Properties

Label 1728.2.c.f.1727.7
Level $1728$
Weight $2$
Character 1728.1727
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,2,Mod(1727,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1728.1727");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.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: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1727.7
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.2.c.f.1727.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.86370i q^{5} -3.73205i q^{7} +O(q^{10})\) \(q+3.86370i q^{5} -3.73205i q^{7} +1.03528 q^{11} -4.46410 q^{13} +1.79315i q^{17} -1.73205i q^{19} -8.76268 q^{23} -9.92820 q^{25} +7.72741i q^{29} +7.46410i q^{31} +14.4195 q^{35} -0.464102 q^{37} -7.72741i q^{41} -0.535898i q^{43} +4.62158 q^{47} -6.92820 q^{49} -3.58630i q^{53} +4.00000i q^{55} -12.3490 q^{59} -11.3923 q^{61} -17.2480i q^{65} +6.26795i q^{67} -11.3137 q^{71} -3.92820 q^{73} -3.86370i q^{77} -4.80385i q^{79} +2.07055 q^{83} -6.92820 q^{85} +1.79315i q^{89} +16.6603i q^{91} +6.69213 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{13} - 24 q^{25} + 24 q^{37} - 8 q^{61} + 24 q^{73} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3.86370i 1.72790i 0.503577 + 0.863950i \(0.332017\pi\)
−0.503577 + 0.863950i \(0.667983\pi\)
\(6\) 0 0
\(7\) − 3.73205i − 1.41058i −0.708918 0.705291i \(-0.750816\pi\)
0.708918 0.705291i \(-0.249184\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.03528 0.312148 0.156074 0.987745i \(-0.450116\pi\)
0.156074 + 0.987745i \(0.450116\pi\)
\(12\) 0 0
\(13\) −4.46410 −1.23812 −0.619060 0.785344i \(-0.712486\pi\)
−0.619060 + 0.785344i \(0.712486\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.79315i 0.434903i 0.976071 + 0.217451i \(0.0697744\pi\)
−0.976071 + 0.217451i \(0.930226\pi\)
\(18\) 0 0
\(19\) − 1.73205i − 0.397360i −0.980064 0.198680i \(-0.936335\pi\)
0.980064 0.198680i \(-0.0636654\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.76268 −1.82715 −0.913573 0.406675i \(-0.866688\pi\)
−0.913573 + 0.406675i \(0.866688\pi\)
\(24\) 0 0
\(25\) −9.92820 −1.98564
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.72741i 1.43494i 0.696588 + 0.717472i \(0.254701\pi\)
−0.696588 + 0.717472i \(0.745299\pi\)
\(30\) 0 0
\(31\) 7.46410i 1.34059i 0.742094 + 0.670296i \(0.233833\pi\)
−0.742094 + 0.670296i \(0.766167\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 14.4195 2.43735
\(36\) 0 0
\(37\) −0.464102 −0.0762978 −0.0381489 0.999272i \(-0.512146\pi\)
−0.0381489 + 0.999272i \(0.512146\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 7.72741i − 1.20682i −0.797432 0.603409i \(-0.793809\pi\)
0.797432 0.603409i \(-0.206191\pi\)
\(42\) 0 0
\(43\) − 0.535898i − 0.0817237i −0.999165 0.0408619i \(-0.986990\pi\)
0.999165 0.0408619i \(-0.0130104\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.62158 0.674126 0.337063 0.941482i \(-0.390566\pi\)
0.337063 + 0.941482i \(0.390566\pi\)
\(48\) 0 0
\(49\) −6.92820 −0.989743
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 3.58630i − 0.492616i −0.969192 0.246308i \(-0.920782\pi\)
0.969192 0.246308i \(-0.0792175\pi\)
\(54\) 0 0
\(55\) 4.00000i 0.539360i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.3490 −1.60770 −0.803850 0.594831i \(-0.797219\pi\)
−0.803850 + 0.594831i \(0.797219\pi\)
\(60\) 0 0
\(61\) −11.3923 −1.45864 −0.729318 0.684175i \(-0.760162\pi\)
−0.729318 + 0.684175i \(0.760162\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 17.2480i − 2.13935i
\(66\) 0 0
\(67\) 6.26795i 0.765752i 0.923800 + 0.382876i \(0.125066\pi\)
−0.923800 + 0.382876i \(0.874934\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.3137 −1.34269 −0.671345 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(72\) 0 0
\(73\) −3.92820 −0.459761 −0.229881 0.973219i \(-0.573834\pi\)
−0.229881 + 0.973219i \(0.573834\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.86370i − 0.440310i
\(78\) 0 0
\(79\) − 4.80385i − 0.540475i −0.962794 0.270238i \(-0.912898\pi\)
0.962794 0.270238i \(-0.0871022\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.07055 0.227273 0.113636 0.993522i \(-0.463750\pi\)
