Properties

Label 4056.2.a.w.1.1
Level $4056$
Weight $2$
Character 4056.1
Self dual yes
Analytic conductor $32.387$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4056,2,Mod(1,4056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4056, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("4056.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.a (trivial)

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: yes
Analytic conductor: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 4056.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -4.60555 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} -4.60555 q^{7} +1.00000 q^{9} +4.60555 q^{11} +1.00000 q^{15} -5.60555 q^{17} -0.605551 q^{19} -4.60555 q^{21} -4.60555 q^{23} -4.00000 q^{25} +1.00000 q^{27} +3.00000 q^{29} +9.21110 q^{31} +4.60555 q^{33} -4.60555 q^{35} +9.60555 q^{37} +4.39445 q^{41} +4.60555 q^{43} +1.00000 q^{45} +8.60555 q^{47} +14.2111 q^{49} -5.60555 q^{51} +3.00000 q^{53} +4.60555 q^{55} -0.605551 q^{57} +5.21110 q^{59} -0.394449 q^{61} -4.60555 q^{63} +3.39445 q^{67} -4.60555 q^{69} +16.6056 q^{71} -7.00000 q^{73} -4.00000 q^{75} -21.2111 q^{77} +12.0000 q^{79} +1.00000 q^{81} -9.81665 q^{83} -5.60555 q^{85} +3.00000 q^{87} -12.4222 q^{89} +9.21110 q^{93} -0.605551 q^{95} +3.21110 q^{97} +4.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{15} - 4 q^{17} + 6 q^{19} - 2 q^{21} - 2 q^{23} - 8 q^{25} + 2 q^{27} + 6 q^{29} + 4 q^{31} + 2 q^{33} - 2 q^{35} + 12 q^{37} + 16 q^{41} + 2 q^{43} + 2 q^{45} + 10 q^{47} + 14 q^{49} - 4 q^{51} + 6 q^{53} + 2 q^{55} + 6 q^{57} - 4 q^{59} - 8 q^{61} - 2 q^{63} + 14 q^{67} - 2 q^{69} + 26 q^{71} - 14 q^{73} - 8 q^{75} - 28 q^{77} + 24 q^{79} + 2 q^{81} + 2 q^{83} - 4 q^{85} + 6 q^{87} + 4 q^{89} + 4 q^{93} + 6 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −4.60555 −1.74073 −0.870367 0.492403i \(-0.836119\pi\)
−0.870367 + 0.492403i \(0.836119\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.60555 1.38863 0.694313 0.719673i \(-0.255708\pi\)
0.694313 + 0.719673i \(0.255708\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −5.60555 −1.35955 −0.679773 0.733423i \(-0.737922\pi\)
−0.679773 + 0.733423i \(0.737922\pi\)
\(18\) 0 0
\(19\) −0.605551 −0.138923 −0.0694615 0.997585i \(-0.522128\pi\)
−0.0694615 + 0.997585i \(0.522128\pi\)
\(20\) 0 0
\(21\) −4.60555 −1.00501
\(22\) 0 0
\(23\) −4.60555 −0.960324 −0.480162 0.877180i \(-0.659422\pi\)
−0.480162 + 0.877180i \(0.659422\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 9.21110 1.65436 0.827181 0.561935i \(-0.189943\pi\)
0.827181 + 0.561935i \(0.189943\pi\)
\(32\) 0 0
\(33\) 4.60555 0.801724
\(34\) 0 0
\(35\) −4.60555 −0.778480
\(36\) 0 0
\(37\) 9.60555 1.57914 0.789571 0.613659i \(-0.210303\pi\)
0.789571 + 0.613659i \(0.210303\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.39445 0.686298 0.343149 0.939281i \(-0.388506\pi\)
0.343149 + 0.939281i \(0.388506\pi\)
\(42\) 0 0
\(43\) 4.60555 0.702340 0.351170 0.936312i \(-0.385784\pi\)
0.351170 + 0.936312i \(0.385784\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 8.60555 1.25525 0.627624 0.778516i \(-0.284027\pi\)
0.627624 + 0.778516i \(0.284027\pi\)
\(48\) 0 0
\(49\) 14.2111 2.03016
\(50\) 0 0
\(51\) −5.60555 −0.784934
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 4.60555 0.621012
\(56\) 0 0
\(57\) −0.605551 −0.0802072
\(58\) 0 0
\(59\) 5.21110 0.678428 0.339214 0.940709i \(-0.389839\pi\)
0.339214 + 0.940709i \(0.389839\pi\)
\(60\) 0 0
\(61\) −0.394449 −0.0505040 −0.0252520 0.999681i \(-0.508039\pi\)
−0.0252520 + 0.999681i \(0.508039\pi\)
\(62\) 0 0
\(63\) −4.60555 −0.580245
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.39445 0.414698 0.207349 0.978267i \(-0.433516\pi\)
0.207349 + 0.978267i \(0.433516\pi\)
\(68\) 0 0
\(69\) −4.60555 −0.554443
\(70\) 0 0
\(71\) 16.6056 1.97072 0.985358 0.170497i \(-0.0545373\pi\)
0.985358 + 0.170497i \(0.0545373\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) −21.2111 −2.41723
