Properties

Label 1080.2.f.d.649.2
Level $1080$
Weight $2$
Character 1080.649
Analytic conductor $8.624$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1080,2,Mod(649,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.f (of order \(2\), degree \(1\), 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: \(8.62384341830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1080.649
Dual form 1080.2.f.d.649.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.00000 + 1.00000i) q^{5} -2.00000i q^{7} +O(q^{10})\) \(q+(2.00000 + 1.00000i) q^{5} -2.00000i q^{7} +2.00000 q^{11} -2.00000i q^{13} -3.00000i q^{17} +1.00000 q^{19} -5.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +10.0000 q^{29} -9.00000 q^{31} +(2.00000 - 4.00000i) q^{35} +8.00000i q^{37} +2.00000 q^{41} -6.00000i q^{43} +3.00000 q^{49} -1.00000i q^{53} +(4.00000 + 2.00000i) q^{55} +6.00000 q^{59} -7.00000 q^{61} +(2.00000 - 4.00000i) q^{65} -8.00000i q^{67} +14.0000 q^{71} +14.0000i q^{73} -4.00000i q^{77} -1.00000 q^{79} +5.00000i q^{83} +(3.00000 - 6.00000i) q^{85} -6.00000 q^{89} -4.00000 q^{91} +(2.00000 + 1.00000i) q^{95} +8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} + 4 q^{11} + 2 q^{19} + 6 q^{25} + 20 q^{29} - 18 q^{31} + 4 q^{35} + 4 q^{41} + 6 q^{49} + 8 q^{55} + 12 q^{59} - 14 q^{61} + 4 q^{65} + 28 q^{71} - 2 q^{79} + 6 q^{85} - 12 q^{89} - 8 q^{91} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(541\) \(1001\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 2.00000 + 1.00000i 0.894427 + 0.447214i
\(6\) 0 0
\(7\) 2.00000i 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000i 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.00000i 1.04257i −0.853382 0.521286i \(-0.825452\pi\)
0.853382 0.521286i \(-0.174548\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 4.00000i 0.338062 0.676123i
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.00000i 0.137361i −0.997639 0.0686803i \(-0.978121\pi\)
0.997639 0.0686803i \(-0.0218788\pi\)
\(54\) 0 0
\(55\) 4.00000 + 2.00000i 0.539360 + 0.269680i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 4.00000i 0.248069 0.496139i
\(66\) 0 0
\(67\) 8.00000i 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 0 0
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.00000i 0.548821i 0.961613 + 0.274411i \(0.0884828\pi\)
−0.961613 + 0.274411i \(0.911517\pi\)
\(84\) 0 0
\(85\) 3.00000 6.00000i 0.325396 0.650791i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.00000 + 1.00000i 0.205196 + 0.102598i
\(96\) 0 0
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 16.0000i 1.57653i −0.615338 0.788263i \(-0.710980\pi\)
0.615338 0.788263i \(-0.289020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) 15.0000 1.43674 0.718370 0.695662i \(-0.244889\pi\)
0.718370 + 0.695662i \(0.244889\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.00000i 0.188144i −0.995565 0.0940721i \(-0.970012\pi\)
0.995565 0.0940721i \(-0.0299884\pi\)
\(114\) 0 0
\(115\) 5.00000 10.0000i 0.466252 0.932505i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.00000 + 11.0000i 0.178885 + 0.983870i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 2.00000i 0.173422i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.0000i 1.45241i −0.687479 0.726204i \(-0.741283\pi\)
0.687479 0.726204i \(-0.258717\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000i 0.334497i
\(144\) 0 0
\(145\) 20.0000 + 10.0000i 1.66091 + 0.830455i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −18.0000 9.00000i −1.44579 0.722897i
\(156\) 0 0
\(157\) 14.0000i 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −10.0000 −0.788110
\(162\) 0 0
\(163\) 20.0000i 1.56652i 0.621694 + 0.783260i \(0.286445\pi\)
−0.621694 + 0.783260i \(0.713555\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.0000i 1.62503i 0.582941 + 0.812514i \(0.301902\pi\)
−0.582941 + 0.812514i \(0.698098\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.00000i 0.684257i 0.939653 + 0.342129i \(0.111148\pi\)
−0.939653 + 0.342129i \(0.888852\pi\)
