Properties

Label 5376.2.c.bf.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 672)
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.bf.2689.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +4.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +4.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} -2.00000i q^{11} -2.00000i q^{13} +4.00000 q^{15} +4.00000i q^{19} -1.00000i q^{21} -6.00000 q^{23} -11.0000 q^{25} +1.00000i q^{27} -10.0000i q^{29} +8.00000 q^{31} -2.00000 q^{33} +4.00000i q^{35} -10.0000i q^{37} -2.00000 q^{39} +4.00000 q^{41} -8.00000i q^{43} -4.00000i q^{45} +4.00000 q^{47} +1.00000 q^{49} -10.0000i q^{53} +8.00000 q^{55} +4.00000 q^{57} +8.00000i q^{59} -6.00000i q^{61} -1.00000 q^{63} +8.00000 q^{65} -4.00000i q^{67} +6.00000i q^{69} +14.0000 q^{71} -6.00000 q^{73} +11.0000i q^{75} -2.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -10.0000 q^{87} -4.00000 q^{89} -2.00000i q^{91} -8.00000i q^{93} -16.0000 q^{95} -2.00000 q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{7} - 2 q^{9} + 8 q^{15} - 12 q^{23} - 22 q^{25} + 16 q^{31} - 4 q^{33} - 4 q^{39} + 8 q^{41} + 8 q^{47} + 2 q^{49} + 16 q^{55} + 8 q^{57} - 2 q^{63} + 16 q^{65} + 28 q^{71} - 12 q^{73} - 8 q^{79} + 2 q^{81} - 20 q^{87} - 8 q^{89} - 32 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 4.00000i 1.78885i 0.447214 + 0.894427i \(0.352416\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 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
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −11.0000 −2.20000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) − 10.0000i − 1.85695i −0.371391 0.928477i \(-0.621119\pi\)
0.371391 0.928477i \(-0.378881\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 4.00000i 0.676123i
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) − 4.00000i − 0.596285i
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 8.00000i 1.04151i 0.853706 + 0.520756i \(0.174350\pi\)
−0.853706 + 0.520756i \(0.825650\pi\)
\(60\) 0 0
\(61\) − 6.00000i − 0.768221i −0.923287 0.384111i \(-0.874508\pi\)
0.923287 0.384111i \(-0.125492\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 6.00000i 0.722315i
\(70\) 0 0
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 11.0000i 1.27017i
\(76\) 0 0
\(77\) − 2.00000i − 0.227921i
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 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
\(85\) 0 0
\(86\) 0 0
\(87\) −10.0000 −1.07211
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) − 2.00000i − 0.209657i
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) −16.0000 −1.64157
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 2.00000i 0.201008i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) − 10.0000i − 0.966736i −0.875417 0.483368i \(-0.839413\pi\)
0.875417 0.483368i \(-0.160587\pi\)
\(108\) 0 0
\(109\) 14.0000i 1.34096i 0.741929 + 0.670478i \(0.233911\pi\)
−0.741929 + 0.670478i \(0.766089\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) − 24.0000i − 2.23801i
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) − 4.00000i − 0.360668i
\(124\) 0 0
\(125\) − 24.0000i − 2.14663i
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 20.0000i 1.74741i 0.486458 + 0.873704i \(0.338289\pi\)
−0.486458 + 0.873704i \(0.661711\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 4.00000i 0.339276i 0.985506 + 0.169638i \(0.0542598\pi\)
−0.985506 + 0.169638i \(0.945740\pi\)
\(140\) 0 0