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.81665 −1.07752 −0.538759 0.842460i \(-0.681107\pi\)
−0.538759 + 0.842460i \(0.681107\pi\)
\(84\) 0 0
\(85\) −5.60555 −0.608007
\(86\) 0 0
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) −12.4222 −1.31675 −0.658376 0.752690i \(-0.728756\pi\)
−0.658376 + 0.752690i \(0.728756\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 9.21110 0.955147
\(94\) 0 0
\(95\) −0.605551 −0.0621282
\(96\) 0 0
\(97\) 3.21110 0.326038 0.163019 0.986623i \(-0.447877\pi\)
0.163019 + 0.986623i \(0.447877\pi\)
\(98\) 0 0
\(99\) 4.60555 0.462875
\(100\) 0 0
\(101\) −10.2111 −1.01604 −0.508021 0.861344i \(-0.669623\pi\)
−0.508021 + 0.861344i \(0.669623\pi\)
\(102\) 0 0
\(103\) 5.81665 0.573132 0.286566 0.958061i \(-0.407486\pi\)
0.286566 + 0.958061i \(0.407486\pi\)
\(104\) 0 0
\(105\) −4.60555 −0.449456
\(106\) 0 0
\(107\) −0.605551 −0.0585409 −0.0292704 0.999572i \(-0.509318\pi\)
−0.0292704 + 0.999572i \(0.509318\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 9.60555 0.911719
\(112\) 0 0
\(113\) −4.39445 −0.413395 −0.206697 0.978405i \(-0.566272\pi\)
−0.206697 + 0.978405i \(0.566272\pi\)
\(114\) 0 0
\(115\) −4.60555 −0.429470
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 25.8167 2.36661
\(120\) 0 0
\(121\) 10.2111 0.928282
\(122\) 0 0
\(123\) 4.39445 0.396234
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 4.60555 0.405496
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 2.78890 0.241828
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −4.81665 −0.411515 −0.205757 0.978603i \(-0.565966\pi\)
−0.205757 + 0.978603i \(0.565966\pi\)
\(138\) 0 0
\(139\) −17.2111 −1.45983 −0.729913 0.683540i \(-0.760440\pi\)
−0.729913 + 0.683540i \(0.760440\pi\)
\(140\) 0 0
\(141\) 8.60555 0.724718
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) 0 0
\(147\) 14.2111 1.17211
\(148\) 0 0
\(149\) −4.21110 −0.344987 −0.172493 0.985011i \(-0.555182\pi\)
−0.172493 + 0.985011i \(0.555182\pi\)
\(150\) 0 0
\(151\) −0.605551 −0.0492791 −0.0246395 0.999696i \(-0.507844\pi\)
−0.0246395 + 0.999696i \(0.507844\pi\)
\(152\) 0 0
\(153\) −5.60555 −0.453182
\(154\) 0 0
\(155\) 9.21110 0.739854
\(156\) 0 0
\(157\) 15.6056 1.24546 0.622729 0.782437i \(-0.286024\pi\)
0.622729 + 0.782437i \(0.286024\pi\)
\(158\) 0 0
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) 21.2111 1.67167
\(162\) 0 0
\(163\) −9.21110 −0.721469 −0.360735 0.932669i \(-0.617474\pi\)
−0.360735 + 0.932669i \(0.617474\pi\)
\(164\) 0 0
\(165\) 4.60555 0.358542
\(166\) 0 0
\(167\) −2.78890 −0.215811 −0.107906 0.994161i \(-0.534414\pi\)
−0.107906 + 0.994161i \(0.534414\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −0.605551 −0.0463077
\(172\) 0 0
\(173\) 21.6333 1.64475 0.822375 0.568946i \(-0.192649\pi\)
0.822375 + 0.568946i \(0.192649\pi\)
\(174\) 0 0
\(175\) 18.4222 1.39259
\(176\) 0 0
\(177\) 5.21110 0.391690
\(178\) 0 0
\(179\) 19.0278 1.42220 0.711101 0.703090i \(-0.248197\pi\)
0.711101 + 0.703090i \(0.248197\pi\)
\(180\) 0 0
\(181\) −3.18335 −0.236616 −0.118308 0.992977i \(-0.537747\pi\)
−0.118308 + 0.992977i \(0.537747\pi\)
\(182\) 0 0
\(183\) −0.394449 −0.0291585
\(184\) 0 0
\(185\) 9.60555 0.706214
\(186\) 0 0
\(187\) −25.8167 −1.88790
\(188\) 0 0
\(189\) −4.60555 −0.335005
\(190\) 0 0
\(191\) 19.6333 1.42062 0.710308 0.703891i \(-0.248556\pi\)
0.710308 + 0.703891i \(0.248556\pi\)
\(192\) 0 0
\(193\) 22.2111 1.59879 0.799395 0.600806i \(-0.205153\pi\)
0.799395 + 0.600806i \(0.205153\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) −1.81665 −0.128779 −0.0643896 0.997925i \(-0.520510\pi\)
−0.0643896 + 0.997925i \(0.520510\pi\)
\(200\) 0 0
\(201\) 3.39445 0.239426
\(202\) 0 0
\(203\) −13.8167 −0.969739
\(204\) 0 0
\(205\) 4.39445 0.306922
\(206\) 0 0
\(207\) −4.60555 −0.320108
\(208\) 0 0
\(209\) −2.78890 −0.192912
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 16.6056 1.13779
\(214\) 0 0
\(215\) 4.60555 0.314096
\(216\) 0 0
