Properties

Label 4800.2.f.n.3649.1
Level $4800$
Weight $2$
Character 4800.3649
Analytic conductor $38.328$
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] = [4800,2,Mod(3649,4800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4800.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.f (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: \(38.3281929702\)
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 3649.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4800.3649
Dual form 4800.2.f.n.3649.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -4.00000i q^{7} -1.00000 q^{9} -6.00000i q^{13} -2.00000i q^{17} -4.00000 q^{19} -4.00000 q^{21} -8.00000i q^{23} +1.00000i q^{27} -6.00000 q^{29} +6.00000i q^{37} -6.00000 q^{39} +10.0000 q^{41} +4.00000i q^{43} -8.00000i q^{47} -9.00000 q^{49} -2.00000 q^{51} +10.0000i q^{53} +4.00000i q^{57} -6.00000 q^{61} +4.00000i q^{63} -4.00000i q^{67} -8.00000 q^{69} +14.0000i q^{73} +16.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +6.00000i q^{87} -2.00000 q^{89} -24.0000 q^{91} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} - 8 q^{19} - 8 q^{21} - 12 q^{29} - 12 q^{39} + 20 q^{41} - 18 q^{49} - 4 q^{51} - 12 q^{61} - 16 q^{69} + 32 q^{79} + 2 q^{81} - 4 q^{89} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(4351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) − 8.00000i − 1.66812i −0.551677 0.834058i \(-0.686012\pi\)
0.551677 0.834058i \(-0.313988\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 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 \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 10.0000i 1.37361i 0.726844 + 0.686803i \(0.240986\pi\)
−0.726844 + 0.686803i \(0.759014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 4.00000i 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 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 \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −24.0000 −2.51588
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(125\) 0 0
\(126\) 0 0
\(127\) 4.00000i 0.354943i 0.984126 + 0.177471i \(0.0567917\pi\)
−0.984126 + 0.177471i \(0.943208\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) 16.0000i 1.38738i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 9.00000i 0.742307i
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 10.0000i − 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) 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 \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 18.0000i 1.36851i 0.729241 + 0.684257i \(0.239873\pi\)
−0.729241 + 0.684257i \(0.760127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −8.00000 −0.597948 −0.298974 0.954261i \(-0.596644\pi\)
−0.298974 + 0.954261i \(0.596644\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 6.00000i 0.443533i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 6.00000i 0.431889i 0.976406 + 0.215945i \(0.0692831\pi\)
−0.976406 + 0.215945i \(0.930717\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 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
\(205\) 0 0
\(206\) 0 0
\(207\) 8.00000i 0.556038i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 12.0000i 0.803579i 0.915732 + 0.401790i \(0.131612\pi\)
−0.915732 + 0.401790i \(0.868388\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.0000i 0.655122i 0.944830 + 0.327561i \(0.106227\pi\)
−0.944830 + 0.327561i \(0.893773\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(245\) 0 0
\(246\) 0 0
\(247\) 24.0000i 1.52708i
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −8.00000 −0.504956 −0.252478 0.967603i \(-0.581245\pi\)
−0.252478 + 0.967603i \(0.581245\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.00000i 0.374270i 0.982334 + 0.187135i \(0.0599201\pi\)
−0.982334 + 0.187135i \(0.940080\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) − 24.0000i − 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.00000i 0.122398i
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 24.0000i 1.45255i
\(274\) 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 \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 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
\(287\) − 40.0000i − 2.36113i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0 0
\(293\) 10.0000i 0.584206i 0.956387 + 0.292103i \(0.0943550\pi\)
−0.956387 + 0.292103i \(0.905645\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −48.0000 −2.77591
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 0 0
\(303\) − 14.0000i − 0.804279i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i 0.918396 + 0.395663i \(0.129485\pi\)
