Properties

Label 3840.2.k.a.1921.2
Level $3840$
Weight $2$
Character 3840.1921
Analytic conductor $30.663$
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] = [3840,2,Mod(1921,3840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3840, 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("3840.1921");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.k (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: \(30.6625543762\)
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 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1921.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3840.1921
Dual form 3840.2.k.a.1921.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{5} -4.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{5} -4.00000 q^{7} -1.00000 q^{9} -6.00000i q^{13} -1.00000 q^{15} -2.00000 q^{17} +4.00000i q^{19} -4.00000i q^{21} +8.00000 q^{23} -1.00000 q^{25} -1.00000i q^{27} -6.00000i q^{29} -4.00000i q^{35} +6.00000i q^{37} +6.00000 q^{39} -10.0000 q^{41} +4.00000i q^{43} -1.00000i q^{45} +8.00000 q^{47} +9.00000 q^{49} -2.00000i q^{51} -10.0000i q^{53} -4.00000 q^{57} +6.00000i q^{61} +4.00000 q^{63} +6.00000 q^{65} -4.00000i q^{67} +8.00000i q^{69} +14.0000 q^{73} -1.00000i q^{75} +16.0000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -2.00000i q^{85} +6.00000 q^{87} -2.00000 q^{89} +24.0000i q^{91} -4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{7} - 2 q^{9} - 2 q^{15} - 4 q^{17} + 16 q^{23} - 2 q^{25} + 12 q^{39} - 20 q^{41} + 16 q^{47} + 18 q^{49} - 8 q^{57} + 8 q^{63} + 12 q^{65} + 28 q^{73} + 32 q^{79} + 2 q^{81} + 12 q^{87} - 4 q^{89} - 8 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 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
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) − 4.00000i − 0.872872i
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 6.00000i − 1.11417i −0.830455 0.557086i \(-0.811919\pi\)
0.830455 0.557086i \(-0.188081\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 4.00000i − 0.676123i
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) − 1.00000i − 0.149071i
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) − 2.00000i − 0.280056i
\(52\) 0 0
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 6.00000i 0.768221i 0.923287 + 0.384111i \(0.125492\pi\)
−0.923287 + 0.384111i \(0.874508\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 8.00000i 0.963087i
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) − 1.00000i − 0.115470i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) − 2.00000i − 0.216930i
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 24.0000i 2.51588i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.0000i 1.39305i 0.717532 + 0.696526i \(0.245272\pi\)
−0.717532 + 0.696526i \(0.754728\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 0 0
\(109\) − 10.0000i − 0.957826i −0.877862 0.478913i \(-0.841031\pi\)
0.877862 0.478913i \(-0.158969\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 8.00000i 0.746004i
\(116\) 0 0
\(117\) 6.00000i 0.554700i
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) − 10.0000i − 0.901670i
\(124\) 0 0
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 16.0000i 1.39793i 0.715158 + 0.698963i \(0.246355\pi\)
−0.715158 + 0.698963i \(0.753645\pi\)
\(132\) 0 0
\(133\) − 16.0000i − 1.38738i
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 12.0000i 1.01783i 0.860818 + 0.508913i \(0.169953\pi\)
−0.860818 + 0.508913i \(0.830047\pi\)
\(140\) 0 0
\(141\) 8.00000i 0.673722i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 9.00000i 0.742307i
\(148\) 0 0
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) −32.0000 −2.52195
\(162\) 0 0
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) − 4.00000i − 0.305888i
