Properties

Label 3040.2.a.g.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +2.82843 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{5} +2.82843 q^{7} -3.00000 q^{9} +4.00000 q^{11} +4.82843 q^{13} +7.65685 q^{17} -1.00000 q^{19} +2.82843 q^{23} +1.00000 q^{25} -3.65685 q^{29} -5.65685 q^{31} +2.82843 q^{35} -6.48528 q^{37} -3.65685 q^{41} +8.48528 q^{43} -3.00000 q^{45} -5.17157 q^{47} +1.00000 q^{49} +7.17157 q^{53} +4.00000 q^{55} -9.65685 q^{59} +6.00000 q^{61} -8.48528 q^{63} +4.82843 q^{65} -11.3137 q^{67} -5.65685 q^{71} +15.6569 q^{73} +11.3137 q^{77} +5.65685 q^{79} +9.00000 q^{81} -5.17157 q^{83} +7.65685 q^{85} -11.6569 q^{89} +13.6569 q^{91} -1.00000 q^{95} +7.17157 q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 6 q^{9}+O(q^{10}) \) 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 \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 4.82843 1.33916 0.669582 0.742738i \(-0.266473\pi\)
0.669582 + 0.742738i \(0.266473\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.65685 1.85706 0.928530 0.371257i \(-0.121073\pi\)
0.928530 + 0.371257i \(0.121073\pi\)
\(18\) 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
\(29\) −3.65685 −0.679061 −0.339530 0.940595i \(-0.610268\pi\)
−0.339530 + 0.940595i \(0.610268\pi\)
\(30\) 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
\(37\) −6.48528 −1.06617 −0.533087 0.846061i \(-0.678968\pi\)
−0.533087 + 0.846061i \(0.678968\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.65685 −0.571105 −0.285552 0.958363i \(-0.592177\pi\)
−0.285552 + 0.958363i \(0.592177\pi\)
\(42\) 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
\(47\) −5.17157 −0.754351 −0.377176 0.926142i \(-0.623105\pi\)
−0.377176 + 0.926142i \(0.623105\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.17157 0.985091 0.492546 0.870287i \(-0.336066\pi\)
0.492546 + 0.870287i \(0.336066\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.65685 −1.25722 −0.628608 0.777723i \(-0.716375\pi\)
−0.628608 + 0.777723i \(0.716375\pi\)
\(60\) 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
\(65\) 4.82843 0.598893
\(66\) 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
\(73\) 15.6569 1.83250 0.916248 0.400611i \(-0.131202\pi\)
0.916248 + 0.400611i \(0.131202\pi\)
\(74\) 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
\(83\) −5.17157 −0.567654 −0.283827 0.958876i \(-0.591604\pi\)
−0.283827 + 0.958876i \(0.591604\pi\)
\(84\) 0 0
\(85\) 7.65685 0.830502
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.6569 −1.23562 −0.617812 0.786326i \(-0.711981\pi\)
−0.617812 + 0.786326i \(0.711981\pi\)
\(90\) 0 0
\(91\) 13.6569 1.43163
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 7.17157 0.728163 0.364081 0.931367i \(-0.381383\pi\)
0.364081 + 0.931367i \(0.381383\pi\)
\(98\) 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
\(103\) 17.6569 1.73978 0.869891 0.493244i \(-0.164189\pi\)
0.869891 + 0.493244i \(0.164189\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.6569 −1.32026 −0.660129 0.751152i \(-0.729498\pi\)
−0.660129 + 0.751152i \(0.729498\pi\)
\(108\) 0 0
\(109\) 5.31371 0.508961 0.254480 0.967078i \(-0.418096\pi\)
0.254480 + 0.967078i \(0.418096\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.4853 0.986372 0.493186 0.869924i \(-0.335832\pi\)
0.493186 + 0.869924i \(0.335832\pi\)