\(141\) − 4.00000i − 0.336861i
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 40.0000 3.32182
\(146\) 0 0
\(147\) − 1.00000i − 0.0824786i
\(148\) 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
\(153\) 0 0
\(154\) 0 0
\(155\) 32.0000i 2.57030i
\(156\) 0 0
\(157\) − 22.0000i − 1.75579i −0.478852 0.877896i \(-0.658947\pi\)
0.478852 0.877896i \(-0.341053\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) 0 0
\(165\) − 8.00000i − 0.622799i
\(166\) 0 0
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) − 4.00000i − 0.305888i
\(172\) 0 0
\(173\) 24.0000i 1.82469i 0.409426 + 0.912343i \(0.365729\pi\)
−0.409426 + 0.912343i \(0.634271\pi\)
\(174\) 0 0
\(175\) −11.0000 −0.831522
\(176\) 0 0
\(177\) 8.00000 0.601317
\(178\) 0 0
\(179\) 6.00000i 0.448461i 0.974536 + 0.224231i \(0.0719869\pi\)
−0.974536 + 0.224231i \(0.928013\pi\)
\(180\) 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
\(185\) 40.0000 2.94086
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000i 0.0727393i
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) − 8.00000i − 0.572892i
\(196\) 0 0
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) − 10.0000i − 0.701862i
\(204\) 0 0
\(205\) 16.0000i 1.11749i
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) − 8.00000i − 0.550743i −0.961338 0.275371i \(-0.911199\pi\)
0.961338 0.275371i \(-0.0888008\pi\)
\(212\) 0 0
\(213\) − 14.0000i − 0.959264i
\(214\) 0 0
\(215\) 32.0000 2.18238
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) 6.00000i 0.405442i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) − 14.0000i − 0.925146i −0.886581 0.462573i \(-0.846926\pi\)
0.886581 0.462573i \(-0.153074\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 16.0000i 1.04372i
\(236\) 0 0
\(237\) 4.00000i 0.259828i
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 4.00000i 0.255551i
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) − 24.0000i − 1.51487i −0.652913 0.757433i \(-0.726453\pi\)
0.652913 0.757433i \(-0.273547\pi\)
\(252\) 0 0
\(253\) 12.0000i 0.754434i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.00000 −0.249513 −0.124757 0.992187i \(-0.539815\pi\)
−0.124757 + 0.992187i \(0.539815\pi\)
\(258\) 0 0
\(259\) − 10.0000i − 0.621370i
\(260\) 0 0
\(261\) 10.0000i 0.618984i
\(262\) 0 0
\(263\) −30.0000 −1.84988 −0.924940 0.380114i \(-0.875885\pi\)
−0.924940 + 0.380114i \(0.875885\pi\)
\(264\) 0 0
\(265\) 40.0000 2.45718
\(266\) 0 0
\(267\) 4.00000i 0.244796i
\(268\) 0 0
\(269\) 12.0000i 0.731653i 0.930683 + 0.365826i \(0.119214\pi\)
−0.930683 + 0.365826i \(0.880786\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) 22.0000i 1.32665i
\(276\) 0 0
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) − 20.0000i − 1.18888i −0.804141 0.594438i \(-0.797374\pi\)
0.804141 0.594438i \(-0.202626\pi\)
\(284\) 0 0
\(285\) 16.0000i 0.947758i
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 2.00000i 0.117242i
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) −32.0000 −1.86311
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) 12.0000i 0.693978i
\(300\) 0 0
\(301\) − 8.00000i − 0.461112i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 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
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) −20.0000 −1.11979
\(320\) 0 0
\(321\) −10.0000 −0.558146
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 22.0000i 1.22034i
\(326\) 0 0
\(327\) 14.0000 0.774202