0.113636 + 0.993522i \(0.463750\pi\)
\(84\) 0 0
\(85\) −6.92820 −0.751469
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.79315i 0.190074i 0.995474 + 0.0950368i \(0.0302969\pi\)
−0.995474 + 0.0950368i \(0.969703\pi\)
\(90\) 0 0
\(91\) 16.6603i 1.74647i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.69213 0.686598
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 4.14110i − 0.412055i −0.978546 0.206028i \(-0.933946\pi\)
0.978546 0.206028i \(-0.0660537\pi\)
\(102\) 0 0
\(103\) − 13.5885i − 1.33891i −0.742852 0.669455i \(-0.766528\pi\)
0.742852 0.669455i \(-0.233472\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.2784 0.993654 0.496827 0.867850i \(-0.334498\pi\)
0.496827 + 0.867850i \(0.334498\pi\)
\(108\) 0 0
\(109\) −12.9282 −1.23830 −0.619149 0.785274i \(-0.712522\pi\)
−0.619149 + 0.785274i \(0.712522\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.93426i 0.558248i 0.960255 + 0.279124i \(0.0900440\pi\)
−0.960255 + 0.279124i \(0.909956\pi\)
\(114\) 0 0
\(115\) − 33.8564i − 3.15713i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.69213 0.613467
\(120\) 0 0
\(121\) −9.92820 −0.902564
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 19.0411i − 1.70309i
\(126\) 0 0
\(127\) 3.46410i 0.307389i 0.988118 + 0.153695i \(0.0491172\pi\)
−0.988118 + 0.153695i \(0.950883\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.07055 −0.180905 −0.0904525 0.995901i \(-0.528831\pi\)
−0.0904525 + 0.995901i \(0.528831\pi\)
\(132\) 0 0
\(133\) −6.46410 −0.560509
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.6617i 1.16719i 0.812043 + 0.583597i \(0.198355\pi\)
−0.812043 + 0.583597i \(0.801645\pi\)
\(138\) 0 0
\(139\) 9.19615i 0.780007i 0.920813 + 0.390004i \(0.127526\pi\)
−0.920813 + 0.390004i \(0.872474\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.62158 −0.386476
\(144\) 0 0
\(145\) −29.8564 −2.47944
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 11.3137i 0.926855i 0.886135 + 0.463428i \(0.153381\pi\)
−0.886135 + 0.463428i \(0.846619\pi\)
\(150\) 0 0
\(151\) − 11.7321i − 0.954741i −0.878702 0.477370i \(-0.841590\pi\)
0.878702 0.477370i \(-0.158410\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −28.8391 −2.31641
\(156\) 0 0
\(157\) 16.9282 1.35102 0.675509 0.737352i \(-0.263924\pi\)
0.675509 + 0.737352i \(0.263924\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 32.7028i 2.57734i
\(162\) 0 0
\(163\) − 15.5885i − 1.22098i −0.792023 0.610491i \(-0.790972\pi\)
0.792023 0.610491i \(-0.209028\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.1469 1.71378 0.856891 0.515498i \(-0.172393\pi\)
0.856891 + 0.515498i \(0.172393\pi\)
\(168\) 0 0
\(169\) 6.92820 0.532939
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 19.0411i 1.44767i 0.689974 + 0.723835i \(0.257622\pi\)
−0.689974 + 0.723835i \(0.742378\pi\)
\(174\) 0 0
\(175\) 37.0526i 2.80091i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −19.5959 −1.46467 −0.732334 0.680946i \(-0.761569\pi\)
−0.732334 + 0.680946i \(0.761569\pi\)
\(180\) 0 0
\(181\) −0.464102 −0.0344964 −0.0172482 0.999851i \(-0.505491\pi\)
−0.0172482 + 0.999851i \(0.505491\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 1.79315i − 0.131835i
\(186\) 0 0
\(187\) 1.85641i 0.135754i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.8332 0.783865 0.391933 0.919994i \(-0.371807\pi\)
0.391933 + 0.919994i \(0.371807\pi\)
\(192\) 0 0
\(193\) 13.9282 1.00257 0.501287 0.865281i \(-0.332860\pi\)
0.501287 + 0.865281i \(0.332860\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 0.277401i − 0.0197640i −0.999951 0.00988202i \(-0.996854\pi\)
0.999951 0.00988202i \(-0.00314559\pi\)
\(198\) 0 0
\(199\) − 5.58846i − 0.396155i −0.980186 0.198078i \(-0.936530\pi\)
0.980186 0.198078i \(-0.0634698\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 28.8391 2.02411
\(204\) 0 0
\(205\) 29.8564 2.08526
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 1.79315i − 0.124035i
\(210\) 0 0
\(211\) 18.2679i 1.25762i 0.777560 + 0.628809i \(0.216457\pi\)