\(114\) 0 0
\(115\) 2.82843 0.263752
\(116\) 0 0
\(117\) −14.4853 −1.33916
\(118\) 0 0
\(119\) 21.6569 1.98528
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −12.9706 −1.15095 −0.575476 0.817819i \(-0.695183\pi\)
−0.575476 + 0.817819i \(0.695183\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.65685 −0.843723 −0.421862 0.906660i \(-0.638623\pi\)
−0.421862 + 0.906660i \(0.638623\pi\)
\(132\) 0 0
\(133\) −2.82843 −0.245256
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.6569 1.33766 0.668828 0.743417i \(-0.266796\pi\)
0.668828 + 0.743417i \(0.266796\pi\)
\(138\) 0 0
\(139\) −12.9706 −1.10015 −0.550074 0.835116i \(-0.685401\pi\)
−0.550074 + 0.835116i \(0.685401\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 19.3137 1.61509
\(144\) 0 0
\(145\) −3.65685 −0.303685
\(146\) 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
\(151\) −3.31371 −0.269666 −0.134833 0.990868i \(-0.543050\pi\)
−0.134833 + 0.990868i \(0.543050\pi\)
\(152\) 0 0
\(153\) −22.9706 −1.85706
\(154\) 0 0
\(155\) −5.65685 −0.454369
\(156\) 0 0
\(157\) −15.6569 −1.24955 −0.624777 0.780804i \(-0.714810\pi\)
−0.624777 + 0.780804i \(0.714810\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) −18.8284 −1.47476 −0.737378 0.675480i \(-0.763936\pi\)
−0.737378 + 0.675480i \(0.763936\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 23.3137 1.80407 0.902034 0.431664i \(-0.142073\pi\)
0.902034 + 0.431664i \(0.142073\pi\)
\(168\) 0 0
\(169\) 10.3137 0.793362
\(170\) 0 0
\(171\) 3.00000 0.229416
\(172\) 0 0
\(173\) −12.1421 −0.923149 −0.461575 0.887101i \(-0.652715\pi\)
−0.461575 + 0.887101i \(0.652715\pi\)
\(174\) 0 0
\(175\) 2.82843 0.213809
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.9706 0.969465 0.484733 0.874662i \(-0.338917\pi\)
0.484733 + 0.874662i \(0.338917\pi\)
\(180\) 0 0
\(181\) 13.3137 0.989600 0.494800 0.869007i \(-0.335241\pi\)
0.494800 + 0.869007i \(0.335241\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.48528 −0.476807
\(186\) 0 0
\(187\) 30.6274 2.23970
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.9706 0.938517 0.469258 0.883061i \(-0.344521\pi\)
0.469258 + 0.883061i \(0.344521\pi\)
\(192\) 0 0
\(193\) −8.82843 −0.635484 −0.317742 0.948177i \(-0.602925\pi\)
−0.317742 + 0.948177i \(0.602925\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.31371 0.0935979 0.0467989 0.998904i \(-0.485098\pi\)
0.0467989 + 0.998904i \(0.485098\pi\)
\(198\) 0 0
\(199\) −7.31371 −0.518455 −0.259228 0.965816i \(-0.583468\pi\)
−0.259228 + 0.965816i \(0.583468\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −10.3431 −0.725947
\(204\) 0 0
\(205\) −3.65685 −0.255406
\(206\) 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 \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 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
\(221\) 36.9706 2.48691
\(222\) 0 0
\(223\) 17.6569 1.18239 0.591195 0.806529i \(-0.298656\pi\)
0.591195 + 0.806529i \(0.298656\pi\)
\(224\) 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
\(229\) 2.68629 0.177515 0.0887576 0.996053i \(-0.471710\pi\)
0.0887576 + 0.996053i \(0.471710\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.31371 −0.0860639 −0.0430320 0.999074i \(-0.513702\pi\)
−0.0430320 + 0.999074i \(0.513702\pi\)
\(234\) 0 0
\(235\) −5.17157 −0.337356
