Properties

Label 1840.1.bq.a.1599.1
Level $1840$
Weight $1$
Character 1840.1599
Analytic conductor $0.918$
Analytic rank $0$
Dimension $20$
Projective image $D_{22}$
CM discriminant -20
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,1,Mod(239,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 0, 11, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.239");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1840.bq (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: \(0.918279623245\)
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: \(\Z[a_1, \ldots, a_{23}]\)
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 1599.1
Root \(-0.540641 - 0.841254i\) of defining polynomial
Character \(\chi\) \(=\) 1840.1599
Dual form 1840.1.bq.a.1359.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.153882 + 1.07028i) q^{3} +(0.959493 - 0.281733i) q^{5} +(0.989821 + 1.14231i) q^{7} +(-0.162317 - 0.0476607i) q^{9} +O(q^{10})\) \(q+(-0.153882 + 1.07028i) q^{3} +(0.959493 - 0.281733i) q^{5} +(0.989821 + 1.14231i) q^{7} +(-0.162317 - 0.0476607i) q^{9} +(0.153882 + 1.07028i) q^{15} +(-1.37491 + 0.883600i) q^{21} +(-0.540641 - 0.841254i) q^{23} +(0.841254 - 0.540641i) q^{25} +(-0.373193 + 0.817178i) q^{27} +(-0.797176 - 1.74557i) q^{29} +(1.27155 + 0.817178i) q^{35} +(-1.25667 + 0.368991i) q^{41} +(0.258908 - 1.80075i) q^{43} -0.169170 q^{45} +0.563465 q^{47} +(-0.182822 + 1.27155i) q^{49} +(0.118239 + 0.822373i) q^{61} +(-0.106222 - 0.232593i) q^{63} +(-1.66538 + 1.07028i) q^{67} +(0.983568 - 0.449181i) q^{69} +(0.449181 + 0.983568i) q^{75} +(-0.959493 - 0.616629i) q^{81} +(-1.89945 - 0.557730i) q^{83} +(1.99091 - 0.584585i) q^{87} +(-0.239446 + 1.66538i) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{5} + 2 q^{9} - 2 q^{25} - 4 q^{29} + 4 q^{41} - 24 q^{45} - 20 q^{49} + 4 q^{61} - 2 q^{81} - 4 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{10}{11}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.153882 + 1.07028i −0.153882 + 1.07028i 0.755750 + 0.654861i \(0.227273\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(4\) 0 0
\(5\) 0.959493 0.281733i 0.959493 0.281733i
\(6\) 0 0
\(7\) 0.989821 + 1.14231i 0.989821 + 1.14231i 0.989821 + 0.142315i \(0.0454545\pi\)
1.00000i \(0.500000\pi\)
\(8\) 0 0
\(9\) −0.162317 0.0476607i −0.162317 0.0476607i
\(10\) 0 0
\(11\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(12\) 0 0
\(13\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(14\) 0 0
\(15\) 0.153882 + 1.07028i 0.153882 + 1.07028i
\(16\) 0 0
\(17\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(18\) 0 0
\(19\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(20\) 0 0
\(21\) −1.37491 + 0.883600i −1.37491 + 0.883600i
\(22\) 0 0
\(23\) −0.540641 0.841254i −0.540641 0.841254i
\(24\) 0 0
\(25\) 0.841254 0.540641i 0.841254 0.540641i
\(26\) 0 0
\(27\) −0.373193 + 0.817178i −0.373193 + 0.817178i
\(28\) 0 0
\(29\) −0.797176 1.74557i −0.797176 1.74557i −0.654861 0.755750i \(-0.727273\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(30\) 0 0
\(31\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.27155 + 0.817178i 1.27155 + 0.817178i
\(36\) 0 0
\(37\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.25667 + 0.368991i −1.25667 + 0.368991i −0.841254 0.540641i \(-0.818182\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(42\) 0 0
\(43\) 0.258908 1.80075i 0.258908 1.80075i −0.281733 0.959493i \(-0.590909\pi\)
0.540641 0.841254i \(-0.318182\pi\)
\(44\) 0 0
\(45\) −0.169170 −0.169170
\(46\) 0 0
\(47\) 0.563465 0.563465 0.281733 0.959493i \(-0.409091\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(48\) 0 0
\(49\) −0.182822 + 1.27155i −0.182822 + 1.27155i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(60\) 0 0
\(61\) 0.118239 + 0.822373i 0.118239 + 0.822373i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(62\) 0 0
\(63\) −0.106222 0.232593i −0.106222 0.232593i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.66538 + 1.07028i −1.66538 + 1.07028i −0.755750 + 0.654861i \(0.772727\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(68\) 0 0
\(69\) 0.983568 0.449181i 0.983568 0.449181i
\(70\) 0 0