\(172\) 0 0
\(173\) 18.0000i 1.36851i 0.729241 + 0.684257i \(0.239873\pi\)
−0.729241 + 0.684257i \(0.760127\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.00000i 0.597948i 0.954261 + 0.298974i \(0.0966444\pi\)
−0.954261 + 0.298974i \(0.903356\pi\)
\(180\) 0 0
\(181\) − 14.0000i − 1.04061i −0.853980 0.520306i \(-0.825818\pi\)
0.853980 0.520306i \(-0.174182\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 4.00000i 0.290957i
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 0 0
\(195\) 6.00000i 0.429669i
\(196\) 0 0
\(197\) − 10.0000i − 0.712470i −0.934396 0.356235i \(-0.884060\pi\)
0.934396 0.356235i \(-0.115940\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 24.0000i 1.68447i
\(204\) 0 0
\(205\) − 10.0000i − 0.698430i
\(206\) 0 0
\(207\) −8.00000 −0.556038
\(208\) 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
\(213\) 0 0
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 14.0000i 0.946032i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) 12.0000 0.803579 0.401790 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) − 6.00000i − 0.396491i −0.980152 0.198246i \(-0.936476\pi\)
0.980152 0.198246i \(-0.0635244\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 8.00000i 0.521862i
\(236\) 0 0
\(237\) 16.0000i 1.03931i
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 9.00000i 0.574989i
\(246\) 0 0
\(247\) 24.0000 1.52708
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 8.00000i 0.504956i 0.967603 + 0.252478i \(0.0812455\pi\)
−0.967603 + 0.252478i \(0.918755\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 2.00000 0.125245
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) − 24.0000i − 1.49129i
\(260\) 0 0
\(261\) 6.00000i 0.371391i
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 0 0
\(267\) − 2.00000i − 0.122398i
\(268\) 0 0
\(269\) − 6.00000i − 0.365826i −0.983129 0.182913i \(-0.941447\pi\)
0.983129 0.182913i \(-0.0585527\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) −24.0000 −1.45255
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) − 4.00000i − 0.236940i
\(286\) 0 0
\(287\) 40.0000 2.36113
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 2.00000i 0.117242i
\(292\) 0 0
\(293\) − 10.0000i − 0.584206i −0.956387 0.292103i \(-0.905645\pi\)
0.956387 0.292103i \(-0.0943550\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 48.0000i − 2.77591i
\(300\) 0 0
\(301\) − 16.0000i − 0.922225i
\(302\) 0 0
\(303\) −14.0000 −0.804279
\(304\) 0 0
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) − 4.00000i − 0.227552i
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 4.00000i 0.225374i
\(316\) 0 0
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) − 8.00000i − 0.445132i
\(324\) 0 0
\(325\) 6.00000i 0.332820i
\(326\) 0 0
\(327\) 10.0000 0.553001
\(328\) 0 0
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) − 28.0000i − 1.53902i −0.638635 0.769510i \(-0.720501\pi\)
0.638635 0.769510i \(-0.279499\pi\)
\(332\) 0 0
\(333\) − 6.00000i − 0.328798i
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 0 0
\(339\) 6.00000i 0.325875i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) −8.00000 −0.430706
\(346\) 0 0
\(347\) 36.0000i 1.93258i 0.257454 + 0.966291i \(0.417117\pi\)
−0.257454 + 0.966291i \(0.582883\pi\)
\(348\) 0 0
\(349\) − 26.0000i − 1.39175i −0.718164 0.695874i \(-0.755017\pi\)
0.718164 0.695874i \(-0.244983\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 8.00000i 0.423405i
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 11.0000i 0.577350i
\(364\) 0 0
\(365\) 14.0000i 0.732793i
\(366\) 0 0
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 40.0000i 2.07670i
\(372\) 0 0
\(373\) − 2.00000i − 0.103556i −0.998659 0.0517780i \(-0.983511\pi\)