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) − 32.0000i − 1.75888i −0.476011 0.879440i \(-0.657918\pi\)
0.476011 0.879440i \(-0.342082\pi\)
\(332\) 0 0
\(333\) 10.0000i 0.547997i
\(334\) 0 0
\(335\) 16.0000 0.874173
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) − 2.00000i − 0.108625i
\(340\) 0 0
\(341\) − 16.0000i − 0.866449i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −24.0000 −1.29212
\(346\) 0 0
\(347\) 2.00000i 0.107366i 0.998558 + 0.0536828i \(0.0170960\pi\)
−0.998558 + 0.0536828i \(0.982904\pi\)
\(348\) 0 0
\(349\) − 14.0000i − 0.749403i −0.927146 0.374701i \(-0.877745\pi\)
0.927146 0.374701i \(-0.122255\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −8.00000 −0.425797 −0.212899 0.977074i \(-0.568290\pi\)
−0.212899 + 0.977074i \(0.568290\pi\)
\(354\) 0 0
\(355\) 56.0000i 2.97217i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 22.0000 1.16112 0.580558 0.814219i \(-0.302835\pi\)
0.580558 + 0.814219i \(0.302835\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) − 7.00000i − 0.367405i
\(364\) 0 0
\(365\) − 24.0000i − 1.25622i
\(366\) 0 0
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 0 0
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) − 10.0000i − 0.519174i
\(372\) 0 0
\(373\) − 14.0000i − 0.724893i −0.932005 0.362446i \(-0.881942\pi\)
0.932005 0.362446i \(-0.118058\pi\)
\(374\) 0 0
\(375\) −24.0000 −1.23935
\(376\) 0 0
\(377\) −20.0000 −1.03005
\(378\) 0 0
\(379\) 32.0000i 1.64373i 0.569683 + 0.821865i \(0.307066\pi\)
−0.569683 + 0.821865i \(0.692934\pi\)
\(380\) 0 0
\(381\) − 20.0000i − 1.02463i
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) 0 0
\(387\) 8.00000i 0.406663i
\(388\) 0 0
\(389\) − 30.0000i − 1.52106i −0.649303 0.760530i \(-0.724939\pi\)
0.649303 0.760530i \(-0.275061\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 20.0000 1.00887
\(394\) 0 0
\(395\) − 16.0000i − 0.805047i
\(396\) 0 0
\(397\) − 38.0000i − 1.90717i −0.301131 0.953583i \(-0.597364\pi\)
0.301131 0.953583i \(-0.402636\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) − 16.0000i − 0.797017i
\(404\) 0 0
\(405\) 4.00000i 0.198762i
\(406\) 0 0
\(407\) −20.0000 −0.991363
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) 6.00000i 0.295958i
\(412\) 0 0
\(413\) 8.00000i 0.393654i
\(414\) 0 0
\(415\) −48.0000 −2.35623
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) − 16.0000i − 0.781651i −0.920465 0.390826i \(-0.872190\pi\)
0.920465 0.390826i \(-0.127810\pi\)
\(420\) 0 0
\(421\) 22.0000i 1.07221i 0.844150 + 0.536107i \(0.180106\pi\)
−0.844150 + 0.536107i \(0.819894\pi\)
\(422\) 0 0
\(423\) −4.00000 −0.194487
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 6.00000i − 0.290360i
\(428\) 0 0
\(429\) 4.00000i 0.193122i
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) − 40.0000i − 1.91785i
\(436\) 0 0
\(437\) − 24.0000i − 1.14808i
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) − 6.00000i − 0.285069i −0.989790 0.142534i \(-0.954475\pi\)
0.989790 0.142534i \(-0.0455251\pi\)
\(444\) 0 0
\(445\) − 16.0000i − 0.758473i
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) − 8.00000i − 0.376705i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 0 0
\(465\) 32.0000 1.48396
\(466\) 0 0
\(467\) 8.00000i 0.370196i 0.982720 + 0.185098i \(0.0592602\pi\)
−0.982720 + 0.185098i \(0.940740\pi\)
\(468\) 0 0
\(469\) − 4.00000i − 0.184703i
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) − 44.0000i − 2.01886i
\(476\) 0 0