−0.918396 + 0.395663i \(0.870515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.00000i − 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 10.0000i 0.553001i
\(328\) 0 0
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) − 6.00000i − 0.328798i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 10.0000i 0.544735i 0.962193 + 0.272367i \(0.0878066\pi\)
−0.962193 + 0.272367i \(0.912193\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 36.0000i − 1.93258i −0.257454 0.966291i \(-0.582883\pi\)
0.257454 0.966291i \(-0.417117\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 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 \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 11.0000i 0.577350i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 28.0000i − 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 40.0000 2.07670
\(372\) 0 0
\(373\) 2.00000i 0.103556i 0.998659 + 0.0517780i \(0.0164888\pi\)
−0.998659 + 0.0517780i \(0.983511\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 36.0000i 1.85409i
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) 8.00000i 0.408781i 0.978889 + 0.204390i \(0.0655212\pi\)
−0.978889 + 0.204390i \(0.934479\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 4.00000i − 0.203331i
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) − 16.0000i − 0.807093i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 2.00000i − 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 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
\(405\) 0 0
\(406\) 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
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 12.0000i − 0.587643i
\(418\) 0 0
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 8.00000i 0.388973i
\(424\) 0 0
\(425\) 0 0
\(426\) 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 \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 0 0
\(433\) − 10.0000i − 0.480569i −0.970702 0.240285i \(-0.922759\pi\)
0.970702 0.240285i \(-0.0772408\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 32.0000i 1.53077i
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 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
\(445\) 0 0
\(446\) 0 0
\(447\) 6.00000i 0.283790i
\(448\) 0 0
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 38.0000i − 1.77757i −0.458329 0.888783i \(-0.651552\pi\)
0.458329 0.888783i \(-0.348448\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) 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
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 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
\(485\) 0 0
\(486\) 0 0
\(487\) 36.0000i 1.63132i 0.578535 + 0.815658i \(0.303625\pi\)
−0.578535 + 0.815658i \(0.696375\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −40.0000 −1.80517 −0.902587 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 23.0000i 1.02147i
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 56.0000 2.47729
\(512\) 0 0
\(513\) − 4.00000i − 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) 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 \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) − 36.0000i − 1.57417i −0.616844 0.787085i \(-0.711589\pi\)
0.616844 0.787085i \(-0.288411\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 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
\(535\) 0 0
\(536\) 0 0
\(537\) 8.00000i 0.345225i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) 14.0000i 0.600798i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 4.00000i 0.171028i 0.996337 + 0.0855138i \(0.0272532\pi\)
−0.996337 + 0.0855138i \(0.972747\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) − 64.0000i − 2.72156i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.0000i 0.593199i 0.955002 + 0.296600i \(0.0958526\pi\)
−0.955002 + 0.296600i \(0.904147\pi\)
\(558\) 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 \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 4.00000i − 0.167984i
\(568\) 0 0
\(569\) 14.0000 0.586911 0.293455 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 8.00000i 0.334205i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 34.0000i 1.41544i 0.706494 + 0.707719i \(0.250276\pi\)
−0.706494 + 0.707719i \(0.749724\pi\)
\(578\) 0 0
\(579\) 6.00000 0.249351
\(580\) 0 0
\(581\) −48.0000 −1.99138
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 0 0
\(593\) − 22.0000i − 0.903432i −0.892162 0.451716i \(-0.850812\pi\)
0.892162 0.451716i \(-0.149188\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 12.0000i 0.487065i 0.969893 + 0.243532i \(0.0783062\pi\)