\(71\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(72\) 0 0
\(73\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(74\) 0 0
\(75\) 0.449181 + 0.983568i 0.449181 + 0.983568i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(80\) 0 0
\(81\) −0.959493 0.616629i −0.959493 0.616629i
\(82\) 0 0
\(83\) −1.89945 0.557730i −1.89945 0.557730i −0.989821 0.142315i \(-0.954545\pi\)
−0.909632 0.415415i \(-0.863636\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.99091 0.584585i 1.99091 0.584585i
\(88\) 0 0
\(89\) −0.239446 + 1.66538i −0.239446 + 1.66538i 0.415415 + 0.909632i \(0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.273100 + 0.0801894i 0.273100 + 0.0801894i 0.415415 0.909632i \(-0.363636\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(102\) 0 0
\(103\) 0.474017 + 0.304632i 0.474017 + 0.304632i 0.755750 0.654861i \(-0.227273\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(104\) 0 0
\(105\) −1.07028 + 1.23516i −1.07028 + 1.23516i
\(106\) 0 0
\(107\) −0.0801894 0.557730i −0.0801894 0.557730i −0.989821 0.142315i \(-0.954545\pi\)
0.909632 0.415415i \(-0.136364\pi\)
\(108\) 0 0
\(109\) −0.698939 1.53046i −0.698939 1.53046i −0.841254 0.540641i \(-0.818182\pi\)
0.142315 0.989821i \(-0.454545\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(114\) 0 0
\(115\) −0.755750 0.654861i −0.755750 0.654861i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(122\) 0 0
\(123\) −0.201543 1.40176i −0.201543 1.40176i
\(124\) 0 0
\(125\) 0.654861 0.755750i 0.654861 0.755750i
\(126\) 0 0
\(127\) 1.66538 + 1.07028i 1.66538 + 1.07028i 0.909632 + 0.415415i \(0.136364\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(128\) 0 0
\(129\) 1.88745 + 0.554206i 1.88745 + 0.554206i
\(130\) 0 0
\(131\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.127850 + 0.889217i −0.127850 + 0.889217i
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −0.0867074 + 0.603063i −0.0867074 + 0.603063i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.25667 1.45027i −1.25667 1.45027i
\(146\) 0 0
\(147\) −1.33278 0.391340i −1.33278 0.391340i
\(148\) 0 0
\(149\) 1.61435 + 1.03748i 1.61435 + 1.03748i 0.959493 + 0.281733i \(0.0909091\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(150\) 0 0
\(151\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.425839 1.45027i 0.425839 1.45027i
\(162\) 0 0
\(163\) −1.53046 + 0.983568i −1.53046 + 0.983568i −0.540641 + 0.841254i \(0.681818\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.755750 1.65486i −0.755750 1.65486i −0.755750 0.654861i \(-0.772727\pi\)
1.00000i \(-0.5\pi\)
\(168\) 0 0
\(169\) −0.142315 0.989821i −0.142315 0.989821i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(174\) 0 0
\(175\) 1.45027 + 0.425839i 1.45027 + 0.425839i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(180\) 0 0
\(181\) 0.0405070 0.281733i 0.0405070 0.281733i −0.959493 0.281733i \(-0.909091\pi\)
1.00000 \(0\)
\(182\) 0 0
\(183\) −0.898361 −0.898361
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.30287 + 0.382557i −1.30287 + 0.382557i
\(190\) 0 0
\(191\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(192\) 0 0
\(193\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(198\) 0 0
\(199\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(200\) 0 0
\(201\) −0.889217 1.94711i −0.889217 1.94711i
\(202\) 0 0
\(203\) 1.20493 2.63843i 1.20493 2.63843i
\(204\) 0 0
\(205\) −1.10181 + 0.708089i −1.10181 + 0.708089i
\(206\) 0 0
\(207\) 0.0476607 + 0.162317i 0.0476607 + 0.162317i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.258908 1.80075i −0.258908 1.80075i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.19136 1.37491i −1.19136 1.37491i −0.909632 0.415415i \(-0.863636\pi\)
−0.281733 0.959493i \(-0.590909\pi\)
\(224\) 0 0
\(225\) −0.162317 + 0.0476607i −0.162317 + 0.0476607i
\(226\) 0 0
\(227\) 0.0801894 0.557730i 0.0801894 0.557730i −0.909632 0.415415i \(-0.863636\pi\)
0.989821 0.142315i \(-0.0454545\pi\)
\(228\) 0 0
\(229\) 1.68251 1.68251 0.841254 0.540641i \(-0.181818\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(234\) 0 0
\(235\) 0.540641 0.158746i 0.540641 0.158746i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(240\) 0 0