\(217\) −42.4222 −2.87981
\(218\) 0 0
\(219\) −7.00000 −0.473016
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) 19.3944 1.28725 0.643627 0.765339i \(-0.277429\pi\)
0.643627 + 0.765339i \(0.277429\pi\)
\(228\) 0 0
\(229\) −0.788897 −0.0521318 −0.0260659 0.999660i \(-0.508298\pi\)
−0.0260659 + 0.999660i \(0.508298\pi\)
\(230\) 0 0
\(231\) −21.2111 −1.39559
\(232\) 0 0
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) 8.60555 0.561364
\(236\) 0 0
\(237\) 12.0000 0.779484
\(238\) 0 0
\(239\) 19.0278 1.23080 0.615402 0.788214i \(-0.288994\pi\)
0.615402 + 0.788214i \(0.288994\pi\)
\(240\) 0 0
\(241\) 28.6333 1.84443 0.922217 0.386673i \(-0.126376\pi\)
0.922217 + 0.386673i \(0.126376\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 14.2111 0.907914
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −9.81665 −0.622105
\(250\) 0 0
\(251\) 14.7889 0.933467 0.466733 0.884398i \(-0.345431\pi\)
0.466733 + 0.884398i \(0.345431\pi\)
\(252\) 0 0
\(253\) −21.2111 −1.33353
\(254\) 0 0
\(255\) −5.60555 −0.351033
\(256\) 0 0
\(257\) −16.3944 −1.02266 −0.511329 0.859385i \(-0.670847\pi\)
−0.511329 + 0.859385i \(0.670847\pi\)
\(258\) 0 0
\(259\) −44.2389 −2.74887
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) 11.3944 0.702612 0.351306 0.936261i \(-0.385738\pi\)
0.351306 + 0.936261i \(0.385738\pi\)
\(264\) 0 0
\(265\) 3.00000 0.184289
\(266\) 0 0
\(267\) −12.4222 −0.760227
\(268\) 0 0
\(269\) −17.6333 −1.07512 −0.537561 0.843225i \(-0.680654\pi\)
−0.537561 + 0.843225i \(0.680654\pi\)
\(270\) 0 0
\(271\) −13.2111 −0.802517 −0.401259 0.915965i \(-0.631427\pi\)
−0.401259 + 0.915965i \(0.631427\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −18.4222 −1.11090
\(276\) 0 0
\(277\) −21.2389 −1.27612 −0.638060 0.769987i \(-0.720263\pi\)
−0.638060 + 0.769987i \(0.720263\pi\)
\(278\) 0 0
\(279\) 9.21110 0.551454
\(280\) 0 0
\(281\) −7.60555 −0.453709 −0.226855 0.973929i \(-0.572844\pi\)
−0.226855 + 0.973929i \(0.572844\pi\)
\(282\) 0 0
\(283\) 7.39445 0.439554 0.219777 0.975550i \(-0.429467\pi\)
0.219777 + 0.975550i \(0.429467\pi\)
\(284\) 0 0
\(285\) −0.605551 −0.0358698
\(286\) 0 0
\(287\) −20.2389 −1.19466
\(288\) 0 0
\(289\) 14.4222 0.848365
\(290\) 0 0
\(291\) 3.21110 0.188238
\(292\) 0 0
\(293\) 14.2111 0.830221 0.415111 0.909771i \(-0.363743\pi\)
0.415111 + 0.909771i \(0.363743\pi\)
\(294\) 0 0
\(295\) 5.21110 0.303402
\(296\) 0 0
\(297\) 4.60555 0.267241
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −21.2111 −1.22259
\(302\) 0 0
\(303\) −10.2111 −0.586613
\(304\) 0 0
\(305\) −0.394449 −0.0225861
\(306\) 0 0
\(307\) −12.6056 −0.719437 −0.359718 0.933061i \(-0.617127\pi\)
−0.359718 + 0.933061i \(0.617127\pi\)
\(308\) 0 0
\(309\) 5.81665 0.330898
\(310\) 0 0
\(311\) 6.18335 0.350625 0.175313 0.984513i \(-0.443906\pi\)
0.175313 + 0.984513i \(0.443906\pi\)
\(312\) 0 0
\(313\) −11.2111 −0.633689 −0.316844 0.948478i \(-0.602623\pi\)
−0.316844 + 0.948478i \(0.602623\pi\)
\(314\) 0 0
\(315\) −4.60555 −0.259493
\(316\) 0 0
\(317\) −12.2111 −0.685844 −0.342922 0.939364i \(-0.611417\pi\)
−0.342922 + 0.939364i \(0.611417\pi\)
\(318\) 0 0
\(319\) 13.8167 0.773584
\(320\) 0 0
\(321\) −0.605551 −0.0337986
\(322\) 0 0
\(323\) 3.39445 0.188872
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) −39.6333 −2.18505
\(330\) 0 0
\(331\) −27.6333 −1.51886 −0.759432 0.650587i \(-0.774523\pi\)
−0.759432 + 0.650587i \(0.774523\pi\)
\(332\) 0 0
\(333\) 9.60555 0.526381
\(334\) 0 0
\(335\) 3.39445 0.185459
\(336\) 0 0
\(337\) −31.4222 −1.71168 −0.855838 0.517243i \(-0.826958\pi\)
−0.855838 + 0.517243i \(0.826958\pi\)
\(338\) 0 0
\(339\) −4.39445 −0.238674
\(340\) 0 0
\(341\) 42.4222 2.29729
\(342\) 0 0
\(343\) −33.2111 −1.79323
\(344\) 0 0
\(345\) −4.60555 −0.247955
\(346\) 0 0
\(347\) −19.0278 −1.02146 −0.510732 0.859740i \(-0.670625\pi\)
−0.510732 + 0.859740i \(0.670625\pi\)
\(348\) 0 0
\(349\) 7.57779 0.405630 0.202815 0.979217i \(-0.434991\pi\)
