Properties

Label 5376.2.c.bb.2689.1
Level $5376$
Weight $2$
Character 5376.2689
Analytic conductor $42.928$
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] = [5376,2,Mod(2689,5376)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5376, 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("5376.2689");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5376.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: \(42.9275761266\)
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 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2689.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5376.2689
Dual form 5376.2.c.bb.2689.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} -6.00000i q^{13} +2.00000 q^{15} -2.00000 q^{17} +4.00000i q^{19} -1.00000i q^{21} -4.00000 q^{23} +1.00000 q^{25} +1.00000i q^{27} +10.0000i q^{29} +8.00000 q^{31} +2.00000i q^{35} +6.00000i q^{37} -6.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} -2.00000i q^{45} -8.00000 q^{47} +1.00000 q^{49} +2.00000i q^{51} -10.0000i q^{53} +4.00000 q^{57} -12.0000i q^{59} +2.00000i q^{61} -1.00000 q^{63} +12.0000 q^{65} +12.0000i q^{67} +4.00000i q^{69} -12.0000 q^{71} +14.0000 q^{73} -1.00000i q^{75} +8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -4.00000i q^{85} +10.0000 q^{87} +2.00000 q^{89} -6.00000i q^{91} -8.00000i q^{93} -8.00000 q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{7} - 2 q^{9} + 4 q^{15} - 4 q^{17} - 8 q^{23} + 2 q^{25} + 16 q^{31} - 12 q^{39} + 4 q^{41} - 16 q^{47} + 2 q^{49} + 8 q^{57} - 2 q^{63} + 24 q^{65} - 24 q^{71} + 28 q^{73} + 16 q^{79} + 2 q^{81} + 20 q^{87} + 4 q^{89} - 16 q^{95} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times\).

\(n\) \(1793\) \(2815\) \(4609\) \(5125\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 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
\(15\) 2.00000 0.516398
\(16\) 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
\(21\) − 1.00000i − 0.218218i
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 10.0000i 1.85695i 0.371391 + 0.928477i \(0.378881\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000i 0.338062i
\(36\) 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
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) − 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
\(49\) 1.00000 0.142857
\(50\) 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
\(59\) − 12.0000i − 1.56227i −0.624364 0.781133i \(-0.714642\pi\)
0.624364 0.781133i \(-0.285358\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i 0.991769 + 0.128037i \(0.0408676\pi\)
−0.991769 + 0.128037i \(0.959132\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 0 0
\(69\) 4.00000i 0.481543i
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 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
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 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
\(85\) − 4.00000i − 0.433861i
\(86\) 0 0
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) − 6.00000i − 0.628971i
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 6.00000i − 0.597022i −0.954406 0.298511i \(-0.903510\pi\)
0.954406 0.298511i \(-0.0964900\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 2.00000i 0.191565i 0.995402 + 0.0957826i \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) − 8.00000i − 0.746004i
\(116\) 0 0
\(117\) 6.00000i 0.554700i
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) − 2.00000i − 0.180334i
\(124\) 0 0
\(125\) 12.0000i 1.07331i
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) − 4.00000i − 0.349482i −0.984614 0.174741i \(-0.944091\pi\)
