Properties

Label 3040.2.a.f.1.2
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
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] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, 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("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +2.82843 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{5} +2.82843 q^{7} -3.00000 q^{9} -4.00000 q^{11} -0.828427 q^{13} -3.65685 q^{17} +1.00000 q^{19} +2.82843 q^{23} +1.00000 q^{25} +7.65685 q^{29} -5.65685 q^{31} +2.82843 q^{35} +10.4853 q^{37} +7.65685 q^{41} +8.48528 q^{43} -3.00000 q^{45} +10.8284 q^{47} +1.00000 q^{49} +12.8284 q^{53} -4.00000 q^{55} -1.65685 q^{59} +6.00000 q^{61} -8.48528 q^{63} -0.828427 q^{65} -11.3137 q^{67} -5.65685 q^{71} +4.34315 q^{73} -11.3137 q^{77} +5.65685 q^{79} +9.00000 q^{81} +10.8284 q^{83} -3.65685 q^{85} -0.343146 q^{89} -2.34315 q^{91} +1.00000 q^{95} +12.8284 q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 6 q^{9} - 8 q^{11} + 4 q^{13} + 4 q^{17} + 2 q^{19} + 2 q^{25} + 4 q^{29} + 4 q^{37} + 4 q^{41} - 6 q^{45} + 16 q^{47} + 2 q^{49} + 20 q^{53} - 8 q^{55} + 8 q^{59} + 12 q^{61} + 4 q^{65} + 20 q^{73} + 18 q^{81} + 16 q^{83} + 4 q^{85} - 12 q^{89} - 16 q^{91} + 2 q^{95} + 20 q^{97} + 24 q^{99}+O(q^{100}) \) 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
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −0.828427 −0.229764 −0.114882 0.993379i \(-0.536649\pi\)
−0.114882 + 0.993379i \(0.536649\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.65685 −0.886917 −0.443459 0.896295i \(-0.646249\pi\)
−0.443459 + 0.896295i \(0.646249\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.65685 1.42184 0.710921 0.703272i \(-0.248278\pi\)
0.710921 + 0.703272i \(0.248278\pi\)
\(30\) 0 0
\(31\) −5.65685 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.82843 0.478091
\(36\) 0 0
\(37\) 10.4853 1.72377 0.861885 0.507104i \(-0.169284\pi\)
0.861885 + 0.507104i \(0.169284\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.65685 1.19580 0.597900 0.801571i \(-0.296002\pi\)
0.597900 + 0.801571i \(0.296002\pi\)
\(42\) 0 0
\(43\) 8.48528 1.29399 0.646997 0.762493i \(-0.276025\pi\)
0.646997 + 0.762493i \(0.276025\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) 10.8284 1.57949 0.789744 0.613436i \(-0.210213\pi\)
0.789744 + 0.613436i \(0.210213\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.8284 1.76212 0.881060 0.473005i \(-0.156831\pi\)
0.881060 + 0.473005i \(0.156831\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.65685 −0.215704 −0.107852 0.994167i \(-0.534397\pi\)
−0.107852 + 0.994167i \(0.534397\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) −8.48528 −1.06904
\(64\) 0 0
\(65\) −0.828427 −0.102754
\(66\) 0 0
\(67\) −11.3137 −1.38219 −0.691095 0.722764i \(-0.742871\pi\)
−0.691095 + 0.722764i \(0.742871\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) 4.34315 0.508327 0.254163 0.967161i \(-0.418200\pi\)
0.254163 + 0.967161i \(0.418200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −11.3137 −1.28932
\(78\) 0 0
\(79\) 5.65685 0.636446 0.318223 0.948016i \(-0.396914\pi\)
0.318223 + 0.948016i \(0.396914\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 10.8284 1.18857 0.594287 0.804253i \(-0.297434\pi\)
0.594287 + 0.804253i \(0.297434\pi\)
\(84\) 0 0
\(85\) −3.65685 −0.396642
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.343146 −0.0363734 −0.0181867 0.999835i \(-0.505789\pi\)
−0.0181867 + 0.999835i \(0.505789\pi\)
\(90\) 0 0
