Properties

Label 2816.2.c.f.1409.1
Level $2816$
Weight $2$
Character 2816.1409
Analytic conductor $22.486$
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] = [2816,2,Mod(1409,2816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2816, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2816.1409");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2816 = 2^{8} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2816.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: \(22.4858732092\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1409.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2816.1409
Dual form 2816.2.c.f.1409.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +1.00000i q^{5} -2.00000 q^{7} +2.00000 q^{9} -1.00000i q^{11} -4.00000i q^{13} +1.00000 q^{15} -2.00000 q^{17} +2.00000i q^{21} -1.00000 q^{23} +4.00000 q^{25} -5.00000i q^{27} -7.00000 q^{31} -1.00000 q^{33} -2.00000i q^{35} +3.00000i q^{37} -4.00000 q^{39} +8.00000 q^{41} +6.00000i q^{43} +2.00000i q^{45} -8.00000 q^{47} -3.00000 q^{49} +2.00000i q^{51} -6.00000i q^{53} +1.00000 q^{55} -5.00000i q^{59} -12.0000i q^{61} -4.00000 q^{63} +4.00000 q^{65} -7.00000i q^{67} +1.00000i q^{69} -3.00000 q^{71} -4.00000 q^{73} -4.00000i q^{75} +2.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -2.00000i q^{85} -15.0000 q^{89} +8.00000i q^{91} +7.00000i q^{93} -7.00000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} + 4 q^{9} + 2 q^{15} - 4 q^{17} - 2 q^{23} + 8 q^{25} - 14 q^{31} - 2 q^{33} - 8 q^{39} + 16 q^{41} - 16 q^{47} - 6 q^{49} + 2 q^{55} - 8 q^{63} + 8 q^{65} - 6 q^{71} - 8 q^{73} + 20 q^{79} + 2 q^{81} - 30 q^{89} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(1541\) \(2047\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 1.00000i − 0.577350i −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i 0.974679 + 0.223607i \(0.0717831\pi\)
−0.974679 + 0.223607i \(0.928217\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) − 1.00000i − 0.301511i
\(12\) 0 0
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) − 2.00000i − 0.338062i
\(36\) 0 0
\(37\) 3.00000i 0.493197i 0.969118 + 0.246598i \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 0 0
\(45\) 2.00000i 0.298142i
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.00000i 0.280056i
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 5.00000i − 0.650945i −0.945552 0.325472i \(-0.894477\pi\)
0.945552 0.325472i \(-0.105523\pi\)
\(60\) 0 0
\(61\) − 12.0000i − 1.53644i −0.640184 0.768221i \(-0.721142\pi\)
0.640184 0.768221i \(-0.278858\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) − 7.00000i − 0.855186i −0.903971 0.427593i \(-0.859362\pi\)
0.903971 0.427593i \(-0.140638\pi\)
\(68\) 0 0
\(69\) 1.00000i 0.120386i
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) − 4.00000i − 0.461880i
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) − 2.00000i − 0.216930i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 8.00000i 0.838628i
\(92\) 0 0
\(93\) 7.00000i 0.725866i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 0 0
\(99\) − 2.00000i − 0.201008i
\(100\) 0 0
\(101\) 2.00000i 0.199007i 0.995037 + 0.0995037i \(0.0317255\pi\)
−0.995037 + 0.0995037i \(0.968274\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) − 18.0000i − 1.74013i −0.492941 0.870063i \(-0.664078\pi\)
0.492941 0.870063i \(-0.335922\pi\)
\(108\) 0 0
\(109\) − 10.0000i − 0.957826i −0.877862 0.478913i \(-0.841031\pi\)
0.877862 0.478913i \(-0.158969\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) − 1.00000i − 0.0932505i
