Properties

Label 2016.2.c.e.1009.3
Level $2016$
Weight $2$
Character 2016.1009
Analytic conductor $16.098$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2016,2,Mod(1009,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.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: \(16.0978410475\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.386672896.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.3
Root \(1.40961 + 0.114062i\) of defining polynomial
Character \(\chi\) \(=\) 2016.1009
Dual form 2016.2.c.e.1009.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.12875i q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-1.12875i q^{5} -1.00000 q^{7} +4.76717i q^{11} -0.456247i q^{13} -0.415006 q^{17} -7.63843i q^{19} +1.58499 q^{23} +3.72593 q^{25} +6.72593i q^{29} +5.89592 q^{31} +1.12875i q^{35} -5.89592i q^{37} +0.415006 q^{41} +9.43967i q^{43} +11.2769 q^{47} +1.00000 q^{49} +7.63843i q^{53} +5.38093 q^{55} -4.00000i q^{59} +1.80125i q^{61} -0.514988 q^{65} -8.09467i q^{67} +10.2068 q^{71} -3.34500 q^{73} -4.76717i q^{77} +4.83001 q^{79} -5.53434i q^{83} +0.468436i q^{85} -4.92999 q^{89} +0.456247i q^{91} -8.62185 q^{95} +16.4468 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} - 4 q^{17} + 12 q^{23} - 24 q^{25} - 8 q^{31} + 4 q^{41} + 8 q^{49} + 8 q^{55} + 16 q^{65} - 28 q^{71} - 8 q^{73} + 40 q^{79} - 20 q^{89} + 40 q^{95} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(5\) − 1.12875i − 0.504791i −0.967624 0.252395i \(-0.918782\pi\)
0.967624 0.252395i \(-0.0812184\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.76717i 1.43736i 0.695343 + 0.718678i \(0.255253\pi\)
−0.695343 + 0.718678i \(0.744747\pi\)
\(12\) 0 0
\(13\) − 0.456247i − 0.126540i −0.997996 0.0632701i \(-0.979847\pi\)
0.997996 0.0632701i \(-0.0201530\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.415006 −0.100654 −0.0503268 0.998733i \(-0.516026\pi\)
−0.0503268 + 0.998733i \(0.516026\pi\)
\(18\) 0 0
\(19\) − 7.63843i − 1.75237i −0.481970 0.876187i \(-0.660079\pi\)
0.481970 0.876187i \(-0.339921\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.58499 0.330494 0.165247 0.986252i \(-0.447158\pi\)
0.165247 + 0.986252i \(0.447158\pi\)
\(24\) 0 0
\(25\) 3.72593 0.745186
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.72593i 1.24897i 0.781035 + 0.624487i \(0.214692\pi\)
−0.781035 + 0.624487i \(0.785308\pi\)
\(30\) 0 0
\(31\) 5.89592 1.05894 0.529469 0.848329i \(-0.322391\pi\)
0.529469 + 0.848329i \(0.322391\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.12875i 0.190793i
\(36\) 0 0
\(37\) − 5.89592i − 0.969283i −0.874713 0.484642i \(-0.838950\pi\)
0.874713 0.484642i \(-0.161050\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.415006 0.0648130 0.0324065 0.999475i \(-0.489683\pi\)
0.0324065 + 0.999475i \(0.489683\pi\)
\(42\) 0 0
\(43\) 9.43967i 1.43954i 0.694214 + 0.719768i \(0.255752\pi\)
−0.694214 + 0.719768i \(0.744248\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.2769 1.64490 0.822449 0.568839i \(-0.192607\pi\)
0.822449 + 0.568839i \(0.192607\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.63843i 1.04922i 0.851343 + 0.524609i \(0.175789\pi\)
−0.851343 + 0.524609i \(0.824211\pi\)
\(54\) 0 0
\(55\) 5.38093 0.725565
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 4.00000i − 0.520756i −0.965507 0.260378i \(-0.916153\pi\)
0.965507 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) 1.80125i 0.230626i 0.993329 + 0.115313i \(0.0367871\pi\)
−0.993329 + 0.115313i \(0.963213\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.514988 −0.0638764
\(66\) 0 0
\(67\) − 8.09467i − 0.988922i −0.869200 0.494461i \(-0.835366\pi\)
0.869200 0.494461i \(-0.164634\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.2068 1.21133 0.605665 0.795720i \(-0.292907\pi\)
0.605665 + 0.795720i \(0.292907\pi\)
\(72\) 0 0
\(73\) −3.34500 −0.391503 −0.195751 0.980654i \(-0.562715\pi\)
−0.195751 + 0.980654i \(0.562715\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 4.76717i − 0.543270i
\(78\) 0 0
\(79\) 4.83001 0.543419 0.271709 0.962379i \(-0.412411\pi\)