0.984614 0.174741i \(-0.0559088\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 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
\(145\) −20.0000 −1.66091
\(146\) 0 0
\(147\) − 1.00000i − 0.0824786i
\(148\) 0 0
\(149\) 22.0000i 1.80231i 0.433497 + 0.901155i \(0.357280\pi\)
−0.433497 + 0.901155i \(0.642720\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 16.0000i 1.28515i
\(156\) 0 0
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) − 4.00000i − 0.305888i
\(172\) 0 0
\(173\) − 2.00000i − 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) − 16.0000i − 1.19590i −0.801535 0.597948i \(-0.795983\pi\)
0.801535 0.597948i \(-0.204017\pi\)
\(180\) 0 0
\(181\) 14.0000i 1.04061i 0.853980 + 0.520306i \(0.174182\pi\)
−0.853980 + 0.520306i \(0.825818\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000i 0.0727393i
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 0 0
\(195\) − 12.0000i − 0.859338i
\(196\) 0 0
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 10.0000i 0.701862i
\(204\) 0 0
\(205\) 4.00000i 0.279372i
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000i 1.37686i 0.725304 + 0.688428i \(0.241699\pi\)
−0.725304 + 0.688428i \(0.758301\pi\)
\(212\) 0 0
\(213\) 12.0000i 0.822226i
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) − 14.0000i − 0.946032i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 0 0
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.00000 −0.131024 −0.0655122 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 0 0
\(235\) − 16.0000i − 1.04372i
\(236\) 0 0
\(237\) − 8.00000i − 0.519656i
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 2.00000i 0.127775i
\(246\) 0 0
\(247\) 24.0000 1.52708
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 20.0000i 1.26239i 0.775625 + 0.631194i \(0.217435\pi\)
−0.775625 + 0.631194i \(0.782565\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 6.00000i 0.372822i
\(260\) 0 0
\(261\) − 10.0000i − 0.618984i
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 20.0000 1.22859
\(266\) 0 0
\(267\) − 2.00000i − 0.122398i
\(268\) 0 0
\(269\) 6.00000i 0.365826i 0.983129 + 0.182913i \(0.0585527\pi\)
−0.983129 + 0.182913i \(0.941447\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) −6.00000 −0.363137
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000i 1.32185i 0.750451 + 0.660926i \(0.229836\pi\)
−0.750451 + 0.660926i \(0.770164\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 0 0
\(285\) 8.00000i 0.473879i
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) − 10.0000i − 0.586210i
\(292\) 0 0
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 24.0000i 1.38796i
\(300\) 0 0
\(301\) 4.00000i 0.230556i
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 0 0
\(315\) − 2.00000i − 0.112687i
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 8.00000i − 0.445132i
\(324\) 0 0
\(325\) − 6.00000i − 0.332820i
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 4.00000i 0.219860i 0.993939 + 0.109930i \(0.0350627\pi\)
−0.993939 + 0.109930i \(0.964937\pi\)
\(332\) 0 0
\(333\) − 6.00000i − 0.328798i
\(334\) 0 0
\(335\) −24.0000 −1.31126
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) − 18.0000i − 0.977626i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −8.00000 −0.430706
\(346\) 0 0
\(347\) − 24.0000i − 1.28839i −0.764862 0.644194i \(-0.777193\pi\)
0.764862 0.644194i \(-0.222807\pi\)
\(348\) 0 0
\(349\) − 14.0000i − 0.749403i −0.927146 0.374701i \(-0.877745\pi\)
0.927146 0.374701i \(-0.122255\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 0 0
\(355\) − 24.0000i − 1.27379i
\(356\) 0 0
\(357\) 2.00000i 0.105851i
\(358\) 0 0
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) − 11.0000i − 0.577350i