\(241\) 1.10181 + 0.708089i 1.10181 + 0.708089i 0.959493 0.281733i \(-0.0909091\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(242\) 0 0
\(243\) 0.219310 0.253098i 0.219310 0.253098i
\(244\) 0 0
\(245\) 0.182822 + 1.27155i 0.182822 + 1.27155i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0.889217 1.94711i 0.889217 1.94711i
\(250\) 0 0
\(251\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0.0462003 + 0.321330i 0.0462003 + 0.321330i
\(262\) 0 0
\(263\) 1.19136 1.37491i 1.19136 1.37491i 0.281733 0.959493i \(-0.409091\pi\)
0.909632 0.415415i \(-0.136364\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −1.74557 0.512546i −1.74557 0.512546i
\(268\) 0 0
\(269\) −0.186393 0.215109i −0.186393 0.215109i 0.654861 0.755750i \(-0.272727\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(270\) 0 0
\(271\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.797176 0.234072i 0.797176 0.234072i 0.142315 0.989821i \(-0.454545\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(282\) 0 0
\(283\) −0.708089 0.817178i −0.708089 0.817178i 0.281733 0.959493i \(-0.409091\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.66538 1.07028i −1.66538 1.07028i
\(288\) 0 0
\(289\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 2.31329 1.48666i 2.31329 1.48666i
\(302\) 0 0
\(303\) −0.127850 + 0.279953i −0.127850 + 0.279953i
\(304\) 0 0
\(305\) 0.345139 + 0.755750i 0.345139 + 0.755750i
\(306\) 0 0
\(307\) 0.215109 + 1.49611i 0.215109 + 1.49611i 0.755750 + 0.654861i \(0.227273\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(308\) 0 0
\(309\) −0.398983 + 0.460451i −0.398983 + 0.460451i
\(310\) 0 0
\(311\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(312\) 0 0
\(313\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(314\) 0 0
\(315\) −0.167448 0.193245i −0.167448 0.193245i
\(316\) 0 0
\(317\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0.609264 0.609264
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.74557 0.512546i 1.74557 0.512546i
\(328\) 0 0
\(329\) 0.557730 + 0.643655i 0.557730 + 0.643655i
\(330\) 0 0
\(331\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.29639 + 1.49611i −1.29639 + 1.49611i
\(336\) 0 0
\(337\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.361922 + 0.232593i −0.361922 + 0.232593i
\(344\) 0 0
\(345\) 0.817178 0.708089i 0.817178 0.708089i
\(346\) 0 0
\(347\) 1.66538 1.07028i 1.66538 1.07028i 0.755750 0.654861i \(-0.227273\pi\)
0.909632 0.415415i \(-0.136364\pi\)
\(348\) 0 0
\(349\) −0.544078 + 1.19136i −0.544078 + 1.19136i 0.415415 + 0.909632i \(0.363636\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(360\) 0 0
\(361\) −0.654861 0.755750i −0.654861 0.755750i
\(362\) 0 0
\(363\) −1.03748 + 0.304632i −1.03748 + 0.304632i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.97964 1.97964 0.989821 0.142315i \(-0.0454545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(368\) 0 0
\(369\) 0.221566 0.221566
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(374\) 0 0
\(375\) 0.708089 + 0.817178i 0.708089 + 0.817178i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(380\) 0 0
\(381\) −1.40176 + 1.61772i −1.40176 + 1.61772i
\(382\) 0 0
\(383\) −0.215109 1.49611i −0.215109 1.49611i −0.755750 0.654861i \(-0.772727\pi\)
0.540641 0.841254i \(-0.318182\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.127850 + 0.279953i −0.127850 + 0.279953i
\(388\) 0 0
\(389\) −1.41542 + 0.909632i −1.41542 + 0.909632i −0.415415 + 0.909632i \(0.636364\pi\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.186393 0.215109i 0.186393 0.215109i −0.654861 0.755750i \(-0.727273\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.09435 0.321330i −1.09435 0.321330i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.84125 + 0.540641i −1.84125 + 0.540641i −0.841254 + 0.540641i \(0.818182\pi\)
−1.00000 \(1.00000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1.97964 −1.97964
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(420\) 0 0
\(421\) −0.857685 0.989821i −0.857685 0.989821i 0.142315 0.989821i \(-0.454545\pi\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) −0.0914602 0.0268551i −0.0914602 0.0268551i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.822373 + 0.949069i −0.822373 + 0.949069i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(432\) 0 0
