Properties

Label 5184.2.c.i.5183.3
Level $5184$
Weight $2$
Character 5184.5183
Analytic conductor $41.394$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5184,2,Mod(5183,5184)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5184, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("5184.5183");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.c (of order \(2\), degree \(1\), not 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: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 2592)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5183.3
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 5184.5183
Dual form 5184.2.c.i.5183.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.517638i q^{5} -2.73205i q^{7} +O(q^{10})\) \(q-0.517638i q^{5} -2.73205i q^{7} +0.378937 q^{11} +2.46410 q^{13} +3.34607i q^{17} -4.73205i q^{19} -1.41421 q^{23} +4.73205 q^{25} -6.83083i q^{29} -0.535898i q^{31} -1.41421 q^{35} +4.26795 q^{37} +5.27792i q^{41} +5.26795i q^{43} +9.52056 q^{47} -0.464102 q^{49} -2.44949i q^{53} -0.196152i q^{55} -13.6617 q^{59} -9.19615 q^{61} -1.27551i q^{65} -11.1244i q^{67} +16.1112 q^{71} -10.2679 q^{73} -1.03528i q^{77} -14.5885i q^{79} +0.757875 q^{83} +1.73205 q^{85} +2.20925i q^{89} -6.73205i q^{91} -2.44949 q^{95} +14.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{13} + 24 q^{25} + 48 q^{37} + 24 q^{49} - 32 q^{61} - 96 q^{73} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) − 0.517638i − 0.231495i −0.993279 0.115747i \(-0.963074\pi\)
0.993279 0.115747i \(-0.0369263\pi\)
\(6\) 0 0
\(7\) − 2.73205i − 1.03262i −0.856402 0.516309i \(-0.827306\pi\)
0.856402 0.516309i \(-0.172694\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.378937 0.114254 0.0571270 0.998367i \(-0.481806\pi\)
0.0571270 + 0.998367i \(0.481806\pi\)
\(12\) 0 0
\(13\) 2.46410 0.683419 0.341709 0.939806i \(-0.388994\pi\)
0.341709 + 0.939806i \(0.388994\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.34607i 0.811540i 0.913975 + 0.405770i \(0.132997\pi\)
−0.913975 + 0.405770i \(0.867003\pi\)
\(18\) 0 0
\(19\) − 4.73205i − 1.08561i −0.839860 0.542803i \(-0.817363\pi\)
0.839860 0.542803i \(-0.182637\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.41421 −0.294884 −0.147442 0.989071i \(-0.547104\pi\)
−0.147442 + 0.989071i \(0.547104\pi\)
\(24\) 0 0
\(25\) 4.73205 0.946410
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.83083i − 1.26845i −0.773147 0.634227i \(-0.781319\pi\)
0.773147 0.634227i \(-0.218681\pi\)
\(30\) 0 0
\(31\) − 0.535898i − 0.0962502i −0.998841 0.0481251i \(-0.984675\pi\)
0.998841 0.0481251i \(-0.0153246\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.41421 −0.239046
\(36\) 0 0
\(37\) 4.26795 0.701647 0.350823 0.936442i \(-0.385902\pi\)
0.350823 + 0.936442i \(0.385902\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.27792i 0.824272i 0.911122 + 0.412136i \(0.135217\pi\)
−0.911122 + 0.412136i \(0.864783\pi\)
\(42\) 0 0
\(43\) 5.26795i 0.803355i 0.915781 + 0.401677i \(0.131573\pi\)
−0.915781 + 0.401677i \(0.868427\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.52056 1.38872 0.694358 0.719630i \(-0.255688\pi\)
0.694358 + 0.719630i \(0.255688\pi\)
\(48\) 0 0
\(49\) −0.464102 −0.0663002
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 2.44949i − 0.336463i −0.985747 0.168232i \(-0.946194\pi\)
0.985747 0.168232i \(-0.0538057\pi\)
\(54\) 0 0
\(55\) − 0.196152i − 0.0264492i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −13.6617 −1.77860 −0.889298 0.457327i \(-0.848807\pi\)
−0.889298 + 0.457327i \(0.848807\pi\)
\(60\) 0 0
\(61\) −9.19615 −1.17745 −0.588723 0.808335i \(-0.700369\pi\)
−0.588723 + 0.808335i \(0.700369\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 1.27551i − 0.158208i
\(66\) 0 0
\(67\) − 11.1244i − 1.35906i −0.733649 0.679528i \(-0.762185\pi\)
0.733649 0.679528i \(-0.237815\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 16.1112 1.91204 0.956021 0.293298i \(-0.0947529\pi\)
0.956021 + 0.293298i \(0.0947529\pi\)
\(72\) 0 0
\(73\) −10.2679 −1.20177 −0.600886 0.799335i \(-0.705186\pi\)