−0.777560 + 0.628809i \(0.783543\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.07055 0.141210
\(216\) 0 0
\(217\) 27.8564 1.89102
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 8.00481i − 0.538462i
\(222\) 0 0
\(223\) − 16.5359i − 1.10733i −0.832741 0.553663i \(-0.813230\pi\)
0.832741 0.553663i \(-0.186770\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.4548 −1.02577 −0.512886 0.858457i \(-0.671424\pi\)
−0.512886 + 0.858457i \(0.671424\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 11.3137i 0.741186i 0.928795 + 0.370593i \(0.120845\pi\)
−0.928795 + 0.370593i \(0.879155\pi\)
\(234\) 0 0
\(235\) 17.8564i 1.16482i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.14110 0.267866 0.133933 0.990990i \(-0.457239\pi\)
0.133933 + 0.990990i \(0.457239\pi\)
\(240\) 0 0
\(241\) −7.92820 −0.510700 −0.255350 0.966849i \(-0.582191\pi\)
−0.255350 + 0.966849i \(0.582191\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 26.7685i − 1.71018i
\(246\) 0 0
\(247\) 7.73205i 0.491979i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −13.3843 −0.844807 −0.422404 0.906408i \(-0.638813\pi\)
−0.422404 + 0.906408i \(0.638813\pi\)
\(252\) 0 0
\(253\) −9.07180 −0.570339
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.72741i 0.482022i 0.970522 + 0.241011i \(0.0774791\pi\)
−0.970522 + 0.241011i \(0.922521\pi\)
\(258\) 0 0
\(259\) 1.73205i 0.107624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4.14110 −0.255351 −0.127676 0.991816i \(-0.540752\pi\)
−0.127676 + 0.991816i \(0.540752\pi\)
\(264\) 0 0
\(265\) 13.8564 0.851192
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 11.5911i − 0.706722i −0.935487 0.353361i \(-0.885039\pi\)
0.935487 0.353361i \(-0.114961\pi\)
\(270\) 0 0
\(271\) 25.0526i 1.52183i 0.648849 + 0.760917i \(0.275251\pi\)
−0.648849 + 0.760917i \(0.724749\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −10.2784 −0.619813
\(276\) 0 0
\(277\) −7.07180 −0.424903 −0.212452 0.977172i \(-0.568145\pi\)
−0.212452 + 0.977172i \(0.568145\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 11.8685i 0.708016i 0.935242 + 0.354008i \(0.115181\pi\)
−0.935242 + 0.354008i \(0.884819\pi\)
\(282\) 0 0
\(283\) 7.46410i 0.443695i 0.975081 + 0.221847i \(0.0712087\pi\)
−0.975081 + 0.221847i \(0.928791\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −28.8391 −1.70232
\(288\) 0 0
\(289\) 13.7846 0.810859
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 3.30890i − 0.193308i −0.995318 0.0966540i \(-0.969186\pi\)
0.995318 0.0966540i \(-0.0308140\pi\)
\(294\) 0 0
\(295\) − 47.7128i − 2.77795i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 39.1175 2.26222
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 44.0165i − 2.52038i
\(306\) 0 0
\(307\) 1.60770i 0.0917560i 0.998947 + 0.0458780i \(0.0146086\pi\)
−0.998947 + 0.0458780i \(0.985391\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.2175 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(312\) 0 0
\(313\) −6.07180 −0.343198 −0.171599 0.985167i \(-0.554893\pi\)
−0.171599 + 0.985167i \(0.554893\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.72741i 0.434014i 0.976170 + 0.217007i \(0.0696295\pi\)
−0.976170 + 0.217007i \(0.930370\pi\)
\(318\) 0 0
\(319\) 8.00000i 0.447914i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.10583 0.172813
\(324\) 0 0
\(325\) 44.3205 2.45846
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 17.2480i − 0.950911i
\(330\) 0 0
\(331\) − 14.8038i − 0.813693i −0.913497 0.406847i \(-0.866628\pi\)
0.913497 0.406847i \(-0.133372\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −24.2175 −1.32314
\(336\) 0 0
\(337\) 12.8564 0.700333 0.350167 0.936687i \(-0.386125\pi\)
0.350167 + 0.936687i \(0.386125\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 7.72741i 0.418463i
\(342\) 0 0
\(343\) − 0.267949i − 0.0144679i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.14110 −0.222306 −0.111153 0.993803i \(-0.535454\pi\)
−0.111153 + 0.993803i \(0.535454\pi\)
\(348\) 0 0
\(349\) 24.3205 1.30185 0.650923 0.759143i \(-0.274382\pi\)