0.271709 + 0.962379i \(0.412411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 5.53434i − 0.607473i −0.952756 0.303737i \(-0.901766\pi\)
0.952756 0.303737i \(-0.0982343\pi\)
\(84\) 0 0
\(85\) 0.468436i 0.0508090i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.92999 −0.522578 −0.261289 0.965261i \(-0.584148\pi\)
−0.261289 + 0.965261i \(0.584148\pi\)
\(90\) 0 0
\(91\) 0.456247i 0.0478277i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.62185 −0.884583
\(96\) 0 0
\(97\) 16.4468 1.66992 0.834962 0.550308i \(-0.185490\pi\)
0.834962 + 0.550308i \(0.185490\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 16.4056i − 1.63242i −0.577757 0.816209i \(-0.696072\pi\)
0.577757 0.816209i \(-0.303928\pi\)
\(102\) 0 0
\(103\) 17.1728 1.69208 0.846042 0.533117i \(-0.178979\pi\)
0.846042 + 0.533117i \(0.178979\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.7672i − 1.23425i −0.786865 0.617125i \(-0.788297\pi\)
0.786865 0.617125i \(-0.211703\pi\)
\(108\) 0 0
\(109\) − 7.27685i − 0.696996i −0.937310 0.348498i \(-0.886692\pi\)
0.937310 0.348498i \(-0.113308\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.34500 0.314671 0.157336 0.987545i \(-0.449710\pi\)
0.157336 + 0.987545i \(0.449710\pi\)
\(114\) 0 0
\(115\) − 1.78906i − 0.166830i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.415006 0.0380435
\(120\) 0 0
\(121\) −11.7259 −1.06599
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 9.84937i − 0.880954i
\(126\) 0 0
\(127\) 10.4468 0.927007 0.463504 0.886095i \(-0.346592\pi\)
0.463504 + 0.886095i \(0.346592\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.25749i 0.546720i 0.961912 + 0.273360i \(0.0881350\pi\)
−0.961912 + 0.273360i \(0.911865\pi\)
\(132\) 0 0
\(133\) 7.63843i 0.662335i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.9618 1.10740 0.553702 0.832715i \(-0.313215\pi\)
0.553702 + 0.832715i \(0.313215\pi\)
\(138\) 0 0
\(139\) 18.3644i 1.55764i 0.627245 + 0.778822i \(0.284183\pi\)
−0.627245 + 0.778822i \(0.715817\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.17501 0.181883
\(144\) 0 0
\(145\) 7.59187 0.630471
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.6384i 1.28115i 0.767896 + 0.640575i \(0.221304\pi\)
−0.767896 + 0.640575i \(0.778696\pi\)
\(150\) 0 0
\(151\) −15.2769 −1.24321 −0.621606 0.783330i \(-0.713520\pi\)
−0.621606 + 0.783330i \(0.713520\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 6.65500i − 0.534543i
\(156\) 0 0
\(157\) 21.7824i 1.73843i 0.494437 + 0.869214i \(0.335374\pi\)
−0.494437 + 0.869214i \(0.664626\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.58499 −0.124915
\(162\) 0 0
\(163\) 4.60966i 0.361056i 0.983570 + 0.180528i \(0.0577807\pi\)
−0.983570 + 0.180528i \(0.942219\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −22.9618 −1.77684 −0.888420 0.459032i \(-0.848196\pi\)
−0.888420 + 0.459032i \(0.848196\pi\)
\(168\) 0 0
\(169\) 12.7918 0.983988
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.216252i 0.0164413i 0.999966 + 0.00822067i \(0.00261675\pi\)
−0.999966 + 0.00822067i \(0.997383\pi\)
\(174\) 0 0
\(175\) −3.72593 −0.281654
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 22.6165i 1.69044i 0.534419 + 0.845220i \(0.320530\pi\)
−0.534419 + 0.845220i \(0.679470\pi\)
\(180\) 0 0
\(181\) 7.73310i 0.574797i 0.957811 + 0.287398i \(0.0927903\pi\)
−0.957811 + 0.287398i \(0.907210\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.65500 −0.489285
\(186\) 0 0
\(187\) − 1.97840i − 0.144675i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.2068 0.738541 0.369271 0.929322i \(-0.379608\pi\)
0.369271 + 0.929322i \(0.379608\pi\)
\(192\) 0 0
\(193\) 5.96407 0.429303 0.214651 0.976691i \(-0.431138\pi\)
0.214651 + 0.976691i \(0.431138\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 18.8084i − 1.34004i −0.742341 0.670022i \(-0.766285\pi\)
0.742341 0.670022i \(-0.233715\pi\)
\(198\) 0 0
\(199\) −3.27685 −0.232290 −0.116145 0.993232i \(-0.537054\pi\)
−0.116145 + 0.993232i \(0.537054\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 6.72593i − 0.472068i
\(204\) 0 0
\(205\) − 0.468436i − 0.0327170i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 36.4137 2.51879