−0.600886 + 0.799335i \(0.705186\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.03528i − 0.117981i
\(78\) 0 0
\(79\) − 14.5885i − 1.64133i −0.571410 0.820665i \(-0.693603\pi\)
0.571410 0.820665i \(-0.306397\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.757875 0.0831876 0.0415938 0.999135i \(-0.486756\pi\)
0.0415938 + 0.999135i \(0.486756\pi\)
\(84\) 0 0
\(85\) 1.73205 0.187867
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.20925i 0.234180i 0.993121 + 0.117090i \(0.0373567\pi\)
−0.993121 + 0.117090i \(0.962643\pi\)
\(90\) 0 0
\(91\) − 6.73205i − 0.705711i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.44949 −0.251312
\(96\) 0 0
\(97\) 14.3923 1.46132 0.730659 0.682743i \(-0.239213\pi\)
0.730659 + 0.682743i \(0.239213\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.48477i 0.346747i 0.984856 + 0.173374i \(0.0554668\pi\)
−0.984856 + 0.173374i \(0.944533\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.69213 0.646953 0.323476 0.946236i \(-0.395148\pi\)
0.323476 + 0.946236i \(0.395148\pi\)
\(108\) 0 0
\(109\) 9.92820 0.950949 0.475475 0.879729i \(-0.342276\pi\)
0.475475 + 0.879729i \(0.342276\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.83032i 0.172182i 0.996287 + 0.0860908i \(0.0274375\pi\)
−0.996287 + 0.0860908i \(0.972562\pi\)
\(114\) 0 0
\(115\) 0.732051i 0.0682641i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 9.14162 0.838011
\(120\) 0 0
\(121\) −10.8564 −0.986946
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 5.03768i − 0.450584i
\(126\) 0 0
\(127\) − 3.12436i − 0.277242i −0.990346 0.138621i \(-0.955733\pi\)
0.990346 0.138621i \(-0.0442669\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −19.6975 −1.72097 −0.860487 0.509472i \(-0.829841\pi\)
−0.860487 + 0.509472i \(0.829841\pi\)
\(132\) 0 0
\(133\) −12.9282 −1.12102
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 22.0082i 1.88029i 0.340773 + 0.940146i \(0.389311\pi\)
−0.340773 + 0.940146i \(0.610689\pi\)
\(138\) 0 0
\(139\) − 8.39230i − 0.711826i −0.934519 0.355913i \(-0.884170\pi\)
0.934519 0.355913i \(-0.115830\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.933740 0.0780833
\(144\) 0 0
\(145\) −3.53590 −0.293640
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 20.2151i − 1.65608i −0.560665 0.828042i \(-0.689455\pi\)
0.560665 0.828042i \(-0.310545\pi\)
\(150\) 0 0
\(151\) − 14.9282i − 1.21484i −0.794381 0.607420i \(-0.792205\pi\)
0.794381 0.607420i \(-0.207795\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.277401 −0.0222814
\(156\) 0 0
\(157\) 6.66025 0.531546 0.265773 0.964036i \(-0.414373\pi\)
0.265773 + 0.964036i \(0.414373\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.86370i 0.304502i
\(162\) 0 0
\(163\) − 22.3923i − 1.75390i −0.480581 0.876950i \(-0.659574\pi\)
0.480581 0.876950i \(-0.340426\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −20.4553 −1.58288 −0.791440 0.611246i \(-0.790668\pi\)
−0.791440 + 0.611246i \(0.790668\pi\)
\(168\) 0 0
\(169\) −6.92820 −0.532939
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 1.17398i − 0.0892558i −0.999004 0.0446279i \(-0.985790\pi\)
0.999004 0.0446279i \(-0.0142102\pi\)
\(174\) 0 0
\(175\) − 12.9282i − 0.977280i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0716 −0.902272 −0.451136 0.892455i \(-0.648981\pi\)
−0.451136 + 0.892455i \(0.648981\pi\)
\(180\) 0 0
\(181\) −11.3205 −0.841447 −0.420723 0.907189i \(-0.638224\pi\)
−0.420723 + 0.907189i \(0.638224\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 2.20925i − 0.162428i
\(186\) 0 0
\(187\) 1.26795i 0.0927216i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.6574 0.771140 0.385570 0.922679i \(-0.374005\pi\)
0.385570 + 0.922679i \(0.374005\pi\)
\(192\) 0 0
\(193\) −1.53590 −0.110556 −0.0552782 0.998471i \(-0.517605\pi\)
−0.0552782 + 0.998471i \(0.517605\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 17.1093i − 1.21898i −0.792792 0.609492i \(-0.791373\pi\)
0.792792 0.609492i \(-0.208627\pi\)
\(198\) 0 0
\(199\) 8.00000i 0.567105i 0.958957 + 0.283552i \(0.0915130\pi\)
−0.958957 + 0.283552i \(0.908487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −18.6622 −1.30983
