186393 0.215109i 0.186393 0.215109i
\(100\) 0.0440780 0.306569i 0.0440780 0.306569i
\(101\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(102\) 0 0
\(103\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1.34125 + 0.393828i −1.34125 + 0.393828i
\(107\) 0.186393 1.29639i 0.186393 1.29639i −0.654861 0.755750i \(-0.727273\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(108\) 0 0
\(109\) 0.345139 0.755750i 0.345139 0.755750i −0.654861 0.755750i \(-0.727273\pi\)
1.00000 \(0\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.0845850 + 0.588302i 0.0845850 + 0.588302i
\(113\) −1.61435 1.03748i −1.61435 1.03748i −0.959493 0.281733i \(-0.909091\pi\)
−0.654861 0.755750i \(-0.727273\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.521109 −0.521109
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.381761 + 0.835939i −0.381761 + 0.835939i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0.698939 0.449181i 0.698939 0.449181i
\(127\) −1.61435 + 1.03748i −1.61435 + 1.03748i −0.654861 + 0.755750i \(0.727273\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(128\) −0.202824 0.234072i −0.202824 0.234072i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0.286752 + 0.627899i 0.286752 + 0.627899i
\(135\) 0 0
\(136\) 0 0
\(137\) 0.830830 0.830830 0.415415 0.909632i \(-0.363636\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.0982369 0.215109i −0.0982369 0.215109i
\(143\) 0 0
\(144\) −0.246902 + 0.540641i −0.246902 + 0.540641i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −0.265644 0.306569i −0.265644 0.306569i
\(149\) −1.61435 + 1.03748i −1.61435 + 1.03748i −0.654861 + 0.755750i \(0.727273\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(150\) 0 0
\(151\) −1.10181 1.27155i −1.10181 1.27155i −0.959493 0.281733i \(-0.909091\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0.154861 0.178719i 0.154861 0.178719i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(158\) −0.198939 1.38365i −0.198939 1.38365i
\(159\) 0 0
\(160\) 0 0
\(161\) 0.841254 0.540641i 0.841254 0.540641i
\(162\) 0.830830 0.830830
\(163\) 1.41542 + 0.909632i 1.41542 + 0.909632i 1.00000 \(0\)
0.415415 + 0.909632i \(0.363636\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(168\) 0 0
\(169\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(170\) 0 0
\(171\) 0 0
\(172\) 0.500000 0.321330i 0.500000 0.321330i
\(173\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(174\) 0 0
\(175\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(176\) −0.0240754 + 0.167448i −0.0240754 + 0.167448i
\(177\) 0 0
\(178\) 0 0
\(179\) 1.84125 + 0.540641i 1.84125 + 0.540641i 1.00000 \(0\)
0.841254 + 0.540641i \(0.181818\pi\)
\(180\) 0 0
\(181\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −0.452036 + 0.989821i −0.452036 + 0.989821i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.857685 0.989821i 0.857685 0.989821i −0.142315 0.989821i \(-0.545455\pi\)
1.00000 \(0\)
\(192\) 0 0
\(193\) 0.273100 0.0801894i 0.273100 0.0801894i −0.142315 0.989821i \(-0.545455\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.260554 + 0.167448i −0.260554 + 0.167448i
\(197\) 1.25667 + 1.45027i 1.25667 + 1.45027i 0.841254 + 0.540641i \(0.181818\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(198\) 0.226900 0.0666238i 0.226900 0.0666238i
\(199\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(200\) 0.712591 0.822373i 0.712591 0.822373i
\(201\) 0 0
\(202\) 0 0
\(203\) 0.698939 + 1.53046i 0.698939 + 1.53046i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000 1.00000