0.650923 + 0.759143i \(0.274382\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 23.7370i − 1.26339i −0.775215 0.631697i \(-0.782359\pi\)
0.775215 0.631697i \(-0.217641\pi\)
\(354\) 0 0
\(355\) − 43.7128i − 2.32004i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.2490 1.43815 0.719073 0.694934i \(-0.244566\pi\)
0.719073 + 0.694934i \(0.244566\pi\)
\(360\) 0 0
\(361\) 16.0000 0.842105
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 15.1774i − 0.794422i
\(366\) 0 0
\(367\) 21.0526i 1.09893i 0.835515 + 0.549467i \(0.185169\pi\)
−0.835515 + 0.549467i \(0.814831\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −13.3843 −0.694876
\(372\) 0 0
\(373\) 35.2487 1.82511 0.912555 0.408955i \(-0.134107\pi\)
0.912555 + 0.408955i \(0.134107\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 34.4959i − 1.77663i
\(378\) 0 0
\(379\) 1.19615i 0.0614422i 0.999528 + 0.0307211i \(0.00978037\pi\)
−0.999528 + 0.0307211i \(0.990220\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −23.7370 −1.21291 −0.606453 0.795120i \(-0.707408\pi\)
−0.606453 + 0.795120i \(0.707408\pi\)
\(384\) 0 0
\(385\) 14.9282 0.760812
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0.277401i 0.0140648i 0.999975 + 0.00703241i \(0.00223850\pi\)
−0.999975 + 0.00703241i \(0.997761\pi\)
\(390\) 0 0
\(391\) − 15.7128i − 0.794631i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 18.5606 0.933887
\(396\) 0 0
\(397\) 14.7846 0.742018 0.371009 0.928629i \(-0.379012\pi\)
0.371009 + 0.928629i \(0.379012\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.6274i 1.12996i 0.825105 + 0.564980i \(0.191116\pi\)
−0.825105 + 0.564980i \(0.808884\pi\)
\(402\) 0 0
\(403\) − 33.3205i − 1.65981i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.480473 −0.0238162
\(408\) 0 0
\(409\) −13.0000 −0.642809 −0.321404 0.946942i \(-0.604155\pi\)
−0.321404 + 0.946942i \(0.604155\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 46.0870i 2.26779i
\(414\) 0 0
\(415\) 8.00000i 0.392705i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.5606 0.906747 0.453373 0.891321i \(-0.350220\pi\)
0.453373 + 0.891321i \(0.350220\pi\)
\(420\) 0 0
\(421\) −16.4641 −0.802411 −0.401206 0.915988i \(-0.631409\pi\)
−0.401206 + 0.915988i \(0.631409\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 17.8028i − 0.863561i
\(426\) 0 0
\(427\) 42.5167i 2.05753i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0764 0.967046 0.483523 0.875332i \(-0.339357\pi\)
0.483523 + 0.875332i \(0.339357\pi\)
\(432\) 0 0
\(433\) 14.7846 0.710503 0.355251 0.934771i \(-0.384395\pi\)
0.355251 + 0.934771i \(0.384395\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 15.1774i 0.726034i
\(438\) 0 0
\(439\) 13.3205i 0.635753i 0.948132 + 0.317877i \(0.102970\pi\)
−0.948132 + 0.317877i \(0.897030\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 21.6665 1.02941 0.514703 0.857369i \(-0.327902\pi\)
0.514703 + 0.857369i \(0.327902\pi\)
\(444\) 0 0
\(445\) −6.92820 −0.328428
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 21.3891i 1.00941i 0.863291 + 0.504706i \(0.168399\pi\)
−0.863291 + 0.504706i \(0.831601\pi\)
\(450\) 0 0
\(451\) − 8.00000i − 0.376705i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −64.3703 −3.01773
\(456\) 0 0
\(457\) 6.78461 0.317371 0.158685 0.987329i \(-0.449274\pi\)
0.158685 + 0.987329i \(0.449274\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 11.0363i − 0.514012i −0.966410 0.257006i \(-0.917264\pi\)
0.966410 0.257006i \(-0.0827360\pi\)
\(462\) 0 0
\(463\) − 20.8038i − 0.966837i −0.875389 0.483418i \(-0.839395\pi\)
0.875389 0.483418i \(-0.160605\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.06678 −0.188188 −0.0940940 0.995563i \(-0.529995\pi\)
−0.0940940 + 0.995563i \(0.529995\pi\)
\(468\) 0 0
\(469\) 23.3923 1.08016
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 0.554803i − 0.0255099i
\(474\) 0 0
\(475\) 17.1962i 0.789014i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −24.6980 −1.12848 −0.564240 0.825611i \(-0.690831\pi\)
−0.564240 + 0.825611i \(0.690831\pi\)
\(480\) 0 0
\(481\) 2.07180 0.0944658