\(210\) 0 0
\(211\) − 10.1628i − 0.699637i −0.936817 0.349819i \(-0.886243\pi\)
0.936817 0.349819i \(-0.113757\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.6550 0.726665
\(216\) 0 0
\(217\) −5.89592 −0.400241
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.189345i 0.0127367i
\(222\) 0 0
\(223\) −21.9668 −1.47101 −0.735504 0.677520i \(-0.763055\pi\)
−0.735504 + 0.677520i \(0.763055\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.91752i 0.525504i 0.964863 + 0.262752i \(0.0846301\pi\)
−0.964863 + 0.262752i \(0.915370\pi\)
\(228\) 0 0
\(229\) − 13.1606i − 0.869676i −0.900509 0.434838i \(-0.856806\pi\)
0.900509 0.434838i \(-0.143194\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −17.2769 −1.13184 −0.565922 0.824459i \(-0.691480\pi\)
−0.565922 + 0.824459i \(0.691480\pi\)
\(234\) 0 0
\(235\) − 12.7287i − 0.830330i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −10.5219 −0.680603 −0.340301 0.940316i \(-0.610529\pi\)
−0.340301 + 0.940316i \(0.610529\pi\)
\(240\) 0 0
\(241\) −6.10686 −0.393378 −0.196689 0.980466i \(-0.563019\pi\)
−0.196689 + 0.980466i \(0.563019\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 1.12875i − 0.0721130i
\(246\) 0 0
\(247\) −3.48501 −0.221746
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 19.8743i − 1.25446i −0.778836 0.627228i \(-0.784189\pi\)
0.778836 0.627228i \(-0.215811\pi\)
\(252\) 0 0
\(253\) 7.55594i 0.475038i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.7550 0.920391 0.460195 0.887818i \(-0.347779\pi\)
0.460195 + 0.887818i \(0.347779\pi\)
\(258\) 0 0
\(259\) 5.89592i 0.366355i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −15.7600 −0.971804 −0.485902 0.874013i \(-0.661509\pi\)
−0.485902 + 0.874013i \(0.661509\pi\)
\(264\) 0 0
\(265\) 8.62185 0.529636
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 21.8331i 1.33119i 0.746315 + 0.665593i \(0.231821\pi\)
−0.746315 + 0.665593i \(0.768179\pi\)
\(270\) 0 0
\(271\) 18.8328 1.14401 0.572005 0.820250i \(-0.306166\pi\)
0.572005 + 0.820250i \(0.306166\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 17.7622i 1.07110i
\(276\) 0 0
\(277\) − 24.4684i − 1.47017i −0.677977 0.735083i \(-0.737143\pi\)
0.677977 0.735083i \(-0.262857\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −14.5150 −0.865892 −0.432946 0.901420i \(-0.642526\pi\)
−0.432946 + 0.901420i \(0.642526\pi\)
\(282\) 0 0
\(283\) 16.5509i 0.983850i 0.870638 + 0.491925i \(0.163707\pi\)
−0.870638 + 0.491925i \(0.836293\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.415006 −0.0244970
\(288\) 0 0
\(289\) −16.8278 −0.989869
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 9.44377i − 0.551711i −0.961199 0.275855i \(-0.911039\pi\)
0.961199 0.275855i \(-0.0889611\pi\)
\(294\) 0 0
\(295\) −4.51499 −0.262873
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 0.723150i − 0.0418208i
\(300\) 0 0
\(301\) − 9.43967i − 0.544094i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.03315 0.116418
\(306\) 0 0
\(307\) − 0.361575i − 0.0206362i −0.999947 0.0103181i \(-0.996716\pi\)
0.999947 0.0103181i \(-0.00328441\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −11.2769 −0.639452 −0.319726 0.947510i \(-0.603591\pi\)
−0.319726 + 0.947510i \(0.603591\pi\)
\(312\) 0 0
\(313\) 25.5837 1.44607 0.723037 0.690809i \(-0.242745\pi\)
0.723037 + 0.690809i \(0.242745\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 0.361575i − 0.0203081i −0.999948 0.0101540i \(-0.996768\pi\)
0.999948 0.0101540i \(-0.00323218\pi\)
\(318\) 0 0
\(319\) −32.0637 −1.79522
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.16999i 0.176383i
\(324\) 0 0
\(325\) − 1.69995i − 0.0942960i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −11.2769 −0.621713
\(330\) 0 0
\(331\) − 7.80403i − 0.428948i −0.976730 0.214474i \(-0.931196\pi\)
0.976730 0.214474i \(-0.0688037\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −9.13684 −0.499199
\(336\) 0 0
\(337\) 8.76186 0.477289 0.238645 0.971107i \(-0.423297\pi\)