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.118239 0.258908i −0.118239 0.258908i 0.841254 0.540641i \(-0.181818\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(212\) 0.500000 + 0.146813i 0.500000 + 0.146813i
\(213\) 0 0
\(214\) 0.712591 0.822373i 0.712591 0.822373i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0.580699 0.373193i 0.580699 0.373193i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(224\) 0.246902 0.540641i 0.246902 0.540641i
\(225\) −0.959493 0.281733i −0.959493 0.281733i
\(226\) −0.662317 1.45027i −0.662317 1.45027i
\(227\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.54019 0.989821i −1.54019 0.989821i
\(233\) 0.186393 + 1.29639i 0.186393 + 1.29639i 0.841254 + 0.540641i \(0.181818\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.25667 0.368991i 1.25667 0.368991i 0.415415 0.909632i \(-0.363636\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(240\) 0 0
\(241\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(242\) −0.642315 + 0.412791i −0.642315 + 0.412791i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(252\) −0.309721 −0.309721
\(253\) 0.273100 0.0801894i 0.273100 0.0801894i
\(254\) −1.59435 −1.59435
\(255\) 0 0
\(256\) 0.118239 + 0.822373i 0.118239 + 0.822373i
\(257\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(258\) 0 0
\(259\) −0.544078 + 1.19136i −0.544078 + 1.19136i
\(260\) 0 0
\(261\) −0.239446 + 1.66538i −0.239446 + 1.66538i
\(262\) 0 0
\(263\) −0.544078 0.627899i −0.544078 0.627899i 0.415415 0.909632i \(-0.363636\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0.0366213 0.254707i 0.0366213 0.254707i
\(269\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(270\) 0 0
\(271\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0.580699 + 0.373193i 0.580699 + 0.373193i
\(275\) −0.284630 −0.284630
\(276\) 0 0
\(277\) −1.30972 −1.30972 −0.654861 0.755750i \(-0.727273\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.61435 0.474017i −1.61435 0.474017i −0.654861 0.755750i \(-0.727273\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(282\) 0 0
\(283\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(284\) −0.0125459 + 0.0872586i −0.0125459 + 0.0872586i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.500000 0.321330i 0.500000 0.321330i
\(289\) −0.654861 0.755750i −0.654861 0.755750i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.202824 1.41067i −0.202824 1.41067i
\(297\) 0 0
\(298\) −1.59435 −1.59435
\(299\) 0 0
\(300\) 0 0
\(301\) −1.61435 1.03748i −1.61435 1.03748i
\(302\) −0.198939 1.38365i −0.198939 1.38365i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(308\) −0.0845850 + 0.0248364i −0.0845850 + 0.0248364i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(312\) 0 0
\(313\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −0.216476 + 0.474017i −0.216476 + 0.474017i
\(317\) 1.25667 + 0.368991i 1.25667 + 0.368991i 0.841254 0.540641i \(-0.181818\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(318\) 0 0
\(319\) 0.0681534 + 0.474017i 0.0681534 + 0.474017i
\(320\) 0 0
\(321\) 0 0
\(322\) 0.830830 0.830830
\(323\) 0 0
\(324\) −0.260554 0.167448i −0.260554 0.167448i
\(325\) 0 0
\(326\) 0.580699 + 1.27155i 0.580699 + 1.27155i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.84125 0.540641i 1.84125 0.540641i 0.841254 0.540641i \(-0.181818\pi\)
1.00000 \(0\)
\(332\) 0 0
\(333\) −1.10181 + 0.708089i −1.10181 + 0.708089i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.0405070 0.281733i 0.0405070 0.281733i −0.959493 0.281733i \(-0.909091\pi\)
1.00000 \(0\)
