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Properties

Label 4.17.b.a.3.1
Level $4$
Weight $17$
Character 4.3
Self dual yes
Analytic conductor $6.493$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4,17,Mod(3,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.49298175427\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 3.1
Character \(\chi\) \(=\) 4.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+256.000 q^{2} +65536.0 q^{4} +329666. q^{5} +1.67772e7 q^{8} +4.30467e7 q^{9} +O(q^{10})\) \(q+256.000 q^{2} +65536.0 q^{4} +329666. q^{5} +1.67772e7 q^{8} +4.30467e7 q^{9} +8.43945e7 q^{10} -1.63123e9 q^{13} +4.29497e9 q^{16} -9.93728e9 q^{17} +1.10200e10 q^{18} +2.16050e10 q^{20} -4.39082e10 q^{25} -4.17596e11 q^{26} +9.81515e11 q^{29} +1.09951e12 q^{32} -2.54394e12 q^{34} +2.82111e12 q^{36} -6.16763e12 q^{37} +5.53088e12 q^{40} -3.16832e12 q^{41} +1.41910e13 q^{45} +3.32329e13 q^{49} -1.12405e13 q^{50} -1.06904e14 q^{52} -3.19627e13 q^{53} +2.51268e14 q^{58} +4.59901e13 q^{61} +2.81475e14 q^{64} -5.37762e14 q^{65} -6.51249e14 q^{68} +7.22204e14 q^{72} +1.38104e15 q^{73} -1.57891e15 q^{74} +1.41590e15 q^{80} +1.85302e15 q^{81} -8.11091e14 q^{82} -3.27598e15 q^{85} -6.95715e15 q^{89} +3.63291e15 q^{90} +1.43857e16 q^{97} +8.50763e15 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 256.000 1.00000
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 65536.0 1.00000
\(5\) 329666. 0.843945 0.421972 − 0.906609i \(-0.361338\pi\)
0.421972 + 0.906609i \(0.361338\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.67772e7 1.00000
\(9\) 4.30467e7 1.00000
\(10\) 8.43945e7 0.843945
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −1.63123e9 −1.99972 −0.999860 − 0.0167355i \(-0.994673\pi\)
−0.999860 + 0.0167355i \(0.994673\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.29497e9 1.00000
\(17\) −9.93728e9 −1.42454 −0.712272 − 0.701903i \(-0.752334\pi\)
−0.712272 + 0.701903i \(0.752334\pi\)
\(18\) 1.10200e10 1.00000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 2.16050e10 0.843945
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −4.39082e10 −0.287757
\(26\) −4.17596e11 −1.99972
\(27\) 0 0
\(28\) 0 0
\(29\) 9.81515e11 1.96206 0.981032 − 0.193848i \(-0.0620969\pi\)
0.981032 + 0.193848i \(0.0620969\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1.09951e12 1.00000
\(33\) 0 0
\(34\) −2.54394e12 −1.42454
\(35\) 0 0
\(36\) 2.82111e12 1.00000
\(37\) −6.16763e12 −1.75592 −0.877959 − 0.478735i \(-0.841095\pi\)
−0.877959 + 0.478735i \(0.841095\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 5.53088e12 0.843945
\(41\) −3.16832e12 −0.396788 −0.198394 − 0.980122i \(-0.563573\pi\)
−0.198394 + 0.980122i \(0.563573\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 1.41910e13 0.843945
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 3.32329e13 1.00000
\(50\) −1.12405e13 −0.287757
\(51\) 0 0
\(52\) −1.06904e14 −1.99972
\(53\) −3.19627e13 −0.513377 −0.256689 − 0.966494i \(-0.582631\pi\)
−0.256689 + 0.966494i \(0.582631\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 2.51268e14 1.96206
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 4.59901e13 0.239897 0.119949 − 0.992780i \(-0.461727\pi\)
0.119949 + 0.992780i \(0.461727\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 2.81475e14 1.00000
\(65\) −5.37762e14 −1.68765
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −6.51249e14 −1.42454
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 7.22204e14 1.00000