0.238645 + 0.971107i \(0.423297\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 28.1069i 1.52207i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.07717i 0.111509i 0.998445 + 0.0557543i \(0.0177563\pi\)
−0.998445 + 0.0557543i \(0.982244\pi\)
\(348\) 0 0
\(349\) − 18.7381i − 1.00303i −0.865149 0.501514i \(-0.832776\pi\)
0.865149 0.501514i \(-0.167224\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 13.2450 0.704961 0.352481 0.935819i \(-0.385338\pi\)
0.352481 + 0.935819i \(0.385338\pi\)
\(354\) 0 0
\(355\) − 11.5209i − 0.611468i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5.69186 −0.300405 −0.150202 0.988655i \(-0.547993\pi\)
−0.150202 + 0.988655i \(0.547993\pi\)
\(360\) 0 0
\(361\) −39.3455 −2.07082
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.77566i 0.197627i
\(366\) 0 0
\(367\) −11.2769 −0.588647 −0.294323 0.955706i \(-0.595094\pi\)
−0.294323 + 0.955706i \(0.595094\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 7.63843i − 0.396567i
\(372\) 0 0
\(373\) − 7.08751i − 0.366977i −0.983022 0.183489i \(-0.941261\pi\)
0.983022 0.183489i \(-0.0587390\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.06869 0.158046
\(378\) 0 0
\(379\) 6.67781i 0.343016i 0.985183 + 0.171508i \(0.0548639\pi\)
−0.985183 + 0.171508i \(0.945136\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −18.6550 −0.953226 −0.476613 0.879113i \(-0.658136\pi\)
−0.476613 + 0.879113i \(0.658136\pi\)
\(384\) 0 0
\(385\) −5.38093 −0.274238
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 20.3428i 1.03142i 0.856764 + 0.515709i \(0.172472\pi\)
−0.856764 + 0.515709i \(0.827528\pi\)
\(390\) 0 0
\(391\) −0.657782 −0.0332654
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 5.45186i − 0.274313i
\(396\) 0 0
\(397\) − 10.9031i − 0.547210i −0.961842 0.273605i \(-0.911784\pi\)
0.961842 0.273605i \(-0.0882161\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −29.1756 −1.45696 −0.728479 0.685068i \(-0.759772\pi\)
−0.728479 + 0.685068i \(0.759772\pi\)
\(402\) 0 0
\(403\) − 2.69000i − 0.133998i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 28.1069 1.39321
\(408\) 0 0
\(409\) −31.7237 −1.56864 −0.784318 0.620359i \(-0.786987\pi\)
−0.784318 + 0.620359i \(0.786987\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.00000i 0.196827i
\(414\) 0 0
\(415\) −6.24687 −0.306647
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 19.2769i 0.941736i 0.882204 + 0.470868i \(0.156059\pi\)
−0.882204 + 0.470868i \(0.843941\pi\)
\(420\) 0 0
\(421\) 19.3234i 0.941765i 0.882196 + 0.470882i \(0.156064\pi\)
−0.882196 + 0.470882i \(0.843936\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.54628 −0.0750057
\(426\) 0 0
\(427\) − 1.80125i − 0.0871684i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.4150 0.694346 0.347173 0.937801i \(-0.387142\pi\)
0.347173 + 0.937801i \(0.387142\pi\)
\(432\) 0 0
\(433\) −9.27685 −0.445817 −0.222908 0.974839i \(-0.571555\pi\)
−0.222908 + 0.974839i \(0.571555\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 12.1069i − 0.579150i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.6465i 0.743388i 0.928355 + 0.371694i \(0.121223\pi\)
−0.928355 + 0.371694i \(0.878777\pi\)
\(444\) 0 0
\(445\) 5.56472i 0.263793i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12.6550 0.597226 0.298613 0.954374i \(-0.403476\pi\)
0.298613 + 0.954374i \(0.403476\pi\)
\(450\) 0 0
\(451\) 1.97840i 0.0931594i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.514988 0.0241430
\(456\) 0 0
\(457\) −23.9237 −1.11910 −0.559551 0.828796i \(-0.689026\pi\)
−0.559551 + 0.828796i \(0.689026\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 6.74558i − 0.314173i −0.987585 0.157086i \(-0.949790\pi\)
0.987585 0.157086i \(-0.0502101\pi\)
\(462\) 0 0
\(463\) 11.1700 0.519113 0.259557 0.965728i \(-0.416424\pi\)
0.259557 + 0.965728i \(0.416424\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.70935i 0.171648i 0.996310 + 0.0858242i \(0.0273523\pi\)
−0.996310 + 0.0858242i \(0.972648\pi\)
\(468\) 0 0
\(469\) 8.09467i 0.373777i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −45.0005 −2.06913
\(474\) 0 0
\(475\) − 28.4602i − 1.30585i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.17501 −0.0993788 −0.0496894 0.998765i \(-0.515823\pi\)