\(338\) −0.544078 + 0.627899i −0.544078 + 0.627899i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(344\) 2.08816 2.08816
\(345\) 0 0
\(346\) 0 0
\(347\) −1.10181 0.708089i −1.10181 0.708089i −0.142315 0.989821i \(-0.545455\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(348\) 0 0
\(349\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(350\) −0.797176 0.234072i −0.797176 0.234072i
\(351\) 0 0
\(352\) 0.110783 0.127850i 0.110783 0.127850i
\(353\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 1.04408 + 1.20493i 1.04408 + 1.20493i
\(359\) 1.84125 0.540641i 1.84125 0.540641i 0.841254 0.540641i \(-0.181818\pi\)
1.00000 \(0\)
\(360\) 0 0
\(361\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −0.500000 + 0.321330i −0.500000 + 0.321330i
\(369\) 0 0
\(370\) 0 0
\(371\) −0.239446 1.66538i −0.239446 1.66538i
\(372\) 0 0
\(373\) 0.273100 + 0.0801894i 0.273100 + 0.0801894i 0.415415 0.909632i \(-0.363636\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.41542 0.909632i 1.41542 0.909632i 0.415415 0.909632i \(-0.363636\pi\)
1.00000 \(0\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.04408 0.306569i 1.04408 0.306569i
\(383\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.226900 + 0.0666238i 0.226900 + 0.0666238i
\(387\) −0.797176 1.74557i −0.797176 1.74557i
\(388\) 0 0
\(389\) −0.239446 0.153882i −0.239446 0.153882i 0.415415 0.909632i \(-0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.08816 −1.08816
\(393\) 0 0
\(394\) 0.226900 + 1.57812i 0.226900 + 1.57812i
\(395\) 0 0
\(396\) −0.0845850 0.0248364i −0.0845850 0.0248364i
\(397\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.570276 0.167448i 0.570276 0.167448i
\(401\) 0.857685 + 0.989821i 0.857685 + 0.989821i 1.00000 \(0\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −0.198939 + 1.38365i −0.198939 + 1.38365i
\(407\) −0.244123 + 0.281733i −0.244123 + 0.281733i
\(408\) 0 0
\(409\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0.698939 + 0.449181i 0.698939 + 0.449181i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(420\) 0 0
\(421\) 1.25667 1.45027i 1.25667 1.45027i 0.415415 0.909632i \(-0.363636\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(422\) 0.0336545 0.234072i 0.0336545 0.234072i
\(423\) 0 0
\(424\) 1.19894 + 1.38365i 1.19894 + 1.38365i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −0.389217 + 0.114284i −0.389217 + 0.114284i
\(429\) 0 0
\(430\) 0 0
\(431\) −0.797176 + 1.74557i −0.797176 + 1.74557i −0.142315 + 0.989821i \(0.545455\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(432\) 0 0
\(433\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.257326 −0.257326
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(440\) 0 0
\(441\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(442\) 0 0
\(443\) −0.118239 + 0.258908i −0.118239 + 0.258908i −0.959493 0.281733i \(-0.909091\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0.915415 0.588302i 0.915415 0.588302i
\(449\) −1.10181 + 0.708089i −1.10181 + 0.708089i −0.959493 0.281733i \(-0.909091\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(450\) −0.544078 0.627899i −0.544078 0.627899i
\(451\) 0 0
\(452\) −0.0845850 + 0.588302i −0.0845850 + 0.588302i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.239446 1.66538i −0.239446 1.66538i −0.654861 0.755750i \(-0.727273\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −0.118239 0.822373i −0.118239 0.822373i −0.959493 0.281733i \(-0.909091\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(464\) −0.415415 0.909632i −0.415415 0.909632i
\(465\) 0 0