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −6.97056 −0.449013 −0.224507 0.974473i \(-0.572077\pi\)
−0.224507 + 0.974473i \(0.572077\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −4.82843 −0.307225
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.9706 −1.32365 −0.661825 0.749658i \(-0.730218\pi\)
−0.661825 + 0.749658i \(0.730218\pi\)
\(252\) 0 0
\(253\) 11.3137 0.711287
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −11.1716 −0.696864 −0.348432 0.937334i \(-0.613286\pi\)
−0.348432 + 0.937334i \(0.613286\pi\)
\(258\) 0 0
\(259\) −18.3431 −1.13979
\(260\) 0 0
\(261\) 10.9706 0.679061
\(262\) 0 0
\(263\) −8.48528 −0.523225 −0.261612 0.965173i \(-0.584254\pi\)
−0.261612 + 0.965173i \(0.584254\pi\)
\(264\) 0 0
\(265\) 7.17157 0.440546
\(266\) 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 \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −18.9706 −1.13983 −0.569915 0.821703i \(-0.693024\pi\)
−0.569915 + 0.821703i \(0.693024\pi\)
\(278\) 0 0
\(279\) 16.9706 1.01600
\(280\) 0 0
\(281\) 15.6569 0.934010 0.467005 0.884255i \(-0.345333\pi\)
0.467005 + 0.884255i \(0.345333\pi\)
\(282\) 0 0
\(283\) 7.51472 0.446704 0.223352 0.974738i \(-0.428300\pi\)
0.223352 + 0.974738i \(0.428300\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −10.3431 −0.610537
\(288\) 0 0
\(289\) 41.6274 2.44867
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 10.4853 0.612557 0.306278 0.951942i \(-0.400916\pi\)
0.306278 + 0.951942i \(0.400916\pi\)
\(294\) 0 0
\(295\) −9.65685 −0.562244
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 13.6569 0.789796
\(300\) 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 \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 0 0
\(313\) 16.6274 0.939837 0.469919 0.882710i \(-0.344283\pi\)
0.469919 + 0.882710i \(0.344283\pi\)
\(314\) 0 0
\(315\) −8.48528 −0.478091
\(316\) 0 0
\(317\) 12.8284 0.720516 0.360258 0.932853i \(-0.382689\pi\)
0.360258 + 0.932853i \(0.382689\pi\)
\(318\) 0 0
\(319\) −14.6274 −0.818978
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −7.65685 −0.426039
\(324\) 0 0
\(325\) 4.82843 0.267833
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −14.6274 −0.806436
\(330\) 0 0
\(331\) −8.68629 −0.477442 −0.238721 0.971088i \(-0.576728\pi\)
−0.238721 + 0.971088i \(0.576728\pi\)
\(332\) 0 0
\(333\) 19.4558 1.06617
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) −16.8284 −0.916703 −0.458351 0.888771i \(-0.651560\pi\)
−0.458351 + 0.888771i \(0.651560\pi\)
\(338\) 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
\(347\) 29.1716 1.56601 0.783006 0.622014i \(-0.213685\pi\)
0.783006 + 0.622014i \(0.213685\pi\)
\(348\) 0 0
\(349\) −35.9411 −1.92388 −0.961942 0.273253i \(-0.911900\pi\)
−0.961942 + 0.273253i \(0.911900\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −19.6569 −1.04623 −0.523114 0.852262i \(-0.675230\pi\)
−0.523114 + 0.852262i \(0.675230\pi\)
\(354\) 0 0
\(355\) −5.65685 −0.300235
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 25.6569 1.35412 0.677058 0.735929i \(-0.263254\pi\)
0.677058 + 0.735929i \(0.263254\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 15.6569 0.819517
\(366\) 0 0
\(367\) 33.4558 1.74638 0.873190 0.487379i \(-0.162047\pi\)
0.873190 + 0.487379i \(0.162047\pi\)
\(368\) 0 0
\(369\) 10.9706 0.571105
\(370\) 0 0
\(371\) 20.2843 1.05311
\(372\) 0 0
\(373\) −12.1421 −0.628696 −0.314348 0.949308i \(-0.601786\pi\)