\(477\) 10.0000i 0.457869i
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) 0 0
\(483\) 6.00000i 0.273009i
\(484\) 0 0
\(485\) − 8.00000i − 0.363261i
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) 10.0000i 0.451294i 0.974209 + 0.225647i \(0.0724495\pi\)
−0.974209 + 0.225647i \(0.927550\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −8.00000 −0.359573
\(496\) 0 0
\(497\) 14.0000 0.627986
\(498\) 0 0
\(499\) − 24.0000i − 1.07439i −0.843459 0.537194i \(-0.819484\pi\)
0.843459 0.537194i \(-0.180516\pi\)
\(500\) 0 0
\(501\) − 20.0000i − 0.893534i
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 9.00000i − 0.399704i
\(508\) 0 0
\(509\) − 36.0000i − 1.59567i −0.602875 0.797836i \(-0.705978\pi\)
0.602875 0.797836i \(-0.294022\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 8.00000i − 0.351840i
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) − 36.0000i − 1.57417i −0.616844 0.787085i \(-0.711589\pi\)
0.616844 0.787085i \(-0.288411\pi\)
\(524\) 0 0
\(525\) 11.0000i 0.480079i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) − 8.00000i − 0.347170i
\(532\) 0 0
\(533\) − 8.00000i − 0.346518i
\(534\) 0 0
\(535\) 40.0000 1.72935
\(536\) 0 0
\(537\) 6.00000 0.258919
\(538\) 0 0
\(539\) − 2.00000i − 0.0861461i
\(540\) 0 0
\(541\) − 14.0000i − 0.601907i −0.953639 0.300954i \(-0.902695\pi\)
0.953639 0.300954i \(-0.0973049\pi\)
\(542\) 0 0
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) −56.0000 −2.39878
\(546\) 0 0
\(547\) − 12.0000i − 0.513083i −0.966533 0.256541i \(-0.917417\pi\)
0.966533 0.256541i \(-0.0825830\pi\)
\(548\) 0 0
\(549\) 6.00000i 0.256074i
\(550\) 0 0
\(551\) 40.0000 1.70406
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 0 0
\(555\) − 40.0000i − 1.69791i
\(556\) 0 0
\(557\) − 22.0000i − 0.932170i −0.884740 0.466085i \(-0.845664\pi\)
0.884740 0.466085i \(-0.154336\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.0000i 1.01148i 0.862686 + 0.505740i \(0.168780\pi\)
−0.862686 + 0.505740i \(0.831220\pi\)
\(564\) 0 0
\(565\) 8.00000i 0.336563i
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 0 0
\(571\) 20.0000i 0.836974i 0.908223 + 0.418487i \(0.137439\pi\)
−0.908223 + 0.418487i \(0.862561\pi\)
\(572\) 0 0
\(573\) 6.00000i 0.250654i
\(574\) 0 0
\(575\) 66.0000 2.75239
\(576\) 0 0
\(577\) −6.00000 −0.249783 −0.124892 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 0 0
\(579\) − 6.00000i − 0.249351i
\(580\) 0 0
\(581\) 12.0000i 0.497844i
\(582\) 0 0
\(583\) −20.0000 −0.828315
\(584\) 0 0
\(585\) −8.00000 −0.330759
\(586\) 0 0
\(587\) 48.0000i 1.98117i 0.136892 + 0.990586i \(0.456289\pi\)
−0.136892 + 0.990586i \(0.543711\pi\)
\(588\) 0 0
\(589\) 32.0000i 1.31854i
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 0 0
\(593\) −24.0000 −0.985562 −0.492781 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.00000i 0.327418i
\(598\) 0 0
\(599\) −10.0000 −0.408589 −0.204294 0.978909i \(-0.565490\pi\)
−0.204294 + 0.978909i \(0.565490\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) 28.0000i 1.13836i
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) −10.0000 −0.405220
\(610\) 0 0
\(611\) − 8.00000i − 0.323645i
\(612\) 0 0
\(613\) − 6.00000i − 0.242338i −0.992632 0.121169i \(-0.961336\pi\)
0.992632 0.121169i \(-0.0386643\pi\)
\(614\) 0 0
\(615\) 16.0000 0.645182
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) 36.0000i 1.44696i 0.690344 + 0.723481i \(0.257459\pi\)
−0.690344 + 0.723481i \(0.742541\pi\)
\(620\) 0 0