\(204\) 0 0
\(205\) 2.73205 0.190815
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 1.79315i − 0.124035i
\(210\) 0 0
\(211\) − 5.12436i − 0.352775i −0.984321 0.176388i \(-0.943559\pi\)
0.984321 0.176388i \(-0.0564412\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.72689 0.185972
\(216\) 0 0
\(217\) −1.46410 −0.0993897
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.24504i 0.554622i
\(222\) 0 0
\(223\) − 20.7321i − 1.38832i −0.719820 0.694160i \(-0.755776\pi\)
0.719820 0.694160i \(-0.244224\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22.2485 1.47668 0.738342 0.674427i \(-0.235609\pi\)
0.738342 + 0.674427i \(0.235609\pi\)
\(228\) 0 0
\(229\) −11.0000 −0.726900 −0.363450 0.931614i \(-0.618401\pi\)
−0.363450 + 0.931614i \(0.618401\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 3.24453i − 0.212556i −0.994336 0.106278i \(-0.966107\pi\)
0.994336 0.106278i \(-0.0338934\pi\)
\(234\) 0 0
\(235\) − 4.92820i − 0.321481i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −9.24316 −0.597890 −0.298945 0.954270i \(-0.596635\pi\)
−0.298945 + 0.954270i \(0.596635\pi\)
\(240\) 0 0
\(241\) 8.12436 0.523336 0.261668 0.965158i \(-0.415727\pi\)
0.261668 + 0.965158i \(0.415727\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.240237i 0.0153482i
\(246\) 0 0
\(247\) − 11.6603i − 0.741924i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 15.3533 0.969090 0.484545 0.874766i \(-0.338985\pi\)
0.484545 + 0.874766i \(0.338985\pi\)
\(252\) 0 0
\(253\) −0.535898 −0.0336916
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.3514i 1.01997i 0.860183 + 0.509986i \(0.170349\pi\)
−0.860183 + 0.509986i \(0.829651\pi\)
\(258\) 0 0
\(259\) − 11.6603i − 0.724533i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.3490 0.761471 0.380736 0.924684i \(-0.375671\pi\)
0.380736 + 0.924684i \(0.375671\pi\)
\(264\) 0 0
\(265\) −1.26795 −0.0778895
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.4176i 0.940031i 0.882658 + 0.470015i \(0.155752\pi\)
−0.882658 + 0.470015i \(0.844248\pi\)
\(270\) 0 0
\(271\) − 16.7321i − 1.01640i −0.861239 0.508200i \(-0.830311\pi\)
0.861239 0.508200i \(-0.169689\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.79315 0.108131
\(276\) 0 0
\(277\) −14.9282 −0.896949 −0.448474 0.893796i \(-0.648032\pi\)
−0.448474 + 0.893796i \(0.648032\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 4.10394i − 0.244821i −0.992480 0.122410i \(-0.960938\pi\)
0.992480 0.122410i \(-0.0390624\pi\)
\(282\) 0 0
\(283\) − 15.3205i − 0.910710i −0.890310 0.455355i \(-0.849512\pi\)
0.890310 0.455355i \(-0.150488\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.4195 0.851158
\(288\) 0 0
\(289\) 5.80385 0.341403
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.5539i 0.674989i 0.941328 + 0.337494i \(0.109579\pi\)
−0.941328 + 0.337494i \(0.890421\pi\)
\(294\) 0 0
\(295\) 7.07180i 0.411736i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.48477 −0.201529
\(300\) 0 0
\(301\) 14.3923 0.829559
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.76028i 0.272573i
\(306\) 0 0
\(307\) 18.0000i 1.02731i 0.857996 + 0.513657i \(0.171710\pi\)
−0.857996 + 0.513657i \(0.828290\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −10.0754 −0.571321 −0.285661 0.958331i \(-0.592213\pi\)
−0.285661 + 0.958331i \(0.592213\pi\)
\(312\) 0 0
\(313\) −6.32051 −0.357256 −0.178628 0.983917i \(-0.557166\pi\)
−0.178628 + 0.983917i \(0.557166\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 19.5588i − 1.09853i −0.835649 0.549265i \(-0.814908\pi\)
0.835649 0.549265i \(-0.185092\pi\)
\(318\) 0 0
\(319\) − 2.58846i − 0.144926i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 15.8338 0.881013
\(324\) 0 0
\(325\) 11.6603 0.646795
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 26.0106i − 1.43401i
\(330\) 0 0
\(331\) − 18.1962i − 1.00015i −0.865982 0.500075i \(-0.833306\pi\)
0.865982 0.500075i \(-0.166694\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5.75839 −0.314614
\(336\) 0 0
\(337\) 8.92820 0.486350 0.243175 0.969982i \(-0.421811\pi\)