0.202815 + 0.979217i \(0.434991\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 20.0278 1.06597 0.532985 0.846125i \(-0.321070\pi\)
0.532985 + 0.846125i \(0.321070\pi\)
\(354\) 0 0
\(355\) 16.6056 0.881331
\(356\) 0 0
\(357\) 25.8167 1.36636
\(358\) 0 0
\(359\) 13.8167 0.729215 0.364608 0.931161i \(-0.381203\pi\)
0.364608 + 0.931161i \(0.381203\pi\)
\(360\) 0 0
\(361\) −18.6333 −0.980700
\(362\) 0 0
\(363\) 10.2111 0.535944
\(364\) 0 0
\(365\) −7.00000 −0.366397
\(366\) 0 0
\(367\) −8.60555 −0.449206 −0.224603 0.974450i \(-0.572109\pi\)
−0.224603 + 0.974450i \(0.572109\pi\)
\(368\) 0 0
\(369\) 4.39445 0.228766
\(370\) 0 0
\(371\) −13.8167 −0.717325
\(372\) 0 0
\(373\) 6.02776 0.312105 0.156053 0.987749i \(-0.450123\pi\)
0.156053 + 0.987749i \(0.450123\pi\)
\(374\) 0 0
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 22.4222 1.15175 0.575876 0.817537i \(-0.304661\pi\)
0.575876 + 0.817537i \(0.304661\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) −10.4222 −0.532550 −0.266275 0.963897i \(-0.585793\pi\)
−0.266275 + 0.963897i \(0.585793\pi\)
\(384\) 0 0
\(385\) −21.2111 −1.08102
\(386\) 0 0
\(387\) 4.60555 0.234113
\(388\) 0 0
\(389\) 6.63331 0.336322 0.168161 0.985760i \(-0.446217\pi\)
0.168161 + 0.985760i \(0.446217\pi\)
\(390\) 0 0
\(391\) 25.8167 1.30560
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 2.78890 0.139620
\(400\) 0 0
\(401\) 38.4500 1.92010 0.960050 0.279829i \(-0.0902779\pi\)
0.960050 + 0.279829i \(0.0902779\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 44.2389 2.19284
\(408\) 0 0
\(409\) −33.4222 −1.65262 −0.826311 0.563214i \(-0.809565\pi\)
−0.826311 + 0.563214i \(0.809565\pi\)
\(410\) 0 0
\(411\) −4.81665 −0.237588
\(412\) 0 0
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) −9.81665 −0.481881
\(416\) 0 0
\(417\) −17.2111 −0.842831
\(418\) 0 0
\(419\) 10.4222 0.509158 0.254579 0.967052i \(-0.418063\pi\)
0.254579 + 0.967052i \(0.418063\pi\)
\(420\) 0 0
\(421\) −31.6056 −1.54036 −0.770180 0.637826i \(-0.779834\pi\)
−0.770180 + 0.637826i \(0.779834\pi\)
\(422\) 0 0
\(423\) 8.60555 0.418416
\(424\) 0 0
\(425\) 22.4222 1.08764
\(426\) 0 0
\(427\) 1.81665 0.0879140
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5.81665 −0.280178 −0.140089 0.990139i \(-0.544739\pi\)
−0.140089 + 0.990139i \(0.544739\pi\)
\(432\) 0 0
\(433\) −6.57779 −0.316109 −0.158054 0.987430i \(-0.550522\pi\)
−0.158054 + 0.987430i \(0.550522\pi\)
\(434\) 0 0
\(435\) 3.00000 0.143839
\(436\) 0 0
\(437\) 2.78890 0.133411
\(438\) 0 0
\(439\) 14.1833 0.676934 0.338467 0.940978i \(-0.390092\pi\)
0.338467 + 0.940978i \(0.390092\pi\)
\(440\) 0 0
\(441\) 14.2111 0.676719
\(442\) 0 0
\(443\) 27.6333 1.31290 0.656449 0.754370i \(-0.272058\pi\)
0.656449 + 0.754370i \(0.272058\pi\)
\(444\) 0 0
\(445\) −12.4222 −0.588869
\(446\) 0 0
\(447\) −4.21110 −0.199178
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 20.2389 0.953011
\(452\) 0 0
\(453\) −0.605551 −0.0284513
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.63331 0.403849 0.201925 0.979401i \(-0.435280\pi\)
0.201925 + 0.979401i \(0.435280\pi\)
\(458\) 0 0
\(459\) −5.60555 −0.261645
\(460\) 0 0
\(461\) 24.6333 1.14729 0.573644 0.819105i \(-0.305529\pi\)
0.573644 + 0.819105i \(0.305529\pi\)
\(462\) 0 0
\(463\) 25.8167 1.19980 0.599901 0.800074i \(-0.295207\pi\)
0.599901 + 0.800074i \(0.295207\pi\)
\(464\) 0 0
\(465\) 9.21110 0.427155
\(466\) 0 0
\(467\) 29.8167 1.37975 0.689875 0.723928i \(-0.257665\pi\)
0.689875 + 0.723928i \(0.257665\pi\)
\(468\) 0 0
\(469\) −15.6333 −0.721879
\(470\) 0 0
\(471\) 15.6056 0.719066
\(472\) 0 0
\(473\) 21.2111 0.975288
\(474\) 0 0
\(475\) 2.42221 0.111138
\(476\) 0 0
\(477\) 3.00000 0.137361
\(478\) 0 0
\(479\) −39.6333 −1.81089 −0.905446 0.424461i \(-0.860463\pi\)
−0.905446 + 0.424461i \(0.860463\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 21.2111 0.965139
\(484\) 0 0
\(485\) 3.21110 0.145809
\(486\) 0 0
\(487\) −23.3944 −1.06010 −0.530052 0.847965i \(-0.677828\pi\)