−0.314348 + 0.949308i \(0.601786\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −17.6569 −0.909374
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) 11.3137 0.576600
\(386\) 0 0
\(387\) −25.4558 −1.29399
\(388\) 0 0
\(389\) 12.6274 0.640235 0.320118 0.947378i \(-0.396278\pi\)
0.320118 + 0.947378i \(0.396278\pi\)
\(390\) 0 0
\(391\) 21.6569 1.09523
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.65685 0.284627
\(396\) 0 0
\(397\) −26.9706 −1.35361 −0.676807 0.736161i \(-0.736637\pi\)
−0.676807 + 0.736161i \(0.736637\pi\)
\(398\) 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
\(403\) −27.3137 −1.36059
\(404\) 0 0
\(405\) 9.00000 0.447214
\(406\) 0 0
\(407\) −25.9411 −1.28585
\(408\) 0 0
\(409\) 13.3137 0.658321 0.329160 0.944274i \(-0.393234\pi\)
0.329160 + 0.944274i \(0.393234\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −27.3137 −1.34402
\(414\) 0 0
\(415\) −5.17157 −0.253863
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.686292 −0.0335275 −0.0167638 0.999859i \(-0.505336\pi\)
−0.0167638 + 0.999859i \(0.505336\pi\)
\(420\) 0 0
\(421\) −22.9706 −1.11952 −0.559758 0.828656i \(-0.689106\pi\)
−0.559758 + 0.828656i \(0.689106\pi\)
\(422\) 0 0
\(423\) 15.5147 0.754351
\(424\) 0 0
\(425\) 7.65685 0.371412
\(426\) 0 0
\(427\) 16.9706 0.821263
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 35.3137 1.70100 0.850501 0.525974i \(-0.176299\pi\)
0.850501 + 0.525974i \(0.176299\pi\)
\(432\) 0 0
\(433\) −8.82843 −0.424267 −0.212134 0.977241i \(-0.568041\pi\)
−0.212134 + 0.977241i \(0.568041\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.82843 −0.135302
\(438\) 0 0
\(439\) −24.9706 −1.19178 −0.595890 0.803066i \(-0.703201\pi\)
−0.595890 + 0.803066i \(0.703201\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −0.485281 −0.0230564 −0.0115282 0.999934i \(-0.503670\pi\)
−0.0115282 + 0.999934i \(0.503670\pi\)
\(444\) 0 0
\(445\) −11.6569 −0.552588
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −33.3137 −1.57217 −0.786086 0.618118i \(-0.787896\pi\)
−0.786086 + 0.618118i \(0.787896\pi\)
\(450\) 0 0
\(451\) −14.6274 −0.688778
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 13.6569 0.640243
\(456\) 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
\(463\) 6.14214 0.285449 0.142725 0.989762i \(-0.454414\pi\)
0.142725 + 0.989762i \(0.454414\pi\)
\(464\) 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
\(477\) −21.5147 −0.985091
\(478\) 0 0
\(479\) −20.9706 −0.958169 −0.479085 0.877769i \(-0.659031\pi\)
−0.479085 + 0.877769i \(0.659031\pi\)
\(480\) 0 0
\(481\) −31.3137 −1.42778
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.17157 0.325644
\(486\) 0 0
\(487\) 18.6274 0.844089 0.422044 0.906575i \(-0.361313\pi\)
0.422044 + 0.906575i \(0.361313\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −28.9706 −1.30742 −0.653712 0.756744i \(-0.726789\pi\)
−0.653712 + 0.756744i \(0.726789\pi\)
\(492\) 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
\(499\) −40.2843 −1.80337 −0.901686 0.432392i \(-0.857670\pi\)
−0.901686 + 0.432392i \(0.857670\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.4853 −0.735042 −0.367521 0.930015i \(-0.619793\pi\)
−0.367521 + 0.930015i \(0.619793\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 10.9706 0.486262 0.243131 0.969994i \(-0.421826\pi\)