0.243175 + 0.969982i \(0.421811\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 0.203072i − 0.0109970i
\(342\) 0 0
\(343\) − 17.8564i − 0.964155i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8.86422 −0.475856 −0.237928 0.971283i \(-0.576468\pi\)
−0.237928 + 0.971283i \(0.576468\pi\)
\(348\) 0 0
\(349\) 29.8564 1.59818 0.799088 0.601214i \(-0.205316\pi\)
0.799088 + 0.601214i \(0.205316\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 30.8081i − 1.63975i −0.572543 0.819875i \(-0.694043\pi\)
0.572543 0.819875i \(-0.305957\pi\)
\(354\) 0 0
\(355\) − 8.33975i − 0.442628i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.44949 0.129279 0.0646396 0.997909i \(-0.479410\pi\)
0.0646396 + 0.997909i \(0.479410\pi\)
\(360\) 0 0
\(361\) −3.39230 −0.178542
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.31508i 0.278204i
\(366\) 0 0
\(367\) 19.4641i 1.01602i 0.861352 + 0.508009i \(0.169618\pi\)
−0.861352 + 0.508009i \(0.830382\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.69213 −0.347438
\(372\) 0 0
\(373\) −16.2487 −0.841326 −0.420663 0.907217i \(-0.638203\pi\)
−0.420663 + 0.907217i \(0.638203\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 16.8319i − 0.866885i
\(378\) 0 0
\(379\) − 6.39230i − 0.328351i −0.986431 0.164175i \(-0.947504\pi\)
0.986431 0.164175i \(-0.0524963\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −24.8738 −1.27099 −0.635497 0.772104i \(-0.719205\pi\)
−0.635497 + 0.772104i \(0.719205\pi\)
\(384\) 0 0
\(385\) −0.535898 −0.0273119
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 33.3591i 1.69137i 0.533679 + 0.845687i \(0.320809\pi\)
−0.533679 + 0.845687i \(0.679191\pi\)
\(390\) 0 0
\(391\) − 4.73205i − 0.239310i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −7.55154 −0.379959
\(396\) 0 0
\(397\) 9.58846 0.481231 0.240615 0.970621i \(-0.422651\pi\)
0.240615 + 0.970621i \(0.422651\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.24453i 0.162024i 0.996713 + 0.0810120i \(0.0258152\pi\)
−0.996713 + 0.0810120i \(0.974185\pi\)
\(402\) 0 0
\(403\) − 1.32051i − 0.0657792i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.61729 0.0801659
\(408\) 0 0
\(409\) 25.1962 1.24587 0.622935 0.782274i \(-0.285940\pi\)
0.622935 + 0.782274i \(0.285940\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 37.3244i 1.83661i
\(414\) 0 0
\(415\) − 0.392305i − 0.0192575i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −32.7028 −1.59764 −0.798818 0.601573i \(-0.794541\pi\)
−0.798818 + 0.601573i \(0.794541\pi\)
\(420\) 0 0
\(421\) 18.8564 0.919005 0.459503 0.888176i \(-0.348028\pi\)
0.459503 + 0.888176i \(0.348028\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 15.8338i 0.768050i
\(426\) 0 0
\(427\) 25.1244i 1.21585i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.48528 −0.408722 −0.204361 0.978896i \(-0.565512\pi\)
−0.204361 + 0.978896i \(0.565512\pi\)
\(432\) 0 0
\(433\) 28.1769 1.35410 0.677048 0.735939i \(-0.263259\pi\)
0.677048 + 0.735939i \(0.263259\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.69213i 0.320128i
\(438\) 0 0
\(439\) 0.535898i 0.0255770i 0.999918 + 0.0127885i \(0.00407082\pi\)
−0.999918 + 0.0127885i \(0.995929\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 36.8439 1.75051 0.875253 0.483666i \(-0.160695\pi\)
0.875253 + 0.483666i \(0.160695\pi\)
\(444\) 0 0
\(445\) 1.14359 0.0542115
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 27.9053i 1.31693i 0.752610 + 0.658467i \(0.228795\pi\)
−0.752610 + 0.658467i \(0.771205\pi\)
\(450\) 0 0
\(451\) 2.00000i 0.0941763i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3.48477 −0.163368
\(456\) 0 0
\(457\) −13.1962 −0.617290 −0.308645 0.951177i \(-0.599876\pi\)
−0.308645 + 0.951177i \(0.599876\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 20.9358i − 0.975078i −0.873101 0.487539i \(-0.837895\pi\)
0.873101 0.487539i \(-0.162105\pi\)
\(462\) 0 0
\(463\) − 2.39230i − 0.111180i −0.998454 0.0555899i \(-0.982296\pi\)
0.998454 0.0555899i \(-0.0177039\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.8695 1.01200 0.506001 0.862533i \(-0.331123\pi\)
0.506001 + 0.862533i \(0.331123\pi\)
\(468\) 0 0
\(469\) −30.3923 −1.40339