−0.530052 + 0.847965i \(0.677828\pi\)
\(488\) 0 0
\(489\) −9.21110 −0.416540
\(490\) 0 0
\(491\) −1.81665 −0.0819844 −0.0409922 0.999159i \(-0.513052\pi\)
−0.0409922 + 0.999159i \(0.513052\pi\)
\(492\) 0 0
\(493\) −16.8167 −0.757384
\(494\) 0 0
\(495\) 4.60555 0.207004
\(496\) 0 0
\(497\) −76.4777 −3.43049
\(498\) 0 0
\(499\) −18.7889 −0.841107 −0.420553 0.907268i \(-0.638164\pi\)
−0.420553 + 0.907268i \(0.638164\pi\)
\(500\) 0 0
\(501\) −2.78890 −0.124599
\(502\) 0 0
\(503\) −5.81665 −0.259352 −0.129676 0.991556i \(-0.541394\pi\)
−0.129676 + 0.991556i \(0.541394\pi\)
\(504\) 0 0
\(505\) −10.2111 −0.454388
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −38.6333 −1.71239 −0.856196 0.516652i \(-0.827178\pi\)
−0.856196 + 0.516652i \(0.827178\pi\)
\(510\) 0 0
\(511\) 32.2389 1.42616
\(512\) 0 0
\(513\) −0.605551 −0.0267357
\(514\) 0 0
\(515\) 5.81665 0.256312
\(516\) 0 0
\(517\) 39.6333 1.74307
\(518\) 0 0
\(519\) 21.6333 0.949597
\(520\) 0 0
\(521\) 27.2389 1.19336 0.596678 0.802481i \(-0.296487\pi\)
0.596678 + 0.802481i \(0.296487\pi\)
\(522\) 0 0
\(523\) −35.0278 −1.53166 −0.765828 0.643045i \(-0.777671\pi\)
−0.765828 + 0.643045i \(0.777671\pi\)
\(524\) 0 0
\(525\) 18.4222 0.804011
\(526\) 0 0
\(527\) −51.6333 −2.24918
\(528\) 0 0
\(529\) −1.78890 −0.0777781
\(530\) 0 0
\(531\) 5.21110 0.226143
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.605551 −0.0261803
\(536\) 0 0
\(537\) 19.0278 0.821108
\(538\) 0 0
\(539\) 65.4500 2.81913
\(540\) 0 0
\(541\) 17.6056 0.756922 0.378461 0.925617i \(-0.376453\pi\)
0.378461 + 0.925617i \(0.376453\pi\)
\(542\) 0 0
\(543\) −3.18335 −0.136610
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −19.0278 −0.813568 −0.406784 0.913524i \(-0.633350\pi\)
−0.406784 + 0.913524i \(0.633350\pi\)
\(548\) 0 0
\(549\) −0.394449 −0.0168347
\(550\) 0 0
\(551\) −1.81665 −0.0773921
\(552\) 0 0
\(553\) −55.2666 −2.35018
\(554\) 0 0
\(555\) 9.60555 0.407733
\(556\) 0 0
\(557\) −27.8444 −1.17981 −0.589903 0.807474i \(-0.700834\pi\)
−0.589903 + 0.807474i \(0.700834\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −25.8167 −1.08998
\(562\) 0 0
\(563\) −15.6333 −0.658865 −0.329433 0.944179i \(-0.606857\pi\)
−0.329433 + 0.944179i \(0.606857\pi\)
\(564\) 0 0
\(565\) −4.39445 −0.184876
\(566\) 0 0
\(567\) −4.60555 −0.193415
\(568\) 0 0
\(569\) −8.78890 −0.368450 −0.184225 0.982884i \(-0.558977\pi\)
−0.184225 + 0.982884i \(0.558977\pi\)
\(570\) 0 0
\(571\) 45.4500 1.90202 0.951011 0.309158i \(-0.100047\pi\)
0.951011 + 0.309158i \(0.100047\pi\)
\(572\) 0 0
\(573\) 19.6333 0.820193
\(574\) 0 0
\(575\) 18.4222 0.768259
\(576\) 0 0
\(577\) 11.7889 0.490778 0.245389 0.969425i \(-0.421084\pi\)
0.245389 + 0.969425i \(0.421084\pi\)
\(578\) 0 0
\(579\) 22.2111 0.923062
\(580\) 0 0
\(581\) 45.2111 1.87567
\(582\) 0 0
\(583\) 13.8167 0.572227
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13.5778 −0.560416 −0.280208 0.959939i \(-0.590403\pi\)
−0.280208 + 0.959939i \(0.590403\pi\)
\(588\) 0 0
\(589\) −5.57779 −0.229829
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 0 0
\(593\) 20.0278 0.822441 0.411221 0.911536i \(-0.365103\pi\)
0.411221 + 0.911536i \(0.365103\pi\)
\(594\) 0 0
\(595\) 25.8167 1.05838
\(596\) 0 0
\(597\) −1.81665 −0.0743507
\(598\) 0 0
\(599\) 2.42221 0.0989686 0.0494843 0.998775i \(-0.484242\pi\)
0.0494843 + 0.998775i \(0.484242\pi\)
\(600\) 0 0
\(601\) −10.5778 −0.431477 −0.215739 0.976451i \(-0.569216\pi\)
−0.215739 + 0.976451i \(0.569216\pi\)
\(602\) 0 0
\(603\) 3.39445 0.138233
\(604\) 0 0
\(605\) 10.2111 0.415140
\(606\) 0 0
\(607\) −42.0555 −1.70698 −0.853490 0.521109i \(-0.825519\pi\)
−0.853490 + 0.521109i \(0.825519\pi\)
\(608\) 0 0
\(609\) −13.8167 −0.559879
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −6.39445 −0.258269 −0.129135 0.991627i \(-0.541220\pi\)
−0.129135 + 0.991627i \(0.541220\pi\)
\(614\) 0 0
\(615\) 4.39445 0.177201
\(616\) 0 0
\(617\) −19.6056 −0.789290 −0.394645 0.918834i \(-0.629132\pi\)