0.243131 + 0.969994i \(0.421826\pi\)
\(510\) 0 0
\(511\) 44.2843 1.95902
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 17.6569 0.778054
\(516\) 0 0
\(517\) −20.6863 −0.909782
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.68629 0.292932 0.146466 0.989216i \(-0.453210\pi\)
0.146466 + 0.989216i \(0.453210\pi\)
\(522\) 0 0
\(523\) −40.9706 −1.79152 −0.895759 0.444540i \(-0.853367\pi\)
−0.895759 + 0.444540i \(0.853367\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −43.3137 −1.88677
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 28.9706 1.25722
\(532\) 0 0
\(533\) −17.6569 −0.764803
\(534\) 0 0
\(535\) −13.6569 −0.590437
\(536\) 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
\(545\) 5.31371 0.227614
\(546\) 0 0
\(547\) 36.2843 1.55140 0.775702 0.631100i \(-0.217396\pi\)
0.775702 + 0.631100i \(0.217396\pi\)
\(548\) 0 0
\(549\) −18.0000 −0.768221
\(550\) 0 0
\(551\) 3.65685 0.155787
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10.9706 −0.464838 −0.232419 0.972616i \(-0.574664\pi\)
−0.232419 + 0.972616i \(0.574664\pi\)
\(558\) 0 0
\(559\) 40.9706 1.73287
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21.6569 −0.912728 −0.456364 0.889793i \(-0.650848\pi\)
−0.456364 + 0.889793i \(0.650848\pi\)
\(564\) 0 0
\(565\) 10.4853 0.441119
\(566\) 0 0
\(567\) 25.4558 1.06904
\(568\) 0 0
\(569\) −31.9411 −1.33904 −0.669521 0.742793i \(-0.733501\pi\)
−0.669521 + 0.742793i \(0.733501\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.82843 0.117954
\(576\) 0 0
\(577\) 8.62742 0.359164 0.179582 0.983743i \(-0.442525\pi\)
0.179582 + 0.983743i \(0.442525\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −14.6274 −0.606848
\(582\) 0 0
\(583\) 28.6863 1.18806
\(584\) 0 0
\(585\) −14.4853 −0.598893
\(586\) 0 0
\(587\) −6.14214 −0.253513 −0.126757 0.991934i \(-0.540457\pi\)
−0.126757 + 0.991934i \(0.540457\pi\)
\(588\) 0 0
\(589\) 5.65685 0.233087
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −36.6274 −1.50411 −0.752054 0.659102i \(-0.770937\pi\)
−0.752054 + 0.659102i \(0.770937\pi\)
\(594\) 0 0
\(595\) 21.6569 0.887844
\(596\) 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
\(601\) −5.02944 −0.205155 −0.102578 0.994725i \(-0.532709\pi\)
−0.102578 + 0.994725i \(0.532709\pi\)
\(602\) 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 \(-0.760758\pi\)
−0.730597 + 0.682808i \(0.760758\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −24.9706 −1.01020
\(612\) 0 0
\(613\) −1.02944 −0.0415786 −0.0207893 0.999784i \(-0.506618\pi\)
−0.0207893 + 0.999784i \(0.506618\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 33.5980 1.35260 0.676302 0.736625i \(-0.263582\pi\)
0.676302 + 0.736625i \(0.263582\pi\)
\(618\) 0 0
\(619\) 35.5980 1.43080 0.715402 0.698713i \(-0.246244\pi\)
0.715402 + 0.698713i \(0.246244\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −32.9706 −1.32094
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −49.6569 −1.97995
\(630\) 0 0
\(631\) 18.6274 0.741546 0.370773 0.928724i \(-0.379093\pi\)
0.370773 + 0.928724i \(0.379093\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −12.9706 −0.514721
\(636\) 0 0
\(637\) 4.82843 0.191309
\(638\) 0 0
\(639\) 16.9706 0.671345
\(640\) 0 0
\(641\) 31.6569 1.25037 0.625185 0.780476i \(-0.285023\pi\)
0.625185 + 0.780476i \(0.285023\pi\)
\(642\) 0 0