\(116\) 0 0
\(117\) − 8.00000i − 0.739600i
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) − 8.00000i − 0.721336i
\(124\) 0 0
\(125\) 9.00000i 0.804984i
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) − 18.0000i − 1.57267i −0.617802 0.786334i \(-0.711977\pi\)
0.617802 0.786334i \(-0.288023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 5.00000 0.430331
\(136\) 0 0
\(137\) 7.00000 0.598050 0.299025 0.954245i \(-0.403339\pi\)
0.299025 + 0.954245i \(0.403339\pi\)
\(138\) 0 0
\(139\) − 10.0000i − 0.848189i −0.905618 0.424094i \(-0.860592\pi\)
0.905618 0.424094i \(-0.139408\pi\)
\(140\) 0 0
\(141\) 8.00000i 0.673722i
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.00000i 0.247436i
\(148\) 0 0
\(149\) − 10.0000i − 0.819232i −0.912258 0.409616i \(-0.865663\pi\)
0.912258 0.409616i \(-0.134337\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) − 7.00000i − 0.562254i
\(156\) 0 0
\(157\) 7.00000i 0.558661i 0.960195 + 0.279330i \(0.0901125\pi\)
−0.960195 + 0.279330i \(0.909888\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 0 0
\(165\) − 1.00000i − 0.0778499i
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) −8.00000 −0.604743
\(176\) 0 0
\(177\) −5.00000 −0.375823
\(178\) 0 0
\(179\) − 15.0000i − 1.12115i −0.828103 0.560576i \(-0.810580\pi\)
0.828103 0.560576i \(-0.189420\pi\)
\(180\) 0 0
\(181\) 7.00000i 0.520306i 0.965567 + 0.260153i \(0.0837730\pi\)
−0.965567 + 0.260153i \(0.916227\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) 2.00000i 0.146254i
\(188\) 0 0
\(189\) 10.0000i 0.727393i
\(190\) 0 0
\(191\) −17.0000 −1.23008 −0.615038 0.788497i \(-0.710860\pi\)
−0.615038 + 0.788497i \(0.710860\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) − 4.00000i − 0.286446i
\(196\) 0 0
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 8.00000i 0.558744i
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000i 0.826114i 0.910705 + 0.413057i \(0.135539\pi\)
−0.910705 + 0.413057i \(0.864461\pi\)
\(212\) 0 0
\(213\) 3.00000i 0.205557i
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 14.0000 0.950382
\(218\) 0 0
\(219\) 4.00000i 0.270295i
\(220\) 0 0
\(221\) 8.00000i 0.538138i
\(222\) 0 0
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) 0 0
\(225\) 8.00000 0.533333
\(226\) 0 0
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 0 0
\(229\) 15.0000i 0.991228i 0.868543 + 0.495614i \(0.165057\pi\)
−0.868543 + 0.495614i \(0.834943\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) − 8.00000i − 0.521862i
\(236\) 0 0
\(237\) − 10.0000i − 0.649570i
\(238\) 0 0
\(239\) 30.0000 1.94054 0.970269 0.242028i \(-0.0778125\pi\)
0.970269 + 0.242028i \(0.0778125\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 0 0
\(243\) − 16.0000i − 1.02640i
\(244\) 0 0
\(245\) − 3.00000i − 0.191663i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 23.0000i 1.45175i 0.687828 + 0.725874i \(0.258564\pi\)
−0.687828 + 0.725874i \(0.741436\pi\)
\(252\) 0 0
\(253\) 1.00000i 0.0628695i
\(254\) 0 0
\(255\) −2.00000 −0.125245
\(256\) 0 0
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) − 6.00000i − 0.372822i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.0000 0.863277 0.431638 0.902047i \(-0.357936\pi\)
0.431638 + 0.902047i \(0.357936\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 15.0000i 0.917985i
\(268\) 0 0
\(269\) − 10.0000i − 0.609711i −0.952399 0.304855i \(-0.901392\pi\)
0.952399 0.304855i \(-0.0986081\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) 0 0