−0.394645 + 0.918834i \(0.629132\pi\)
\(618\) 0 0
\(619\) 23.6333 0.949903 0.474951 0.880012i \(-0.342466\pi\)
0.474951 + 0.880012i \(0.342466\pi\)
\(620\) 0 0
\(621\) −4.60555 −0.184814
\(622\) 0 0
\(623\) 57.2111 2.29211
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −2.78890 −0.111378
\(628\) 0 0
\(629\) −53.8444 −2.14692
\(630\) 0 0
\(631\) 41.2111 1.64059 0.820294 0.571942i \(-0.193810\pi\)
0.820294 + 0.571942i \(0.193810\pi\)
\(632\) 0 0
\(633\) −8.00000 −0.317971
\(634\) 0 0
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 16.6056 0.656905
\(640\) 0 0
\(641\) −33.6056 −1.32734 −0.663670 0.748026i \(-0.731002\pi\)
−0.663670 + 0.748026i \(0.731002\pi\)
\(642\) 0 0
\(643\) −0.366692 −0.0144609 −0.00723047 0.999974i \(-0.502302\pi\)
−0.00723047 + 0.999974i \(0.502302\pi\)
\(644\) 0 0
\(645\) 4.60555 0.181343
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) −42.4222 −1.66266
\(652\) 0 0
\(653\) 21.6333 0.846577 0.423288 0.905995i \(-0.360876\pi\)
0.423288 + 0.905995i \(0.360876\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −7.00000 −0.273096
\(658\) 0 0
\(659\) 40.0000 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) −0.816654 −0.0317642 −0.0158821 0.999874i \(-0.505056\pi\)
−0.0158821 + 0.999874i \(0.505056\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.78890 0.108149
\(666\) 0 0
\(667\) −13.8167 −0.534983
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) −1.81665 −0.0701311
\(672\) 0 0
\(673\) 15.0000 0.578208 0.289104 0.957298i \(-0.406643\pi\)
0.289104 + 0.957298i \(0.406643\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −44.0555 −1.69319 −0.846595 0.532237i \(-0.821352\pi\)
−0.846595 + 0.532237i \(0.821352\pi\)
\(678\) 0 0
\(679\) −14.7889 −0.567546
\(680\) 0 0
\(681\) 19.3944 0.743197
\(682\) 0 0
\(683\) 10.4222 0.398795 0.199397 0.979919i \(-0.436102\pi\)
0.199397 + 0.979919i \(0.436102\pi\)
\(684\) 0 0
\(685\) −4.81665 −0.184035
\(686\) 0 0
\(687\) −0.788897 −0.0300983
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −0.605551 −0.0230363 −0.0115181 0.999934i \(-0.503666\pi\)
−0.0115181 + 0.999934i \(0.503666\pi\)
\(692\) 0 0
\(693\) −21.2111 −0.805743
\(694\) 0 0
\(695\) −17.2111 −0.652854
\(696\) 0 0
\(697\) −24.6333 −0.933053
\(698\) 0 0
\(699\) −14.0000 −0.529529
\(700\) 0 0
\(701\) −49.6333 −1.87462 −0.937312 0.348491i \(-0.886694\pi\)
−0.937312 + 0.348491i \(0.886694\pi\)
\(702\) 0 0
\(703\) −5.81665 −0.219379
\(704\) 0 0
\(705\) 8.60555 0.324104
\(706\) 0 0
\(707\) 47.0278 1.76866
\(708\) 0 0
\(709\) −39.6056 −1.48742 −0.743709 0.668504i \(-0.766935\pi\)
−0.743709 + 0.668504i \(0.766935\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 0 0
\(713\) −42.4222 −1.58872
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 19.0278 0.710605
\(718\) 0 0
\(719\) −29.2111 −1.08939 −0.544695 0.838634i \(-0.683355\pi\)
−0.544695 + 0.838634i \(0.683355\pi\)
\(720\) 0 0
\(721\) −26.7889 −0.997671
\(722\) 0 0
\(723\) 28.6333 1.06488
\(724\) 0 0
\(725\) −12.0000 −0.445669
\(726\) 0 0
\(727\) 5.81665 0.215728 0.107864 0.994166i \(-0.465599\pi\)
0.107864 + 0.994166i \(0.465599\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −25.8167 −0.954863
\(732\) 0 0
\(733\) 17.2389 0.636732 0.318366 0.947968i \(-0.396866\pi\)
0.318366 + 0.947968i \(0.396866\pi\)
\(734\) 0 0
\(735\) 14.2111 0.524184
\(736\) 0 0
\(737\) 15.6333 0.575860
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −25.2111 −0.924906 −0.462453 0.886644i \(-0.653031\pi\)
−0.462453 + 0.886644i \(0.653031\pi\)
\(744\) 0 0
\(745\) −4.21110 −0.154283
\(746\) 0 0
\(747\) −9.81665 −0.359173
\(748\) 0 0
\(749\) 2.78890 0.101904
\(750\) 0 0
\(751\) 27.0278 0.986257 0.493128 0.869957i \(-0.335853\pi\)
0.493128 + 0.869957i \(0.335853\pi\)
\(752\) 0 0
\(753\) 14.7889 0.538937
\(754\) 0 0
\(755\) −0.605551 −0.0220383
\(756\) 0 0
\(757\) 0.422205 0.0153453 0.00767265 0.999971i \(-0.497558\pi\)
0.00767265 + 0.999971i \(0.497558\pi\)
\(758\) 0 0
\(759\) −21.2111 −0.769914