\(433\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(434\) 0 0
\(435\) 1.74557 1.12181i 1.74557 1.12181i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(440\) 0 0
\(441\) 0.0902783 0.197682i 0.0902783 0.197682i
\(442\) 0 0
\(443\) 0.755750 + 1.65486i 0.755750 + 1.65486i 0.755750 + 0.654861i \(0.227273\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0.239446 + 1.66538i 0.239446 + 1.66538i
\(446\) 0 0
\(447\) −1.35881 + 1.56815i −1.35881 + 1.56815i
\(448\) 0 0
\(449\) −1.41542 0.909632i −1.41542 0.909632i −0.415415 0.909632i \(-0.636364\pi\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.68251 1.68251 0.841254 0.540641i \(-0.181818\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(462\) 0 0
\(463\) −0.215109 + 1.49611i −0.215109 + 1.49611i 0.540641 + 0.841254i \(0.318182\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(468\) 0 0
\(469\) −2.87102 0.843008i −2.87102 0.843008i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 1.48666 + 0.678936i 1.48666 + 0.678936i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −0.627899 + 1.37491i −0.627899 + 1.37491i 0.281733 + 0.959493i \(0.409091\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(488\) 0 0
\(489\) −0.817178 1.78937i −0.817178 1.78937i
\(490\) 0 0
\(491\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(500\) 0 0
\(501\) 1.88745 0.554206i 1.88745 0.554206i
\(502\) 0 0
\(503\) −0.153882 + 1.07028i −0.153882 + 1.07028i 0.755750 + 0.654861i \(0.227273\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(504\) 0 0
\(505\) 0.284630 0.284630
\(506\) 0 0
\(507\) 1.08128 1.08128
\(508\) 0 0
\(509\) −0.118239 + 0.822373i −0.118239 + 0.822373i 0.841254 + 0.540641i \(0.181818\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.540641 + 0.158746i 0.540641 + 0.158746i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.273100 + 1.89945i 0.273100 + 1.89945i 0.415415 + 0.909632i \(0.363636\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(522\) 0 0
\(523\) 0.822373 + 1.80075i 0.822373 + 1.80075i 0.540641 + 0.841254i \(0.318182\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(524\) 0 0
\(525\) −0.678936 + 1.48666i −0.678936 + 1.48666i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.234072 0.512546i −0.234072 0.512546i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −0.239446 0.153882i −0.239446 0.153882i 0.415415 0.909632i \(-0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(542\) 0 0
\(543\) 0.295298 + 0.0867074i 0.295298 + 0.0867074i
\(544\) 0 0
\(545\) −1.10181 1.27155i −1.10181 1.27155i
\(546\) 0 0
\(547\) −0.540641 + 0.158746i −0.540641 + 0.158746i −0.540641 0.841254i \(-0.681818\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 0.0200026 0.139121i 0.0200026 0.139121i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.27155 + 0.817178i 1.27155 + 0.817178i 0.989821 0.142315i \(-0.0454545\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.245343 1.70640i −0.245343 1.70640i
\(568\) 0 0
\(569\) 0.118239 + 0.258908i 0.118239 + 0.258908i 0.959493 0.281733i \(-0.0909091\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(570\) 0 0
\(571\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.909632 0.415415i −0.909632 0.415415i
\(576\) 0 0
\(577\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.24302 2.72183i −1.24302 2.72183i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.53046 0.983568i −1.53046 0.983568i −0.989821 0.142315i \(-0.954545\pi\)
−0.540641 0.841254i \(-0.681818\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0.239446 1.66538i 0.239446 1.66538i −0.415415 0.909632i \(-0.636364\pi\)
0.654861 0.755750i \(-0.272727\pi\)
\(602\) 0 0
\(603\) 0.321330 0.0943511i 0.321330 0.0943511i
\(604\) 0 0
\(605\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(606\) 0 0
\(607\) 1.74557 + 0.512546i 1.74557 + 0.512546i 0.989821 0.142315i \(-0.0454545\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(608\) 0 0
\(609\) 2.63843 + 1.69562i 2.63843 + 1.69562i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(614\) 0 0