\(275\) − 4.00000i − 0.241209i
\(276\) 0 0
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 0 0
\(279\) −14.0000 −0.838158
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −16.0000 −0.944450
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 7.00000i 0.410347i
\(292\) 0 0
\(293\) 24.0000i 1.40209i 0.713115 + 0.701047i \(0.247284\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(294\) 0 0
\(295\) 5.00000 0.291111
\(296\) 0 0
\(297\) −5.00000 −0.290129
\(298\) 0 0
\(299\) 4.00000i 0.231326i
\(300\) 0 0
\(301\) − 12.0000i − 0.691669i
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) 8.00000i 0.456584i 0.973593 + 0.228292i \(0.0733141\pi\)
−0.973593 + 0.228292i \(0.926686\pi\)
\(308\) 0 0
\(309\) 16.0000i 0.910208i
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 0 0
\(315\) − 4.00000i − 0.225374i
\(316\) 0 0
\(317\) − 13.0000i − 0.730153i −0.930978 0.365076i \(-0.881043\pi\)
0.930978 0.365076i \(-0.118957\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) − 16.0000i − 0.887520i
\(326\) 0 0
\(327\) −10.0000 −0.553001
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) − 7.00000i − 0.384755i −0.981321 0.192377i \(-0.938380\pi\)
0.981321 0.192377i \(-0.0616198\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 7.00000 0.382451
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) − 9.00000i − 0.488813i
\(340\) 0 0
\(341\) 7.00000i 0.379071i
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) − 28.0000i − 1.50312i −0.659665 0.751559i \(-0.729302\pi\)
0.659665 0.751559i \(-0.270698\pi\)
\(348\) 0 0
\(349\) − 30.0000i − 1.60586i −0.596071 0.802932i \(-0.703272\pi\)
0.596071 0.802932i \(-0.296728\pi\)
\(350\) 0 0
\(351\) −20.0000 −1.06752
\(352\) 0 0
\(353\) −21.0000 −1.11772 −0.558859 0.829263i \(-0.688761\pi\)
−0.558859 + 0.829263i \(0.688761\pi\)
\(354\) 0 0
\(355\) − 3.00000i − 0.159223i
\(356\) 0 0
\(357\) − 4.00000i − 0.211702i
\(358\) 0 0
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 1.00000i 0.0524864i
\(364\) 0 0
\(365\) − 4.00000i − 0.209370i
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 0 0
\(369\) 16.0000 0.832927
\(370\) 0 0
\(371\) 12.0000i 0.623009i
\(372\) 0 0
\(373\) − 26.0000i − 1.34623i −0.739538 0.673114i \(-0.764956\pi\)
0.739538 0.673114i \(-0.235044\pi\)
\(374\) 0 0
\(375\) 9.00000 0.464758
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 5.00000i 0.256833i 0.991720 + 0.128416i \(0.0409894\pi\)
−0.991720 + 0.128416i \(0.959011\pi\)
\(380\) 0 0
\(381\) 8.00000i 0.409852i
\(382\) 0 0
\(383\) 1.00000 0.0510976 0.0255488 0.999674i \(-0.491867\pi\)
0.0255488 + 0.999674i \(0.491867\pi\)
\(384\) 0 0
\(385\) −2.00000 −0.101929
\(386\) 0 0
\(387\) 12.0000i 0.609994i
\(388\) 0 0
\(389\) − 15.0000i − 0.760530i −0.924878 0.380265i \(-0.875833\pi\)
0.924878 0.380265i \(-0.124167\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) −18.0000 −0.907980
\(394\) 0 0
\(395\) 10.0000i 0.503155i
\(396\) 0 0
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 28.0000i 1.39478i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 3.00000 0.148704
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 0 0
\(411\) − 7.00000i − 0.345285i
\(412\) 0 0
\(413\) 10.0000i 0.492068i
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) −10.0000 −0.489702
\(418\) 0 0
\(419\) 20.0000i 0.977064i 0.872546 + 0.488532i \(0.162467\pi\)
−0.872546 + 0.488532i \(0.837533\pi\)
\(420\) 0 0
\(421\) 22.0000i 1.07221i 0.844150 + 0.536107i \(0.180106\pi\)