\(760\) 0 0
\(761\) 40.4222 1.46530 0.732652 0.680604i \(-0.238282\pi\)
0.732652 + 0.680604i \(0.238282\pi\)
\(762\) 0 0
\(763\) −9.21110 −0.333464
\(764\) 0 0
\(765\) −5.60555 −0.202669
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −49.6333 −1.78982 −0.894911 0.446244i \(-0.852761\pi\)
−0.894911 + 0.446244i \(0.852761\pi\)
\(770\) 0 0
\(771\) −16.3944 −0.590432
\(772\) 0 0
\(773\) −21.6333 −0.778096 −0.389048 0.921217i \(-0.627196\pi\)
−0.389048 + 0.921217i \(0.627196\pi\)
\(774\) 0 0
\(775\) −36.8444 −1.32349
\(776\) 0 0
\(777\) −44.2389 −1.58706
\(778\) 0 0
\(779\) −2.66106 −0.0953425
\(780\) 0 0
\(781\) 76.4777 2.73659
\(782\) 0 0
\(783\) 3.00000 0.107211
\(784\) 0 0
\(785\) 15.6056 0.556986
\(786\) 0 0
\(787\) −6.78890 −0.241998 −0.120999 0.992653i \(-0.538610\pi\)
−0.120999 + 0.992653i \(0.538610\pi\)
\(788\) 0 0
\(789\) 11.3944 0.405653
\(790\) 0 0
\(791\) 20.2389 0.719611
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 3.00000 0.106399
\(796\) 0 0
\(797\) −54.8444 −1.94269 −0.971344 0.237677i \(-0.923614\pi\)
−0.971344 + 0.237677i \(0.923614\pi\)
\(798\) 0 0
\(799\) −48.2389 −1.70657
\(800\) 0 0
\(801\) −12.4222 −0.438917
\(802\) 0 0
\(803\) −32.2389 −1.13768
\(804\) 0 0
\(805\) 21.2111 0.747593
\(806\) 0 0
\(807\) −17.6333 −0.620722
\(808\) 0 0
\(809\) −24.3944 −0.857663 −0.428832 0.903384i \(-0.641075\pi\)
−0.428832 + 0.903384i \(0.641075\pi\)
\(810\) 0 0
\(811\) 19.6333 0.689419 0.344709 0.938709i \(-0.387977\pi\)
0.344709 + 0.938709i \(0.387977\pi\)
\(812\) 0 0
\(813\) −13.2111 −0.463334
\(814\) 0 0
\(815\) −9.21110 −0.322651
\(816\) 0 0
\(817\) −2.78890 −0.0975712
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 7.21110 0.251669 0.125835 0.992051i \(-0.459839\pi\)
0.125835 + 0.992051i \(0.459839\pi\)
\(822\) 0 0
\(823\) −23.6333 −0.823805 −0.411903 0.911228i \(-0.635136\pi\)
−0.411903 + 0.911228i \(0.635136\pi\)
\(824\) 0 0
\(825\) −18.4222 −0.641379
\(826\) 0 0
\(827\) −13.2111 −0.459395 −0.229698 0.973262i \(-0.573774\pi\)
−0.229698 + 0.973262i \(0.573774\pi\)
\(828\) 0 0
\(829\) 25.1833 0.874654 0.437327 0.899303i \(-0.355925\pi\)
0.437327 + 0.899303i \(0.355925\pi\)
\(830\) 0 0
\(831\) −21.2389 −0.736768
\(832\) 0 0
\(833\) −79.6611 −2.76009
\(834\) 0 0
\(835\) −2.78890 −0.0965138
\(836\) 0 0
\(837\) 9.21110 0.318382
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −7.60555 −0.261949
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −47.0278 −1.61589
\(848\) 0 0
\(849\) 7.39445 0.253777
\(850\) 0 0
\(851\) −44.2389 −1.51649
\(852\) 0 0
\(853\) 15.1833 0.519868 0.259934 0.965626i \(-0.416299\pi\)
0.259934 + 0.965626i \(0.416299\pi\)
\(854\) 0 0
\(855\) −0.605551 −0.0207094
\(856\) 0 0
\(857\) 11.6056 0.396438 0.198219 0.980158i \(-0.436484\pi\)
0.198219 + 0.980158i \(0.436484\pi\)
\(858\) 0 0
\(859\) −26.1833 −0.893364 −0.446682 0.894693i \(-0.647395\pi\)
−0.446682 + 0.894693i \(0.647395\pi\)
\(860\) 0 0
\(861\) −20.2389 −0.689738
\(862\) 0 0
\(863\) −3.02776 −0.103066 −0.0515330 0.998671i \(-0.516411\pi\)
−0.0515330 + 0.998671i \(0.516411\pi\)
\(864\) 0 0
\(865\) 21.6333 0.735555
\(866\) 0 0
\(867\) 14.4222 0.489804
\(868\) 0 0
\(869\) 55.2666 1.87479
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 3.21110 0.108679
\(874\) 0 0
\(875\) 41.4500 1.40126
\(876\) 0 0
\(877\) 2.81665 0.0951116 0.0475558 0.998869i \(-0.484857\pi\)
0.0475558 + 0.998869i \(0.484857\pi\)
\(878\) 0 0
\(879\) 14.2111 0.479329
\(880\) 0 0
\(881\) −7.18335 −0.242013 −0.121007 0.992652i \(-0.538612\pi\)
−0.121007 + 0.992652i \(0.538612\pi\)
\(882\) 0 0
\(883\) −5.21110 −0.175368 −0.0876838 0.996148i \(-0.527947\pi\)
−0.0876838 + 0.996148i \(0.527947\pi\)
\(884\) 0 0
\(885\) 5.21110 0.175169
\(886\) 0 0
\(887\) −39.6333 −1.33076 −0.665378 0.746507i \(-0.731730\pi\)
−0.665378 + 0.746507i \(0.731730\pi\)
\(888\) 0 0
\(889\) −18.4222 −0.617861
\(890\) 0 0
\(891\) 4.60555 0.154292
\(892\) 0 0
\(893\) −5.21110 −0.174383
\(894\) 0 0
\(895\) 19.0278 0.636028