\(174\) 0 0
\(175\) 8.00000 6.00000i 0.604743 0.453557i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −23.0000 −1.70958 −0.854788 0.518977i \(-0.826313\pi\)
−0.854788 + 0.518977i \(0.826313\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.00000 + 16.0000i −0.588172 + 1.17634i
\(186\) 0 0
\(187\) 6.00000i 0.438763i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.00000i 0.213741i 0.994273 + 0.106871i \(0.0340831\pi\)
−0.994273 + 0.106871i \(0.965917\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 20.0000i 1.40372i
\(204\) 0 0
\(205\) 4.00000 + 2.00000i 0.279372 + 0.139686i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −23.0000 −1.58339 −0.791693 0.610920i \(-0.790800\pi\)
−0.791693 + 0.610920i \(0.790800\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.00000 12.0000i 0.409197 0.818393i
\(216\) 0 0
\(217\) 18.0000i 1.22192i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) 12.0000i 0.803579i 0.915732 + 0.401790i \(0.131612\pi\)
−0.915732 + 0.401790i \(0.868388\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 23.0000i 1.52656i 0.646066 + 0.763282i \(0.276413\pi\)
−0.646066 + 0.763282i \(0.723587\pi\)
\(228\) 0 0
\(229\) −15.0000 −0.991228 −0.495614 0.868543i \(-0.665057\pi\)
−0.495614 + 0.868543i \(0.665057\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 30.0000i 1.96537i 0.185296 + 0.982683i \(0.440675\pi\)
−0.185296 + 0.982683i \(0.559325\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157 −0.0322078 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.00000 + 3.00000i 0.383326 + 0.191663i
\(246\) 0 0
\(247\) 2.00000i 0.127257i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 10.0000i 0.628695i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.00000i 0.187135i 0.995613 + 0.0935674i \(0.0298271\pi\)
−0.995613 + 0.0935674i \(0.970173\pi\)
\(258\) 0 0
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) 0 0
\(265\) 1.00000 2.00000i 0.0614295 0.122859i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.00000 + 8.00000i 0.361814 + 0.482418i
\(276\) 0 0
\(277\) 30.0000i 1.80253i −0.433273 0.901263i \(-0.642641\pi\)
0.433273 0.901263i \(-0.357359\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 18.0000i 1.06999i −0.844856 0.534994i \(-0.820314\pi\)
0.844856 0.534994i \(-0.179686\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000i 0.236113i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 23.0000i 1.34367i −0.740699 0.671837i \(-0.765505\pi\)
0.740699 0.671837i \(-0.234495\pi\)
\(294\) 0 0
\(295\) 12.0000 + 6.00000i 0.698667 + 0.349334i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −10.0000 −0.578315
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −14.0000 7.00000i −0.801638 0.400819i
\(306\) 0 0
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) 4.00000i 0.226093i −0.993590 0.113047i \(-0.963939\pi\)
0.993590 0.113047i \(-0.0360610\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 27.0000i 1.51647i −0.651981 0.758236i \(-0.726062\pi\)
0.651981 0.758236i \(-0.273938\pi\)
\(318\) 0 0
\(319\) 20.0000 1.11979
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.00000i 0.166924i
\(324\) 0 0
\(325\) 8.00000 6.00000i 0.443760 0.332820i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.00000 16.0000i 0.437087 0.874173i
\(336\) 0 0
\(337\) 10.0000i 0.544735i 0.962193 + 0.272367i \(0.0878066\pi\)
−0.962193 + 0.272367i \(0.912193\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −18.0000 −0.974755
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.00000i 0.429463i 0.976673 + 0.214731i \(0.0688876\pi\)
−0.976673 + 0.214731i \(0.931112\pi\)
\(348\) 0 0
\(349\) −5.00000 −0.267644 −0.133822 0.991005i \(-0.542725\pi\)
−0.133822 + 0.991005i \(0.542725\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 30.0000i 1.59674i 0.602168 + 0.798369i \(0.294304\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(354\) 0 0
\(355\) 28.0000 + 14.0000i 1.48609 + 0.743043i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −14.0000 + 28.0000i −0.732793 + 1.46559i