\(364\) 0 0
\(365\) 28.0000i 1.46559i
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) − 10.0000i − 0.519174i
\(372\) 0 0
\(373\) − 10.0000i − 0.517780i −0.965907 0.258890i \(-0.916643\pi\)
0.965907 0.258890i \(-0.0833568\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 60.0000 3.09016
\(378\) 0 0
\(379\) − 12.0000i − 0.616399i −0.951322 0.308199i \(-0.900274\pi\)
0.951322 0.308199i \(-0.0997264\pi\)
\(380\) 0 0
\(381\) 8.00000i 0.409852i
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 4.00000i − 0.203331i
\(388\) 0 0
\(389\) − 2.00000i − 0.101404i −0.998714 0.0507020i \(-0.983854\pi\)
0.998714 0.0507020i \(-0.0161459\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 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
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) − 48.0000i − 2.39105i
\(404\) 0 0
\(405\) 2.00000i 0.0993808i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 2.00000i 0.0986527i
\(412\) 0 0
\(413\) − 12.0000i − 0.590481i
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 20.0000i 0.977064i 0.872546 + 0.488532i \(0.162467\pi\)
−0.872546 + 0.488532i \(0.837533\pi\)
\(420\) 0 0
\(421\) − 26.0000i − 1.26716i −0.773676 0.633581i \(-0.781584\pi\)
0.773676 0.633581i \(-0.218416\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 2.00000i 0.0967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 20.0000i 0.958927i
\(436\) 0 0
\(437\) − 16.0000i − 0.765384i
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) − 8.00000i − 0.380091i −0.981775 0.190046i \(-0.939136\pi\)
0.981775 0.190046i \(-0.0608636\pi\)
\(444\) 0 0
\(445\) 4.00000i 0.189618i
\(446\) 0 0
\(447\) 22.0000 1.04056
\(448\) 0 0
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 8.00000i 0.375873i
\(454\) 0 0
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) − 2.00000i − 0.0933520i
\(460\) 0 0
\(461\) − 18.0000i − 0.838344i −0.907907 0.419172i \(-0.862320\pi\)
0.907907 0.419172i \(-0.137680\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 0 0
\(465\) 16.0000 0.741982
\(466\) 0 0
\(467\) 36.0000i 1.66588i 0.553362 + 0.832941i \(0.313345\pi\)
−0.553362 + 0.832941i \(0.686655\pi\)
\(468\) 0 0
\(469\) 12.0000i 0.554109i
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000i 0.183533i
\(476\) 0 0
\(477\) 10.0000i 0.457869i
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 36.0000 1.64146
\(482\) 0 0
\(483\) 4.00000i 0.182006i
\(484\) 0 0
\(485\) 20.0000i 0.908153i
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) 24.0000i 1.08310i 0.840667 + 0.541552i \(0.182163\pi\)
−0.840667 + 0.541552i \(0.817837\pi\)
\(492\) 0 0
\(493\) − 20.0000i − 0.900755i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 4.00000i 0.179065i 0.995984 + 0.0895323i \(0.0285372\pi\)
−0.995984 + 0.0895323i \(0.971463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 23.0000i 1.02147i
\(508\) 0 0
\(509\) − 42.0000i − 1.86162i −0.365507 0.930809i \(-0.619104\pi\)
0.365507 0.930809i \(-0.380896\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) 0 0
\(525\) − 1.00000i − 0.0436436i
\(526\) 0 0
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 12.0000i 0.520756i
\(532\) 0 0
\(533\) − 12.0000i − 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −16.0000 −0.690451
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.00000i 0.0859867i 0.999075 + 0.0429934i \(0.0136894\pi\)
−0.999075 + 0.0429934i \(0.986311\pi\)
\(542\) 0 0
\(543\) 14.0000 0.600798
\(544\) 0 0
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 0 0
\(549\) − 2.00000i − 0.0853579i
\(550\) 0 0
\(551\) −40.0000 −1.70406
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 12.0000i 0.509372i