0.998659 0.0517780i \(-0.0164888\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) − 4.00000i − 0.205466i −0.994709 0.102733i \(-0.967241\pi\)
0.994709 0.102733i \(-0.0327588\pi\)
\(380\) 0 0
\(381\) − 4.00000i − 0.204926i
\(382\) 0 0
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 4.00000i − 0.203331i
\(388\) 0 0
\(389\) 6.00000i 0.304212i 0.988364 + 0.152106i \(0.0486055\pi\)
−0.988364 + 0.152106i \(0.951394\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) −16.0000 −0.807093
\(394\) 0 0
\(395\) 16.0000i 0.805047i
\(396\) 0 0
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 38.0000 1.87898 0.939490 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 0 0
\(411\) 18.0000i 0.887875i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) − 16.0000i − 0.781651i −0.920465 0.390826i \(-0.872190\pi\)
0.920465 0.390826i \(-0.127810\pi\)
\(420\) 0 0
\(421\) − 6.00000i − 0.292422i −0.989253 0.146211i \(-0.953292\pi\)
0.989253 0.146211i \(-0.0467079\pi\)
\(422\) 0 0
\(423\) −8.00000 −0.388973
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) − 24.0000i − 1.16144i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) 0 0
\(435\) 6.00000i 0.287678i
\(436\) 0 0
\(437\) 32.0000i 1.53077i
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −9.00000 −0.428571
\(442\) 0 0
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 0 0
\(445\) − 2.00000i − 0.0948091i
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −24.0000 −1.12514
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 0 0
\(459\) 2.00000i 0.0933520i
\(460\) 0 0
\(461\) 2.00000i 0.0931493i 0.998915 + 0.0465746i \(0.0148305\pi\)
−0.998915 + 0.0465746i \(0.985169\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 20.0000i − 0.925490i −0.886492 0.462745i \(-0.846865\pi\)
0.886492 0.462745i \(-0.153135\pi\)
\(468\) 0 0
\(469\) 16.0000i 0.738811i
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) − 4.00000i − 0.183533i
\(476\) 0 0
\(477\) 10.0000i 0.457869i
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 36.0000 1.64146
\(482\) 0 0
\(483\) − 32.0000i − 1.45605i
\(484\) 0 0
\(485\) 2.00000i 0.0908153i
\(486\) 0 0
\(487\) 36.0000 1.63132 0.815658 0.578535i \(-0.196375\pi\)
0.815658 + 0.578535i \(0.196375\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 40.0000i 1.80517i 0.430507 + 0.902587i \(0.358335\pi\)
−0.430507 + 0.902587i \(0.641665\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 20.0000i 0.895323i 0.894203 + 0.447661i \(0.147743\pi\)
−0.894203 + 0.447661i \(0.852257\pi\)
\(500\) 0 0
\(501\) 8.00000i 0.357414i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −14.0000 −0.622992
\(506\) 0 0
\(507\) − 23.0000i − 1.02147i
\(508\) 0 0
\(509\) − 6.00000i − 0.265945i −0.991120 0.132973i \(-0.957548\pi\)
0.991120 0.132973i \(-0.0424523\pi\)
\(510\) 0 0
\(511\) −56.0000 −2.47729
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) − 4.00000i − 0.176261i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) − 36.0000i − 1.57417i −0.616844 0.787085i \(-0.711589\pi\)
0.616844 0.787085i \(-0.288411\pi\)
\(524\) 0 0
\(525\) 4.00000i 0.174574i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 60.0000i 2.59889i
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 10.0000i − 0.429934i −0.976621 0.214967i \(-0.931036\pi\)
0.976621 0.214967i \(-0.0689643\pi\)
\(542\) 0 0
\(543\) 14.0000 0.600798
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) 4.00000i 0.171028i 0.996337 + 0.0855138i \(0.0272532\pi\)
−0.996337 + 0.0855138i \(0.972747\pi\)
\(548\) 0 0
\(549\) − 6.00000i − 0.256074i
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) −64.0000 −2.72156
\(554\) 0 0
\(555\) − 6.00000i − 0.254686i
\(556\) 0 0
\(557\) − 14.0000i − 0.593199i −0.955002 0.296600i \(-0.904147\pi\)