\(615\) −0.588302 1.28820i −0.588302 1.28820i
\(616\) 0 0
\(617\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(618\) 0 0
\(619\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(620\) 0 0
\(621\) 0.889217 0.127850i 0.889217 0.127850i
\(622\) 0 0
\(623\) −2.13940 + 1.37491i −2.13940 + 1.37491i
\(624\) 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
\(631\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.89945 + 0.557730i 1.89945 + 0.557730i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.273100 + 1.89945i −0.273100 + 1.89945i 0.142315 + 0.989821i \(0.454545\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(642\) 0 0
\(643\) 1.81926 1.81926 0.909632 0.415415i \(-0.136364\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(644\) 0 0
\(645\) 1.96714 1.96714
\(646\) 0 0
\(647\) 0.281733 1.95949i 0.281733 1.95949i 1.00000i \(-0.5\pi\)
0.281733 0.959493i \(-0.409091\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(660\) 0 0
\(661\) 0.797176 + 1.74557i 0.797176 + 1.74557i 0.654861 + 0.755750i \(0.272727\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.03748 + 1.61435i −1.03748 + 1.61435i
\(668\) 0 0
\(669\) 1.65486 1.06351i 1.65486 1.06351i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(674\) 0 0
\(675\) 0.127850 + 0.889217i 0.127850 + 0.889217i
\(676\) 0 0
\(677\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0.584585 + 0.171650i 0.584585 + 0.171650i
\(682\) 0 0
\(683\) 0.989821 + 1.14231i 0.989821 + 1.14231i 0.989821 + 0.142315i \(0.0454545\pi\)
1.00000i \(0.500000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −0.258908 + 1.80075i −0.258908 + 1.80075i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −0.698939 0.449181i −0.698939 0.449181i 0.142315 0.989821i \(-0.454545\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0.0867074 + 0.603063i 0.0867074 + 0.603063i
\(706\) 0 0
\(707\) 0.178719 + 0.391340i 0.178719 + 0.391340i
\(708\) 0 0
\(709\) 0.830830 1.81926i 0.830830 1.81926i 0.415415 0.909632i \(-0.363636\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(720\) 0 0
\(721\) 0.121206 + 0.843008i 0.121206 + 0.843008i
\(722\) 0 0
\(723\) −0.927399 + 1.07028i −0.927399 + 1.07028i
\(724\) 0 0
\(725\) −1.61435 1.03748i −1.61435 1.03748i
\(726\) 0 0
\(727\) 0.540641 + 0.158746i 0.540641 + 0.158746i 0.540641 0.841254i \(-0.318182\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −0.509766 0.588302i −0.509766 0.588302i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(734\) 0 0
\(735\) −1.38905 −1.38905
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0.368991 + 0.425839i 0.368991 + 0.425839i 0.909632 0.415415i \(-0.136364\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(744\) 0 0
\(745\) 1.84125 + 0.540641i 1.84125 + 0.540641i
\(746\) 0 0
\(747\) 0.281733 + 0.181059i 0.281733 + 0.181059i
\(748\) 0 0
\(749\) 0.557730 0.643655i 0.557730 0.643655i
\(750\) 0 0
\(751\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.10181 + 0.708089i −1.10181 + 0.708089i −0.959493 0.281733i \(-0.909091\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(762\) 0 0
\(763\) 1.05645 2.31329i 1.05645 2.31329i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −0.544078 + 0.627899i −0.544078 + 0.627899i −0.959493 0.281733i \(-0.909091\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 1.72394 1.72394
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.03748 0.304632i 1.03748 0.304632i 0.281733 0.959493i \(-0.409091\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(788\) 0 0
\(789\) 1.28820 + 1.48666i 1.28820 + 1.48666i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0.118239 0.258908i 0.118239 0.258908i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 1.51150i 1.51150i
\(806\) 0 0
\(807\) 0.258908 0.166390i 0.258908 0.166390i
\(808\) 0 0
\(809\) 0.698939 1.53046i 0.698939 1.53046i −0.142315 0.989821i \(-0.545455\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(810\) 0 0
\(811\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.19136 + 1.37491i −1.19136 + 1.37491i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.25667 1.45027i −1.25667 1.45027i −0.841254 0.540641i \(-0.818182\pi\)