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 27.0459i 1.22809i
\(486\) 0 0
\(487\) − 35.4449i − 1.60616i −0.595871 0.803080i \(-0.703193\pi\)
0.595871 0.803080i \(-0.296807\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −18.5606 −0.837630 −0.418815 0.908072i \(-0.637554\pi\)
−0.418815 + 0.908072i \(0.637554\pi\)
\(492\) 0 0
\(493\) −13.8564 −0.624061
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 42.2233i 1.89398i
\(498\) 0 0
\(499\) 19.4641i 0.871333i 0.900108 + 0.435666i \(0.143487\pi\)
−0.900108 + 0.435666i \(0.856513\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −14.9743 −0.667673 −0.333836 0.942631i \(-0.608343\pi\)
−0.333836 + 0.942631i \(0.608343\pi\)
\(504\) 0 0
\(505\) 16.0000 0.711991
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15.7322i 0.697318i 0.937250 + 0.348659i \(0.113363\pi\)
−0.937250 + 0.348659i \(0.886637\pi\)
\(510\) 0 0
\(511\) 14.6603i 0.648531i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 52.5018 2.31350
\(516\) 0 0
\(517\) 4.78461 0.210427
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.48906i 0.284291i 0.989846 + 0.142145i \(0.0454001\pi\)
−0.989846 + 0.142145i \(0.954600\pi\)
\(522\) 0 0
\(523\) 21.1962i 0.926843i 0.886138 + 0.463422i \(0.153378\pi\)
−0.886138 + 0.463422i \(0.846622\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −13.3843 −0.583028
\(528\) 0 0
\(529\) 53.7846 2.33846
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 34.4959i 1.49418i
\(534\) 0 0
\(535\) 39.7128i 1.71693i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −7.17260 −0.308946
\(540\) 0 0
\(541\) −35.3923 −1.52163 −0.760817 0.648966i \(-0.775202\pi\)
−0.760817 + 0.648966i \(0.775202\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 49.9507i − 2.13965i
\(546\) 0 0
\(547\) 2.26795i 0.0969705i 0.998824 + 0.0484853i \(0.0154394\pi\)
−0.998824 + 0.0484853i \(0.984561\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 13.3843 0.570189
\(552\) 0 0
\(553\) −17.9282 −0.762385
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.1774i 0.643088i 0.946895 + 0.321544i \(0.104202\pi\)
−0.946895 + 0.321544i \(0.895798\pi\)
\(558\) 0 0
\(559\) 2.39230i 0.101184i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.24316 0.389553 0.194776 0.980848i \(-0.437602\pi\)
0.194776 + 0.980848i \(0.437602\pi\)
\(564\) 0 0
\(565\) −22.9282 −0.964597
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 43.4617i 1.82201i 0.412397 + 0.911004i \(0.364692\pi\)
−0.412397 + 0.911004i \(0.635308\pi\)
\(570\) 0 0
\(571\) 15.0526i 0.629930i 0.949103 + 0.314965i \(0.101993\pi\)
−0.949103 + 0.314965i \(0.898007\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 86.9977 3.62805
\(576\) 0 0
\(577\) −39.9282 −1.66223 −0.831116 0.556098i \(-0.812298\pi\)
−0.831116 + 0.556098i \(0.812298\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 7.72741i − 0.320587i
\(582\) 0 0
\(583\) − 3.71281i − 0.153769i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −14.4195 −0.595158 −0.297579 0.954697i \(-0.596179\pi\)
−0.297579 + 0.954697i \(0.596179\pi\)
\(588\) 0 0
\(589\) 12.9282 0.532697
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 42.7781i − 1.75669i −0.478029 0.878344i \(-0.658649\pi\)
0.478029 0.878344i \(-0.341351\pi\)
\(594\) 0 0
\(595\) 25.8564i 1.06001i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −17.5254 −0.716067 −0.358034 0.933709i \(-0.616553\pi\)
−0.358034 + 0.933709i \(0.616553\pi\)
\(600\) 0 0
\(601\) −34.7846 −1.41889 −0.709447 0.704759i \(-0.751055\pi\)
−0.709447 + 0.704759i \(0.751055\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 38.3596i − 1.55954i
\(606\) 0 0
\(607\) − 6.66025i − 0.270331i −0.990823 0.135166i \(-0.956843\pi\)
0.990823 0.135166i \(-0.0431567\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −20.6312 −0.834649
\(612\) 0 0
\(613\) 16.6077 0.670778 0.335389 0.942080i \(-0.391132\pi\)
0.335389 + 0.942080i \(0.391132\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 44.0165i 1.77204i 0.463650 + 0.886018i \(0.346540\pi\)
−0.463650 + 0.886018i \(0.653460\pi\)
\(618\) 0 0