0.955002 0.296600i \(-0.0958526\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 36.0000i 1.51722i 0.651546 + 0.758610i \(0.274121\pi\)
−0.651546 + 0.758610i \(0.725879\pi\)
\(564\) 0 0
\(565\) 6.00000i 0.252422i
\(566\) 0 0
\(567\) −4.00000 −0.167984
\(568\) 0 0
\(569\) 14.0000 0.586911 0.293455 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(570\) 0 0
\(571\) 20.0000i 0.836974i 0.908223 + 0.418487i \(0.137439\pi\)
−0.908223 + 0.418487i \(0.862561\pi\)
\(572\) 0 0
\(573\) 8.00000i 0.334205i
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 0 0
\(579\) − 6.00000i − 0.249351i
\(580\) 0 0
\(581\) − 48.0000i − 1.99138i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) − 36.0000i − 1.48588i −0.669359 0.742940i \(-0.733431\pi\)
0.669359 0.742940i \(-0.266569\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 0 0
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 0 0
\(595\) 8.00000i 0.327968i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) 11.0000i 0.447214i
\(606\) 0 0
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) − 48.0000i − 1.94187i
\(612\) 0 0
\(613\) − 26.0000i − 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) 0 0
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 0 0
\(619\) 28.0000i 1.12542i 0.826656 + 0.562708i \(0.190240\pi\)
−0.826656 + 0.562708i \(0.809760\pi\)
\(620\) 0 0
\(621\) − 8.00000i − 0.321029i
\(622\) 0 0
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 12.0000i − 0.478471i
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) − 4.00000i − 0.158735i
\(636\) 0 0
\(637\) − 54.0000i − 2.13956i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) − 36.0000i − 1.41970i −0.704352 0.709851i \(-0.748762\pi\)
0.704352 0.709851i \(-0.251238\pi\)
\(644\) 0 0
\(645\) − 4.00000i − 0.157500i
\(646\) 0 0
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.0000i 0.704394i 0.935926 + 0.352197i \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) 0 0
\(655\) −16.0000 −0.625172
\(656\) 0 0
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) − 16.0000i − 0.623272i −0.950202 0.311636i \(-0.899123\pi\)
0.950202 0.311636i \(-0.100877\pi\)
\(660\) 0 0
\(661\) 2.00000i 0.0777910i 0.999243 + 0.0388955i \(0.0123839\pi\)
−0.999243 + 0.0388955i \(0.987616\pi\)
\(662\) 0 0
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) − 48.0000i − 1.85857i
\(668\) 0 0
\(669\) 12.0000i 0.463947i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 0 0
\(685\) 18.0000i 0.687745i
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) 0 0
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) 4.00000i 0.152167i 0.997101 + 0.0760836i \(0.0242416\pi\)
−0.997101 + 0.0760836i \(0.975758\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) 0 0
\(699\) 10.0000i 0.378235i
\(700\) 0 0
\(701\) − 38.0000i − 1.43524i −0.696435 0.717620i \(-0.745231\pi\)
0.696435 0.717620i \(-0.254769\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) − 56.0000i − 2.10610i
\(708\) 0 0
\(709\) − 22.0000i − 0.826227i −0.910679 0.413114i \(-0.864441\pi\)
0.910679 0.413114i \(-0.135559\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 24.0000i − 0.896296i
\(718\) 0 0
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 2.00000i 0.0743808i
\(724\) 0 0
\(725\) 6.00000i 0.222834i
\(726\) 0 0
\(727\) 4.00000 0.148352 0.0741759 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 8.00000i − 0.295891i
\(732\) 0 0
\(733\) − 46.0000i − 1.69905i −0.527549 0.849524i \(-0.676889\pi\)
0.527549 0.849524i \(-0.323111\pi\)
\(734\) 0 0
\(735\) −9.00000 −0.331970
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 4.00000i 0.147142i 0.997290 + 0.0735712i \(0.0234396\pi\)
−0.997290 + 0.0735712i \(0.976560\pi\)
\(740\) 0 0
\(741\) 24.0000i 0.881662i
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) 16.0000i 0.584627i