\(556\) 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
\(561\) 0 0
\(562\) 0 0
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 0 0
\(565\) 36.0000i 1.51453i
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) − 20.0000i − 0.836974i −0.908223 0.418487i \(-0.862561\pi\)
0.908223 0.418487i \(-0.137439\pi\)
\(572\) 0 0
\(573\) − 20.0000i − 0.835512i
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 0 0
\(579\) − 18.0000i − 0.748054i
\(580\) 0 0
\(581\) 12.0000i 0.497844i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −12.0000 −0.496139
\(586\) 0 0
\(587\) − 28.0000i − 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) 0 0
\(589\) 32.0000i 1.31854i
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) − 4.00000i − 0.163984i
\(596\) 0 0
\(597\) − 8.00000i − 0.327418i
\(598\) 0 0
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) − 12.0000i − 0.488678i
\(604\) 0 0
\(605\) 22.0000i 0.894427i
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 0 0
\(609\) 10.0000 0.405220
\(610\) 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
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 12.0000i 0.482321i 0.970485 + 0.241160i \(0.0775280\pi\)
−0.970485 + 0.241160i \(0.922472\pi\)
\(620\) 0 0
\(621\) − 4.00000i − 0.160514i
\(622\) 0 0
\(623\) 2.00000 0.0801283
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 12.0000i − 0.478471i
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 0 0
\(635\) − 16.0000i − 0.634941i
\(636\) 0 0
\(637\) − 6.00000i − 0.237729i
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 0 0
\(645\) 8.00000i 0.315000i
\(646\) 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
\(651\) − 8.00000i − 0.313545i
\(652\) 0 0
\(653\) 10.0000i 0.391330i 0.980671 + 0.195665i \(0.0626866\pi\)
−0.980671 + 0.195665i \(0.937313\pi\)
\(654\) 0 0
\(655\) 8.00000 0.312586
\(656\) 0 0
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) 8.00000i 0.311636i 0.987786 + 0.155818i \(0.0498013\pi\)
−0.987786 + 0.155818i \(0.950199\pi\)
\(660\) 0 0
\(661\) − 10.0000i − 0.388955i −0.980907 0.194477i \(-0.937699\pi\)
0.980907 0.194477i \(-0.0623011\pi\)
\(662\) 0 0
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) − 40.0000i − 1.54881i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 0 0
\(685\) − 4.00000i − 0.152832i
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 0 0
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) 44.0000i 1.67384i 0.547326 + 0.836919i \(0.315646\pi\)
−0.547326 + 0.836919i \(0.684354\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −24.0000 −0.910372
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0 0
\(699\) 2.00000i 0.0756469i
\(700\) 0 0
\(701\) − 22.0000i − 0.830929i −0.909610 0.415464i \(-0.863619\pi\)
0.909610 0.415464i \(-0.136381\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) −16.0000 −0.602595
\(706\) 0 0
\(707\) − 6.00000i − 0.225653i
\(708\) 0 0
\(709\) 22.0000i 0.826227i 0.910679 + 0.413114i \(0.135559\pi\)
−0.910679 + 0.413114i \(0.864441\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 20.0000i 0.746914i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 10.0000i − 0.371904i
\(724\) 0 0
\(725\) 10.0000i 0.371391i
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 8.00000i − 0.295891i
\(732\) 0 0
\(733\) 26.0000i 0.960332i 0.877178 + 0.480166i \(0.159424\pi\)
−0.877178 + 0.480166i \(0.840576\pi\)
\(734\) 0 0
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 20.0000i − 0.735712i −0.929883 0.367856i \(-0.880092\pi\)
0.929883 0.367856i \(-0.119908\pi\)
\(740\) 0 0
\(741\) − 24.0000i − 0.881662i
\(742\) 0 0
\(743\) 20.0000 0.733729 0.366864 0.930274i \(-0.380431\pi\)