\(366\) 0 0
\(367\) 14.0000i 0.730794i 0.930852 + 0.365397i \(0.119067\pi\)
−0.930852 + 0.365397i \(0.880933\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) 20.0000i 1.03556i −0.855514 0.517780i \(-0.826758\pi\)
0.855514 0.517780i \(-0.173242\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.0000i 1.03005i
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 21.0000i 1.07305i −0.843884 0.536525i \(-0.819737\pi\)
0.843884 0.536525i \(-0.180263\pi\)
\(384\) 0 0
\(385\) 4.00000 8.00000i 0.203859 0.407718i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −15.0000 −0.758583
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.00000 1.00000i −0.100631 0.0503155i
\(396\) 0 0
\(397\) 30.0000i 1.50566i −0.658217 0.752828i \(-0.728689\pi\)
0.658217 0.752828i \(-0.271311\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.00000 0.199750 0.0998752 0.995000i \(-0.468156\pi\)
0.0998752 + 0.995000i \(0.468156\pi\)
\(402\) 0 0
\(403\) 18.0000i 0.896644i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16.0000i 0.793091i
\(408\) 0 0
\(409\) 19.0000 0.939490 0.469745 0.882802i \(-0.344346\pi\)
0.469745 + 0.882802i \(0.344346\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 12.0000i 0.590481i
\(414\) 0 0
\(415\) −5.00000 + 10.0000i −0.245440 + 0.490881i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −17.0000 −0.828529 −0.414265 0.910156i \(-0.635961\pi\)
−0.414265 + 0.910156i \(0.635961\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 12.0000 9.00000i 0.582086 0.436564i
\(426\) 0 0
\(427\) 14.0000i 0.677507i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 34.0000 1.63772 0.818861 0.573992i \(-0.194606\pi\)
0.818861 + 0.573992i \(0.194606\pi\)
\(432\) 0 0
\(433\) 6.00000i 0.288342i −0.989553 0.144171i \(-0.953949\pi\)
0.989553 0.144171i \(-0.0460515\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.00000i 0.239182i
\(438\) 0 0
\(439\) −15.0000 −0.715911 −0.357955 0.933739i \(-0.616526\pi\)
−0.357955 + 0.933739i \(0.616526\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9.00000i 0.427603i 0.976877 + 0.213801i \(0.0685846\pi\)
−0.976877 + 0.213801i \(0.931415\pi\)
\(444\) 0 0
\(445\) −12.0000 6.00000i −0.568855 0.284427i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.00000 4.00000i −0.375046 0.187523i
\(456\) 0 0
\(457\) 36.0000i 1.68401i 0.539471 + 0.842004i \(0.318624\pi\)
−0.539471 + 0.842004i \(0.681376\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 0 0
\(463\) 40.0000i 1.85896i 0.368875 + 0.929479i \(0.379743\pi\)
−0.368875 + 0.929479i \(0.620257\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.0000i 1.24941i 0.780860 + 0.624705i \(0.214781\pi\)
−0.780860 + 0.624705i \(0.785219\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) 3.00000 + 4.00000i 0.137649 + 0.183533i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −26.0000 −1.18797 −0.593985 0.804476i \(-0.702446\pi\)
−0.593985 + 0.804476i \(0.702446\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.00000 + 16.0000i −0.363261 + 0.726523i
\(486\) 0 0
\(487\) 2.00000i 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 30.0000i 1.35113i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 28.0000i 1.25597i
\(498\) 0 0
\(499\) 13.0000 0.581960 0.290980 0.956729i \(-0.406019\pi\)
0.290980 + 0.956729i \(0.406019\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 25.0000i 1.11469i −0.830279 0.557347i \(-0.811819\pi\)
0.830279 0.557347i \(-0.188181\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8.00000 0.354594 0.177297 0.984157i \(-0.443265\pi\)
0.177297 + 0.984157i \(0.443265\pi\)
\(510\) 0 0
\(511\) 28.0000 1.23865
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.0000 32.0000i 0.705044 1.41009i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) 2.00000i 0.0874539i −0.999044 0.0437269i \(-0.986077\pi\)
0.999044 0.0437269i \(-0.0139232\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 27.0000i 1.17614i
\(528\) 0 0