−0.969893 + 0.243532i \(0.921694\pi\)
\(608\) 0 0
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 26.0000i − 1.04672i −0.852111 0.523360i \(-0.824678\pi\)
0.852111 0.523360i \(-0.175322\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) 8.00000i 0.320513i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) − 12.0000i − 0.476957i
\(634\) 0 0
\(635\) 0 0
\(636\) 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 \(0.251238\pi\)
−0.704352 + 0.709851i \(0.748762\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 48.0000i − 1.88707i −0.331266 0.943537i \(-0.607476\pi\)
0.331266 0.943537i \(-0.392524\pi\)
\(648\) 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
\(655\) 0 0
\(656\) 0 0
\(657\) − 14.0000i − 0.546192i
\(658\) 0 0
\(659\) 16.0000 0.623272 0.311636 0.950202i \(-0.399123\pi\)
0.311636 + 0.950202i \(0.399123\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 0 0
\(663\) 12.0000i 0.466041i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 48.0000i 1.85857i
\(668\) 0 0
\(669\) 12.0000 0.463947
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 22.0000i 0.848038i 0.905653 + 0.424019i \(0.139381\pi\)
−0.905653 + 0.424019i \(0.860619\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 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
\(685\) 0 0
\(686\) 0 0
\(687\) − 6.00000i − 0.228914i
\(688\) 0 0
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 20.0000i − 0.757554i
\(698\) 0 0
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 38.0000 1.43524 0.717620 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) 0 0
\(703\) − 24.0000i − 0.905177i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 56.0000i − 2.10610i
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 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
\(725\) 0 0
\(726\) 0 0
\(727\) 4.00000i 0.148352i 0.997245 + 0.0741759i \(0.0236326\pi\)
−0.997245 + 0.0741759i \(0.976367\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) − 46.0000i − 1.69905i −0.527549 0.849524i \(-0.676889\pi\)
0.527549 0.849524i \(-0.323111\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 24.0000 0.881662
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 0 0
\(753\) 8.00000i 0.291536i
\(754\) 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 \(-0.138933\pi\)
0.906249 + 0.422744i \(0.138933\pi\)
\(762\) 0 0
\(763\) 40.0000i 1.44810i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) − 38.0000i − 1.36677i −0.730061 0.683383i \(-0.760508\pi\)
0.730061 0.683383i \(-0.239492\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 24.0000i − 0.860995i
\(778\) 0 0
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 6.00000i − 0.214423i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 12.0000i 0.427754i 0.976861 + 0.213877i \(0.0686091\pi\)
−0.976861 + 0.213877i \(0.931391\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) 36.0000i 1.27840i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 34.0000i − 1.20434i −0.798367 0.602171i \(-0.794303\pi\)
0.798367 0.602171i \(-0.205697\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.00000i 0.211210i
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) 8.00000i 0.280572i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 16.0000i − 0.559769i
\(818\) 0 0
\(819\) 24.0000 0.838628
\(820\) 0 0
\(821\) −50.0000 −1.74501 −0.872506 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 0 0
\(823\) − 4.00000i − 0.139431i −0.997567 0.0697156i \(-0.977791\pi\)
0.997567 0.0697156i \(-0.0222092\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 20.0000i − 0.695468i −0.937593 0.347734i \(-0.886951\pi\)
0.937593 0.347734i \(-0.113049\pi\)
\(828\) 0 0
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 18.0000i 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) − 2.00000i − 0.0688837i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 44.0000i 1.51186i
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 48.0000 1.64542
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 38.0000i 1.29806i 0.760765 + 0.649028i \(0.224824\pi\)
−0.760765 + 0.649028i \(0.775176\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) −40.0000 −1.36320
\(862\) 0 0
\(863\) 16.0000i 0.544646i 0.962206 + 0.272323i \(0.0877920\pi\)
−0.962206 + 0.272323i \(0.912208\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) − 2.00000i − 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 10.0000i − 0.337676i −0.985644 0.168838i \(-0.945999\pi\)
0.985644 0.168838i \(-0.0540015\pi\)