\(466\) −0.452036 + 0.989821i −0.452036 + 0.989821i
\(467\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(468\) 0 0
\(469\) −0.797176 + 0.234072i −0.797176 + 0.234072i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.357685 0.412791i −0.357685 0.412791i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.698939 1.53046i 0.698939 1.53046i
\(478\) 1.04408 + 0.306569i 1.04408 + 0.306569i
\(479\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0.284630 0.284630
\(485\) 0 0
\(486\) 0 0
\(487\) −0.797176 1.74557i −0.797176 1.74557i −0.654861 0.755750i \(-0.727273\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −0.118239 + 0.822373i −0.118239 + 0.822373i 0.841254 + 0.540641i \(0.181818\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.273100 0.0801894i 0.273100 0.0801894i
\(498\) 0 0
\(499\) −0.544078 + 0.627899i −0.544078 + 0.627899i −0.959493 0.281733i \(-0.909091\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(504\) −0.915415 0.588302i −0.915415 0.588302i
\(505\) 0 0
\(506\) 0.226900 + 0.0666238i 0.226900 + 0.0666238i
\(507\) 0 0
\(508\) 0.500000 + 0.321330i 0.500000 + 0.321330i
\(509\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −0.915415 + 0.588302i −0.915415 + 0.588302i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(522\) −0.915415 + 1.05645i −0.915415 + 1.05645i
\(523\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −0.0982369 0.683252i −0.0982369 0.683252i
\(527\) 0 0
\(528\) 0 0
\(529\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0.592042 0.683252i 0.592042 0.683252i
\(537\) 0 0
\(538\) 0 0
\(539\) 0.186393 + 0.215109i 0.186393 + 0.215109i
\(540\) 0 0
\(541\) 0.698939 0.449181i 0.698939 0.449181i −0.142315 0.989821i \(-0.545455\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.273100 + 0.0801894i 0.273100 + 0.0801894i 0.415415 0.909632i \(-0.363636\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(548\) −0.106897 0.234072i −0.106897 0.234072i
\(549\) 0 0
\(550\) −0.198939 0.127850i −0.198939 0.127850i
\(551\) 0 0
\(552\) 0 0
\(553\) 1.68251 1.68251
\(554\) −0.915415 0.588302i −0.915415 0.588302i
\(555\) 0 0
\(556\) 0 0
\(557\) −1.91899 0.563465i −1.91899 0.563465i −0.959493 0.281733i \(-0.909091\pi\)
−0.959493 0.281733i \(-0.909091\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −0.915415 1.05645i −0.915415 1.05645i
\(563\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(568\) −0.202824 + 0.234072i −0.202824 + 0.234072i
\(569\) −0.544078 + 1.19136i −0.544078 + 1.19136i 0.415415 + 0.909632i \(0.363636\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(570\) 0 0
\(571\) −0.544078 1.19136i −0.544078 1.19136i −0.959493 0.281733i \(-0.909091\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.654861 0.755750i −0.654861 0.755750i
\(576\) 1.08816 1.08816
\(577\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(578\) −0.118239 0.822373i −0.118239 0.822373i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.0681534 0.474017i 0.0681534 0.474017i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.323373 0.708089i 0.323373 0.708089i
\(593\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.500000 + 0.321330i 0.500000 + 0.321330i
\(597\) 0 0
\(598\) 0 0
\(599\) 0.830830 0.830830 0.415415 0.909632i \(-0.363636\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(600\) 0 0
\(601\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(602\) −0.662317 1.45027i −0.662317 1.45027i
\(603\) −0.797176 0.234072i −0.797176 0.234072i