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 27.6333 0.921622
\(900\) 0 0
\(901\) −16.8167 −0.560244
\(902\) 0 0
\(903\) −21.2111 −0.705861
\(904\) 0 0
\(905\) −3.18335 −0.105818
\(906\) 0 0
\(907\) −25.2111 −0.837121 −0.418560 0.908189i \(-0.637465\pi\)
−0.418560 + 0.908189i \(0.637465\pi\)
\(908\) 0 0
\(909\) −10.2111 −0.338681
\(910\) 0 0
\(911\) 23.6333 0.783006 0.391503 0.920177i \(-0.371955\pi\)
0.391503 + 0.920177i \(0.371955\pi\)
\(912\) 0 0
\(913\) −45.2111 −1.49627
\(914\) 0 0
\(915\) −0.394449 −0.0130401
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 39.6333 1.30738 0.653691 0.756761i \(-0.273220\pi\)
0.653691 + 0.756761i \(0.273220\pi\)
\(920\) 0 0
\(921\) −12.6056 −0.415367
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −38.4222 −1.26331
\(926\) 0 0
\(927\) 5.81665 0.191044
\(928\) 0 0
\(929\) −27.6056 −0.905709 −0.452854 0.891584i \(-0.649594\pi\)
−0.452854 + 0.891584i \(0.649594\pi\)
\(930\) 0 0
\(931\) −8.60555 −0.282036
\(932\) 0 0
\(933\) 6.18335 0.202434
\(934\) 0 0
\(935\) −25.8167 −0.844295
\(936\) 0 0
\(937\) 45.0555 1.47190 0.735950 0.677036i \(-0.236736\pi\)
0.735950 + 0.677036i \(0.236736\pi\)
\(938\) 0 0
\(939\) −11.2111 −0.365861
\(940\) 0 0
\(941\) −32.7889 −1.06889 −0.534444 0.845204i \(-0.679479\pi\)
−0.534444 + 0.845204i \(0.679479\pi\)
\(942\) 0 0
\(943\) −20.2389 −0.659068
\(944\) 0 0
\(945\) −4.60555 −0.149819
\(946\) 0 0
\(947\) −39.6333 −1.28791 −0.643955 0.765064i \(-0.722707\pi\)
−0.643955 + 0.765064i \(0.722707\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −12.2111 −0.395972
\(952\) 0 0
\(953\) 23.2111 0.751881 0.375941 0.926644i \(-0.377320\pi\)
0.375941 + 0.926644i \(0.377320\pi\)
\(954\) 0 0
\(955\) 19.6333 0.635319
\(956\) 0 0
\(957\) 13.8167 0.446629
\(958\) 0 0
\(959\) 22.1833 0.716338
\(960\) 0 0
\(961\) 53.8444 1.73692
\(962\) 0 0
\(963\) −0.605551 −0.0195136
\(964\) 0 0
\(965\) 22.2111 0.715001
\(966\) 0 0
\(967\) −39.0278 −1.25505 −0.627524 0.778597i \(-0.715932\pi\)
−0.627524 + 0.778597i \(0.715932\pi\)
\(968\) 0 0
\(969\) 3.39445 0.109045
\(970\) 0 0
\(971\) 43.2666 1.38849 0.694246 0.719738i \(-0.255738\pi\)
0.694246 + 0.719738i \(0.255738\pi\)
\(972\) 0 0
\(973\) 79.2666 2.54117
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 14.4500 0.462295 0.231148 0.972919i \(-0.425752\pi\)
0.231148 + 0.972919i \(0.425752\pi\)
\(978\) 0 0
\(979\) −57.2111 −1.82847
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 22.7889 0.726853 0.363426 0.931623i \(-0.381607\pi\)
0.363426 + 0.931623i \(0.381607\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 0 0
\(987\) −39.6333 −1.26154
\(988\) 0 0
\(989\) −21.2111 −0.674474
\(990\) 0 0
\(991\) −28.6056 −0.908685 −0.454343 0.890827i \(-0.650126\pi\)
−0.454343 + 0.890827i \(0.650126\pi\)
\(992\) 0 0
\(993\) −27.6333 −0.876917
\(994\) 0 0
\(995\) −1.81665 −0.0575918
\(996\) 0 0
\(997\) −6.44996 −0.204272 −0.102136 0.994770i \(-0.532568\pi\)
−0.102136 + 0.994770i \(0.532568\pi\)
\(998\) 0 0
\(999\) 9.60555 0.303906
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 4056.2.a.w.1.1 2
4.3 odd 2 8112.2.a.bn.1.2 2
13.4 even 6 312.2.q.d.289.1 yes 4
13.5 odd 4 4056.2.c.l.337.1 4
13.8 odd 4 4056.2.c.l.337.4 4
13.10 even 6 312.2.q.d.217.1 4
13.12 even 2 4056.2.a.v.1.2 2
39.17 odd 6 936.2.t.e.289.1 4
39.23 odd 6 936.2.t.e.217.1 4
52.23 odd 6 624.2.q.i.529.2 4
52.43 odd 6 624.2.q.i.289.2 4
52.51 odd 2 8112.2.a.bl.1.1 2
156.23 even 6 1872.2.t.q.1153.2 4
156.95 even 6 1872.2.t.q.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.q.d.217.1 4 13.10 even 6
312.2.q.d.289.1 yes 4 13.4 even 6
624.2.q.i.289.2 4 52.43 odd 6
624.2.q.i.529.2 4 52.23 odd 6
936.2.t.e.217.1 4 39.23 odd 6
936.2.t.e.289.1 4 39.17 odd 6
1872.2.t.q.289.2 4 156.95 even 6
1872.2.t.q.1153.2 4 156.23 even 6
4056.2.a.v.1.2 2 13.12 even 2
4056.2.a.w.1.1 2 1.1 even 1 trivial
4056.2.c.l.337.1 4 13.5 odd 4
4056.2.c.l.337.4 4 13.8 odd 4
8112.2.a.bl.1.1 2 52.51 odd 2
8112.2.a.bn.1.2 2 4.3 odd 2