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.99622i 0.0917864i
\(474\) 0 0
\(475\) − 22.3923i − 1.02743i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.00429 −0.137270 −0.0686348 0.997642i \(-0.521864\pi\)
−0.0686348 + 0.997642i \(0.521864\pi\)
\(480\) 0 0
\(481\) 10.5167 0.479518
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 7.45001i − 0.338287i
\(486\) 0 0
\(487\) − 23.0718i − 1.04548i −0.852491 0.522741i \(-0.824909\pi\)
0.852491 0.522741i \(-0.175091\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 33.8396 1.52716 0.763580 0.645714i \(-0.223440\pi\)
0.763580 + 0.645714i \(0.223440\pi\)
\(492\) 0 0
\(493\) 22.8564 1.02940
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 44.0165i − 1.97441i
\(498\) 0 0
\(499\) 22.4449i 1.00477i 0.864644 + 0.502385i \(0.167544\pi\)
−0.864644 + 0.502385i \(0.832456\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −14.9743 −0.667673 −0.333836 0.942631i \(-0.608343\pi\)
−0.333836 + 0.942631i \(0.608343\pi\)
\(504\) 0 0
\(505\) 1.80385 0.0802702
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 2.65256i − 0.117573i −0.998271 0.0587864i \(-0.981277\pi\)
0.998271 0.0587864i \(-0.0187231\pi\)
\(510\) 0 0
\(511\) 28.0526i 1.24097i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.07055 0.0912394
\(516\) 0 0
\(517\) 3.60770 0.158666
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 29.9759i − 1.31327i −0.754210 0.656634i \(-0.771980\pi\)
0.754210 0.656634i \(-0.228020\pi\)
\(522\) 0 0
\(523\) 5.80385i 0.253785i 0.991917 + 0.126892i \(0.0405003\pi\)
−0.991917 + 0.126892i \(0.959500\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.79315 0.0781109
\(528\) 0 0
\(529\) −21.0000 −0.913043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 13.0053i 0.563323i
\(534\) 0 0
\(535\) − 3.46410i − 0.149766i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.175865 −0.00757506
\(540\) 0 0
\(541\) −7.00000 −0.300954 −0.150477 0.988614i \(-0.548081\pi\)
−0.150477 + 0.988614i \(0.548081\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 5.13922i − 0.220140i
\(546\) 0 0
\(547\) − 33.3205i − 1.42468i −0.701834 0.712341i \(-0.747635\pi\)
0.701834 0.712341i \(-0.252365\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −32.3238 −1.37704
\(552\) 0 0
\(553\) −39.8564 −1.69487
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 19.1798i − 0.812675i −0.913723 0.406337i \(-0.866806\pi\)
0.913723 0.406337i \(-0.133194\pi\)
\(558\) 0 0
\(559\) 12.9808i 0.549028i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 45.3292 1.91040 0.955198 0.295967i \(-0.0956418\pi\)
0.955198 + 0.295967i \(0.0956418\pi\)
\(564\) 0 0
\(565\) 0.947441 0.0398591
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 33.9783i 1.42444i 0.701954 + 0.712222i \(0.252311\pi\)
−0.701954 + 0.712222i \(0.747689\pi\)
\(570\) 0 0
\(571\) − 12.1436i − 0.508194i −0.967179 0.254097i \(-0.918222\pi\)
0.967179 0.254097i \(-0.0817782\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.69213 −0.279081
\(576\) 0 0
\(577\) 11.5359 0.480246 0.240123 0.970743i \(-0.422812\pi\)
0.240123 + 0.970743i \(0.422812\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 2.07055i − 0.0859010i
\(582\) 0 0
\(583\) − 0.928203i − 0.0384422i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.0053 0.536787 0.268394 0.963309i \(-0.413507\pi\)
0.268394 + 0.963309i \(0.413507\pi\)
\(588\) 0 0
\(589\) −2.53590 −0.104490
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 25.3915i − 1.04270i −0.853342 0.521351i \(-0.825428\pi\)
0.853342 0.521351i \(-0.174572\pi\)
\(594\) 0 0
\(595\) − 4.73205i − 0.193995i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0.277401 0.0113343 0.00566716 0.999984i \(-0.498196\pi\)
0.00566716 + 0.999984i \(0.498196\pi\)
\(600\) 0 0
\(601\) −37.7846 −1.54127 −0.770633 0.637279i \(-0.780060\pi\)
−0.770633 + 0.637279i \(0.780060\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.61969i 0.228473i
\(606\) 0 0
\(607\) − 3.26795i − 0.132642i −0.997798 0.0663210i \(-0.978874\pi\)
0.997798 0.0663210i \(-0.0211261\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 23.4596 0.949075