\(604\) −0.216476 + 0.474017i −0.216476 + 0.474017i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −0.118239 + 0.822373i −0.118239 + 0.822373i 0.841254 + 0.540641i \(0.181818\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −0.297176 0.0872586i −0.297176 0.0872586i
\(617\) 0.345139 + 0.755750i 0.345139 + 0.755750i 1.00000 \(0\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(618\) 0 0
\(619\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.25667 + 1.45027i 1.25667 + 1.45027i 0.841254 + 0.540641i \(0.181818\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(632\) −1.54019 + 0.989821i −1.54019 + 0.989821i
\(633\) 0 0
\(634\) 0.712591 + 0.822373i 0.712591 + 0.822373i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −0.165284 + 0.361922i −0.165284 + 0.361922i
\(639\) 0.273100 + 0.0801894i 0.273100 + 0.0801894i
\(640\) 0 0
\(641\) −0.284630 1.97964i −0.284630 1.97964i −0.142315 0.989821i \(-0.545455\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −0.260554 0.167448i −0.260554 0.167448i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(648\) −0.452036 0.989821i −0.452036 0.989821i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0.0741615 0.515804i 0.0741615 0.515804i
\(653\) −0.797176 + 0.234072i −0.797176 + 0.234072i −0.654861 0.755750i \(-0.727273\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −0.239446 + 1.66538i −0.239446 + 1.66538i 0.415415 + 0.909632i \(0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(660\) 0 0
\(661\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(662\) 1.52977 + 0.449181i 1.52977 + 0.449181i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −1.08816 −1.08816
\(667\) −1.10181 + 1.27155i −1.10181 + 1.27155i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.345139 0.755750i 0.345139 0.755750i −0.654861 0.755750i \(-0.727273\pi\)
1.00000 \(0\)
\(674\) 0.154861 0.178719i 0.154861 0.178719i
\(675\) 0 0
\(676\) 0.297176 0.0872586i 0.297176 0.0872586i
\(677\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.857685 0.989821i 0.857685 0.989821i −0.142315 0.989821i \(-0.545455\pi\)
1.00000 \(0\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.345139 + 0.755750i 0.345139 + 0.755750i
\(687\) 0 0
\(688\) 0.959493 + 0.616629i 0.959493 + 0.616629i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0.0405070 + 0.281733i 0.0405070 + 0.281733i
\(694\) −0.452036 0.989821i −0.452036 0.989821i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.202824 + 0.234072i 0.202824 + 0.234072i
\(701\) 0.698939 0.449181i 0.698939 0.449181i −0.142315 0.989821i \(-0.545455\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.297176 0.0872586i 0.297176 0.0872586i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.118239 0.258908i −0.118239 0.258908i 0.841254 0.540641i \(-0.181818\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(710\) 0 0
\(711\) 1.41542 + 0.909632i 1.41542 + 0.909632i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −0.0845850 0.588302i −0.0845850 0.588302i
\(717\) 0 0
\(718\) 1.52977 + 0.449181i 1.52977 + 0.449181i
\(719\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −0.797176 + 0.234072i −0.797176 + 0.234072i
\(723\) 0 0
\(724\) 0 0
\(725\) 1.41542 0.909632i 1.41542 0.909632i
\(726\) 0 0
\(727\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(728\) 0 0
\(729\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0.594351 0.594351
\(737\) −0.236479 −0.236479
\(738\) 0 0
\(739\) −0.118239 0.822373i −0.118239 0.822373i −0.959493 0.281733i \(-0.909091\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.580699 1.27155i 0.580699 1.27155i