\(750\) 0 0
\(751\) −48.0000 −1.75154 −0.875772 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) 0 0
\(753\) −8.00000 −0.291536
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 42.0000i − 1.52652i −0.646094 0.763258i \(-0.723599\pi\)
0.646094 0.763258i \(-0.276401\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −50.0000 −1.81250 −0.906249 0.422744i \(-0.861067\pi\)
−0.906249 + 0.422744i \(0.861067\pi\)
\(762\) 0 0
\(763\) 40.0000i 1.44810i
\(764\) 0 0
\(765\) 2.00000i 0.0723102i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) 6.00000i 0.216085i
\(772\) 0 0
\(773\) 38.0000i 1.36677i 0.730061 + 0.683383i \(0.239492\pi\)
−0.730061 + 0.683383i \(0.760508\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 24.0000 0.860995
\(778\) 0 0
\(779\) − 40.0000i − 1.43315i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) 12.0000i 0.427754i 0.976861 + 0.213877i \(0.0686091\pi\)
−0.976861 + 0.213877i \(0.931391\pi\)
\(788\) 0 0
\(789\) 24.0000i 0.854423i
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) 36.0000 1.27840
\(794\) 0 0
\(795\) 10.0000i 0.354663i
\(796\) 0 0
\(797\) 34.0000i 1.20434i 0.798367 + 0.602171i \(0.205697\pi\)
−0.798367 + 0.602171i \(0.794303\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) − 32.0000i − 1.12785i
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 12.0000i 0.421377i 0.977553 + 0.210688i \(0.0675706\pi\)
−0.977553 + 0.210688i \(0.932429\pi\)
\(812\) 0 0
\(813\) 8.00000i 0.280572i
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) − 24.0000i − 0.838628i
\(820\) 0 0
\(821\) − 50.0000i − 1.74501i −0.488603 0.872506i \(-0.662493\pi\)
0.488603 0.872506i \(-0.337507\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20.0000i 0.695468i 0.937593 + 0.347734i \(0.113049\pi\)
−0.937593 + 0.347734i \(0.886951\pi\)
\(828\) 0 0
\(829\) 38.0000i 1.31979i 0.751356 + 0.659897i \(0.229400\pi\)
−0.751356 + 0.659897i \(0.770600\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 8.00000i 0.276851i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) − 2.00000i − 0.0688837i
\(844\) 0 0
\(845\) − 23.0000i − 0.791224i
\(846\) 0 0
\(847\) −44.0000 −1.51186
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 48.0000i 1.64542i
\(852\) 0 0
\(853\) − 10.0000i − 0.342393i −0.985237 0.171197i \(-0.945237\pi\)
0.985237 0.171197i \(-0.0547634\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) −38.0000 −1.29806 −0.649028 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) 0 0
\(859\) − 36.0000i − 1.22830i −0.789188 0.614152i \(-0.789498\pi\)
0.789188 0.614152i \(-0.210502\pi\)
\(860\) 0 0
\(861\) 40.0000i 1.36320i
\(862\) 0 0
\(863\) 16.0000 0.544646 0.272323 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 4.00000i 0.135225i
\(876\) 0 0
\(877\) 10.0000i 0.337676i 0.985644 + 0.168838i \(0.0540015\pi\)
−0.985644 + 0.168838i \(0.945999\pi\)
\(878\) 0 0
\(879\) 10.0000 0.337292
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 0 0
\(883\) 28.0000i 0.942275i 0.882060 + 0.471138i \(0.156156\pi\)
−0.882060 + 0.471138i \(0.843844\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 32.0000i 1.07084i
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) 48.0000 1.60267
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 20.0000i 0.666297i
\(902\) 0 0
\(903\) 16.0000 0.532447
\(904\) 0 0
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) 0 0
\(909\) − 14.0000i − 0.464351i
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) − 6.00000i − 0.198354i
\(916\) 0 0
\(917\) − 64.0000i − 2.11347i
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 6.00000i − 0.197279i
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 36.0000i 1.17985i
\(932\) 0 0
\(933\) − 16.0000i − 0.523816i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) 14.0000i 0.456873i