\(643\) −47.1127 −1.85794 −0.928972 0.370151i \(-0.879306\pi\)
−0.928972 + 0.370151i \(0.879306\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −22.1421 −0.870497 −0.435249 0.900310i \(-0.643340\pi\)
−0.435249 + 0.900310i \(0.643340\pi\)
\(648\) 0 0
\(649\) −38.6274 −1.51626
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.343146 0.0134283 0.00671417 0.999977i \(-0.497863\pi\)
0.00671417 + 0.999977i \(0.497863\pi\)
\(654\) 0 0
\(655\) −9.65685 −0.377325
\(656\) 0 0
\(657\) −46.9706 −1.83250
\(658\) 0 0
\(659\) 1.65685 0.0645419 0.0322709 0.999479i \(-0.489726\pi\)
0.0322709 + 0.999479i \(0.489726\pi\)
\(660\) 0 0
\(661\) 23.6569 0.920145 0.460072 0.887881i \(-0.347823\pi\)
0.460072 + 0.887881i \(0.347823\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.82843 −0.109682
\(666\) 0 0
\(667\) −10.3431 −0.400488
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −37.1127 −1.43059 −0.715295 0.698823i \(-0.753708\pi\)
−0.715295 + 0.698823i \(0.753708\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 44.4264 1.70745 0.853723 0.520728i \(-0.174339\pi\)
0.853723 + 0.520728i \(0.174339\pi\)
\(678\) 0 0
\(679\) 20.2843 0.778439
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.2843 1.69449 0.847245 0.531202i \(-0.178259\pi\)
0.847245 + 0.531202i \(0.178259\pi\)
\(684\) 0 0
\(685\) 15.6569 0.598218
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 34.6274 1.31920
\(690\) 0 0
\(691\) −16.6863 −0.634776 −0.317388 0.948296i \(-0.602806\pi\)
−0.317388 + 0.948296i \(0.602806\pi\)
\(692\) 0 0
\(693\) −33.9411 −1.28932
\(694\) 0 0
\(695\) −12.9706 −0.492001
\(696\) 0 0
\(697\) −28.0000 −1.06058
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 41.3137 1.56040 0.780199 0.625532i \(-0.215118\pi\)
0.780199 + 0.625532i \(0.215118\pi\)
\(702\) 0 0
\(703\) 6.48528 0.244597
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 39.5980 1.48924
\(708\) 0 0
\(709\) 23.9411 0.899128 0.449564 0.893248i \(-0.351579\pi\)
0.449564 + 0.893248i \(0.351579\pi\)
\(710\) 0 0
\(711\) −16.9706 −0.636446
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 19.3137 0.722292
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −40.2843 −1.50235 −0.751175 0.660103i \(-0.770513\pi\)
−0.751175 + 0.660103i \(0.770513\pi\)
\(720\) 0 0
\(721\) 49.9411 1.85990
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.65685 −0.135812
\(726\) 0 0
\(727\) 15.5147 0.575409 0.287705 0.957719i \(-0.407108\pi\)
0.287705 + 0.957719i \(0.407108\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 64.9706 2.40302
\(732\) 0 0
\(733\) −6.68629 −0.246964 −0.123482 0.992347i \(-0.539406\pi\)
−0.123482 + 0.992347i \(0.539406\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −45.2548 −1.66698
\(738\) 0 0
\(739\) −8.68629 −0.319530 −0.159765 0.987155i \(-0.551074\pi\)
−0.159765 + 0.987155i \(0.551074\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5.37258 −0.197101 −0.0985505 0.995132i \(-0.531421\pi\)
−0.0985505 + 0.995132i \(0.531421\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) 15.5147 0.567654
\(748\) 0 0
\(749\) −38.6274 −1.41142
\(750\) 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
\(755\) −3.31371 −0.120598
\(756\) 0 0
\(757\) 1.31371 0.0477475 0.0238738 0.999715i \(-0.492400\pi\)
0.0238738 + 0.999715i \(0.492400\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24.6274 0.892743 0.446372 0.894848i \(-0.352716\pi\)