\(91\) −2.34315 −0.245628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 12.8284 1.30253 0.651265 0.758851i \(-0.274239\pi\)
0.651265 + 0.758851i \(0.274239\pi\)
\(98\) 0 0
\(99\) 12.0000 1.20605
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) −6.34315 −0.625009 −0.312504 0.949916i \(-0.601168\pi\)
−0.312504 + 0.949916i \(0.601168\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.34315 0.226520 0.113260 0.993565i \(-0.463871\pi\)
0.113260 + 0.993565i \(0.463871\pi\)
\(108\) 0 0
\(109\) −17.3137 −1.65835 −0.829176 0.558987i \(-0.811190\pi\)
−0.829176 + 0.558987i \(0.811190\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.48528 −0.610084 −0.305042 0.952339i \(-0.598670\pi\)
−0.305042 + 0.952339i \(0.598670\pi\)
\(114\) 0 0
\(115\) 2.82843 0.263752
\(116\) 0 0
\(117\) 2.48528 0.229764
\(118\) 0 0
\(119\) −10.3431 −0.948155
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −20.9706 −1.86084 −0.930418 0.366499i \(-0.880556\pi\)
−0.930418 + 0.366499i \(0.880556\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.65685 −0.144760 −0.0723800 0.997377i \(-0.523059\pi\)
−0.0723800 + 0.997377i \(0.523059\pi\)
\(132\) 0 0
\(133\) 2.82843 0.245256
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.34315 0.371060 0.185530 0.982639i \(-0.440600\pi\)
0.185530 + 0.982639i \(0.440600\pi\)
\(138\) 0 0
\(139\) −20.9706 −1.77870 −0.889350 0.457227i \(-0.848843\pi\)
−0.889350 + 0.457227i \(0.848843\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.31371 0.277106
\(144\) 0 0
\(145\) 7.65685 0.635867
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −19.3137 −1.57173 −0.785864 0.618400i \(-0.787781\pi\)
−0.785864 + 0.618400i \(0.787781\pi\)
\(152\) 0 0
\(153\) 10.9706 0.886917
\(154\) 0 0
\(155\) −5.65685 −0.454369
\(156\) 0 0
\(157\) −4.34315 −0.346621 −0.173310 0.984867i \(-0.555446\pi\)
−0.173310 + 0.984867i \(0.555446\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 13.1716 1.03168 0.515839 0.856686i \(-0.327480\pi\)
0.515839 + 0.856686i \(0.327480\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.686292 −0.0531068 −0.0265534 0.999647i \(-0.508453\pi\)
−0.0265534 + 0.999647i \(0.508453\pi\)
\(168\) 0 0
\(169\) −12.3137 −0.947208
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) 0 0
\(173\) 16.1421 1.22726 0.613632 0.789592i \(-0.289708\pi\)
0.613632 + 0.789592i \(0.289708\pi\)
\(174\) 0 0
\(175\) 2.82843 0.213809
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 20.9706 1.56741 0.783707 0.621131i \(-0.213327\pi\)
0.783707 + 0.621131i \(0.213327\pi\)
\(180\) 0 0
\(181\) −9.31371 −0.692283 −0.346141 0.938182i \(-0.612508\pi\)
−0.346141 + 0.938182i \(0.612508\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.4853 0.770893
\(186\) 0 0
\(187\) 14.6274 1.06966
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.9706 1.51738 0.758688 0.651454i \(-0.225841\pi\)
0.758688 + 0.651454i \(0.225841\pi\)
\(192\) 0 0
\(193\) −3.17157 −0.228295 −0.114147 0.993464i \(-0.536414\pi\)
−0.114147 + 0.993464i \(0.536414\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −21.3137 −1.51854 −0.759269 0.650776i \(-0.774444\pi\)
−0.759269 + 0.650776i \(0.774444\pi\)
\(198\) 0 0
\(199\) −15.3137 −1.08556 −0.542780 0.839875i \(-0.682628\pi\)
−0.542780 + 0.839875i \(0.682628\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 21.6569 1.52001
\(204\) 0 0
\(205\) 7.65685 0.534778
\(206\) 0 0
\(207\) −8.48528 −0.589768
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.48528 0.578691
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.02944 0.203782
\(222\) 0 0
\(223\) −6.34315 −0.424768 −0.212384 0.977186i \(-0.568123\pi\)