\(940\) 0 0
\(941\) − 14.0000i − 0.456387i −0.973616 0.228193i \(-0.926718\pi\)
0.973616 0.228193i \(-0.0732819\pi\)
\(942\) 0 0
\(943\) −80.0000 −2.60516
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) − 44.0000i − 1.42981i −0.699223 0.714904i \(-0.746470\pi\)
0.699223 0.714904i \(-0.253530\pi\)
\(948\) 0 0
\(949\) − 84.0000i − 2.72676i
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) 8.00000i 0.258874i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −72.0000 −2.32500
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 4.00000i 0.128898i
\(964\) 0 0
\(965\) − 6.00000i − 0.193147i
\(966\) 0 0
\(967\) −20.0000 −0.643157 −0.321578 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 56.0000i 1.79713i 0.438845 + 0.898563i \(0.355388\pi\)
−0.438845 + 0.898563i \(0.644612\pi\)
\(972\) 0 0
\(973\) − 48.0000i − 1.53881i
\(974\) 0 0
\(975\) −6.00000 −0.192154
\(976\) 0 0
\(977\) −50.0000 −1.59964 −0.799821 0.600239i \(-0.795072\pi\)
−0.799821 + 0.600239i \(0.795072\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000i 0.319275i
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 0 0
\(987\) − 32.0000i − 1.01857i
\(988\) 0 0
\(989\) 32.0000i 1.01754i
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 0 0
\(993\) 28.0000 0.888553
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 30.0000i 0.950110i 0.879956 + 0.475055i \(0.157572\pi\)
−0.879956 + 0.475055i \(0.842428\pi\)
\(998\) 0 0
\(999\) 6.00000 0.189832
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 3840.2.k.a.1921.2 2
4.3 odd 2 3840.2.k.z.1921.1 2
8.3 odd 2 3840.2.k.z.1921.2 2
8.5 even 2 inner 3840.2.k.a.1921.1 2
16.3 odd 4 960.2.a.n.1.1 1
16.5 even 4 120.2.a.a.1.1 1
16.11 odd 4 240.2.a.a.1.1 1
16.13 even 4 960.2.a.g.1.1 1
48.5 odd 4 360.2.a.e.1.1 1
48.11 even 4 720.2.a.f.1.1 1
48.29 odd 4 2880.2.a.r.1.1 1
48.35 even 4 2880.2.a.b.1.1 1
80.3 even 4 4800.2.f.n.3649.2 2
80.13 odd 4 4800.2.f.u.3649.1 2
80.19 odd 4 4800.2.a.bh.1.1 1
80.27 even 4 1200.2.f.f.49.2 2
80.29 even 4 4800.2.a.bl.1.1 1
80.37 odd 4 600.2.f.c.49.1 2
80.43 even 4 1200.2.f.f.49.1 2
80.53 odd 4 600.2.f.c.49.2 2
80.59 odd 4 1200.2.a.r.1.1 1
80.67 even 4 4800.2.f.n.3649.1 2
80.69 even 4 600.2.a.a.1.1 1
80.77 odd 4 4800.2.f.u.3649.2 2
112.69 odd 4 5880.2.a.p.1.1 1
144.5 odd 12 3240.2.q.a.2161.1 2
144.85 even 12 3240.2.q.m.2161.1 2
144.101 odd 12 3240.2.q.a.1081.1 2
144.133 even 12 3240.2.q.m.1081.1 2
240.53 even 4 1800.2.f.g.649.1 2
240.59 even 4 3600.2.a.bo.1.1 1
240.107 odd 4 3600.2.f.l.2449.1 2
240.149 odd 4 1800.2.a.c.1.1 1
240.197 even 4 1800.2.f.g.649.2 2
240.203 odd 4 3600.2.f.l.2449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.a.a.1.1 1 16.5 even 4
240.2.a.a.1.1 1 16.11 odd 4
360.2.a.e.1.1 1 48.5 odd 4
600.2.a.a.1.1 1 80.69 even 4
600.2.f.c.49.1 2 80.37 odd 4
600.2.f.c.49.2 2 80.53 odd 4
720.2.a.f.1.1 1 48.11 even 4
960.2.a.g.1.1 1 16.13 even 4
960.2.a.n.1.1 1 16.3 odd 4
1200.2.a.r.1.1 1 80.59 odd 4
1200.2.f.f.49.1 2 80.43 even 4
1200.2.f.f.49.2 2 80.27 even 4
1800.2.a.c.1.1 1 240.149 odd 4
1800.2.f.g.649.1 2 240.53 even 4
1800.2.f.g.649.2 2 240.197 even 4
2880.2.a.b.1.1 1 48.35 even 4
2880.2.a.r.1.1 1 48.29 odd 4
3240.2.q.a.1081.1 2 144.101 odd 12
3240.2.q.a.2161.1 2 144.5 odd 12
3240.2.q.m.1081.1 2 144.133 even 12
3240.2.q.m.2161.1 2 144.85 even 12
3600.2.a.bo.1.1 1 240.59 even 4
3600.2.f.l.2449.1 2 240.107 odd 4
3600.2.f.l.2449.2 2 240.203 odd 4
3840.2.k.a.1921.1 2 8.5 even 2 inner
3840.2.k.a.1921.2 2 1.1 even 1 trivial
3840.2.k.z.1921.1 2 4.3 odd 2
3840.2.k.z.1921.2 2 8.3 odd 2
4800.2.a.bh.1.1 1 80.19 odd 4
4800.2.a.bl.1.1 1 80.29 even 4
4800.2.f.n.3649.1 2 80.67 even 4
4800.2.f.n.3649.2 2 80.3 even 4
4800.2.f.u.3649.1 2 80.13 odd 4
4800.2.f.u.3649.2 2 80.77 odd 4
5880.2.a.p.1.1 1 112.69 odd 4