\(612\) 0 0
\(613\) −34.3923 −1.38909 −0.694546 0.719448i \(-0.744395\pi\)
−0.694546 + 0.719448i \(0.744395\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 7.86611i − 0.316678i −0.987385 0.158339i \(-0.949386\pi\)
0.987385 0.158339i \(-0.0506138\pi\)
\(618\) 0 0
\(619\) 8.92820i 0.358855i 0.983771 + 0.179427i \(0.0574245\pi\)
−0.983771 + 0.179427i \(0.942576\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.03579 0.241819
\(624\) 0 0
\(625\) 21.0526 0.842102
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 14.2808i 0.569414i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.61729 −0.0641800
\(636\) 0 0
\(637\) −1.14359 −0.0453108
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 4.55721i − 0.179999i −0.995942 0.0899994i \(-0.971313\pi\)
0.995942 0.0899994i \(-0.0286865\pi\)
\(642\) 0 0
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −42.1218 −1.65598 −0.827989 0.560744i \(-0.810515\pi\)
−0.827989 + 0.560744i \(0.810515\pi\)
\(648\) 0 0
\(649\) −5.17691 −0.203212
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 45.4307i 1.77784i 0.458062 + 0.888920i \(0.348544\pi\)
−0.458062 + 0.888920i \(0.651456\pi\)
\(654\) 0 0
\(655\) 10.1962i 0.398397i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −7.62587 −0.297062 −0.148531 0.988908i \(-0.547454\pi\)
−0.148531 + 0.988908i \(0.547454\pi\)
\(660\) 0 0
\(661\) −36.5167 −1.42033 −0.710167 0.704034i \(-0.751380\pi\)
−0.710167 + 0.704034i \(0.751380\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.69213i 0.259510i
\(666\) 0 0
\(667\) 9.66025i 0.374047i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −3.48477 −0.134528
\(672\) 0 0
\(673\) −4.32051 −0.166543 −0.0832717 0.996527i \(-0.526537\pi\)
−0.0832717 + 0.996527i \(0.526537\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 44.4698i 1.70911i 0.519360 + 0.854556i \(0.326170\pi\)
−0.519360 + 0.854556i \(0.673830\pi\)
\(678\) 0 0
\(679\) − 39.3205i − 1.50898i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −20.0764 −0.768202 −0.384101 0.923291i \(-0.625489\pi\)
−0.384101 + 0.923291i \(0.625489\pi\)
\(684\) 0 0
\(685\) 11.3923 0.435278
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 6.03579i − 0.229945i
\(690\) 0 0
\(691\) 28.5885i 1.08756i 0.839229 + 0.543778i \(0.183007\pi\)
−0.839229 + 0.543778i \(0.816993\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.34418 −0.164784
\(696\) 0 0
\(697\) −17.6603 −0.668930
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 30.1146i 1.13741i 0.822541 + 0.568706i \(0.192556\pi\)
−0.822541 + 0.568706i \(0.807444\pi\)
\(702\) 0 0
\(703\) − 20.1962i − 0.761712i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.52056 0.358057
\(708\) 0 0
\(709\) 13.0000 0.488225 0.244113 0.969747i \(-0.421503\pi\)
0.244113 + 0.969747i \(0.421503\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0.757875i 0.0283826i
\(714\) 0 0
\(715\) − 0.483340i − 0.0180759i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 31.4916 1.17444 0.587220 0.809427i \(-0.300222\pi\)
0.587220 + 0.809427i \(0.300222\pi\)
\(720\) 0 0
\(721\) 10.9282 0.406988
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 32.3238i − 1.20048i
\(726\) 0 0
\(727\) 39.9090i 1.48014i 0.672529 + 0.740071i \(0.265208\pi\)
−0.672529 + 0.740071i \(0.734792\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −17.6269 −0.651955
\(732\) 0 0
\(733\) −4.67949 −0.172841 −0.0864205 0.996259i \(-0.527543\pi\)
−0.0864205 + 0.996259i \(0.527543\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 4.21543i − 0.155278i
\(738\) 0 0
\(739\) 32.0000i 1.17714i 0.808447 + 0.588570i \(0.200309\pi\)
−0.808447 + 0.588570i \(0.799691\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 49.3959 1.81216 0.906081 0.423105i \(-0.139060\pi\)
0.906081 + 0.423105i \(0.139060\pi\)
\(744\) 0 0
\(745\) −10.4641 −0.383375
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 18.2832i − 0.668055i
\(750\) 0 0
\(751\) 40.5885i 1.48109i 0.672005 + 0.740547i \(0.265433\pi\)
−0.672005 + 0.740547i \(0.734567\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −7.72741 −0.281229
\(756\) 0 0