−0.212384 + 0.977186i \(0.568123\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) 5.65685 0.375459 0.187729 0.982221i \(-0.439887\pi\)
0.187729 + 0.982221i \(0.439887\pi\)
\(228\) 0 0
\(229\) 25.3137 1.67278 0.836388 0.548137i \(-0.184663\pi\)
0.836388 + 0.548137i \(0.184663\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.3137 1.39631 0.698154 0.715948i \(-0.254005\pi\)
0.698154 + 0.715948i \(0.254005\pi\)
\(234\) 0 0
\(235\) 10.8284 0.706369
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 26.9706 1.73733 0.868663 0.495403i \(-0.164980\pi\)
0.868663 + 0.495403i \(0.164980\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −0.828427 −0.0527116
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.9706 −0.818695 −0.409347 0.912379i \(-0.634244\pi\)
−0.409347 + 0.912379i \(0.634244\pi\)
\(252\) 0 0
\(253\) −11.3137 −0.711287
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −16.8284 −1.04973 −0.524864 0.851186i \(-0.675884\pi\)
−0.524864 + 0.851186i \(0.675884\pi\)
\(258\) 0 0
\(259\) 29.6569 1.84279
\(260\) 0 0
\(261\) −22.9706 −1.42184
\(262\) 0 0
\(263\) −8.48528 −0.523225 −0.261612 0.965173i \(-0.584254\pi\)
−0.261612 + 0.965173i \(0.584254\pi\)
\(264\) 0 0
\(265\) 12.8284 0.788044
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 14.9706 0.899494 0.449747 0.893156i \(-0.351514\pi\)
0.449747 + 0.893156i \(0.351514\pi\)
\(278\) 0 0
\(279\) 16.9706 1.01600
\(280\) 0 0
\(281\) 4.34315 0.259090 0.129545 0.991574i \(-0.458648\pi\)
0.129545 + 0.991574i \(0.458648\pi\)
\(282\) 0 0
\(283\) −24.4853 −1.45550 −0.727749 0.685843i \(-0.759434\pi\)
−0.727749 + 0.685843i \(0.759434\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 21.6569 1.27836
\(288\) 0 0
\(289\) −3.62742 −0.213377
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6.48528 −0.378874 −0.189437 0.981893i \(-0.560666\pi\)
−0.189437 + 0.981893i \(0.560666\pi\)
\(294\) 0 0
\(295\) −1.65685 −0.0964658
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.34315 −0.135508
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.00000 0.343559
\(306\) 0 0
\(307\) 5.65685 0.322854 0.161427 0.986885i \(-0.448390\pi\)
0.161427 + 0.986885i \(0.448390\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) 0 0
\(313\) −28.6274 −1.61812 −0.809059 0.587728i \(-0.800023\pi\)
−0.809059 + 0.587728i \(0.800023\pi\)
\(314\) 0 0
\(315\) −8.48528 −0.478091
\(316\) 0 0
\(317\) 7.17157 0.402796 0.201398 0.979510i \(-0.435452\pi\)
0.201398 + 0.979510i \(0.435452\pi\)
\(318\) 0 0
\(319\) −30.6274 −1.71481
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.65685 −0.203473
\(324\) 0 0
\(325\) −0.828427 −0.0459529
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 30.6274 1.68854
\(330\) 0 0
\(331\) 31.3137 1.72116 0.860579 0.509318i \(-0.170102\pi\)
0.860579 + 0.509318i \(0.170102\pi\)
\(332\) 0 0
\(333\) −31.4558 −1.72377
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) −11.1716 −0.608554 −0.304277 0.952584i \(-0.598415\pi\)
−0.304277 + 0.952584i \(0.598415\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 22.6274 1.22534
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −34.8284 −1.86969 −0.934844 0.355059i \(-0.884461\pi\)
−0.934844 + 0.355059i \(0.884461\pi\)
\(348\) 0 0
\(349\) 31.9411 1.70977 0.854885 0.518818i \(-0.173628\pi\)
0.854885 + 0.518818i \(0.173628\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8.34315 −0.444061 −0.222030 0.975040i \(-0.571268\pi\)
−0.222030 + 0.975040i \(0.571268\pi\)
\(354\) 0 0
\(355\) −5.65685 −0.300235