−0.844150 + 0.536107i \(0.819894\pi\)
\(422\) 0 0
\(423\) −16.0000 −0.777947
\(424\) 0 0
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) 24.0000i 1.16144i
\(428\) 0 0
\(429\) 4.00000i 0.193122i
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) −11.0000 −0.528626 −0.264313 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 11.0000i 0.522626i 0.965254 + 0.261313i \(0.0841554\pi\)
−0.965254 + 0.261313i \(0.915845\pi\)
\(444\) 0 0
\(445\) − 15.0000i − 0.711068i
\(446\) 0 0
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) 35.0000 1.65175 0.825876 0.563852i \(-0.190681\pi\)
0.825876 + 0.563852i \(0.190681\pi\)
\(450\) 0 0
\(451\) − 8.00000i − 0.376705i
\(452\) 0 0
\(453\) − 2.00000i − 0.0939682i
\(454\) 0 0
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) 12.0000 0.561336 0.280668 0.959805i \(-0.409444\pi\)
0.280668 + 0.959805i \(0.409444\pi\)
\(458\) 0 0
\(459\) 10.0000i 0.466760i
\(460\) 0 0
\(461\) − 12.0000i − 0.558896i −0.960161 0.279448i \(-0.909849\pi\)
0.960161 0.279448i \(-0.0901514\pi\)
\(462\) 0 0
\(463\) 11.0000 0.511213 0.255607 0.966781i \(-0.417725\pi\)
0.255607 + 0.966781i \(0.417725\pi\)
\(464\) 0 0
\(465\) −7.00000 −0.324617
\(466\) 0 0
\(467\) − 27.0000i − 1.24941i −0.780860 0.624705i \(-0.785219\pi\)
0.780860 0.624705i \(-0.214781\pi\)
\(468\) 0 0
\(469\) 14.0000i 0.646460i
\(470\) 0 0
\(471\) 7.00000 0.322543
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 12.0000i − 0.549442i
\(478\) 0 0
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 0 0
\(483\) − 2.00000i − 0.0910032i
\(484\) 0 0
\(485\) − 7.00000i − 0.317854i
\(486\) 0 0
\(487\) 23.0000 1.04223 0.521115 0.853487i \(-0.325516\pi\)
0.521115 + 0.853487i \(0.325516\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 8.00000i 0.361035i 0.983572 + 0.180517i \(0.0577772\pi\)
−0.983572 + 0.180517i \(0.942223\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) 20.0000i 0.895323i 0.894203 + 0.447661i \(0.147743\pi\)
−0.894203 + 0.447661i \(0.852257\pi\)
\(500\) 0 0
\(501\) 12.0000i 0.536120i
\(502\) 0 0
\(503\) −26.0000 −1.15928 −0.579641 0.814872i \(-0.696807\pi\)
−0.579641 + 0.814872i \(0.696807\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) 3.00000i 0.133235i
\(508\) 0 0
\(509\) − 15.0000i − 0.664863i −0.943127 0.332432i \(-0.892131\pi\)
0.943127 0.332432i \(-0.107869\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 16.0000i − 0.705044i
\(516\) 0 0
\(517\) 8.00000i 0.351840i
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) 16.0000i 0.699631i 0.936819 + 0.349816i \(0.113756\pi\)
−0.936819 + 0.349816i \(0.886244\pi\)
\(524\) 0 0
\(525\) 8.00000i 0.349149i
\(526\) 0 0
\(527\) 14.0000 0.609850
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) − 10.0000i − 0.433963i
\(532\) 0 0
\(533\) − 32.0000i − 1.38607i
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 0 0
\(537\) −15.0000 −0.647298
\(538\) 0 0
\(539\) 3.00000i 0.129219i
\(540\) 0 0
\(541\) 8.00000i 0.343947i 0.985102 + 0.171973i \(0.0550143\pi\)
−0.985102 + 0.171973i \(0.944986\pi\)
\(542\) 0 0
\(543\) 7.00000 0.300399
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 0 0
\(549\) − 24.0000i − 1.02430i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −20.0000 −0.850487
\(554\) 0 0
\(555\) 3.00000i 0.127343i
\(556\) 0 0
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) 0 0
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 0 0
\(565\) 9.00000i 0.378633i
\(566\) 0 0
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 28.0000i 1.17176i 0.810397 + 0.585882i \(0.199252\pi\)