−0.0496894 + 0.998765i \(0.515823\pi\)
\(480\) 0 0
\(481\) −2.69000 −0.122653
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 18.5643i − 0.842962i
\(486\) 0 0
\(487\) 4.51499 0.204594 0.102297 0.994754i \(-0.467381\pi\)
0.102297 + 0.994754i \(0.467381\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 1.91221i − 0.0862967i −0.999069 0.0431483i \(-0.986261\pi\)
0.999069 0.0431483i \(-0.0137388\pi\)
\(492\) 0 0
\(493\) − 2.79130i − 0.125714i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10.2068 −0.457840
\(498\) 0 0
\(499\) − 27.4065i − 1.22688i −0.789740 0.613442i \(-0.789784\pi\)
0.789740 0.613442i \(-0.210216\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 17.1368 0.764094 0.382047 0.924143i \(-0.375219\pi\)
0.382047 + 0.924143i \(0.375219\pi\)
\(504\) 0 0
\(505\) −18.5178 −0.824030
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 21.6437i − 0.959342i −0.877449 0.479671i \(-0.840756\pi\)
0.877449 0.479671i \(-0.159244\pi\)
\(510\) 0 0
\(511\) 3.34500 0.147974
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 19.3837i − 0.854148i
\(516\) 0 0
\(517\) 53.7587i 2.36430i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −28.3137 −1.24045 −0.620223 0.784426i \(-0.712958\pi\)
−0.620223 + 0.784426i \(0.712958\pi\)
\(522\) 0 0
\(523\) 15.0687i 0.658908i 0.944172 + 0.329454i \(0.106865\pi\)
−0.944172 + 0.329454i \(0.893135\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.44684 −0.106586
\(528\) 0 0
\(529\) −20.4878 −0.890774
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 0.189345i − 0.00820145i
\(534\) 0 0
\(535\) −14.4109 −0.623038
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.76717i 0.205337i
\(540\) 0 0
\(541\) 46.2990i 1.99055i 0.0971017 + 0.995274i \(0.469043\pi\)
−0.0971017 + 0.995274i \(0.530957\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8.21372 −0.351837
\(546\) 0 0
\(547\) − 2.03714i − 0.0871020i −0.999051 0.0435510i \(-0.986133\pi\)
0.999051 0.0435510i \(-0.0138671\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 51.3755 2.18867
\(552\) 0 0
\(553\) −4.83001 −0.205393
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 11.1234i − 0.471315i −0.971836 0.235658i \(-0.924276\pi\)
0.971836 0.235658i \(-0.0757244\pi\)
\(558\) 0 0
\(559\) 4.30683 0.182159
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 37.2437i − 1.56963i −0.619727 0.784817i \(-0.712757\pi\)
0.619727 0.784817i \(-0.287243\pi\)
\(564\) 0 0
\(565\) − 3.77566i − 0.158843i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −22.9369 −0.961564 −0.480782 0.876840i \(-0.659647\pi\)
−0.480782 + 0.876840i \(0.659647\pi\)
\(570\) 0 0
\(571\) 35.1634i 1.47154i 0.677231 + 0.735770i \(0.263180\pi\)
−0.677231 + 0.735770i \(0.736820\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5.90558 0.246280
\(576\) 0 0
\(577\) 13.7918 0.574162 0.287081 0.957906i \(-0.407315\pi\)
0.287081 + 0.957906i \(0.407315\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.53434i 0.229603i
\(582\) 0 0
\(583\) −36.4137 −1.50810
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 26.9862i 1.11384i 0.830566 + 0.556920i \(0.188017\pi\)
−0.830566 + 0.556920i \(0.811983\pi\)
\(588\) 0 0
\(589\) − 45.0355i − 1.85566i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 31.8337 1.30725 0.653627 0.756817i \(-0.273247\pi\)
0.653627 + 0.756817i \(0.273247\pi\)
\(594\) 0 0
\(595\) − 0.468436i − 0.0192040i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −34.9006 −1.42600 −0.712999 0.701165i \(-0.752664\pi\)
−0.712999 + 0.701165i \(0.752664\pi\)
\(600\) 0 0
\(601\) −0.175010 −0.00713881 −0.00356941 0.999994i \(-0.501136\pi\)
−0.00356941 + 0.999994i \(0.501136\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 13.2356i 0.538104i
\(606\) 0 0
\(607\) 32.1705 1.30576 0.652881 0.757461i \(-0.273560\pi\)
0.652881 + 0.757461i \(0.273560\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 5.14503i − 0.208146i
\(612\) 0 0
\(613\) 8.37869i 0.338412i 0.985581 + 0.169206i \(0.0541203\pi\)
−0.985581 + 0.169206i \(0.945880\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −28.9618 −1.16596 −0.582980 0.812487i \(-0.698113\pi\)