\(529\) −2.00000 −0.0869565
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.00000i 0.173259i
\(534\) 0 0
\(535\) −4.00000 + 8.00000i −0.172935 + 0.345870i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −26.0000 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 30.0000 + 15.0000i 1.28506 + 0.642529i
\(546\) 0 0
\(547\) 2.00000i 0.0855138i −0.999086 0.0427569i \(-0.986386\pi\)
0.999086 0.0427569i \(-0.0136141\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.0000 0.426014
\(552\) 0 0
\(553\) 2.00000i 0.0850487i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000i 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 36.0000i 1.51722i −0.651546 0.758610i \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) 0 0
\(565\) 2.00000 4.00000i 0.0841406 0.168281i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 7.00000 0.292941 0.146470 0.989215i \(-0.453209\pi\)
0.146470 + 0.989215i \(0.453209\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 20.0000 15.0000i 0.834058 0.625543i
\(576\) 0 0
\(577\) 38.0000i 1.58196i −0.611842 0.790980i \(-0.709571\pi\)
0.611842 0.790980i \(-0.290429\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10.0000 0.414870
\(582\) 0 0
\(583\) 2.00000i 0.0828315i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.0000i 0.949312i −0.880172 0.474656i \(-0.842573\pi\)
0.880172 0.474656i \(-0.157427\pi\)
\(588\) 0 0
\(589\) −9.00000 −0.370839
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 29.0000i 1.19089i 0.803397 + 0.595444i \(0.203024\pi\)
−0.803397 + 0.595444i \(0.796976\pi\)
\(594\) 0 0
\(595\) −12.0000 6.00000i −0.491952 0.245976i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.00000 0.0817178 0.0408589 0.999165i \(-0.486991\pi\)
0.0408589 + 0.999165i \(0.486991\pi\)
\(600\) 0 0
\(601\) −3.00000 −0.122373 −0.0611863 0.998126i \(-0.519488\pi\)
−0.0611863 + 0.998126i \(0.519488\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −14.0000 7.00000i −0.569181 0.284590i
\(606\) 0 0
\(607\) 14.0000i 0.568242i 0.958788 + 0.284121i \(0.0917018\pi\)
−0.958788 + 0.284121i \(0.908298\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 32.0000i 1.29247i −0.763139 0.646234i \(-0.776343\pi\)
0.763139 0.646234i \(-0.223657\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19.0000i 0.764911i −0.923974 0.382456i \(-0.875078\pi\)
0.923974 0.382456i \(-0.124922\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 12.0000i 0.480770i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 5.00000 0.199047 0.0995234 0.995035i \(-0.468268\pi\)
0.0995234 + 0.995035i \(0.468268\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.00000i 0.237729i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 30.0000i 1.18308i 0.806274 + 0.591542i \(0.201481\pi\)
−0.806274 + 0.591542i \(0.798519\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.00000i 0.0393141i 0.999807 + 0.0196570i \(0.00625743\pi\)
−0.999807 + 0.0196570i \(0.993743\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 33.0000i 1.29139i −0.763596 0.645695i \(-0.776568\pi\)
0.763596 0.645695i \(-0.223432\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.00000 4.00000i 0.0775567 0.155113i
\(666\) 0 0
\(667\) 50.0000i 1.93601i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −14.0000 −0.540464
\(672\) 0 0
\(673\) 38.0000i 1.46479i 0.680879 + 0.732396i \(0.261598\pi\)
−0.680879 + 0.732396i \(0.738402\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000i 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) 0 0
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 25.0000i 0.956598i 0.878197 + 0.478299i \(0.158747\pi\)
−0.878197 + 0.478299i \(0.841253\pi\)
\(684\) 0 0
\(685\) 17.0000 34.0000i 0.649537 1.29907i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) 21.0000 0.798878 0.399439 0.916760i \(-0.369205\pi\)
0.399439 + 0.916760i \(0.369205\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.00000 4.00000i −0.303457 0.151729i
\(696\) 0 0