\(619\) − 16.6603i − 0.669632i −0.942284 0.334816i \(-0.891326\pi\)
0.942284 0.334816i \(-0.108674\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.69213 0.268115
\(624\) 0 0
\(625\) 23.9282 0.957128
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 0.832204i − 0.0331822i
\(630\) 0 0
\(631\) 31.9808i 1.27313i 0.771221 + 0.636567i \(0.219646\pi\)
−0.771221 + 0.636567i \(0.780354\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −13.3843 −0.531138
\(636\) 0 0
\(637\) 30.9282 1.22542
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 3.03150i − 0.119737i −0.998206 0.0598685i \(-0.980932\pi\)
0.998206 0.0598685i \(-0.0190681\pi\)
\(642\) 0 0
\(643\) − 38.3923i − 1.51404i −0.653389 0.757022i \(-0.726653\pi\)
0.653389 0.757022i \(-0.273347\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.21166 0.244205 0.122103 0.992517i \(-0.461036\pi\)
0.122103 + 0.992517i \(0.461036\pi\)
\(648\) 0 0
\(649\) −12.7846 −0.501840
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 38.6370i 1.51198i 0.654581 + 0.755992i \(0.272845\pi\)
−0.654581 + 0.755992i \(0.727155\pi\)
\(654\) 0 0
\(655\) − 8.00000i − 0.312586i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5.10205 0.198748 0.0993739 0.995050i \(-0.468316\pi\)
0.0993739 + 0.995050i \(0.468316\pi\)
\(660\) 0 0
\(661\) −5.24871 −0.204151 −0.102076 0.994777i \(-0.532548\pi\)
−0.102076 + 0.994777i \(0.532548\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 24.9754i − 0.968503i
\(666\) 0 0
\(667\) − 67.7128i − 2.62185i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −11.7942 −0.455309
\(672\) 0 0
\(673\) −6.85641 −0.264295 −0.132148 0.991230i \(-0.542187\pi\)
−0.132148 + 0.991230i \(0.542187\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 26.4911i − 1.01814i −0.860726 0.509068i \(-0.829990\pi\)
0.860726 0.509068i \(-0.170010\pi\)
\(678\) 0 0
\(679\) − 26.1244i − 1.00256i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −33.9411 −1.29872 −0.649361 0.760481i \(-0.724963\pi\)
−0.649361 + 0.760481i \(0.724963\pi\)
\(684\) 0 0
\(685\) −52.7846 −2.01680
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 16.0096i 0.609918i
\(690\) 0 0
\(691\) 9.60770i 0.365494i 0.983160 + 0.182747i \(0.0584989\pi\)
−0.983160 + 0.182747i \(0.941501\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −35.5312 −1.34778
\(696\) 0 0
\(697\) 13.8564 0.524849
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 24.0144i − 0.907012i −0.891253 0.453506i \(-0.850173\pi\)
0.891253 0.453506i \(-0.149827\pi\)
\(702\) 0 0
\(703\) 0.803848i 0.0303177i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −15.4548 −0.581238
\(708\) 0 0
\(709\) 50.1769 1.88443 0.942217 0.335004i \(-0.108738\pi\)
0.942217 + 0.335004i \(0.108738\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 65.4056i − 2.44946i
\(714\) 0 0
\(715\) − 17.8564i − 0.667792i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −13.3843 −0.499149 −0.249574 0.968356i \(-0.580291\pi\)
−0.249574 + 0.968356i \(0.580291\pi\)
\(720\) 0 0
\(721\) −50.7128 −1.88864
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 76.7193i − 2.84928i
\(726\) 0 0
\(727\) 13.3205i 0.494030i 0.969012 + 0.247015i \(0.0794497\pi\)
−0.969012 + 0.247015i \(0.920550\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0.960947 0.0355419
\(732\) 0 0
\(733\) 19.0718 0.704433 0.352216 0.935919i \(-0.385428\pi\)
0.352216 + 0.935919i \(0.385428\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.48906i 0.239028i
\(738\) 0 0
\(739\) − 6.39230i − 0.235145i −0.993064 0.117572i \(-0.962489\pi\)
0.993064 0.117572i \(-0.0375112\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28.3586 1.04038 0.520188 0.854052i \(-0.325862\pi\)
0.520188 + 0.854052i \(0.325862\pi\)
\(744\) 0 0
\(745\) −43.7128 −1.60151
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 38.3596i − 1.40163i
\(750\) 0 0
\(751\) 7.19615i 0.262591i 0.991343 + 0.131296i \(0.0419137\pi\)
−0.991343 + 0.131296i \(0.958086\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 45.3292 1.64970
\(756\) 0 0
\(757\) −15.3923 −0.559443 −0.279721 0.960081i \(-0.590242\pi\)