−0.582980 + 0.812487i \(0.698113\pi\)
\(618\) 0 0
\(619\) 6.85497i 0.275524i 0.990465 + 0.137762i \(0.0439910\pi\)
−0.990465 + 0.137762i \(0.956009\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.92999 0.197516
\(624\) 0 0
\(625\) 7.51221 0.300488
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.44684i 0.0975619i
\(630\) 0 0
\(631\) 40.2055 1.60056 0.800278 0.599629i \(-0.204685\pi\)
0.800278 + 0.599629i \(0.204685\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 11.7918i − 0.467945i
\(636\) 0 0
\(637\) − 0.456247i − 0.0180772i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 24.0737 0.950854 0.475427 0.879755i \(-0.342294\pi\)
0.475427 + 0.879755i \(0.342294\pi\)
\(642\) 0 0
\(643\) 23.7559i 0.936841i 0.883506 + 0.468421i \(0.155177\pi\)
−0.883506 + 0.468421i \(0.844823\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −32.6218 −1.28250 −0.641249 0.767333i \(-0.721583\pi\)
−0.641249 + 0.767333i \(0.721583\pi\)
\(648\) 0 0
\(649\) 19.0687 0.748512
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 14.4109i − 0.563942i −0.959423 0.281971i \(-0.909012\pi\)
0.959423 0.281971i \(-0.0909883\pi\)
\(654\) 0 0
\(655\) 7.06313 0.275979
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 23.3315i − 0.908866i −0.890781 0.454433i \(-0.849842\pi\)
0.890781 0.454433i \(-0.150158\pi\)
\(660\) 0 0
\(661\) − 44.1468i − 1.71711i −0.512721 0.858555i \(-0.671362\pi\)
0.512721 0.858555i \(-0.328638\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.62185 0.334341
\(666\) 0 0
\(667\) 10.6606i 0.412779i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −8.58685 −0.331492
\(672\) 0 0
\(673\) 11.6960 0.450846 0.225423 0.974261i \(-0.427624\pi\)
0.225423 + 0.974261i \(0.427624\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0.648756i 0.0249337i 0.999922 + 0.0124669i \(0.00396843\pi\)
−0.999922 + 0.0124669i \(0.996032\pi\)
\(678\) 0 0
\(679\) −16.4468 −0.631172
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 22.4909i − 0.860589i −0.902689 0.430294i \(-0.858410\pi\)
0.902689 0.430294i \(-0.141590\pi\)
\(684\) 0 0
\(685\) − 14.6306i − 0.559007i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.48501 0.132768
\(690\) 0 0
\(691\) 45.2681i 1.72208i 0.508538 + 0.861039i \(0.330186\pi\)
−0.508538 + 0.861039i \(0.669814\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 20.7287 0.786285
\(696\) 0 0
\(697\) −0.172230 −0.00652366
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 1.03091i − 0.0389370i −0.999810 0.0194685i \(-0.993803\pi\)
0.999810 0.0194685i \(-0.00619740\pi\)
\(702\) 0 0
\(703\) −45.0355 −1.69855
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 16.4056i 0.616996i
\(708\) 0 0
\(709\) 10.5481i 0.396144i 0.980188 + 0.198072i \(0.0634679\pi\)
−0.980188 + 0.198072i \(0.936532\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 9.34500 0.349973
\(714\) 0 0
\(715\) − 2.45504i − 0.0918131i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −7.00502 −0.261243 −0.130622 0.991432i \(-0.541697\pi\)
−0.130622 + 0.991432i \(0.541697\pi\)
\(720\) 0 0
\(721\) −17.1728 −0.639547
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 25.0603i 0.930718i
\(726\) 0 0
\(727\) 17.8028 0.660270 0.330135 0.943934i \(-0.392906\pi\)
0.330135 + 0.943934i \(0.392906\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 3.91752i − 0.144895i
\(732\) 0 0
\(733\) 34.3300i 1.26801i 0.773330 + 0.634004i \(0.218590\pi\)
−0.773330 + 0.634004i \(0.781410\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 38.5887 1.42143
\(738\) 0 0
\(739\) 0.0946726i 0.00348259i 0.999998 + 0.00174129i \(0.000554271\pi\)
−0.999998 + 0.00174129i \(0.999446\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −13.2755 −0.487032 −0.243516 0.969897i \(-0.578301\pi\)
−0.243516 + 0.969897i \(0.578301\pi\)
\(744\) 0 0
\(745\) 17.6518 0.646713
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12.7672i 0.466502i
\(750\) 0 0
\(751\) −15.4187 −0.562637 −0.281318 0.959615i \(-0.590772\pi\)
−0.281318 + 0.959615i \(0.590772\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 17.2437i 0.627562i