\(757\) −16.7846 −0.610047 −0.305024 0.952345i \(-0.598664\pi\)
−0.305024 + 0.952345i \(0.598664\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 34.6346i − 1.25550i −0.778413 0.627752i \(-0.783975\pi\)
0.778413 0.627752i \(-0.216025\pi\)
\(762\) 0 0
\(763\) − 27.1244i − 0.981968i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −33.6637 −1.21553
\(768\) 0 0
\(769\) 50.0333 1.80425 0.902124 0.431477i \(-0.142007\pi\)
0.902124 + 0.431477i \(0.142007\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 53.1209i − 1.91063i −0.295594 0.955314i \(-0.595517\pi\)
0.295594 0.955314i \(-0.404483\pi\)
\(774\) 0 0
\(775\) − 2.53590i − 0.0910922i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.9754 0.894836
\(780\) 0 0
\(781\) 6.10512 0.218458
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 3.44760i − 0.123050i
\(786\) 0 0
\(787\) 21.2679i 0.758121i 0.925372 + 0.379060i \(0.123753\pi\)
−0.925372 + 0.379060i \(0.876247\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.00052 0.177798
\(792\) 0 0
\(793\) −22.6603 −0.804689
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 9.58493i − 0.339516i −0.985486 0.169758i \(-0.945701\pi\)
0.985486 0.169758i \(-0.0542985\pi\)
\(798\) 0 0
\(799\) 31.8564i 1.12700i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.89091 −0.137307
\(804\) 0 0
\(805\) 2.00000 0.0704907
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 23.2466i − 0.817307i −0.912690 0.408653i \(-0.865999\pi\)
0.912690 0.408653i \(-0.134001\pi\)
\(810\) 0 0
\(811\) 22.6410i 0.795034i 0.917595 + 0.397517i \(0.130128\pi\)
−0.917595 + 0.397517i \(0.869872\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −11.5911 −0.406019
\(816\) 0 0
\(817\) 24.9282 0.872127
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 44.7372i 1.56134i 0.624944 + 0.780669i \(0.285122\pi\)
−0.624944 + 0.780669i \(0.714878\pi\)
\(822\) 0 0
\(823\) 24.3923i 0.850262i 0.905132 + 0.425131i \(0.139772\pi\)
−0.905132 + 0.425131i \(0.860228\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −22.5259 −0.783302 −0.391651 0.920114i \(-0.628096\pi\)
−0.391651 + 0.920114i \(0.628096\pi\)
\(828\) 0 0
\(829\) 39.7128 1.37928 0.689642 0.724151i \(-0.257768\pi\)
0.689642 + 0.724151i \(0.257768\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 1.55291i − 0.0538053i
\(834\) 0 0
\(835\) 10.5885i 0.366429i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 34.6718 1.19700 0.598502 0.801122i \(-0.295763\pi\)
0.598502 + 0.801122i \(0.295763\pi\)
\(840\) 0 0
\(841\) −17.6603 −0.608974
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.58630i 0.123373i
\(846\) 0 0
\(847\) 29.6603i 1.01914i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −6.03579 −0.206904
\(852\) 0 0
\(853\) −39.5692 −1.35482 −0.677412 0.735604i \(-0.736899\pi\)
−0.677412 + 0.735604i \(0.736899\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 26.9072i 0.919133i 0.888143 + 0.459567i \(0.151995\pi\)
−0.888143 + 0.459567i \(0.848005\pi\)
\(858\) 0 0
\(859\) 43.4641i 1.48298i 0.670966 + 0.741488i \(0.265880\pi\)
−0.670966 + 0.741488i \(0.734120\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −30.0502 −1.02292 −0.511461 0.859307i \(-0.670895\pi\)
−0.511461 + 0.859307i \(0.670895\pi\)
\(864\) 0 0
\(865\) −0.607695 −0.0206623
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 5.52811i − 0.187528i
\(870\) 0 0
\(871\) − 27.4115i − 0.928805i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −13.7632 −0.465281
\(876\) 0 0
\(877\) 20.1244 0.679551 0.339776 0.940507i \(-0.389649\pi\)
0.339776 + 0.940507i \(0.389649\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 37.7033i 1.27026i 0.772407 + 0.635128i \(0.219053\pi\)
−0.772407 + 0.635128i \(0.780947\pi\)
\(882\) 0 0
\(883\) 35.3205i 1.18863i 0.804232 + 0.594315i \(0.202577\pi\)
−0.804232 + 0.594315i \(0.797423\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 4.79744 0.161082 0.0805412 0.996751i \(-0.474335\pi\)
0.0805412 + 0.996751i \(0.474335\pi\)
\(888\) 0 0
\(889\) −8.53590 −0.286285
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 45.0518i − 1.50760i
\(894\) 0 0