−0.279721 + 0.960081i \(0.590242\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 29.6713i 1.07558i 0.843078 + 0.537791i \(0.180741\pi\)
−0.843078 + 0.537791i \(0.819259\pi\)
\(762\) 0 0
\(763\) 48.2487i 1.74672i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 55.1271 1.99053
\(768\) 0 0
\(769\) −14.8564 −0.535736 −0.267868 0.963456i \(-0.586319\pi\)
−0.267868 + 0.963456i \(0.586319\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 30.3548i − 1.09179i −0.837855 0.545894i \(-0.816190\pi\)
0.837855 0.545894i \(-0.183810\pi\)
\(774\) 0 0
\(775\) − 74.1051i − 2.66193i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −13.3843 −0.479541
\(780\) 0 0
\(781\) −11.7128 −0.419117
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 65.4056i 2.33442i
\(786\) 0 0
\(787\) − 33.7321i − 1.20242i −0.799092 0.601209i \(-0.794686\pi\)
0.799092 0.601209i \(-0.205314\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 22.1469 0.787455
\(792\) 0 0
\(793\) 50.8564 1.80596
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 42.2233i − 1.49563i −0.663909 0.747814i \(-0.731104\pi\)
0.663909 0.747814i \(-0.268896\pi\)
\(798\) 0 0
\(799\) 8.28719i 0.293180i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −4.06678 −0.143513
\(804\) 0 0
\(805\) −126.354 −4.45339
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 19.5959i 0.688956i 0.938794 + 0.344478i \(0.111944\pi\)
−0.938794 + 0.344478i \(0.888056\pi\)
\(810\) 0 0
\(811\) − 8.53590i − 0.299736i −0.988706 0.149868i \(-0.952115\pi\)
0.988706 0.149868i \(-0.0478849\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 60.2292 2.10974
\(816\) 0 0
\(817\) −0.928203 −0.0324737
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26.4911i 0.924546i 0.886738 + 0.462273i \(0.152966\pi\)
−0.886738 + 0.462273i \(0.847034\pi\)
\(822\) 0 0
\(823\) 11.9808i 0.417623i 0.977956 + 0.208812i \(0.0669595\pi\)
−0.977956 + 0.208812i \(0.933040\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.9134 −1.00542 −0.502709 0.864456i \(-0.667663\pi\)
−0.502709 + 0.864456i \(0.667663\pi\)
\(828\) 0 0
\(829\) −50.3205 −1.74770 −0.873852 0.486192i \(-0.838385\pi\)
−0.873852 + 0.486192i \(0.838385\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 12.4233i − 0.430442i
\(834\) 0 0
\(835\) 85.5692i 2.96124i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −43.1843 −1.49089 −0.745443 0.666569i \(-0.767762\pi\)
−0.745443 + 0.666569i \(0.767762\pi\)
\(840\) 0 0
\(841\) −30.7128 −1.05906
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 26.7685i 0.920865i
\(846\) 0 0
\(847\) 37.0526i 1.27314i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.06678 0.139407
\(852\) 0 0
\(853\) −47.3923 −1.62268 −0.811341 0.584573i \(-0.801262\pi\)
−0.811341 + 0.584573i \(0.801262\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 18.4863i − 0.631480i −0.948846 0.315740i \(-0.897747\pi\)
0.948846 0.315740i \(-0.102253\pi\)
\(858\) 0 0
\(859\) − 26.5167i − 0.904737i −0.891831 0.452368i \(-0.850579\pi\)
0.891831 0.452368i \(-0.149421\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −53.0566 −1.80607 −0.903033 0.429570i \(-0.858665\pi\)
−0.903033 + 0.429570i \(0.858665\pi\)
\(864\) 0 0
\(865\) −73.5692 −2.50143
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 4.97331i − 0.168708i
\(870\) 0 0
\(871\) − 27.9808i − 0.948092i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −71.0624 −2.40235
\(876\) 0 0
\(877\) 23.2487 0.785053 0.392527 0.919741i \(-0.371601\pi\)
0.392527 + 0.919741i \(0.371601\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 21.3891i − 0.720616i −0.932833 0.360308i \(-0.882672\pi\)
0.932833 0.360308i \(-0.117328\pi\)
\(882\) 0 0
\(883\) 0.124356i 0.00418490i 0.999998 + 0.00209245i \(0.000666048\pi\)
−0.999998 + 0.00209245i \(0.999334\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 17.5254 0.588444 0.294222 0.955737i \(-0.404939\pi\)
0.294222 + 0.955737i \(0.404939\pi\)
\(888\) 0 0
\(889\) 12.9282 0.433598
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 8.00481i − 0.267871i
\(894\) 0 0