\(756\) 0 0
\(757\) 9.96685i 0.362251i 0.983460 + 0.181126i \(0.0579741\pi\)
−0.983460 + 0.181126i \(0.942026\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −23.4837 −0.851283 −0.425642 0.904892i \(-0.639952\pi\)
−0.425642 + 0.904892i \(0.639952\pi\)
\(762\) 0 0
\(763\) 7.27685i 0.263440i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.82499 −0.0658966
\(768\) 0 0
\(769\) 22.9369 0.827125 0.413562 0.910476i \(-0.364284\pi\)
0.413562 + 0.910476i \(0.364284\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 35.0956i − 1.26230i −0.775660 0.631150i \(-0.782583\pi\)
0.775660 0.631150i \(-0.217417\pi\)
\(774\) 0 0
\(775\) 21.9678 0.789106
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 3.16999i − 0.113577i
\(780\) 0 0
\(781\) 48.6578i 1.74111i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 24.5869 0.877542
\(786\) 0 0
\(787\) − 17.9480i − 0.639778i −0.947455 0.319889i \(-0.896354\pi\)
0.947455 0.319889i \(-0.103646\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.34500 −0.118934
\(792\) 0 0
\(793\) 0.821814 0.0291835
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 34.4199i 1.21922i 0.792703 + 0.609608i \(0.208673\pi\)
−0.792703 + 0.609608i \(0.791327\pi\)
\(798\) 0 0
\(799\) −4.67996 −0.165565
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 15.9462i − 0.562729i
\(804\) 0 0
\(805\) 1.78906i 0.0630560i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.89314 −0.0665592 −0.0332796 0.999446i \(-0.510595\pi\)
−0.0332796 + 0.999446i \(0.510595\pi\)
\(810\) 0 0
\(811\) − 10.3400i − 0.363086i −0.983383 0.181543i \(-0.941891\pi\)
0.983383 0.181543i \(-0.0581091\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.20314 0.182258
\(816\) 0 0
\(817\) 72.1042 2.52261
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 2.40029i − 0.0837706i −0.999122 0.0418853i \(-0.986664\pi\)
0.999122 0.0418853i \(-0.0133364\pi\)
\(822\) 0 0
\(823\) −25.8655 −0.901616 −0.450808 0.892621i \(-0.648864\pi\)
−0.450808 + 0.892621i \(0.648864\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17.0928i 0.594375i 0.954819 + 0.297188i \(0.0960487\pi\)
−0.954819 + 0.297188i \(0.903951\pi\)
\(828\) 0 0
\(829\) − 41.8893i − 1.45488i −0.686174 0.727438i \(-0.740711\pi\)
0.686174 0.727438i \(-0.259289\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.415006 −0.0143791
\(834\) 0 0
\(835\) 25.9181i 0.896933i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −24.5500 −0.847560 −0.423780 0.905765i \(-0.639297\pi\)
−0.423780 + 0.905765i \(0.639297\pi\)
\(840\) 0 0
\(841\) −16.2381 −0.559936
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 14.4387i − 0.496708i
\(846\) 0 0
\(847\) 11.7259 0.402908
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 9.34500i − 0.320342i
\(852\) 0 0
\(853\) − 38.7193i − 1.32572i −0.748742 0.662862i \(-0.769342\pi\)
0.748742 0.662862i \(-0.230658\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 45.2074 1.54425 0.772127 0.635468i \(-0.219193\pi\)
0.772127 + 0.635468i \(0.219193\pi\)
\(858\) 0 0
\(859\) − 57.1890i − 1.95126i −0.219418 0.975631i \(-0.570416\pi\)
0.219418 0.975631i \(-0.429584\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −45.1106 −1.53558 −0.767791 0.640701i \(-0.778644\pi\)
−0.767791 + 0.640701i \(0.778644\pi\)
\(864\) 0 0
\(865\) 0.244094 0.00829944
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 23.0255i 0.781086i
\(870\) 0 0
\(871\) −3.69317 −0.125138
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.84937i 0.332969i
\(876\) 0 0
\(877\) − 17.6600i − 0.596337i −0.954513 0.298168i \(-0.903624\pi\)
0.954513 0.298168i \(-0.0963757\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −6.78998 −0.228760 −0.114380 0.993437i \(-0.536488\pi\)
−0.114380 + 0.993437i \(0.536488\pi\)
\(882\) 0 0
\(883\) − 28.6015i − 0.962516i −0.876579 0.481258i \(-0.840180\pi\)
0.876579 0.481258i \(-0.159820\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 23.5919 0.792138 0.396069 0.918221i \(-0.370374\pi\)
0.396069 + 0.918221i \(0.370374\pi\)
\(888\) 0 0
\(889\) −10.4468 −0.350376
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 86.1374i − 2.88248i