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −14.3431 −0.757002 −0.378501 0.925601i \(-0.623560\pi\)
−0.378501 + 0.925601i \(0.623560\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.34315 0.227331
\(366\) 0 0
\(367\) 17.4558 0.911188 0.455594 0.890188i \(-0.349427\pi\)
0.455594 + 0.890188i \(0.349427\pi\)
\(368\) 0 0
\(369\) −22.9706 −1.19580
\(370\) 0 0
\(371\) 36.2843 1.88379
\(372\) 0 0
\(373\) 16.1421 0.835808 0.417904 0.908491i \(-0.362765\pi\)
0.417904 + 0.908491i \(0.362765\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.34315 −0.326689
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.00000 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) 0 0
\(385\) −11.3137 −0.576600
\(386\) 0 0
\(387\) −25.4558 −1.29399
\(388\) 0 0
\(389\) −32.6274 −1.65428 −0.827138 0.561999i \(-0.810032\pi\)
−0.827138 + 0.561999i \(0.810032\pi\)
\(390\) 0 0
\(391\) −10.3431 −0.523075
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.65685 0.284627
\(396\) 0 0
\(397\) 6.97056 0.349843 0.174921 0.984582i \(-0.444033\pi\)
0.174921 + 0.984582i \(0.444033\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.0000 1.29838 0.649189 0.760627i \(-0.275108\pi\)
0.649189 + 0.760627i \(0.275108\pi\)
\(402\) 0 0
\(403\) 4.68629 0.233441
\(404\) 0 0
\(405\) 9.00000 0.447214
\(406\) 0 0
\(407\) −41.9411 −2.07894
\(408\) 0 0
\(409\) −9.31371 −0.460533 −0.230267 0.973128i \(-0.573960\pi\)
−0.230267 + 0.973128i \(0.573960\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4.68629 −0.230597
\(414\) 0 0
\(415\) 10.8284 0.531547
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 23.3137 1.13895 0.569475 0.822009i \(-0.307147\pi\)
0.569475 + 0.822009i \(0.307147\pi\)
\(420\) 0 0
\(421\) 10.9706 0.534673 0.267336 0.963603i \(-0.413857\pi\)
0.267336 + 0.963603i \(0.413857\pi\)
\(422\) 0 0
\(423\) −32.4853 −1.57949
\(424\) 0 0
\(425\) −3.65685 −0.177383
\(426\) 0 0
\(427\) 16.9706 0.821263
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.6863 −0.611077 −0.305539 0.952180i \(-0.598836\pi\)
−0.305539 + 0.952180i \(0.598836\pi\)
\(432\) 0 0
\(433\) −3.17157 −0.152416 −0.0762080 0.997092i \(-0.524281\pi\)
−0.0762080 + 0.997092i \(0.524281\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.82843 0.135302
\(438\) 0 0
\(439\) −8.97056 −0.428142 −0.214071 0.976818i \(-0.568672\pi\)
−0.214071 + 0.976818i \(0.568672\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −16.4853 −0.783239 −0.391620 0.920127i \(-0.628085\pi\)
−0.391620 + 0.920127i \(0.628085\pi\)
\(444\) 0 0
\(445\) −0.343146 −0.0162667
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10.6863 −0.504317 −0.252159 0.967686i \(-0.581140\pi\)
−0.252159 + 0.967686i \(0.581140\pi\)
\(450\) 0 0
\(451\) −30.6274 −1.44219
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.34315 −0.109848
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 22.1421 1.02903 0.514516 0.857481i \(-0.327972\pi\)
0.514516 + 0.857481i \(0.327972\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −14.1421 −0.654420 −0.327210 0.944952i \(-0.606108\pi\)
−0.327210 + 0.944952i \(0.606108\pi\)
\(468\) 0 0
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −33.9411 −1.56061
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −38.4853 −1.76212
\(478\) 0 0
\(479\) −12.9706 −0.592640 −0.296320 0.955089i \(-0.595760\pi\)
−0.296320 + 0.955089i \(0.595760\pi\)
\(480\) 0 0
\(481\) −8.68629 −0.396061
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.8284 0.582509
\(486\) 0 0
\(487\) 26.6274 1.20660 0.603302 0.797513i \(-0.293851\pi\)