0.446372 + 0.894848i \(0.352716\pi\)
\(762\) 0 0
\(763\) 15.0294 0.544102
\(764\) 0 0
\(765\) −22.9706 −0.830502
\(766\) 0 0
\(767\) −46.6274 −1.68362
\(768\) 0 0
\(769\) −51.2548 −1.84830 −0.924148 0.382034i \(-0.875224\pi\)
−0.924148 + 0.382034i \(0.875224\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25.5147 0.917701 0.458850 0.888514i \(-0.348261\pi\)
0.458850 + 0.888514i \(0.348261\pi\)
\(774\) 0 0
\(775\) −5.65685 −0.203200
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.65685 0.131020
\(780\) 0 0
\(781\) −22.6274 −0.809673
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −15.6569 −0.558817
\(786\) 0 0
\(787\) −44.2843 −1.57856 −0.789282 0.614031i \(-0.789547\pi\)
−0.789282 + 0.614031i \(0.789547\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 29.6569 1.05448
\(792\) 0 0
\(793\) 28.9706 1.02877
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.8284 −0.596093 −0.298047 0.954551i \(-0.596335\pi\)
−0.298047 + 0.954551i \(0.596335\pi\)
\(798\) 0 0
\(799\) −39.5980 −1.40088
\(800\) 0 0
\(801\) 34.9706 1.23562
\(802\) 0 0
\(803\) 62.6274 2.21007
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 48.6274 1.70965 0.854824 0.518917i \(-0.173665\pi\)
0.854824 + 0.518917i \(0.173665\pi\)
\(810\) 0 0
\(811\) 22.3431 0.784574 0.392287 0.919843i \(-0.371684\pi\)
0.392287 + 0.919843i \(0.371684\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −18.8284 −0.659531
\(816\) 0 0
\(817\) −8.48528 −0.296862
\(818\) 0 0
\(819\) −40.9706 −1.43163
\(820\) 0 0
\(821\) 9.31371 0.325051 0.162525 0.986704i \(-0.448036\pi\)
0.162525 + 0.986704i \(0.448036\pi\)
\(822\) 0 0
\(823\) 35.7990 1.24787 0.623937 0.781475i \(-0.285532\pi\)
0.623937 + 0.781475i \(0.285532\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16.0000 0.556375 0.278187 0.960527i \(-0.410266\pi\)
0.278187 + 0.960527i \(0.410266\pi\)
\(828\) 0 0
\(829\) −51.2548 −1.78015 −0.890077 0.455810i \(-0.849350\pi\)
−0.890077 + 0.455810i \(0.849350\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7.65685 0.265294
\(834\) 0 0
\(835\) 23.3137 0.806804
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 37.6569 1.30006 0.650029 0.759909i \(-0.274757\pi\)
0.650029 + 0.759909i \(0.274757\pi\)
\(840\) 0 0
\(841\) −15.6274 −0.538876
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 10.3137 0.354802
\(846\) 0 0
\(847\) 14.1421 0.485930
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −18.3431 −0.628795
\(852\) 0 0
\(853\) 44.6274 1.52801 0.764007 0.645208i \(-0.223229\pi\)
0.764007 + 0.645208i \(0.223229\pi\)
\(854\) 0 0
\(855\) 3.00000 0.102598
\(856\) 0 0
\(857\) −17.7990 −0.608002 −0.304001 0.952672i \(-0.598323\pi\)
−0.304001 + 0.952672i \(0.598323\pi\)
\(858\) 0 0
\(859\) −23.3137 −0.795453 −0.397727 0.917504i \(-0.630201\pi\)
−0.397727 + 0.917504i \(0.630201\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −52.0000 −1.77010 −0.885050 0.465495i \(-0.845876\pi\)
−0.885050 + 0.465495i \(0.845876\pi\)
\(864\) 0 0
\(865\) −12.1421 −0.412845
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 22.6274 0.767583
\(870\) 0 0
\(871\) −54.6274 −1.85098
\(872\) 0 0
\(873\) −21.5147 −0.728163
\(874\) 0 0
\(875\) 2.82843 0.0956183
\(876\) 0 0
\(877\) 29.7990 1.00624 0.503120 0.864216i \(-0.332185\pi\)