\(894\) 0 0
\(895\) 25.5284 0.853319
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 39.6555i 1.32259i
\(900\) 0 0
\(901\) − 3.16999i − 0.105608i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8.72871 0.290152
\(906\) 0 0
\(907\) − 2.02096i − 0.0671050i −0.999437 0.0335525i \(-0.989318\pi\)
0.999437 0.0335525i \(-0.0106821\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 52.5524 1.74114 0.870569 0.492046i \(-0.163751\pi\)
0.870569 + 0.492046i \(0.163751\pi\)
\(912\) 0 0
\(913\) 26.3832 0.873156
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 6.25749i − 0.206641i
\(918\) 0 0
\(919\) −21.5237 −0.710002 −0.355001 0.934866i \(-0.615520\pi\)
−0.355001 + 0.934866i \(0.615520\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 4.65685i − 0.153282i
\(924\) 0 0
\(925\) − 21.9678i − 0.722296i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 15.7500 0.516739 0.258370 0.966046i \(-0.416815\pi\)
0.258370 + 0.966046i \(0.416815\pi\)
\(930\) 0 0
\(931\) − 7.63843i − 0.250339i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −2.23312 −0.0730307
\(936\) 0 0
\(937\) 17.0687 0.557610 0.278805 0.960348i \(-0.410062\pi\)
0.278805 + 0.960348i \(0.410062\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 26.5456i 0.865362i 0.901547 + 0.432681i \(0.142432\pi\)
−0.901547 + 0.432681i \(0.857568\pi\)
\(942\) 0 0
\(943\) 0.657782 0.0214203
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 20.3265i − 0.660522i −0.943890 0.330261i \(-0.892863\pi\)
0.943890 0.330261i \(-0.107137\pi\)
\(948\) 0 0
\(949\) 1.52615i 0.0495408i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 38.6855 1.25315 0.626573 0.779362i \(-0.284457\pi\)
0.626573 + 0.779362i \(0.284457\pi\)
\(954\) 0 0
\(955\) − 11.5209i − 0.372809i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.9618 −0.418559
\(960\) 0 0
\(961\) 3.76186 0.121350
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 6.73192i − 0.216708i
\(966\) 0 0
\(967\) 26.6606 0.857346 0.428673 0.903460i \(-0.358981\pi\)
0.428673 + 0.903460i \(0.358981\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 3.27685i − 0.105159i −0.998617 0.0525796i \(-0.983256\pi\)
0.998617 0.0525796i \(-0.0167443\pi\)
\(972\) 0 0
\(973\) − 18.3644i − 0.588734i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12.0387 −0.385153 −0.192576 0.981282i \(-0.561684\pi\)
−0.192576 + 0.981282i \(0.561684\pi\)
\(978\) 0 0
\(979\) − 23.5021i − 0.751131i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4.42188 0.141036 0.0705181 0.997510i \(-0.477535\pi\)
0.0705181 + 0.997510i \(0.477535\pi\)
\(984\) 0 0
\(985\) −21.2299 −0.676442
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 14.9618i 0.475758i
\(990\) 0 0
\(991\) 9.10184 0.289129 0.144565 0.989495i \(-0.453822\pi\)
0.144565 + 0.989495i \(0.453822\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3.69873i 0.117258i
\(996\) 0 0
\(997\) 42.8556i 1.35725i 0.734485 + 0.678625i \(0.237424\pi\)
−0.734485 + 0.678625i \(0.762576\pi\)
\(998\) 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 2016.2.c.e.1009.3 8
3.2 odd 2 672.2.c.b.337.3 8
4.3 odd 2 504.2.c.f.253.8 8
8.3 odd 2 504.2.c.f.253.7 8
8.5 even 2 inner 2016.2.c.e.1009.6 8
12.11 even 2 168.2.c.b.85.1 8
21.20 even 2 4704.2.c.c.2353.6 8
24.5 odd 2 672.2.c.b.337.6 8
24.11 even 2 168.2.c.b.85.2 yes 8
48.5 odd 4 5376.2.a.bl.1.2 4
48.11 even 4 5376.2.a.bp.1.2 4
48.29 odd 4 5376.2.a.bq.1.3 4
48.35 even 4 5376.2.a.bm.1.3 4
84.83 odd 2 1176.2.c.c.589.1 8
168.83 odd 2 1176.2.c.c.589.2 8
168.125 even 2 4704.2.c.c.2353.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.c.b.85.1 8 12.11 even 2
168.2.c.b.85.2 yes 8 24.11 even 2
504.2.c.f.253.7 8 8.3 odd 2
504.2.c.f.253.8 8 4.3 odd 2
672.2.c.b.337.3 8 3.2 odd 2
672.2.c.b.337.6 8 24.5 odd 2
1176.2.c.c.589.1 8 84.83 odd 2
1176.2.c.c.589.2 8 168.83 odd 2
2016.2.c.e.1009.3 8 1.1 even 1 trivial
2016.2.c.e.1009.6 8 8.5 even 2 inner
4704.2.c.c.2353.3 8 168.125 even 2
4704.2.c.c.2353.6 8 21.20 even 2
5376.2.a.bl.1.2 4 48.5 odd 4
5376.2.a.bm.1.3 4 48.35 even 4
5376.2.a.bp.1.2 4 48.11 even 4
5376.2.a.bq.1.3 4 48.29 odd 4