\(878\) 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 \(-0.843844\pi\)
0.882060 0.471138i \(-0.156156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 24.0000i − 0.805841i −0.915235 0.402921i \(-0.867995\pi\)
0.915235 0.402921i \(-0.132005\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 32.0000i 1.07084i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 48.0000i 1.60267i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 20.0000 0.666297
\(902\) 0 0
\(903\) − 16.0000i − 0.532447i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 28.0000i 0.929725i 0.885383 + 0.464862i \(0.153896\pi\)
−0.885383 + 0.464862i \(0.846104\pi\)
\(908\) 0 0
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 64.0000i − 2.11347i
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 4.00000i − 0.131377i
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) 0 0
\(933\) 16.0000i 0.523816i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 26.0000i 0.849383i 0.905338 + 0.424691i \(0.139617\pi\)
−0.905338 + 0.424691i \(0.860383\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) 0 0
\(943\) − 80.0000i − 2.60516i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 44.0000i − 1.42981i −0.699223 0.714904i \(-0.746470\pi\)
0.699223 0.714904i \(-0.253530\pi\)
\(948\) 0 0
\(949\) 84.0000 2.72676
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) 18.0000i 0.583077i 0.956559 + 0.291539i \(0.0941672\pi\)
−0.956559 + 0.291539i \(0.905833\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 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
\(965\) 0 0
\(966\) 0 0
\(967\) − 20.0000i − 0.643157i −0.946883 0.321578i \(-0.895787\pi\)
0.946883 0.321578i \(-0.104213\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) −56.0000 −1.79713 −0.898563 0.438845i \(-0.855388\pi\)
−0.898563 + 0.438845i \(0.855388\pi\)
\(972\) 0 0
\(973\) − 48.0000i − 1.53881i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 50.0000i − 1.59964i −0.600239 0.799821i \(-0.704928\pi\)
0.600239 0.799821i \(-0.295072\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 32.0000i 1.01857i
\(988\) 0 0
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) −56.0000 −1.77890 −0.889449 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) 0 0
\(993\) − 28.0000i − 0.888553i
\(994\) 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 4800.2.f.n.3649.1 2
4.3 odd 2 4800.2.f.u.3649.2 2
5.2 odd 4 4800.2.a.bh.1.1 1
5.3 odd 4 960.2.a.n.1.1 1
5.4 even 2 inner 4800.2.f.n.3649.2 2
8.3 odd 2 600.2.f.c.49.1 2
8.5 even 2 1200.2.f.f.49.2 2
15.8 even 4 2880.2.a.b.1.1 1
20.3 even 4 960.2.a.g.1.1 1
20.7 even 4 4800.2.a.bl.1.1 1
20.19 odd 2 4800.2.f.u.3649.1 2
24.5 odd 2 3600.2.f.l.2449.1 2
24.11 even 2 1800.2.f.g.649.2 2
40.3 even 4 120.2.a.a.1.1 1
40.13 odd 4 240.2.a.a.1.1 1
40.19 odd 2 600.2.f.c.49.2 2
40.27 even 4 600.2.a.a.1.1 1
40.29 even 2 1200.2.f.f.49.1 2
40.37 odd 4 1200.2.a.r.1.1 1
60.23 odd 4 2880.2.a.r.1.1 1
80.3 even 4 3840.2.k.a.1921.1 2
80.13 odd 4 3840.2.k.z.1921.2 2
80.43 even 4 3840.2.k.a.1921.2 2
80.53 odd 4 3840.2.k.z.1921.1 2
120.29 odd 2 3600.2.f.l.2449.2 2
120.53 even 4 720.2.a.f.1.1 1
120.59 even 2 1800.2.f.g.649.1 2
120.77 even 4 3600.2.a.bo.1.1 1
120.83 odd 4 360.2.a.e.1.1 1
120.107 odd 4 1800.2.a.c.1.1 1
280.83 odd 4 5880.2.a.p.1.1 1
360.43 even 12 3240.2.q.m.1081.1 2
360.83 odd 12 3240.2.q.a.1081.1 2
360.203 odd 12 3240.2.q.a.2161.1 2
360.283 even 12 3240.2.q.m.2161.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.a.a.1.1 1 40.3 even 4
240.2.a.a.1.1 1 40.13 odd 4
360.2.a.e.1.1 1 120.83 odd 4
600.2.a.a.1.1 1 40.27 even 4
600.2.f.c.49.1 2 8.3 odd 2
600.2.f.c.49.2 2 40.19 odd 2
720.2.a.f.1.1 1 120.53 even 4
960.2.a.g.1.1 1 20.3 even 4
960.2.a.n.1.1 1 5.3 odd 4
1200.2.a.r.1.1 1 40.37 odd 4
1200.2.f.f.49.1 2 40.29 even 2
1200.2.f.f.49.2 2 8.5 even 2
1800.2.a.c.1.1 1 120.107 odd 4
1800.2.f.g.649.1 2 120.59 even 2
1800.2.f.g.649.2 2 24.11 even 2
2880.2.a.b.1.1 1 15.8 even 4
2880.2.a.r.1.1 1 60.23 odd 4
3240.2.q.a.1081.1 2 360.83 odd 12
3240.2.q.a.2161.1 2 360.203 odd 12
3240.2.q.m.1081.1 2 360.43 even 12
3240.2.q.m.2161.1 2 360.283 even 12
3600.2.a.bo.1.1 1 120.77 even 4
3600.2.f.l.2449.1 2 24.5 odd 2
3600.2.f.l.2449.2 2 120.29 odd 2
3840.2.k.a.1921.1 2 80.3 even 4
3840.2.k.a.1921.2 2 80.43 even 4
3840.2.k.z.1921.1 2 80.53 odd 4
3840.2.k.z.1921.2 2 80.13 odd 4
4800.2.a.bh.1.1 1 5.2 odd 4
4800.2.a.bl.1.1 1 20.7 even 4
4800.2.f.n.3649.1 2 1.1 even 1 trivial
4800.2.f.n.3649.2 2 5.4 even 2 inner
4800.2.f.u.3649.1 2 20.19 odd 2
4800.2.f.u.3649.2 2 4.3 odd 2
5880.2.a.p.1.1 1 280.83 odd 4