\(697\) 6.00000i 0.227266i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 28.0000 1.05755 0.528773 0.848763i \(-0.322652\pi\)
0.528773 + 0.848763i \(0.322652\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 45.0000i 1.68526i
\(714\) 0 0
\(715\) 4.00000 8.00000i 0.149592 0.299183i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 32.0000 1.19340 0.596699 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 30.0000 + 40.0000i 1.11417 + 1.48556i
\(726\) 0 0
\(727\) 12.0000i 0.445055i −0.974926 0.222528i \(-0.928569\pi\)
0.974926 0.222528i \(-0.0714308\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −18.0000 −0.665754
\(732\) 0 0
\(733\) 24.0000i 0.886460i 0.896408 + 0.443230i \(0.146168\pi\)
−0.896408 + 0.443230i \(0.853832\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) −9.00000 −0.331070 −0.165535 0.986204i \(-0.552935\pi\)
−0.165535 + 0.986204i \(0.552935\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36.0000i 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 0 0
\(745\) −12.0000 6.00000i −0.439646 0.219823i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) −19.0000 −0.693320 −0.346660 0.937991i \(-0.612684\pi\)
−0.346660 + 0.937991i \(0.612684\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −40.0000 20.0000i −1.45575 0.727875i
\(756\) 0 0
\(757\) 20.0000i 0.726912i 0.931611 + 0.363456i \(0.118403\pi\)
−0.931611 + 0.363456i \(0.881597\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) 30.0000i 1.08607i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000i 0.433295i
\(768\) 0 0
\(769\) −25.0000 −0.901523 −0.450762 0.892644i \(-0.648848\pi\)
−0.450762 + 0.892644i \(0.648848\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 7.00000i 0.251773i 0.992045 + 0.125886i \(0.0401774\pi\)
−0.992045 + 0.125886i \(0.959823\pi\)
\(774\) 0 0
\(775\) −27.0000 36.0000i −0.969869 1.29316i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) 28.0000 1.00192
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 14.0000 28.0000i 0.499681 0.999363i
\(786\) 0 0
\(787\) 50.0000i 1.78231i 0.453701 + 0.891154i \(0.350103\pi\)
−0.453701 + 0.891154i \(0.649897\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) 14.0000i 0.497155i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 39.0000i 1.38145i 0.723117 + 0.690725i \(0.242709\pi\)
−0.723117 + 0.690725i \(0.757291\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 28.0000i 0.988099i
\(804\) 0 0
\(805\) −20.0000 10.0000i −0.704907 0.352454i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 48.0000 1.68551 0.842754 0.538299i \(-0.180933\pi\)
0.842754 + 0.538299i \(0.180933\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −20.0000 + 40.0000i −0.700569 + 1.40114i
\(816\) 0 0
\(817\) 6.00000i 0.209913i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −36.0000 −1.25641 −0.628204 0.778048i \(-0.716210\pi\)
−0.628204 + 0.778048i \(0.716210\pi\)
\(822\) 0 0
\(823\) 10.0000i 0.348578i 0.984695 + 0.174289i \(0.0557627\pi\)
−0.984695 + 0.174289i \(0.944237\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.00000i 0.312961i 0.987681 + 0.156480i \(0.0500148\pi\)
−0.987681 + 0.156480i \(0.949985\pi\)
\(828\) 0 0
\(829\) −22.0000 −0.764092 −0.382046 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 9.00000i 0.311832i
\(834\) 0 0
\(835\) −21.0000 + 42.0000i −0.726735 + 1.45347i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 18.0000 + 9.00000i 0.619219 + 0.309609i
\(846\) 0 0
\(847\) 14.0000i 0.481046i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 40.0000 1.37118
\(852\) 0 0
\(853\) 28.0000i 0.958702i 0.877623 + 0.479351i \(0.159128\pi\)
−0.877623 + 0.479351i \(0.840872\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.00000i 0.239115i −0.992827 0.119558i \(-0.961852\pi\)
0.992827 0.119558i \(-0.0381477\pi\)
\(858\) 0 0
\(859\) −37.0000 −1.26242 −0.631212 0.775610i \(-0.717442\pi\)
−0.631212 + 0.775610i \(0.717442\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.00000i 0.102121i 0.998696 + 0.0510606i \(0.0162602\pi\)
−0.998696 + 0.0510606i \(0.983740\pi\)