\(743\) −0.544078 + 0.627899i −0.544078 + 0.627899i −0.959493 0.281733i \(-0.909091\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.154861 + 0.178719i 0.154861 + 0.178719i
\(747\) 0 0
\(748\) 0 0
\(749\) 0.857685 + 0.989821i 0.857685 + 0.989821i
\(750\) 0 0
\(751\) −0.239446 + 1.66538i −0.239446 + 1.66538i 0.415415 + 0.909632i \(0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.239446 0.153882i −0.239446 0.153882i 0.415415 0.909632i \(-0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(758\) 1.39788 1.39788
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(762\) 0 0
\(763\) 0.345139 + 0.755750i 0.345139 + 0.755750i
\(764\) −0.389217 0.114284i −0.389217 0.114284i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.0577299 0.0666238i −0.0577299 0.0666238i
\(773\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(774\) 0.226900 1.57812i 0.226900 1.57812i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −0.0982369 0.215109i −0.0982369 0.215109i
\(779\) 0 0
\(780\) 0 0
\(781\) 0.0810141 0.0810141
\(782\) 0 0
\(783\) 0 0
\(784\) −0.500000 0.321330i −0.500000 0.321330i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(788\) 0.246902 0.540641i 0.246902 0.540641i
\(789\) 0 0
\(790\) 0 0
\(791\) 1.84125 0.540641i 1.84125 0.540641i
\(792\) −0.202824 0.234072i −0.202824 0.234072i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.570276 0.167448i −0.570276 0.167448i
\(801\) 0 0
\(802\) 0.154861 + 1.07708i 0.154861 + 1.07708i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.698939 + 1.53046i 0.698939 + 1.53046i 0.841254 + 0.540641i \(0.181818\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(810\) 0 0
\(811\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(812\) 0.341254 0.393828i 0.341254 0.393828i
\(813\) 0 0
\(814\) −0.297176 + 0.0872586i −0.297176 + 0.0872586i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0.186393 0.215109i 0.186393 0.215109i −0.654861 0.755750i \(-0.727273\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(822\) 0 0
\(823\) −1.61435 0.474017i −1.61435 0.474017i −0.654861 0.755750i \(-0.727273\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.30972 −1.30972 −0.654861 0.755750i \(-0.727273\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(828\) −0.128663 0.281733i −0.128663 0.281733i
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(840\) 0 0
\(841\) −1.19894 1.38365i −1.19894 1.38365i
\(842\) 1.52977 0.449181i 1.52977 0.449181i
\(843\) 0 0
\(844\) −0.0577299 + 0.0666238i −0.0577299 + 0.0666238i
\(845\) 0 0
\(846\) 0 0
\(847\) −0.381761 0.835939i −0.381761 0.835939i
\(848\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(849\) 0 0
\(850\) 0 0
\(851\) −1.30972 −1.30972
\(852\) 0 0
\(853\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.36745 0.401520i −1.36745 0.401520i
\(857\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(858\) 0 0
\(859\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.34125 + 0.861971i −1.34125 + 0.861971i
\(863\) −0.239446 + 0.153882i −0.239446 + 0.153882i −0.654861 0.755750i \(-0.727273\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0.459493 + 0.134919i 0.459493 + 0.134919i
\(870\) 0 0
\(871\) 0 0
\(872\) −0.760554 0.488779i −0.760554 0.488779i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.0405070 + 0.281733i 0.0405070 + 0.281733i 1.00000 \(0\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(882\) −0.118239 + 0.822373i −0.118239 + 0.822373i
\(883\) −1.61435 + 0.474017i −1.61435 + 0.474017i −0.959493 0.281733i \(-0.909091\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.198939 + 0.127850i −0.198939 + 0.127850i