−0.415415 0.909632i \(-0.636364\pi\)
\(822\) 0 0
\(823\) −1.45027 + 0.425839i −1.45027 + 0.425839i −0.909632 0.415415i \(-0.863636\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 1.30972 1.30972 0.654861 0.755750i \(-0.272727\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1.19136 1.37491i −1.19136 1.37491i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(840\) 0 0
\(841\) −1.75667 + 2.02730i −1.75667 + 2.02730i
\(842\) 0 0
\(843\) 0.127850 + 0.889217i 0.127850 + 0.889217i
\(844\) 0 0
\(845\) −0.415415 0.909632i −0.415415 0.909632i
\(846\) 0 0
\(847\) −0.627899 + 1.37491i −0.627899 + 1.37491i
\(848\) 0 0
\(849\) 0.983568 0.632101i 0.983568 0.632101i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(858\) 0 0
\(859\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(860\) 0 0
\(861\) 1.40176 1.61772i 1.40176 1.61772i
\(862\) 0 0
\(863\) −1.27155 0.817178i −1.27155 0.817178i −0.281733 0.959493i \(-0.590909\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.708089 0.817178i −0.708089 0.817178i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.51150 1.51150
\(876\) 0 0
\(877\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.186393 0.215109i −0.186393 0.215109i 0.654861 0.755750i \(-0.272727\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(882\) 0 0
\(883\) 1.03748 + 0.304632i 1.03748 + 0.304632i 0.755750 0.654861i \(-0.227273\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0.708089 0.817178i 0.708089 0.817178i −0.281733 0.959493i \(-0.590909\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(888\) 0 0
\(889\) 0.425839 + 2.96177i 0.425839 + 2.96177i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 1.23516 + 2.70463i 1.23516 + 2.70463i
\(904\) 0 0
\(905\) −0.0405070 0.281733i −0.0405070 0.281733i
\(906\) 0 0
\(907\) 0.708089 0.817178i 0.708089 0.817178i −0.281733 0.959493i \(-0.590909\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(908\) 0 0
\(909\) −0.0405070 0.0260323i −0.0405070 0.0260323i
\(910\) 0 0
\(911\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −0.861971 + 0.253098i −0.861971 + 0.253098i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −1.63436 −1.63436
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −0.0624222 0.0720391i −0.0624222 0.0720391i
\(928\) 0 0
\(929\) −1.84125 0.540641i −1.84125 0.540641i −0.841254 0.540641i \(-0.818182\pi\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.698939 0.449181i 0.698939 0.449181i −0.142315 0.989821i \(-0.545455\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(942\) 0 0
\(943\) 0.989821 + 0.857685i 0.989821 + 0.857685i
\(944\) 0 0
\(945\) −1.14231 + 0.734121i −1.14231 + 0.734121i
\(946\) 0 0
\(947\) −0.449181 + 0.983568i −0.449181 + 0.983568i 0.540641 + 0.841254i \(0.318182\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(962\) 0 0
\(963\) −0.0135656 + 0.0943511i −0.0135656 + 0.0943511i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.08128 −1.08128 −0.540641 0.841254i \(-0.681818\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0.0405070 + 0.281733i 0.0405070 + 0.281733i
\(982\) 0 0
\(983\) 0.449181 + 0.983568i 0.449181 + 0.983568i 0.989821 + 0.142315i \(0.0454545\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −0.774713 + 0.497878i −0.774713 + 0.497878i
\(988\) 0 0
\(989\) −1.65486 + 0.755750i −1.65486 + 0.755750i
\(990\) 0 0
\(991\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1840.1.bq.a.1599.1 yes 20
4.3 odd 2 inner 1840.1.bq.a.1599.2 yes 20
5.4 even 2 inner 1840.1.bq.a.1599.2 yes 20
20.19 odd 2 CM 1840.1.bq.a.1599.1 yes 20
23.2 even 11 inner 1840.1.bq.a.1359.1 20
92.71 odd 22 inner 1840.1.bq.a.1359.2 yes 20
115.94 even 22 inner 1840.1.bq.a.1359.2 yes 20
460.439 odd 22 inner 1840.1.bq.a.1359.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1840.1.bq.a.1359.1 20 23.2 even 11 inner
1840.1.bq.a.1359.1 20 460.439 odd 22 inner
1840.1.bq.a.1359.2 yes 20 92.71 odd 22 inner
1840.1.bq.a.1359.2 yes 20 115.94 even 22 inner
1840.1.bq.a.1599.1 yes 20 1.1 even 1 trivial
1840.1.bq.a.1599.1 yes 20 20.19 odd 2 CM
1840.1.bq.a.1599.2 yes 20 4.3 odd 2 inner
1840.1.bq.a.1599.2 yes 20 5.4 even 2 inner