0.366864 + 0.930274i \(0.380431\pi\)
\(744\) 0 0
\(745\) −44.0000 −1.61204
\(746\) 0 0
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) 20.0000 0.728841
\(754\) 0 0
\(755\) − 16.0000i − 0.582300i
\(756\) 0 0
\(757\) − 26.0000i − 0.944986i −0.881334 0.472493i \(-0.843354\pi\)
0.881334 0.472493i \(-0.156646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 2.00000i 0.0724049i
\(764\) 0 0
\(765\) 4.00000i 0.144620i
\(766\) 0 0
\(767\) −72.0000 −2.59977
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) − 6.00000i − 0.216085i
\(772\) 0 0
\(773\) 10.0000i 0.359675i 0.983696 + 0.179838i \(0.0575572\pi\)
−0.983696 + 0.179838i \(0.942443\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 0 0
\(779\) 8.00000i 0.286630i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −10.0000 −0.357371
\(784\) 0 0
\(785\) −36.0000 −1.28490
\(786\) 0 0
\(787\) 28.0000i 0.998092i 0.866575 + 0.499046i \(0.166316\pi\)
−0.866575 + 0.499046i \(0.833684\pi\)
\(788\) 0 0
\(789\) − 12.0000i − 0.427211i
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 0 0
\(795\) − 20.0000i − 0.709327i
\(796\) 0 0
\(797\) − 18.0000i − 0.637593i −0.947823 0.318796i \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) − 8.00000i − 0.281963i
\(806\) 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
\(811\) − 36.0000i − 1.26413i −0.774915 0.632065i \(-0.782207\pi\)
0.774915 0.632065i \(-0.217793\pi\)
\(812\) 0 0
\(813\) 8.00000i 0.280572i
\(814\) 0 0
\(815\) 40.0000 1.40114
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 6.00000i 0.209657i
\(820\) 0 0
\(821\) 6.00000i 0.209401i 0.994504 + 0.104701i \(0.0333885\pi\)
−0.994504 + 0.104701i \(0.966612\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) − 38.0000i − 1.31979i −0.751356 0.659897i \(-0.770600\pi\)
0.751356 0.659897i \(-0.229400\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000i 0.276520i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −71.0000 −2.44828
\(842\) 0 0
\(843\) 18.0000i 0.619953i
\(844\) 0 0
\(845\) − 46.0000i − 1.58245i
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) − 24.0000i − 0.822709i
\(852\) 0 0
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 20.0000i 0.682391i 0.939992 + 0.341196i \(0.110832\pi\)
−0.939992 + 0.341196i \(0.889168\pi\)
\(860\) 0 0
\(861\) − 2.00000i − 0.0681598i
\(862\) 0 0
\(863\) −20.0000 −0.680808 −0.340404 0.940279i \(-0.610564\pi\)
−0.340404 + 0.940279i \(0.610564\pi\)
\(864\) 0 0
\(865\) 4.00000 0.136004
\(866\) 0 0
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 72.0000 2.43963
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 12.0000i 0.405674i
\(876\) 0 0
\(877\) − 14.0000i − 0.472746i −0.971662 0.236373i \(-0.924041\pi\)
0.971662 0.236373i \(-0.0759588\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 4.00000i 0.134611i 0.997732 + 0.0673054i \(0.0214402\pi\)
−0.997732 + 0.0673054i \(0.978560\pi\)
\(884\) 0 0
\(885\) − 24.0000i − 0.806751i
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 32.0000i − 1.07084i
\(894\) 0 0
\(895\) 32.0000 1.06964
\(896\) 0 0
\(897\) 24.0000 0.801337
\(898\) 0 0
\(899\) 80.0000i 2.66815i
\(900\) 0 0
\(901\) 20.0000i 0.666297i
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) −28.0000 −0.930751
\(906\) 0 0
\(907\) − 44.0000i − 1.46100i −0.682915 0.730498i \(-0.739288\pi\)
0.682915 0.730498i \(-0.260712\pi\)
\(908\) 0 0
\(909\) 6.00000i 0.199007i
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 4.00000i 0.132236i
\(916\) 0 0
\(917\) − 4.00000i − 0.132092i
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) 72.0000i 2.36991i
\(924\) 0 0
\(925\) 6.00000i 0.197279i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 0 0