\(621\) − 6.00000i − 0.240772i
\(622\) 0 0
\(623\) −4.00000 −0.160257
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) − 8.00000i − 0.319489i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) −8.00000 −0.317971
\(634\) 0 0
\(635\) 80.0000i 3.17470i
\(636\) 0 0
\(637\) − 2.00000i − 0.0792429i
\(638\) 0 0
\(639\) −14.0000 −0.553831
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) 12.0000i 0.473234i 0.971603 + 0.236617i \(0.0760386\pi\)
−0.971603 + 0.236617i \(0.923961\pi\)
\(644\) 0 0
\(645\) − 32.0000i − 1.26000i
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) − 8.00000i − 0.313545i
\(652\) 0 0
\(653\) − 26.0000i − 1.01746i −0.860927 0.508729i \(-0.830115\pi\)
0.860927 0.508729i \(-0.169885\pi\)
\(654\) 0 0
\(655\) −80.0000 −3.12586
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) − 6.00000i − 0.233727i −0.993148 0.116863i \(-0.962716\pi\)
0.993148 0.116863i \(-0.0372840\pi\)
\(660\) 0 0
\(661\) 22.0000i 0.855701i 0.903850 + 0.427850i \(0.140729\pi\)
−0.903850 + 0.427850i \(0.859271\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −16.0000 −0.620453
\(666\) 0 0
\(667\) 60.0000i 2.32321i
\(668\) 0 0
\(669\) − 16.0000i − 0.618596i
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) − 11.0000i − 0.423390i
\(676\) 0 0
\(677\) − 16.0000i − 0.614930i −0.951559 0.307465i \(-0.900519\pi\)
0.951559 0.307465i \(-0.0994807\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 18.0000i 0.688751i 0.938832 + 0.344375i \(0.111909\pi\)
−0.938832 + 0.344375i \(0.888091\pi\)
\(684\) 0 0
\(685\) − 24.0000i − 0.916993i
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 0 0
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) 28.0000i 1.06517i 0.846376 + 0.532585i \(0.178779\pi\)
−0.846376 + 0.532585i \(0.821221\pi\)
\(692\) 0 0
\(693\) 2.00000i 0.0759737i
\(694\) 0 0
\(695\) −16.0000 −0.606915
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) − 18.0000i − 0.680823i
\(700\) 0 0
\(701\) − 18.0000i − 0.679851i −0.940452 0.339925i \(-0.889598\pi\)
0.940452 0.339925i \(-0.110402\pi\)
\(702\) 0 0
\(703\) 40.0000 1.50863
\(704\) 0 0
\(705\) 16.0000 0.602595
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 26.0000i 0.976450i 0.872718 + 0.488225i \(0.162356\pi\)
−0.872718 + 0.488225i \(0.837644\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) −48.0000 −1.79761
\(714\) 0 0
\(715\) − 16.0000i − 0.598366i
\(716\) 0 0
\(717\) − 6.00000i − 0.224074i
\(718\) 0 0
\(719\) −32.0000 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 2.00000i 0.0743808i
\(724\) 0 0
\(725\) 110.000i 4.08530i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 38.0000i 1.40356i 0.712393 + 0.701781i \(0.247612\pi\)
−0.712393 + 0.701781i \(0.752388\pi\)
\(734\) 0 0
\(735\) 4.00000 0.147542
\(736\) 0 0
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) − 20.0000i − 0.735712i −0.929883 0.367856i \(-0.880092\pi\)
0.929883 0.367856i \(-0.119908\pi\)
\(740\) 0 0
\(741\) − 8.00000i − 0.293887i
\(742\) 0 0
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 0 0
\(745\) −24.0000 −0.879292
\(746\) 0 0
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) − 10.0000i − 0.365392i
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) −24.0000 −0.874609
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 42.0000i 1.52652i 0.646094 + 0.763258i \(0.276401\pi\)
−0.646094 + 0.763258i \(0.723599\pi\)
\(758\) 0 0