0.603302 + 0.797513i \(0.293851\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4.97056 −0.224318 −0.112159 0.993690i \(-0.535777\pi\)
−0.112159 + 0.993690i \(0.535777\pi\)
\(492\) 0 0
\(493\) −28.0000 −1.26106
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) 0 0
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −16.2843 −0.728984 −0.364492 0.931207i \(-0.618757\pi\)
−0.364492 + 0.931207i \(0.618757\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −0.485281 −0.0216376 −0.0108188 0.999941i \(-0.503444\pi\)
−0.0108188 + 0.999941i \(0.503444\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −22.9706 −1.01815 −0.509076 0.860721i \(-0.670013\pi\)
−0.509076 + 0.860721i \(0.670013\pi\)
\(510\) 0 0
\(511\) 12.2843 0.543424
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.34315 −0.279512
\(516\) 0 0
\(517\) −43.3137 −1.90493
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 29.3137 1.28426 0.642128 0.766597i \(-0.278052\pi\)
0.642128 + 0.766597i \(0.278052\pi\)
\(522\) 0 0
\(523\) 7.02944 0.307376 0.153688 0.988119i \(-0.450885\pi\)
0.153688 + 0.988119i \(0.450885\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.6863 0.901109
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 4.97056 0.215704
\(532\) 0 0
\(533\) −6.34315 −0.274752
\(534\) 0 0
\(535\) 2.34315 0.101303
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −17.3137 −0.741638
\(546\) 0 0
\(547\) 20.2843 0.867293 0.433646 0.901083i \(-0.357227\pi\)
0.433646 + 0.901083i \(0.357227\pi\)
\(548\) 0 0
\(549\) −18.0000 −0.768221
\(550\) 0 0
\(551\) 7.65685 0.326193
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 22.9706 0.973294 0.486647 0.873599i \(-0.338220\pi\)
0.486647 + 0.873599i \(0.338220\pi\)
\(558\) 0 0
\(559\) −7.02944 −0.297314
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.3431 0.435912 0.217956 0.975959i \(-0.430061\pi\)
0.217956 + 0.975959i \(0.430061\pi\)
\(564\) 0 0
\(565\) −6.48528 −0.272838
\(566\) 0 0
\(567\) 25.4558 1.06904
\(568\) 0 0
\(569\) 35.9411 1.50673 0.753365 0.657602i \(-0.228429\pi\)
0.753365 + 0.657602i \(0.228429\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.82843 0.117954
\(576\) 0 0
\(577\) −36.6274 −1.52482 −0.762410 0.647095i \(-0.775984\pi\)
−0.762410 + 0.647095i \(0.775984\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 30.6274 1.27064
\(582\) 0 0
\(583\) −51.3137 −2.12520
\(584\) 0 0
\(585\) 2.48528 0.102754
\(586\) 0 0
\(587\) −22.1421 −0.913904 −0.456952 0.889491i \(-0.651059\pi\)
−0.456952 + 0.889491i \(0.651059\pi\)
\(588\) 0 0
\(589\) −5.65685 −0.233087
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8.62742 0.354286 0.177143 0.984185i \(-0.443315\pi\)
0.177143 + 0.984185i \(0.443315\pi\)
\(594\) 0 0
\(595\) −10.3431 −0.424028
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 5.65685 0.231133 0.115566 0.993300i \(-0.463132\pi\)
0.115566 + 0.993300i \(0.463132\pi\)
\(600\) 0 0
\(601\) −38.9706 −1.58964 −0.794821 0.606844i \(-0.792435\pi\)
−0.794821 + 0.606844i \(0.792435\pi\)
\(602\) 0 0
\(603\) 33.9411 1.38219
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) 36.0000 1.46119 0.730597 0.682808i \(-0.239242\pi\)
0.730597 + 0.682808i \(0.239242\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.97056 −0.362910
\(612\) 0 0
\(613\) −34.9706 −1.41245 −0.706224 0.707989i \(-0.749603\pi\)
−0.706224 + 0.707989i \(0.749603\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −45.5980 −1.83571 −0.917853 0.396921i \(-0.870079\pi\)
−0.917853 + 0.396921i \(0.870079\pi\)
\(618\) 0 0
\(619\) 43.5980 1.75235 0.876175 0.481992i \(-0.160087\pi\)