−0.810397 + 0.585882i \(0.800748\pi\)
\(572\) 0 0
\(573\) 17.0000i 0.710185i
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 33.0000 1.37381 0.686904 0.726748i \(-0.258969\pi\)
0.686904 + 0.726748i \(0.258969\pi\)
\(578\) 0 0
\(579\) − 4.00000i − 0.166234i
\(580\) 0 0
\(581\) 12.0000i 0.497844i
\(582\) 0 0
\(583\) −6.00000 −0.248495
\(584\) 0 0
\(585\) 8.00000 0.330759
\(586\) 0 0
\(587\) − 28.0000i − 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 0 0
\(593\) 44.0000 1.80686 0.903432 0.428732i \(-0.141040\pi\)
0.903432 + 0.428732i \(0.141040\pi\)
\(594\) 0 0
\(595\) 4.00000i 0.163984i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 0 0
\(603\) − 14.0000i − 0.570124i
\(604\) 0 0
\(605\) − 1.00000i − 0.0406558i
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 32.0000i 1.29458i
\(612\) 0 0
\(613\) − 16.0000i − 0.646234i −0.946359 0.323117i \(-0.895269\pi\)
0.946359 0.323117i \(-0.104731\pi\)
\(614\) 0 0
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 25.0000i 1.00483i 0.864625 + 0.502417i \(0.167556\pi\)
−0.864625 + 0.502417i \(0.832444\pi\)
\(620\) 0 0
\(621\) 5.00000i 0.200643i
\(622\) 0 0
\(623\) 30.0000 1.20192
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 6.00000i − 0.239236i
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 0 0
\(635\) − 8.00000i − 0.317470i
\(636\) 0 0
\(637\) 12.0000i 0.475457i
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) 29.0000i 1.14365i 0.820376 + 0.571824i \(0.193764\pi\)
−0.820376 + 0.571824i \(0.806236\pi\)
\(644\) 0 0
\(645\) 6.00000i 0.236250i
\(646\) 0 0
\(647\) −7.00000 −0.275198 −0.137599 0.990488i \(-0.543939\pi\)
−0.137599 + 0.990488i \(0.543939\pi\)
\(648\) 0 0
\(649\) −5.00000 −0.196267
\(650\) 0 0
\(651\) − 14.0000i − 0.548703i
\(652\) 0 0
\(653\) 41.0000i 1.60445i 0.597019 + 0.802227i \(0.296352\pi\)
−0.597019 + 0.802227i \(0.703648\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) 0 0
\(657\) −8.00000 −0.312110
\(658\) 0 0
\(659\) 10.0000i 0.389545i 0.980848 + 0.194772i \(0.0623968\pi\)
−0.980848 + 0.194772i \(0.937603\pi\)
\(660\) 0 0
\(661\) 37.0000i 1.43913i 0.694423 + 0.719567i \(0.255660\pi\)
−0.694423 + 0.719567i \(0.744340\pi\)
\(662\) 0 0
\(663\) 8.00000 0.310694
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 19.0000i 0.734582i
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) − 20.0000i − 0.769800i
\(676\) 0 0
\(677\) − 42.0000i − 1.61419i −0.590421 0.807096i \(-0.701038\pi\)
0.590421 0.807096i \(-0.298962\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) 16.0000i 0.612223i 0.951996 + 0.306111i \(0.0990280\pi\)
−0.951996 + 0.306111i \(0.900972\pi\)
\(684\) 0 0
\(685\) 7.00000i 0.267456i
\(686\) 0 0
\(687\) 15.0000 0.572286
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 17.0000i 0.646710i 0.946278 + 0.323355i \(0.104811\pi\)
−0.946278 + 0.323355i \(0.895189\pi\)
\(692\) 0 0
\(693\) 4.00000i 0.151947i
\(694\) 0 0
\(695\) 10.0000 0.379322
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) 0 0
\(699\) 24.0000i 0.907763i
\(700\) 0 0
\(701\) − 2.00000i − 0.0755390i −0.999286 0.0377695i \(-0.987975\pi\)
0.999286 0.0377695i \(-0.0120253\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) − 4.00000i − 0.150435i
\(708\) 0 0
\(709\) − 25.0000i − 0.938895i −0.882960 0.469447i \(-0.844453\pi\)
0.882960 0.469447i \(-0.155547\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) 7.00000 0.262152
\(714\) 0 0
\(715\) − 4.00000i − 0.149592i
\(716\) 0 0
\(717\) − 30.0000i − 1.12037i
\(718\) 0 0