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) 14.0000i 0.506834i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 16.0000 0.577727
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 4.00000i 0.144056i
\(772\) 0 0
\(773\) − 24.0000i − 0.863220i −0.902060 0.431610i \(-0.857946\pi\)
0.902060 0.431610i \(-0.142054\pi\)
\(774\) 0 0
\(775\) −88.0000 −3.16105
\(776\) 0 0
\(777\) −10.0000 −0.358748
\(778\) 0 0
\(779\) 16.0000i 0.573259i
\(780\) 0 0
\(781\) − 28.0000i − 1.00192i
\(782\) 0 0
\(783\) 10.0000 0.357371
\(784\) 0 0
\(785\) 88.0000 3.14085
\(786\) 0 0
\(787\) − 36.0000i − 1.28326i −0.767014 0.641631i \(-0.778258\pi\)
0.767014 0.641631i \(-0.221742\pi\)
\(788\) 0 0
\(789\) 30.0000i 1.06803i
\(790\) 0 0
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 0 0
\(795\) − 40.0000i − 1.41865i
\(796\) 0 0
\(797\) 24.0000i 0.850124i 0.905164 + 0.425062i \(0.139748\pi\)
−0.905164 + 0.425062i \(0.860252\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) 0 0
\(803\) 12.0000i 0.423471i
\(804\) 0 0
\(805\) − 24.0000i − 0.845889i
\(806\) 0 0
\(807\) 12.0000 0.422420
\(808\) 0 0
\(809\) 14.0000 0.492214 0.246107 0.969243i \(-0.420849\pi\)
0.246107 + 0.969243i \(0.420849\pi\)
\(810\) 0 0
\(811\) 20.0000i 0.702295i 0.936320 + 0.351147i \(0.114208\pi\)
−0.936320 + 0.351147i \(0.885792\pi\)
\(812\) 0 0
\(813\) − 24.0000i − 0.841717i
\(814\) 0 0
\(815\) 80.0000 2.80228
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 0 0
\(819\) 2.00000i 0.0698857i
\(820\) 0 0
\(821\) − 2.00000i − 0.0698005i −0.999391 0.0349002i \(-0.988889\pi\)
0.999391 0.0349002i \(-0.0111113\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 22.0000 0.765942
\(826\) 0 0
\(827\) − 26.0000i − 0.904109i −0.891990 0.452054i \(-0.850691\pi\)
0.891990 0.452054i \(-0.149309\pi\)
\(828\) 0 0
\(829\) 14.0000i 0.486240i 0.969996 + 0.243120i \(0.0781709\pi\)
−0.969996 + 0.243120i \(0.921829\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 80.0000i 2.76851i
\(836\) 0 0
\(837\) 8.00000i 0.276520i
\(838\) 0 0
\(839\) 44.0000 1.51905 0.759524 0.650479i \(-0.225432\pi\)
0.759524 + 0.650479i \(0.225432\pi\)
\(840\) 0 0
\(841\) −71.0000 −2.44828
\(842\) 0 0
\(843\) 6.00000i 0.206651i
\(844\) 0 0
\(845\) 36.0000i 1.23844i
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 60.0000i 2.05677i
\(852\) 0 0
\(853\) − 42.0000i − 1.43805i −0.694983 0.719026i \(-0.744588\pi\)
0.694983 0.719026i \(-0.255412\pi\)
\(854\) 0 0
\(855\) 16.0000 0.547188
\(856\) 0 0
\(857\) −28.0000 −0.956462 −0.478231 0.878234i \(-0.658722\pi\)
−0.478231 + 0.878234i \(0.658722\pi\)
\(858\) 0 0
\(859\) − 20.0000i − 0.682391i −0.939992 0.341196i \(-0.889168\pi\)
0.939992 0.341196i \(-0.110832\pi\)
\(860\) 0 0
\(861\) − 4.00000i − 0.136320i
\(862\) 0 0
\(863\) 22.0000 0.748889 0.374444 0.927249i \(-0.377833\pi\)
0.374444 + 0.927249i \(0.377833\pi\)
\(864\) 0 0
\(865\) −96.0000 −3.26410
\(866\) 0 0
\(867\) 17.0000i 0.577350i
\(868\) 0 0
\(869\) 8.00000i 0.271381i
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) − 24.0000i − 0.811348i
\(876\) 0 0
\(877\) − 50.0000i − 1.68838i −0.536044 0.844190i \(-0.680082\pi\)
0.536044 0.844190i \(-0.319918\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 48.0000 1.61716 0.808581 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) 0 0
\(883\) 8.00000i 0.269221i 0.990899 + 0.134611i \(0.0429784\pi\)
−0.990899 + 0.134611i \(0.957022\pi\)
\(884\) 0 0
\(885\) 32.0000i 1.07567i
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 20.0000 0.670778
\(890\) 0 0