0.876175 + 0.481992i \(0.160087\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −0.970563 −0.0388848
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −38.3431 −1.52884
\(630\) 0 0
\(631\) 26.6274 1.06002 0.530010 0.847991i \(-0.322188\pi\)
0.530010 + 0.847991i \(0.322188\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −20.9706 −0.832191
\(636\) 0 0
\(637\) −0.828427 −0.0328235
\(638\) 0 0
\(639\) 16.9706 0.671345
\(640\) 0 0
\(641\) 20.3431 0.803506 0.401753 0.915748i \(-0.368401\pi\)
0.401753 + 0.915748i \(0.368401\pi\)
\(642\) 0 0
\(643\) −15.1127 −0.595987 −0.297993 0.954568i \(-0.596317\pi\)
−0.297993 + 0.954568i \(0.596317\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.14214 −0.241472 −0.120736 0.992685i \(-0.538525\pi\)
−0.120736 + 0.992685i \(0.538525\pi\)
\(648\) 0 0
\(649\) 6.62742 0.260149
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11.6569 0.456168 0.228084 0.973641i \(-0.426754\pi\)
0.228084 + 0.973641i \(0.426754\pi\)
\(654\) 0 0
\(655\) −1.65685 −0.0647387
\(656\) 0 0
\(657\) −13.0294 −0.508327
\(658\) 0 0
\(659\) 9.65685 0.376178 0.188089 0.982152i \(-0.439771\pi\)
0.188089 + 0.982152i \(0.439771\pi\)
\(660\) 0 0
\(661\) 12.3431 0.480093 0.240046 0.970761i \(-0.422837\pi\)
0.240046 + 0.970761i \(0.422837\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.82843 0.109682
\(666\) 0 0
\(667\) 21.6569 0.838557
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 25.1127 0.968023 0.484012 0.875062i \(-0.339179\pi\)
0.484012 + 0.875062i \(0.339179\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −40.4264 −1.55371 −0.776857 0.629678i \(-0.783187\pi\)
−0.776857 + 0.629678i \(0.783187\pi\)
\(678\) 0 0
\(679\) 36.2843 1.39246
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.2843 0.470045 0.235022 0.971990i \(-0.424484\pi\)
0.235022 + 0.971990i \(0.424484\pi\)
\(684\) 0 0
\(685\) 4.34315 0.165943
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −10.6274 −0.404872
\(690\) 0 0
\(691\) 39.3137 1.49556 0.747782 0.663944i \(-0.231119\pi\)
0.747782 + 0.663944i \(0.231119\pi\)
\(692\) 0 0
\(693\) 33.9411 1.28932
\(694\) 0 0
\(695\) −20.9706 −0.795459
\(696\) 0 0
\(697\) −28.0000 −1.06058
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 18.6863 0.705771 0.352886 0.935666i \(-0.385200\pi\)
0.352886 + 0.935666i \(0.385200\pi\)
\(702\) 0 0
\(703\) 10.4853 0.395460
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 39.5980 1.48924
\(708\) 0 0
\(709\) −43.9411 −1.65024 −0.825122 0.564955i \(-0.808894\pi\)
−0.825122 + 0.564955i \(0.808894\pi\)
\(710\) 0 0
\(711\) −16.9706 −0.636446
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 3.31371 0.123926
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −16.2843 −0.607301 −0.303650 0.952784i \(-0.598205\pi\)
−0.303650 + 0.952784i \(0.598205\pi\)
\(720\) 0 0
\(721\) −17.9411 −0.668162
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.65685 0.284368
\(726\) 0 0
\(727\) −32.4853 −1.20481 −0.602406 0.798190i \(-0.705791\pi\)
−0.602406 + 0.798190i \(0.705791\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −31.0294 −1.14767
\(732\) 0 0
\(733\) −29.3137 −1.08273 −0.541363 0.840789i \(-0.682092\pi\)
−0.541363 + 0.840789i \(0.682092\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 45.2548 1.66698
\(738\) 0 0
\(739\) 31.3137 1.15189 0.575947 0.817487i \(-0.304634\pi\)
0.575947 + 0.817487i \(0.304634\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 50.6274 1.85734 0.928670 0.370907i \(-0.120953\pi\)
0.928670 + 0.370907i \(0.120953\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) −32.4853 −1.18857
\(748\) 0 0