\(895\) − 75.7128i − 2.53080i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −57.6781 −1.92367
\(900\) 0 0
\(901\) 6.43078 0.214240
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 1.79315i − 0.0596064i
\(906\) 0 0
\(907\) 9.48334i 0.314889i 0.987528 + 0.157445i \(0.0503256\pi\)
−0.987528 + 0.157445i \(0.949674\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 28.8391 0.955481 0.477741 0.878501i \(-0.341456\pi\)
0.477741 + 0.878501i \(0.341456\pi\)
\(912\) 0 0
\(913\) 2.14359 0.0709426
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.72741i 0.255181i
\(918\) 0 0
\(919\) 33.0333i 1.08967i 0.838544 + 0.544834i \(0.183407\pi\)
−0.838544 + 0.544834i \(0.816593\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 50.5055 1.66241
\(924\) 0 0
\(925\) 4.60770 0.151500
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 22.6274i 0.742381i 0.928557 + 0.371191i \(0.121050\pi\)
−0.928557 + 0.371191i \(0.878950\pi\)
\(930\) 0 0
\(931\) 12.0000i 0.393284i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −7.17260 −0.234569
\(936\) 0 0
\(937\) 47.7846 1.56106 0.780528 0.625121i \(-0.214951\pi\)
0.780528 + 0.625121i \(0.214951\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 12.1459i − 0.395945i −0.980208 0.197973i \(-0.936564\pi\)
0.980208 0.197973i \(-0.0634357\pi\)
\(942\) 0 0
\(943\) 67.7128i 2.20503i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14.4195 0.468572 0.234286 0.972168i \(-0.424725\pi\)
0.234286 + 0.972168i \(0.424725\pi\)
\(948\) 0 0
\(949\) 17.5359 0.569239
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 21.3891i 0.692860i 0.938076 + 0.346430i \(0.112606\pi\)
−0.938076 + 0.346430i \(0.887394\pi\)
\(954\) 0 0
\(955\) 41.8564i 1.35444i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 50.9860 1.64642
\(960\) 0 0
\(961\) −24.7128 −0.797188
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 53.8144i 1.73235i
\(966\) 0 0
\(967\) 8.26795i 0.265879i 0.991124 + 0.132940i \(0.0424417\pi\)
−0.991124 + 0.132940i \(0.957558\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −26.8429 −0.861428 −0.430714 0.902488i \(-0.641738\pi\)
−0.430714 + 0.902488i \(0.641738\pi\)
\(972\) 0 0
\(973\) 34.3205 1.10026
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 19.0411i − 0.609179i −0.952484 0.304590i \(-0.901481\pi\)
0.952484 0.304590i \(-0.0985193\pi\)
\(978\) 0 0
\(979\) 1.85641i 0.0593310i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 9.72363 0.310136 0.155068 0.987904i \(-0.450440\pi\)
0.155068 + 0.987904i \(0.450440\pi\)
\(984\) 0 0
\(985\) 1.07180 0.0341503
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.69591i 0.149321i
\(990\) 0 0
\(991\) − 52.5167i − 1.66825i −0.551578 0.834123i \(-0.685974\pi\)
0.551578 0.834123i \(-0.314026\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 21.5921 0.684517
\(996\) 0 0
\(997\) −34.7846 −1.10164 −0.550820 0.834624i \(-0.685685\pi\)
−0.550820 + 0.834624i \(0.685685\pi\)
\(998\) 0 0
\(999\) 0 0
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 1728.2.c.f.1727.7 8
3.2 odd 2 inner 1728.2.c.f.1727.1 8
4.3 odd 2 inner 1728.2.c.f.1727.8 8
8.3 odd 2 864.2.c.b.863.2 yes 8
8.5 even 2 864.2.c.b.863.1 8
12.11 even 2 inner 1728.2.c.f.1727.2 8
24.5 odd 2 864.2.c.b.863.7 yes 8
24.11 even 2 864.2.c.b.863.8 yes 8
72.5 odd 6 2592.2.s.g.1727.4 8
72.11 even 6 2592.2.s.g.863.1 8
72.13 even 6 2592.2.s.g.1727.1 8
72.29 odd 6 2592.2.s.c.863.1 8
72.43 odd 6 2592.2.s.g.863.4 8
72.59 even 6 2592.2.s.c.1727.4 8
72.61 even 6 2592.2.s.c.863.4 8
72.67 odd 6 2592.2.s.c.1727.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.c.b.863.1 8 8.5 even 2
864.2.c.b.863.2 yes 8 8.3 odd 2
864.2.c.b.863.7 yes 8 24.5 odd 2
864.2.c.b.863.8 yes 8 24.11 even 2
1728.2.c.f.1727.1 8 3.2 odd 2 inner
1728.2.c.f.1727.2 8 12.11 even 2 inner
1728.2.c.f.1727.7 8 1.1 even 1 trivial
1728.2.c.f.1727.8 8 4.3 odd 2 inner
2592.2.s.c.863.1 8 72.29 odd 6
2592.2.s.c.863.4 8 72.61 even 6
2592.2.s.c.1727.1 8 72.67 odd 6
2592.2.s.c.1727.4 8 72.59 even 6
2592.2.s.g.863.1 8 72.11 even 6
2592.2.s.g.863.4 8 72.43 odd 6
2592.2.s.g.1727.1 8 72.13 even 6
2592.2.s.g.1727.4 8 72.5 odd 6