\(891\) − 2.00000i − 0.0670025i
\(892\) 0 0
\(893\) 16.0000i 0.535420i
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) 0 0
\(899\) − 80.0000i − 2.66815i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) 0 0
\(905\) 56.0000 1.86150
\(906\) 0 0
\(907\) 32.0000i 1.06254i 0.847202 + 0.531271i \(0.178286\pi\)
−0.847202 + 0.531271i \(0.821714\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 0 0
\(913\) 24.0000 0.794284
\(914\) 0 0
\(915\) − 24.0000i − 0.793416i
\(916\) 0 0
\(917\) 20.0000i 0.660458i
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) − 28.0000i − 0.921631i
\(924\) 0 0
\(925\) 110.000i 3.61678i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −36.0000 −1.18112 −0.590561 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) 0 0
\(931\) 4.00000i 0.131095i
\(932\) 0 0
\(933\) 12.0000i 0.392862i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) − 14.0000i − 0.456873i
\(940\) 0 0
\(941\) 4.00000i 0.130396i 0.997872 + 0.0651981i \(0.0207679\pi\)
−0.997872 + 0.0651981i \(0.979232\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) 18.0000i 0.584921i 0.956278 + 0.292461i \(0.0944741\pi\)
−0.956278 + 0.292461i \(0.905526\pi\)
\(948\) 0 0
\(949\) 12.0000i 0.389536i
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) − 24.0000i − 0.776622i
\(956\) 0 0
\(957\) 20.0000i 0.646508i
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 10.0000i 0.322245i
\(964\) 0 0
\(965\) 24.0000i 0.772587i
\(966\) 0 0
\(967\) 12.0000 0.385894 0.192947 0.981209i \(-0.438195\pi\)
0.192947 + 0.981209i \(0.438195\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 20.0000i − 0.641831i −0.947108 0.320915i \(-0.896010\pi\)
0.947108 0.320915i \(-0.103990\pi\)
\(972\) 0 0
\(973\) 4.00000i 0.128234i
\(974\) 0 0
\(975\) 22.0000 0.704564
\(976\) 0 0
\(977\) 14.0000 0.447900 0.223950 0.974601i \(-0.428105\pi\)
0.223950 + 0.974601i \(0.428105\pi\)
\(978\) 0 0
\(979\) 8.00000i 0.255681i
\(980\) 0 0
\(981\) − 14.0000i − 0.446986i
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 8.00000 0.254901
\(986\) 0 0
\(987\) − 4.00000i − 0.127321i
\(988\) 0 0
\(989\) 48.0000i 1.52631i
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −32.0000 −1.01549
\(994\) 0 0
\(995\) − 32.0000i − 1.01447i
\(996\) 0 0
\(997\) 6.00000i 0.190022i 0.995476 + 0.0950110i \(0.0302886\pi\)
−0.995476 + 0.0950110i \(0.969711\pi\)
\(998\) 0 0
\(999\) 10.0000 0.316386
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.bf.2689.1 2
4.3 odd 2 5376.2.c.b.2689.2 2
8.3 odd 2 5376.2.c.b.2689.1 2
8.5 even 2 inner 5376.2.c.bf.2689.2 2
16.3 odd 4 1344.2.a.j.1.1 1
16.5 even 4 672.2.a.a.1.1 1
16.11 odd 4 672.2.a.e.1.1 yes 1
16.13 even 4 1344.2.a.t.1.1 1
48.5 odd 4 2016.2.a.m.1.1 1
48.11 even 4 2016.2.a.n.1.1 1
48.29 odd 4 4032.2.a.b.1.1 1
48.35 even 4 4032.2.a.c.1.1 1
112.13 odd 4 9408.2.a.c.1.1 1
112.27 even 4 4704.2.a.q.1.1 1
112.69 odd 4 4704.2.a.bg.1.1 1
112.83 even 4 9408.2.a.bq.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.a.a.1.1 1 16.5 even 4
672.2.a.e.1.1 yes 1 16.11 odd 4
1344.2.a.j.1.1 1 16.3 odd 4
1344.2.a.t.1.1 1 16.13 even 4
2016.2.a.m.1.1 1 48.5 odd 4
2016.2.a.n.1.1 1 48.11 even 4
4032.2.a.b.1.1 1 48.29 odd 4
4032.2.a.c.1.1 1 48.35 even 4
4704.2.a.q.1.1 1 112.27 even 4
4704.2.a.bg.1.1 1 112.69 odd 4
5376.2.c.b.2689.1 2 8.3 odd 2
5376.2.c.b.2689.2 2 4.3 odd 2
5376.2.c.bf.2689.1 2 1.1 even 1 trivial
5376.2.c.bf.2689.2 2 8.5 even 2 inner
9408.2.a.c.1.1 1 112.13 odd 4
9408.2.a.bq.1.1 1 112.83 even 4