\(864\) 0 0
\(865\) −9.00000 + 18.0000i −0.306009 + 0.612018i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2.00000 −0.0678454
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 22.0000 4.00000i 0.743736 0.135225i
\(876\) 0 0
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −38.0000 −1.28025 −0.640126 0.768270i \(-0.721118\pi\)
−0.640126 + 0.768270i \(0.721118\pi\)
\(882\) 0 0
\(883\) 46.0000i 1.54802i −0.633171 0.774012i \(-0.718247\pi\)
0.633171 0.774012i \(-0.281753\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 57.0000i 1.91387i 0.290298 + 0.956936i \(0.406246\pi\)
−0.290298 + 0.956936i \(0.593754\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 8.00000 + 4.00000i 0.267411 + 0.133705i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −90.0000 −3.00167
\(900\) 0 0
\(901\) −3.00000 −0.0999445
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −46.0000 23.0000i −1.52909 0.764546i
\(906\) 0 0
\(907\) 8.00000i 0.265636i 0.991140 + 0.132818i \(0.0424025\pi\)
−0.991140 + 0.132818i \(0.957597\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) 10.0000i 0.330952i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 28.0000i 0.921631i
\(924\) 0 0
\(925\) −32.0000 + 24.0000i −1.05215 + 0.789115i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −50.0000 −1.64045 −0.820223 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 6.00000 12.0000i 0.196221 0.392442i
\(936\) 0 0
\(937\) 20.0000i 0.653372i −0.945133 0.326686i \(-0.894068\pi\)
0.945133 0.326686i \(-0.105932\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −4.00000 −0.130396 −0.0651981 0.997872i \(-0.520768\pi\)
−0.0651981 + 0.997872i \(0.520768\pi\)
\(942\) 0 0
\(943\) 10.0000i 0.325645i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.0000i 0.877382i 0.898638 + 0.438691i \(0.144558\pi\)
−0.898638 + 0.438691i \(0.855442\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.00000i 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) 0 0
\(955\) −48.0000 24.0000i −1.55324 0.776622i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −34.0000 −1.09792
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −14.0000 + 28.0000i −0.450676 + 0.901352i
\(966\) 0 0
\(967\) 44.0000i 1.41494i −0.706741 0.707472i \(-0.749835\pi\)
0.706741 0.707472i \(-0.250165\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 0 0
\(973\) 8.00000i 0.256468i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 38.0000i 1.21573i −0.794041 0.607864i \(-0.792027\pi\)
0.794041 0.607864i \(-0.207973\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 7.00000i 0.223265i 0.993750 + 0.111633i \(0.0356080\pi\)
−0.993750 + 0.111633i \(0.964392\pi\)
\(984\) 0 0
\(985\) −3.00000 + 6.00000i −0.0955879 + 0.191176i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −30.0000 −0.953945
\(990\) 0 0
\(991\) 49.0000 1.55654 0.778268 0.627932i \(-0.216098\pi\)
0.778268 + 0.627932i \(0.216098\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 16.0000 + 8.00000i 0.507234 + 0.253617i
\(996\) 0 0
\(997\) 6.00000i 0.190022i 0.995476 + 0.0950110i \(0.0302886\pi\)
−0.995476 + 0.0950110i \(0.969711\pi\)
\(998\) 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 1080.2.f.d.649.2 yes 2
3.2 odd 2 1080.2.f.a.649.1 2
4.3 odd 2 2160.2.f.g.1729.2 2
5.2 odd 4 5400.2.a.bj.1.1 1
5.3 odd 4 5400.2.a.o.1.1 1
5.4 even 2 inner 1080.2.f.d.649.1 yes 2
12.11 even 2 2160.2.f.b.1729.1 2
15.2 even 4 5400.2.a.bg.1.1 1
15.8 even 4 5400.2.a.l.1.1 1
15.14 odd 2 1080.2.f.a.649.2 yes 2
20.19 odd 2 2160.2.f.g.1729.1 2
60.59 even 2 2160.2.f.b.1729.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.2.f.a.649.1 2 3.2 odd 2
1080.2.f.a.649.2 yes 2 15.14 odd 2
1080.2.f.d.649.1 yes 2 5.4 even 2 inner
1080.2.f.d.649.2 yes 2 1.1 even 1 trivial
2160.2.f.b.1729.1 2 12.11 even 2
2160.2.f.b.1729.2 2 60.59 even 2
2160.2.f.g.1729.1 2 20.19 odd 2
2160.2.f.g.1729.2 2 4.3 odd 2
5400.2.a.l.1.1 1 15.8 even 4
5400.2.a.o.1.1 1 5.3 odd 4
5400.2.a.bg.1.1 1 15.2 even 4
5400.2.a.bj.1.1 1 5.2 odd 4