\(749\) 6.62742 0.242161
\(750\) 0 0
\(751\) −22.6274 −0.825686 −0.412843 0.910802i \(-0.635464\pi\)
−0.412843 + 0.910802i \(0.635464\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −19.3137 −0.702898
\(756\) 0 0
\(757\) −21.3137 −0.774660 −0.387330 0.921941i \(-0.626603\pi\)
−0.387330 + 0.921941i \(0.626603\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −20.6274 −0.747743 −0.373872 0.927480i \(-0.621970\pi\)
−0.373872 + 0.927480i \(0.621970\pi\)
\(762\) 0 0
\(763\) −48.9706 −1.77285
\(764\) 0 0
\(765\) 10.9706 0.396642
\(766\) 0 0
\(767\) 1.37258 0.0495611
\(768\) 0 0
\(769\) 39.2548 1.41557 0.707783 0.706430i \(-0.249696\pi\)
0.707783 + 0.706430i \(0.249696\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 42.4853 1.52809 0.764045 0.645163i \(-0.223211\pi\)
0.764045 + 0.645163i \(0.223211\pi\)
\(774\) 0 0
\(775\) −5.65685 −0.203200
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.65685 0.274335
\(780\) 0 0
\(781\) 22.6274 0.809673
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.34315 −0.155014
\(786\) 0 0
\(787\) −12.2843 −0.437887 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −18.3431 −0.652207
\(792\) 0 0
\(793\) −4.97056 −0.176510
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −11.1716 −0.395717 −0.197859 0.980231i \(-0.563399\pi\)
−0.197859 + 0.980231i \(0.563399\pi\)
\(798\) 0 0
\(799\) −39.5980 −1.40088
\(800\) 0 0
\(801\) 1.02944 0.0363734
\(802\) 0 0
\(803\) −17.3726 −0.613065
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.37258 0.118574 0.0592869 0.998241i \(-0.481117\pi\)
0.0592869 + 0.998241i \(0.481117\pi\)
\(810\) 0 0
\(811\) −33.6569 −1.18185 −0.590926 0.806726i \(-0.701237\pi\)
−0.590926 + 0.806726i \(0.701237\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 13.1716 0.461380
\(816\) 0 0
\(817\) 8.48528 0.296862
\(818\) 0 0
\(819\) 7.02944 0.245628
\(820\) 0 0
\(821\) −13.3137 −0.464652 −0.232326 0.972638i \(-0.574634\pi\)
−0.232326 + 0.972638i \(0.574634\pi\)
\(822\) 0 0
\(823\) 3.79899 0.132424 0.0662122 0.997806i \(-0.478909\pi\)
0.0662122 + 0.997806i \(0.478909\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −16.0000 −0.556375 −0.278187 0.960527i \(-0.589734\pi\)
−0.278187 + 0.960527i \(0.589734\pi\)
\(828\) 0 0
\(829\) 39.2548 1.36338 0.681688 0.731643i \(-0.261246\pi\)
0.681688 + 0.731643i \(0.261246\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.65685 −0.126702
\(834\) 0 0
\(835\) −0.686292 −0.0237501
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −26.3431 −0.909466 −0.454733 0.890628i \(-0.650265\pi\)
−0.454733 + 0.890628i \(0.650265\pi\)
\(840\) 0 0
\(841\) 29.6274 1.02164
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −12.3137 −0.423604
\(846\) 0 0
\(847\) 14.1421 0.485930
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 29.6569 1.01662
\(852\) 0 0
\(853\) −0.627417 −0.0214823 −0.0107412 0.999942i \(-0.503419\pi\)
−0.0107412 + 0.999942i \(0.503419\pi\)
\(854\) 0 0
\(855\) −3.00000 −0.102598
\(856\) 0 0
\(857\) 21.7990 0.744639 0.372320 0.928105i \(-0.378563\pi\)
0.372320 + 0.928105i \(0.378563\pi\)
\(858\) 0 0
\(859\) 0.686292 0.0234160 0.0117080 0.999931i \(-0.496273\pi\)
0.0117080 + 0.999931i \(0.496273\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 52.0000 1.77010 0.885050 0.465495i \(-0.154124\pi\)
0.885050 + 0.465495i \(0.154124\pi\)
\(864\) 0 0
\(865\) 16.1421 0.548849
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −22.6274 −0.767583
\(870\) 0 0
\(871\) 9.37258 0.317578
\(872\) 0 0
\(873\) −38.4853 −1.30253
\(874\) 0 0
\(875\) 2.82843 0.0956183