\(887\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(888\) 0 0
\(889\) 0.273100 1.89945i 0.273100 1.89945i
\(890\) 0 0
\(891\) −0.118239 + 0.258908i −0.118239 + 0.258908i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0.309721 0.309721
\(897\) 0 0
\(898\) −1.08816 −1.08816
\(899\) 0 0
\(900\) 0.0440780 + 0.306569i 0.0440780 + 0.306569i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −1.36745 + 1.57812i −1.36745 + 1.57812i
\(905\) 0 0
\(906\) 0 0
\(907\) −0.544078 0.627899i −0.544078 0.627899i 0.415415 0.909632i \(-0.363636\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −0.797176 + 0.234072i −0.797176 + 0.234072i −0.654861 0.755750i \(-0.727273\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0.580699 1.27155i 0.580699 1.27155i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −1.30972 −1.30972 −0.654861 0.755750i \(-0.727273\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.25667 + 0.368991i 1.25667 + 0.368991i
\(926\) 0.286752 0.627899i 0.286752 0.627899i
\(927\) 0 0
\(928\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(929\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0.341254 0.219310i 0.341254 0.219310i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(938\) −0.662317 0.194474i −0.662317 0.194474i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −0.0645824 0.449181i −0.0645824 0.449181i
\(947\) −0.544078 1.19136i −0.544078 1.19136i −0.959493 0.281733i \(-0.909091\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.857685 + 0.989821i 0.857685 + 0.989821i 1.00000 \(0\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(954\) 1.17597 0.755750i 1.17597 0.755750i
\(955\) 0 0
\(956\) −0.265644 0.306569i −0.265644 0.306569i
\(957\) 0 0
\(958\) 0 0
\(959\) −0.544078 + 0.627899i −0.544078 + 0.627899i
\(960\) 0 0
\(961\) −0.959493 0.281733i −0.959493 0.281733i
\(962\) 0 0
\(963\) 0.186393 + 1.29639i 0.186393 + 1.29639i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −0.284630 −0.284630 −0.142315 0.989821i \(-0.545455\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(968\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0.226900 1.57812i 0.226900 1.57812i
\(975\) 0 0
\(976\) 0 0
\(977\) 1.68251 1.08128i 1.68251 1.08128i 0.841254 0.540641i \(-0.181818\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −0.118239 + 0.822373i −0.118239 + 0.822373i
\(982\) −0.452036 + 0.521678i −0.452036 + 0.521678i
\(983\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0.273100 1.89945i 0.273100 1.89945i
\(990\) 0 0
\(991\) 0.698939 + 0.449181i 0.698939 + 0.449181i 0.841254 0.540641i \(-0.181818\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0.226900 + 0.0666238i 0.226900 + 0.0666238i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(998\) −0.662317 + 0.194474i −0.662317 + 0.194474i
\(999\) 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 161.1.l.a.48.1 10
3.2 odd 2 1449.1.bq.a.370.1 10
4.3 odd 2 2576.1.cj.a.209.1 10
7.2 even 3 1127.1.v.a.1060.1 20
7.3 odd 6 1127.1.v.a.117.1 20
7.4 even 3 1127.1.v.a.117.1 20
7.5 odd 6 1127.1.v.a.1060.1 20
7.6 odd 2 CM 161.1.l.a.48.1 10
21.20 even 2 1449.1.bq.a.370.1 10
23.2 even 11 3703.1.l.c.3044.1 10
23.3 even 11 3703.1.l.a.2617.1 10
23.4 even 11 3703.1.l.a.699.1 10
23.5 odd 22 3703.1.l.i.2603.1 10
23.6 even 11 3703.1.l.c.118.1 10
23.7 odd 22 3703.1.l.i.2582.1 10
23.8 even 11 3703.1.l.b.2911.1 10
23.9 even 11 3703.1.b.c.1588.4 5
23.10 odd 22 3703.1.l.d.706.1 10
23.11 odd 22 3703.1.l.f.1392.1 10
23.12 even 11 inner 161.1.l.a.104.1 yes 10