\(719\) −15.0000 −0.559406 −0.279703 0.960087i \(-0.590236\pi\)
−0.279703 + 0.960087i \(0.590236\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 0 0
\(723\) 8.00000i 0.297523i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.00000 0.111264 0.0556319 0.998451i \(-0.482283\pi\)
0.0556319 + 0.998451i \(0.482283\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) − 12.0000i − 0.443836i
\(732\) 0 0
\(733\) 36.0000i 1.32969i 0.746981 + 0.664845i \(0.231502\pi\)
−0.746981 + 0.664845i \(0.768498\pi\)
\(734\) 0 0
\(735\) −3.00000 −0.110657
\(736\) 0 0
\(737\) −7.00000 −0.257848
\(738\) 0 0
\(739\) 50.0000i 1.83928i 0.392763 + 0.919640i \(0.371519\pi\)
−0.392763 + 0.919640i \(0.628481\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 0 0
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) 36.0000i 1.31541i
\(750\) 0 0
\(751\) 23.0000 0.839282 0.419641 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) 0 0
\(753\) 23.0000 0.838167
\(754\) 0 0
\(755\) 2.00000i 0.0727875i
\(756\) 0 0
\(757\) − 22.0000i − 0.799604i −0.916602 0.399802i \(-0.869079\pi\)
0.916602 0.399802i \(-0.130921\pi\)
\(758\) 0 0
\(759\) 1.00000 0.0362977
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) 20.0000i 0.724049i
\(764\) 0 0
\(765\) − 4.00000i − 0.144620i
\(766\) 0 0
\(767\) −20.0000 −0.722158
\(768\) 0 0
\(769\) 20.0000 0.721218 0.360609 0.932717i \(-0.382569\pi\)
0.360609 + 0.932717i \(0.382569\pi\)
\(770\) 0 0
\(771\) 2.00000i 0.0720282i
\(772\) 0 0
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 0 0
\(775\) −28.0000 −1.00579
\(776\) 0 0
\(777\) −6.00000 −0.215249
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 3.00000i 0.107348i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.00000 −0.249841
\(786\) 0 0
\(787\) − 32.0000i − 1.14068i −0.821410 0.570338i \(-0.806812\pi\)
0.821410 0.570338i \(-0.193188\pi\)
\(788\) 0 0
\(789\) − 14.0000i − 0.498413i
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) −48.0000 −1.70453
\(794\) 0 0
\(795\) − 6.00000i − 0.212798i
\(796\) 0 0
\(797\) − 53.0000i − 1.87736i −0.344795 0.938678i \(-0.612051\pi\)
0.344795 0.938678i \(-0.387949\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) 4.00000i 0.141157i
\(804\) 0 0
\(805\) 2.00000i 0.0704907i
\(806\) 0 0
\(807\) −10.0000 −0.352017
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 38.0000i 1.33436i 0.744896 + 0.667180i \(0.232499\pi\)
−0.744896 + 0.667180i \(0.767501\pi\)
\(812\) 0 0
\(813\) − 28.0000i − 0.982003i
\(814\) 0 0
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 16.0000i 0.559085i
\(820\) 0 0
\(821\) 22.0000i 0.767805i 0.923374 + 0.383903i \(0.125420\pi\)
−0.923374 + 0.383903i \(0.874580\pi\)
\(822\) 0 0
\(823\) 39.0000 1.35945 0.679727 0.733465i \(-0.262098\pi\)
0.679727 + 0.733465i \(0.262098\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 52.0000i 1.80822i 0.427303 + 0.904109i \(0.359464\pi\)
−0.427303 + 0.904109i \(0.640536\pi\)
\(828\) 0 0
\(829\) − 25.0000i − 0.868286i −0.900844 0.434143i \(-0.857051\pi\)
0.900844 0.434143i \(-0.142949\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) − 12.0000i − 0.415277i
\(836\) 0 0
\(837\) 35.0000i 1.20978i
\(838\) 0 0
\(839\) −5.00000 −0.172619 −0.0863096 0.996268i \(-0.527507\pi\)
−0.0863096 + 0.996268i \(0.527507\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) − 18.0000i − 0.619953i
\(844\) 0 0
\(845\) − 3.00000i − 0.103203i
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) − 3.00000i − 0.102839i
\(852\) 0 0