\(931\) 4.00000i 0.131095i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 54.0000 1.76410 0.882052 0.471153i \(-0.156162\pi\)
0.882052 + 0.471153i \(0.156162\pi\)
\(938\) 0 0
\(939\) − 22.0000i − 0.717943i
\(940\) 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
\(945\) −2.00000 −0.0650600
\(946\) 0 0
\(947\) − 56.0000i − 1.81976i −0.414876 0.909878i \(-0.636175\pi\)
0.414876 0.909878i \(-0.363825\pi\)
\(948\) 0 0
\(949\) − 84.0000i − 2.72676i
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −58.0000 −1.87880 −0.939402 0.342817i \(-0.888619\pi\)
−0.939402 + 0.342817i \(0.888619\pi\)
\(954\) 0 0
\(955\) 40.0000i 1.29437i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 36.0000i 1.15888i
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) − 4.00000i − 0.128366i −0.997938 0.0641831i \(-0.979556\pi\)
0.997938 0.0641831i \(-0.0204442\pi\)
\(972\) 0 0
\(973\) 12.0000i 0.384702i
\(974\) 0 0
\(975\) −6.00000 −0.192154
\(976\) 0 0
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) − 2.00000i − 0.0638551i
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) −12.0000 −0.382352
\(986\) 0 0
\(987\) 8.00000i 0.254643i
\(988\) 0 0
\(989\) − 16.0000i − 0.508770i
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 16.0000i 0.507234i
\(996\) 0 0
\(997\) − 26.0000i − 0.823428i −0.911313 0.411714i \(-0.864930\pi\)
0.911313 0.411714i \(-0.135070\pi\)
\(998\) 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 5376.2.c.bb.2689.1 2
4.3 odd 2 5376.2.c.d.2689.2 2
8.3 odd 2 5376.2.c.d.2689.1 2
8.5 even 2 inner 5376.2.c.bb.2689.2 2
16.3 odd 4 168.2.a.a.1.1 1
16.5 even 4 1344.2.a.b.1.1 1
16.11 odd 4 1344.2.a.m.1.1 1
16.13 even 4 336.2.a.e.1.1 1
48.5 odd 4 4032.2.a.bc.1.1 1
48.11 even 4 4032.2.a.bh.1.1 1
48.29 odd 4 1008.2.a.b.1.1 1
48.35 even 4 504.2.a.e.1.1 1
80.3 even 4 4200.2.t.j.1849.1 2
80.19 odd 4 4200.2.a.t.1.1 1
80.29 even 4 8400.2.a.y.1.1 1
80.67 even 4 4200.2.t.j.1849.2 2
112.3 even 12 1176.2.q.d.961.1 2
112.13 odd 4 2352.2.a.c.1.1 1
112.19 even 12 1176.2.q.d.361.1 2
112.27 even 4 9408.2.a.be.1.1 1
112.45 odd 12 2352.2.q.w.961.1 2
112.51 odd 12 1176.2.q.f.361.1 2
112.61 odd 12 2352.2.q.w.1537.1 2
112.67 odd 12 1176.2.q.f.961.1 2
112.69 odd 4 9408.2.a.da.1.1 1
112.83 even 4 1176.2.a.f.1.1 1
112.93 even 12 2352.2.q.d.1537.1 2
112.109 even 12 2352.2.q.d.961.1 2
336.83 odd 4 3528.2.a.v.1.1 1
336.125 even 4 7056.2.a.bq.1.1 1
336.131 odd 12 3528.2.s.g.361.1 2
336.179 even 12 3528.2.s.w.3313.1 2
336.227 odd 12 3528.2.s.g.3313.1 2
336.275 even 12 3528.2.s.w.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.a.a.1.1 1 16.3 odd 4
336.2.a.e.1.1 1 16.13 even 4
504.2.a.e.1.1 1 48.35 even 4
1008.2.a.b.1.1 1 48.29 odd 4
1176.2.a.f.1.1 1 112.83 even 4
1176.2.q.d.361.1 2 112.19 even 12
1176.2.q.d.961.1 2 112.3 even 12
1176.2.q.f.361.1 2 112.51 odd 12
1176.2.q.f.961.1 2 112.67 odd 12
1344.2.a.b.1.1 1 16.5 even 4
1344.2.a.m.1.1 1 16.11 odd 4
2352.2.a.c.1.1 1 112.13 odd 4
2352.2.q.d.961.1 2 112.109 even 12
2352.2.q.d.1537.1 2 112.93 even 12
2352.2.q.w.961.1 2 112.45 odd 12
2352.2.q.w.1537.1 2 112.61 odd 12
3528.2.a.v.1.1 1 336.83 odd 4
3528.2.s.g.361.1 2 336.131 odd 12
3528.2.s.g.3313.1 2 336.227 odd 12
3528.2.s.w.361.1 2 336.275 even 12
3528.2.s.w.3313.1 2 336.179 even 12
4032.2.a.bc.1.1 1 48.5 odd 4
4032.2.a.bh.1.1 1 48.11 even 4
4200.2.a.t.1.1 1 80.19 odd 4
4200.2.t.j.1849.1 2 80.3 even 4
4200.2.t.j.1849.2 2 80.67 even 4
5376.2.c.d.2689.1 2 8.3 odd 2
5376.2.c.d.2689.2 2 4.3 odd 2
5376.2.c.bb.2689.1 2 1.1 even 1 trivial
5376.2.c.bb.2689.2 2 8.5 even 2 inner
7056.2.a.bq.1.1 1 336.125 even 4
8400.2.a.y.1.1 1 80.29 even 4
9408.2.a.be.1.1 1 112.27 even 4
9408.2.a.da.1.1 1 112.69 odd 4