\(876\) 0 0
\(877\) −9.79899 −0.330888 −0.165444 0.986219i \(-0.552906\pi\)
−0.165444 + 0.986219i \(0.552906\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 29.3137 0.987604 0.493802 0.869574i \(-0.335607\pi\)
0.493802 + 0.869574i \(0.335607\pi\)
\(882\) 0 0
\(883\) −10.8284 −0.364406 −0.182203 0.983261i \(-0.558323\pi\)
−0.182203 + 0.983261i \(0.558323\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 25.6569 0.861473 0.430736 0.902478i \(-0.358254\pi\)
0.430736 + 0.902478i \(0.358254\pi\)
\(888\) 0 0
\(889\) −59.3137 −1.98932
\(890\) 0 0
\(891\) −36.0000 −1.20605
\(892\) 0 0
\(893\) 10.8284 0.362359
\(894\) 0 0
\(895\) 20.9706 0.700969
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −43.3137 −1.44459
\(900\) 0 0
\(901\) −46.9117 −1.56285
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9.31371 −0.309598
\(906\) 0 0
\(907\) −12.2843 −0.407893 −0.203946 0.978982i \(-0.565377\pi\)
−0.203946 + 0.978982i \(0.565377\pi\)
\(908\) 0 0
\(909\) −42.0000 −1.39305
\(910\) 0 0
\(911\) −45.2548 −1.49936 −0.749680 0.661801i \(-0.769792\pi\)
−0.749680 + 0.661801i \(0.769792\pi\)
\(912\) 0 0
\(913\) −43.3137 −1.43347
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4.68629 −0.154755
\(918\) 0 0
\(919\) −13.3726 −0.441121 −0.220560 0.975373i \(-0.570789\pi\)
−0.220560 + 0.975373i \(0.570789\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 4.68629 0.154251
\(924\) 0 0
\(925\) 10.4853 0.344754
\(926\) 0 0
\(927\) 19.0294 0.625009
\(928\) 0 0
\(929\) −38.0000 −1.24674 −0.623370 0.781927i \(-0.714237\pi\)
−0.623370 + 0.781927i \(0.714237\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 14.6274 0.478368
\(936\) 0 0
\(937\) −43.6569 −1.42621 −0.713104 0.701059i \(-0.752711\pi\)
−0.713104 + 0.701059i \(0.752711\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 21.6569 0.705244
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −42.8284 −1.39174 −0.695868 0.718169i \(-0.744980\pi\)
−0.695868 + 0.718169i \(0.744980\pi\)
\(948\) 0 0
\(949\) −3.59798 −0.116795
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 37.7990 1.22443 0.612215 0.790692i \(-0.290279\pi\)
0.612215 + 0.790692i \(0.290279\pi\)
\(954\) 0 0
\(955\) 20.9706 0.678591
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 12.2843 0.396680
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −7.02944 −0.226520
\(964\) 0 0
\(965\) −3.17157 −0.102097
\(966\) 0 0
\(967\) 23.1127 0.743254 0.371627 0.928382i \(-0.378800\pi\)
0.371627 + 0.928382i \(0.378800\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.97056 −0.159513 −0.0797565 0.996814i \(-0.525414\pi\)
−0.0797565 + 0.996814i \(0.525414\pi\)
\(972\) 0 0
\(973\) −59.3137 −1.90151
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 45.7990 1.46524 0.732620 0.680638i \(-0.238297\pi\)
0.732620 + 0.680638i \(0.238297\pi\)
\(978\) 0 0
\(979\) 1.37258 0.0438679
\(980\) 0 0
\(981\) 51.9411 1.65835
\(982\) 0 0
\(983\) −7.31371 −0.233271 −0.116636 0.993175i \(-0.537211\pi\)
−0.116636 + 0.993175i \(0.537211\pi\)
\(984\) 0 0
\(985\) −21.3137 −0.679111
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −15.3137 −0.485477
\(996\) 0 0
\(997\) −23.6569 −0.749220 −0.374610 0.927182i \(-0.622223\pi\)
−0.374610 + 0.927182i \(0.622223\pi\)
\(998\) 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 3040.2.a.f.1.2 2
4.3 odd 2 3040.2.a.g.1.1 yes 2
8.3 odd 2 6080.2.a.bd.1.1 2
8.5 even 2 6080.2.a.be.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.f.1.2 2 1.1 even 1 trivial
3040.2.a.g.1.1 yes 2 4.3 odd 2
6080.2.a.bd.1.1 2 8.3 odd 2
6080.2.a.be.1.2 2 8.5 even 2