\(853\) 14.0000i 0.479351i 0.970853 + 0.239675i \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −8.00000 −0.273275 −0.136637 0.990621i \(-0.543630\pi\)
−0.136637 + 0.990621i \(0.543630\pi\)
\(858\) 0 0
\(859\) 15.0000i 0.511793i 0.966704 + 0.255897i \(0.0823707\pi\)
−0.966704 + 0.255897i \(0.917629\pi\)
\(860\) 0 0
\(861\) 16.0000i 0.545279i
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) − 10.0000i − 0.339227i
\(870\) 0 0
\(871\) −28.0000 −0.948744
\(872\) 0 0
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) − 18.0000i − 0.608511i
\(876\) 0 0
\(877\) 12.0000i 0.405211i 0.979260 + 0.202606i \(0.0649409\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) 0 0
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) −43.0000 −1.44871 −0.724353 0.689429i \(-0.757862\pi\)
−0.724353 + 0.689429i \(0.757862\pi\)
\(882\) 0 0
\(883\) 4.00000i 0.134611i 0.997732 + 0.0673054i \(0.0214402\pi\)
−0.997732 + 0.0673054i \(0.978560\pi\)
\(884\) 0 0
\(885\) − 5.00000i − 0.168073i
\(886\) 0 0
\(887\) −22.0000 −0.738688 −0.369344 0.929293i \(-0.620418\pi\)
−0.369344 + 0.929293i \(0.620418\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) − 1.00000i − 0.0335013i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 15.0000 0.501395
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000i 0.399778i
\(902\) 0 0
\(903\) −12.0000 −0.399335
\(904\) 0 0
\(905\) −7.00000 −0.232688
\(906\) 0 0
\(907\) 12.0000i 0.398453i 0.979953 + 0.199227i \(0.0638430\pi\)
−0.979953 + 0.199227i \(0.936157\pi\)
\(908\) 0 0
\(909\) 4.00000i 0.132672i
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) −6.00000 −0.198571
\(914\) 0 0
\(915\) − 12.0000i − 0.396708i
\(916\) 0 0
\(917\) 36.0000i 1.18882i
\(918\) 0 0
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) 0 0
\(923\) 12.0000i 0.394985i
\(924\) 0 0
\(925\) 12.0000i 0.394558i
\(926\) 0 0
\(927\) −32.0000 −1.05102
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 12.0000i − 0.392862i
\(934\) 0 0
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) −8.00000 −0.261349 −0.130674 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) 0 0
\(939\) − 1.00000i − 0.0326338i
\(940\) 0 0
\(941\) − 42.0000i − 1.36916i −0.728937 0.684580i \(-0.759985\pi\)
0.728937 0.684580i \(-0.240015\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) −10.0000 −0.325300
\(946\) 0 0
\(947\) − 27.0000i − 0.877382i −0.898638 0.438691i \(-0.855442\pi\)
0.898638 0.438691i \(-0.144558\pi\)
\(948\) 0 0
\(949\) 16.0000i 0.519382i
\(950\) 0 0
\(951\) −13.0000 −0.421554
\(952\) 0 0
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) − 17.0000i − 0.550107i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −14.0000 −0.452084
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) − 36.0000i − 1.16008i
\(964\) 0 0
\(965\) 4.00000i 0.128765i
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 47.0000i − 1.50830i −0.656701 0.754151i \(-0.728049\pi\)
0.656701 0.754151i \(-0.271951\pi\)
\(972\) 0 0
\(973\) 20.0000i 0.641171i
\(974\) 0 0
\(975\) −16.0000 −0.512410
\(976\) 0 0
\(977\) −27.0000 −0.863807 −0.431903 0.901920i \(-0.642158\pi\)
−0.431903 + 0.901920i \(0.642158\pi\)
\(978\) 0 0
\(979\) 15.0000i 0.479402i
\(980\) 0 0
\(981\) − 20.0000i − 0.638551i
\(982\) 0 0
\(983\) 39.0000 1.24391 0.621953 0.783054i \(-0.286339\pi\)
0.621953 + 0.783054i \(0.286339\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) − 16.0000i − 0.509286i
\(988\) 0 0
\(989\) − 6.00000i − 0.190789i
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) −7.00000 −0.222138