\(895\) 6.24871i 0.208871i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.66063 −0.122089
\(900\) 0 0
\(901\) 8.19615 0.273053
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.85993i 0.194791i
\(906\) 0 0
\(907\) 19.4641i 0.646295i 0.946349 + 0.323147i \(0.104741\pi\)
−0.946349 + 0.323147i \(0.895259\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −33.3591 −1.10524 −0.552618 0.833434i \(-0.686371\pi\)
−0.552618 + 0.833434i \(0.686371\pi\)
\(912\) 0 0
\(913\) 0.287187 0.00950451
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 53.8144i 1.77711i
\(918\) 0 0
\(919\) 20.4449i 0.674414i 0.941431 + 0.337207i \(0.109482\pi\)
−0.941431 + 0.337207i \(0.890518\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 39.6995 1.30673
\(924\) 0 0
\(925\) 20.1962 0.664045
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 17.9687i − 0.589533i −0.955569 0.294767i \(-0.904758\pi\)
0.955569 0.294767i \(-0.0952419\pi\)
\(930\) 0 0
\(931\) 2.19615i 0.0719760i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.656339 0.0214646
\(936\) 0 0
\(937\) −13.3923 −0.437508 −0.218754 0.975780i \(-0.570199\pi\)
−0.218754 + 0.975780i \(0.570199\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 32.6384i 1.06398i 0.846750 + 0.531991i \(0.178556\pi\)
−0.846750 + 0.531991i \(0.821444\pi\)
\(942\) 0 0
\(943\) − 7.46410i − 0.243065i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −43.8406 −1.42463 −0.712314 0.701861i \(-0.752353\pi\)
−0.712314 + 0.701861i \(0.752353\pi\)
\(948\) 0 0
\(949\) −25.3013 −0.821314
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 29.9387i − 0.969810i −0.874567 0.484905i \(-0.838854\pi\)
0.874567 0.484905i \(-0.161146\pi\)
\(954\) 0 0
\(955\) − 5.51666i − 0.178515i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 60.1276 1.94162
\(960\) 0 0
\(961\) 30.7128 0.990736
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0.795040i 0.0255932i
\(966\) 0 0
\(967\) 55.6603i 1.78991i 0.446153 + 0.894957i \(0.352794\pi\)
−0.446153 + 0.894957i \(0.647206\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 20.6584 0.662960 0.331480 0.943462i \(-0.392452\pi\)
0.331480 + 0.943462i \(0.392452\pi\)
\(972\) 0 0
\(973\) −22.9282 −0.735044
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 20.5297i 0.656802i 0.944538 + 0.328401i \(0.106510\pi\)
−0.944538 + 0.328401i \(0.893490\pi\)
\(978\) 0 0
\(979\) 0.837169i 0.0267560i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −38.5627 −1.22996 −0.614980 0.788543i \(-0.710836\pi\)
−0.614980 + 0.788543i \(0.710836\pi\)
\(984\) 0 0
\(985\) −8.85641 −0.282189
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 7.45001i − 0.236896i
\(990\) 0 0
\(991\) 32.4449i 1.03065i 0.856996 + 0.515323i \(0.172328\pi\)
−0.856996 + 0.515323i \(0.827672\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.14110 0.131282
\(996\) 0 0
\(997\) −7.19615 −0.227904 −0.113952 0.993486i \(-0.536351\pi\)
−0.113952 + 0.993486i \(0.536351\pi\)
\(998\) 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 5184.2.c.i.5183.3 8
3.2 odd 2 inner 5184.2.c.i.5183.5 8
4.3 odd 2 inner 5184.2.c.i.5183.4 8
8.3 odd 2 2592.2.c.a.2591.6 yes 8
8.5 even 2 2592.2.c.a.2591.5 yes 8
12.11 even 2 inner 5184.2.c.i.5183.6 8
24.5 odd 2 2592.2.c.a.2591.3 8
24.11 even 2 2592.2.c.a.2591.4 yes 8
72.5 odd 6 2592.2.s.f.1727.2 8
72.11 even 6 2592.2.s.f.863.3 8
72.13 even 6 2592.2.s.f.1727.3 8
72.29 odd 6 2592.2.s.b.863.3 8
72.43 odd 6 2592.2.s.f.863.2 8
72.59 even 6 2592.2.s.b.1727.2 8
72.61 even 6 2592.2.s.b.863.2 8
72.67 odd 6 2592.2.s.b.1727.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2592.2.c.a.2591.3 8 24.5 odd 2
2592.2.c.a.2591.4 yes 8 24.11 even 2
2592.2.c.a.2591.5 yes 8 8.5 even 2
2592.2.c.a.2591.6 yes 8 8.3 odd 2
2592.2.s.b.863.2 8 72.61 even 6
2592.2.s.b.863.3 8 72.29 odd 6
2592.2.s.b.1727.2 8 72.59 even 6
2592.2.s.b.1727.3 8 72.67 odd 6
2592.2.s.f.863.2 8 72.43 odd 6
2592.2.s.f.863.3 8 72.11 even 6
2592.2.s.f.1727.2 8 72.5 odd 6
2592.2.s.f.1727.3 8 72.13 even 6
5184.2.c.i.5183.3 8 1.1 even 1 trivial
5184.2.c.i.5183.4 8 4.3 odd 2 inner
5184.2.c.i.5183.5 8 3.2 odd 2 inner
5184.2.c.i.5183.6 8 12.11 even 2 inner