23.13 even 11 3703.1.l.b.706.1 10
23.14 odd 22 3703.1.b.b.1588.4 5
23.15 odd 22 3703.1.l.d.2911.1 10
23.16 even 11 3703.1.l.h.2582.1 10
23.17 odd 22 3703.1.l.g.118.1 10
23.18 even 11 3703.1.l.h.2603.1 10
23.19 odd 22 3703.1.l.e.699.1 10
23.20 odd 22 3703.1.l.e.2617.1 10
23.21 odd 22 3703.1.l.g.3044.1 10
23.22 odd 2 3703.1.l.f.3429.1 10
28.27 even 2 2576.1.cj.a.209.1 10
69.35 odd 22 1449.1.bq.a.748.1 10
92.35 odd 22 2576.1.cj.a.1553.1 10
161.6 odd 22 3703.1.l.c.118.1 10
161.12 odd 66 1127.1.v.a.472.1 20
161.13 odd 22 3703.1.l.b.706.1 10
161.20 even 22 3703.1.l.e.2617.1 10
161.27 odd 22 3703.1.l.a.699.1 10
161.34 even 22 3703.1.l.f.1392.1 10
161.41 odd 22 3703.1.l.h.2603.1 10
161.48 odd 22 3703.1.l.c.3044.1 10
161.55 odd 22 3703.1.b.c.1588.4 5
161.58 even 33 1127.1.v.a.472.1 20
161.62 odd 22 3703.1.l.h.2582.1 10
161.76 even 22 3703.1.l.i.2582.1 10
161.81 even 33 1127.1.v.a.656.1 20
161.83 even 22 3703.1.b.b.1588.4 5
161.90 even 22 3703.1.l.g.3044.1 10
161.97 even 22 3703.1.l.i.2603.1 10
161.104 odd 22 inner 161.1.l.a.104.1 yes 10
161.111 even 22 3703.1.l.e.699.1 10
161.118 odd 22 3703.1.l.a.2617.1 10
161.125 even 22 3703.1.l.d.706.1 10
161.132 even 22 3703.1.l.g.118.1 10
161.146 odd 22 3703.1.l.b.2911.1 10
161.150 odd 66 1127.1.v.a.656.1 20
161.153 even 22 3703.1.l.d.2911.1 10
161.160 even 2 3703.1.l.f.3429.1 10
483.104 even 22 1449.1.bq.a.748.1 10
644.587 even 22 2576.1.cj.a.1553.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.1.l.a.48.1 10 1.1 even 1 trivial
161.1.l.a.48.1 10 7.6 odd 2 CM
161.1.l.a.104.1 yes 10 23.12 even 11 inner
161.1.l.a.104.1 yes 10 161.104 odd 22 inner
1127.1.v.a.117.1 20 7.3 odd 6
1127.1.v.a.117.1 20 7.4 even 3
1127.1.v.a.472.1 20 161.12 odd 66
1127.1.v.a.472.1 20 161.58 even 33
1127.1.v.a.656.1 20 161.81 even 33
1127.1.v.a.656.1 20 161.150 odd 66
1127.1.v.a.1060.1 20 7.2 even 3
1127.1.v.a.1060.1 20 7.5 odd 6
1449.1.bq.a.370.1 10 3.2 odd 2
1449.1.bq.a.370.1 10 21.20 even 2
1449.1.bq.a.748.1 10 69.35 odd 22
1449.1.bq.a.748.1 10 483.104 even 22
2576.1.cj.a.209.1 10 4.3 odd 2
2576.1.cj.a.209.1 10 28.27 even 2
2576.1.cj.a.1553.1 10 92.35 odd 22
2576.1.cj.a.1553.1 10 644.587 even 22
3703.1.b.b.1588.4 5 23.14 odd 22
3703.1.b.b.1588.4 5 161.83 even 22
3703.1.b.c.1588.4 5 23.9 even 11
3703.1.b.c.1588.4 5 161.55 odd 22
3703.1.l.a.699.1 10 23.4 even 11
3703.1.l.a.699.1 10 161.27 odd 22
3703.1.l.a.2617.1 10 23.3 even 11
3703.1.l.a.2617.1 10 161.118 odd 22
3703.1.l.b.706.1 10 23.13 even 11
3703.1.l.b.706.1 10 161.13 odd 22
3703.1.l.b.2911.1 10 23.8 even 11
3703.1.l.b.2911.1 10 161.146 odd 22
3703.1.l.c.118.1 10 23.6 even 11
3703.1.l.c.118.1 10 161.6 odd 22
3703.1.l.c.3044.1 10 23.2 even 11
3703.1.l.c.3044.1 10 161.48 odd 22
3703.1.l.d.706.1 10 23.10 odd 22
3703.1.l.d.706.1 10 161.125 even 22
3703.1.l.d.2911.1 10 23.15 odd 22
3703.1.l.d.2911.1 10 161.153 even 22
3703.1.l.e.699.1 10 23.19 odd 22
3703.1.l.e.699.1 10 161.111 even 22
3703.1.l.e.2617.1 10 23.20 odd 22
3703.1.l.e.2617.1 10 161.20 even 22
3703.1.l.f.1392.1 10 23.11 odd 22
3703.1.l.f.1392.1 10 161.34 even 22
3703.1.l.f.3429.1 10 23.22 odd 2
3703.1.l.f.3429.1 10 161.160 even 2
3703.1.l.g.118.1 10 23.17 odd 22
3703.1.l.g.118.1 10 161.132 even 22
3703.1.l.g.3044.1 10 23.21 odd 22
3703.1.l.g.3044.1 10 161.90 even 22
3703.1.l.h.2582.1 10 23.16 even 11
3703.1.l.h.2582.1 10 161.62 odd 22
3703.1.l.h.2603.1 10 23.18 even 11
3703.1.l.h.2603.1 10 161.41 odd 22
3703.1.l.i.2582.1 10 23.7 odd 22
3703.1.l.i.2582.1 10 161.76 even 22
3703.1.l.i.2603.1 10 23.5 odd 22
3703.1.l.i.2603.1 10 161.97 even 22