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 38.0000i 1.20347i 0.798695 + 0.601736i \(0.205524\pi\)
−0.798695 + 0.601736i \(0.794476\pi\)
\(998\) 0 0
\(999\) 15.0000 0.474579
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 2816.2.c.f.1409.1 2
4.3 odd 2 2816.2.c.j.1409.2 2
8.3 odd 2 2816.2.c.j.1409.1 2
8.5 even 2 inner 2816.2.c.f.1409.2 2
16.3 odd 4 11.2.a.a.1.1 1
16.5 even 4 704.2.a.c.1.1 1
16.11 odd 4 704.2.a.h.1.1 1
16.13 even 4 176.2.a.b.1.1 1
48.5 odd 4 6336.2.a.bu.1.1 1
48.11 even 4 6336.2.a.br.1.1 1
48.29 odd 4 1584.2.a.g.1.1 1
48.35 even 4 99.2.a.d.1.1 1
80.3 even 4 275.2.b.a.199.2 2
80.13 odd 4 4400.2.b.h.4049.2 2
80.19 odd 4 275.2.a.b.1.1 1
80.29 even 4 4400.2.a.i.1.1 1
80.67 even 4 275.2.b.a.199.1 2
80.77 odd 4 4400.2.b.h.4049.1 2
112.3 even 12 539.2.e.g.177.1 2
112.13 odd 4 8624.2.a.j.1.1 1
112.19 even 12 539.2.e.g.67.1 2
112.51 odd 12 539.2.e.h.67.1 2
112.67 odd 12 539.2.e.h.177.1 2
112.83 even 4 539.2.a.a.1.1 1
144.67 odd 12 891.2.e.k.298.1 2
144.83 even 12 891.2.e.b.595.1 2
144.115 odd 12 891.2.e.k.595.1 2
144.131 even 12 891.2.e.b.298.1 2
176.3 odd 20 121.2.c.e.9.1 4
176.19 even 20 121.2.c.a.9.1 4
176.21 odd 4 7744.2.a.k.1.1 1
176.35 even 20 121.2.c.a.81.1 4
176.43 even 4 7744.2.a.x.1.1 1
176.51 even 20 121.2.c.a.27.1 4
176.83 even 20 121.2.c.a.3.1 4
176.109 odd 4 1936.2.a.i.1.1 1
176.115 odd 20 121.2.c.e.3.1 4
176.131 even 4 121.2.a.d.1.1 1
176.147 odd 20 121.2.c.e.27.1 4
176.163 odd 20 121.2.c.e.81.1 4
208.51 odd 4 1859.2.a.b.1.1 1
240.83 odd 4 2475.2.c.a.199.1 2
240.179 even 4 2475.2.a.a.1.1 1
240.227 odd 4 2475.2.c.a.199.2 2
272.67 odd 4 3179.2.a.a.1.1 1
304.227 even 4 3971.2.a.b.1.1 1
336.83 odd 4 4851.2.a.t.1.1 1
368.275 even 4 5819.2.a.a.1.1 1
464.115 odd 4 9251.2.a.d.1.1 1
528.131 odd 4 1089.2.a.b.1.1 1
880.659 even 4 3025.2.a.a.1.1 1
1232.307 odd 4 5929.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.2.a.a.1.1 1 16.3 odd 4
99.2.a.d.1.1 1 48.35 even 4
121.2.a.d.1.1 1 176.131 even 4
121.2.c.a.3.1 4 176.83 even 20
121.2.c.a.9.1 4 176.19 even 20
121.2.c.a.27.1 4 176.51 even 20
121.2.c.a.81.1 4 176.35 even 20
121.2.c.e.3.1 4 176.115 odd 20
121.2.c.e.9.1 4 176.3 odd 20
121.2.c.e.27.1 4 176.147 odd 20
121.2.c.e.81.1 4 176.163 odd 20
176.2.a.b.1.1 1 16.13 even 4
275.2.a.b.1.1 1 80.19 odd 4
275.2.b.a.199.1 2 80.67 even 4
275.2.b.a.199.2 2 80.3 even 4
539.2.a.a.1.1 1 112.83 even 4
539.2.e.g.67.1 2 112.19 even 12
539.2.e.g.177.1 2 112.3 even 12
539.2.e.h.67.1 2 112.51 odd 12
539.2.e.h.177.1 2 112.67 odd 12
704.2.a.c.1.1 1 16.5 even 4
704.2.a.h.1.1 1 16.11 odd 4
891.2.e.b.298.1 2 144.131 even 12
891.2.e.b.595.1 2 144.83 even 12
891.2.e.k.298.1 2 144.67 odd 12
891.2.e.k.595.1 2 144.115 odd 12
1089.2.a.b.1.1 1 528.131 odd 4
1584.2.a.g.1.1 1 48.29 odd 4
1859.2.a.b.1.1 1 208.51 odd 4
1936.2.a.i.1.1 1 176.109 odd 4
2475.2.a.a.1.1 1 240.179 even 4
2475.2.c.a.199.1 2 240.83 odd 4
2475.2.c.a.199.2 2 240.227 odd 4
2816.2.c.f.1409.1 2 1.1 even 1 trivial
2816.2.c.f.1409.2 2 8.5 even 2 inner
2816.2.c.j.1409.1 2 8.3 odd 2
2816.2.c.j.1409.2 2 4.3 odd 2
3025.2.a.a.1.1 1 880.659 even 4
3179.2.a.a.1.1 1 272.67 odd 4
3971.2.a.b.1.1 1 304.227 even 4
4400.2.a.i.1.1 1 80.29 even 4
4400.2.b.h.4049.1 2 80.77 odd 4
4400.2.b.h.4049.2 2 80.13 odd 4
4851.2.a.t.1.1 1 336.83 odd 4
5819.2.a.a.1.1 1 368.275 even 4
5929.2.a.h.1.1 1 1232.307 odd 4
6336.2.a.br.1.1 1 48.11 even 4
6336.2.a.bu.1.1 1 48.5 odd 4
7744.2.a.k.1.1 1 176.21 odd 4
7744.2.a.x.1.1 1 176.43 even 4
8624.2.a.j.1.1 1 112.13 odd 4
9251.2.a.d.1.1 1 464.115 odd 4