0.503120 + 0.864216i \(0.332185\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 6.68629 0.225267 0.112633 0.993637i \(-0.464071\pi\)
0.112633 + 0.993637i \(0.464071\pi\)
\(882\) 0 0
\(883\) 5.17157 0.174037 0.0870186 0.996207i \(-0.472266\pi\)
0.0870186 + 0.996207i \(0.472266\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −14.3431 −0.481596 −0.240798 0.970575i \(-0.577409\pi\)
−0.240798 + 0.970575i \(0.577409\pi\)
\(888\) 0 0
\(889\) −36.6863 −1.23042
\(890\) 0 0
\(891\) 36.0000 1.20605
\(892\) 0 0
\(893\) 5.17157 0.173060
\(894\) 0 0
\(895\) 12.9706 0.433558
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 20.6863 0.689926
\(900\) 0 0
\(901\) 54.9117 1.82937
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 13.3137 0.442563
\(906\) 0 0
\(907\) −44.2843 −1.47044 −0.735218 0.677831i \(-0.762920\pi\)
−0.735218 + 0.677831i \(0.762920\pi\)
\(908\) 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
\(913\) −20.6863 −0.684616
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −27.3137 −0.901978
\(918\) 0 0
\(919\) 58.6274 1.93394 0.966970 0.254890i \(-0.0820393\pi\)
0.966970 + 0.254890i \(0.0820393\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −27.3137 −0.899042
\(924\) 0 0
\(925\) −6.48528 −0.213235
\(926\) 0 0
\(927\) −52.9706 −1.73978
\(928\) 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
\(935\) 30.6274 1.00162
\(936\) 0 0
\(937\) −32.3431 −1.05660 −0.528302 0.849056i \(-0.677171\pi\)
−0.528302 + 0.849056i \(0.677171\pi\)
\(938\) 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
\(943\) −10.3431 −0.336819
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 37.1716 1.20791 0.603957 0.797017i \(-0.293590\pi\)
0.603957 + 0.797017i \(0.293590\pi\)
\(948\) 0 0
\(949\) 75.5980 2.45401
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.79899 −0.0582750 −0.0291375 0.999575i \(-0.509276\pi\)
−0.0291375 + 0.999575i \(0.509276\pi\)
\(954\) 0 0
\(955\) 12.9706 0.419718
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 44.2843 1.43001
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 40.9706 1.32026
\(964\) 0 0
\(965\) −8.82843 −0.284197
\(966\) 0 0
\(967\) 39.1127 1.25778 0.628890 0.777494i \(-0.283510\pi\)
0.628890 + 0.777494i \(0.283510\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −28.9706 −0.929710 −0.464855 0.885387i \(-0.653893\pi\)
−0.464855 + 0.885387i \(0.653893\pi\)
\(972\) 0 0
\(973\) −36.6863 −1.17611
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.20101 0.198388 0.0991939 0.995068i \(-0.468374\pi\)
0.0991939 + 0.995068i \(0.468374\pi\)
\(978\) 0 0
\(979\) −46.6274 −1.49022
\(980\) 0 0
\(981\) −15.9411 −0.508961
\(982\) 0 0
\(983\) −15.3137 −0.488431 −0.244216 0.969721i \(-0.578531\pi\)
−0.244216 + 0.969721i \(0.578531\pi\)
\(984\) 0 0
\(985\) 1.31371 0.0418582
\(986\) 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 \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −7.31371 −0.231860
\(996\) 0 0
\(997\) −12.3431 −0.390911 −0.195456 0.980713i \(-0.562619\pi\)
−0.195456 + 0.980713i \(0.562619\pi\)
\(998\) 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.g.1.2 yes 2
4.3 odd 2 3040.2.a.f.1.1 2
8.3 odd 2 6080.2.a.be.1.1 2
8.5 even 2 6080.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.f.1.1 2 4.3 odd 2
3040.2.a.g.1.2 yes 2 1.1 even 1 trivial
6080.2.a.bd.1.2 2 8.5 even 2
6080.2.a.be.1.1 2 8.3 odd 2