\(73\) 1.38104e15 1.71248 0.856238 − 0.516582i \(-0.172796\pi\)
0.856238 + 0.516582i \(0.172796\pi\)
\(74\) −1.57891e15 −1.75592
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.41590e15 0.843945
\(81\) 1.85302e15 1.00000
\(82\) −8.11091e14 −0.396788
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −3.27598e15 −1.20224
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.95715e15 −1.76730 −0.883652 − 0.468144i \(-0.844923\pi\)
−0.883652 + 0.468144i \(0.844923\pi\)
\(90\) 3.63291e15 0.843945
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.43857e16 1.83551 0.917756 − 0.397145i \(-0.129999\pi\)
0.917756 + 0.397145i \(0.129999\pi\)
\(98\) 8.50763e15 1.00000
\(99\) 0 0
\(100\) −2.87757e15 −0.287757
\(101\) −5.17602e14 −0.0477997 −0.0238998 − 0.999714i \(-0.507608\pi\)
−0.0238998 + 0.999714i \(0.507608\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −2.73675e16 −1.99972
\(105\) 0 0
\(106\) −8.18245e15 −0.513377
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −1.95513e15 −0.0981211 −0.0490606 − 0.998796i \(-0.515623\pi\)
−0.0490606 + 0.998796i \(0.515623\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.57563e16 0.968850 0.484425 − 0.874833i \(-0.339029\pi\)
0.484425 + 0.874833i \(0.339029\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.43246e16 1.96206
\(117\) −7.02192e16 −1.99972
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 4.59497e16 1.00000
\(122\) 1.17735e16 0.239897
\(123\) 0 0
\(124\) 0 0
\(125\) −6.47781e16 −1.08680
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 7.20576e16 1.00000
\(129\) 0 0
\(130\) −1.37667e17 −1.68765
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −1.66720e17 −1.42454
\(137\) 1.89374e17 1.52601 0.763004 − 0.646394i \(-0.223724\pi\)
0.763004 + 0.646394i \(0.223724\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.84884e17 1.00000
\(145\) 3.23572e17 1.65587
\(146\) 3.53547e17 1.71248
\(147\) 0 0
\(148\) −4.04202e17 −1.75592
\(149\) −4.56938e17 −1.88090 −0.940452 − 0.339925i \(-0.889598\pi\)
−0.940452 + 0.339925i \(0.889598\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −4.27767e17 −1.42454
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −9.97807e16 −0.270302 −0.135151 − 0.990825i \(-0.543152\pi\)
−0.135151 + 0.990825i \(0.543152\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 3.62472e17 0.843945
\(161\) 0 0
\(162\) 4.74373e17 1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −2.07639e17 −0.396788
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.99550e18 2.99888
\(170\) −8.38652e17 −1.20224
\(171\) 0 0
\(172\) 0 0
\(173\) −1.22818e18 −1.53071 −0.765355 − 0.643608i \(-0.777437\pi\)
−0.765355 + 0.643608i \(0.777437\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −1.78103e18 −1.76730
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 9.30024e17 0.843945
\(181\) 1.53547e18 1.33295 0.666473 − 0.745530i \(-0.267803\pi\)
0.666473 + 0.745530i \(0.267803\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.03326e18 −1.48190
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −2.15876e18 −1.12136 −0.560682 − 0.828031i \(-0.689461\pi\)
−0.560682 + 0.828031i \(0.689461\pi\)
\(194\) 3.68274e18 1.83551
\(195\) 0 0
\(196\) 2.17795e18 1.00000
\(197\) 1.89080e18 0.833518 0.416759 − 0.909017i \(-0.363166\pi\)
0.416759 + 0.909017i \(0.363166\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −7.36658e17 −0.287757
\(201\) 0 0
\(202\) −1.32506e17 −0.0477997
\(203\) 0 0
\(204\) 0 0
\(205\) −1.04449e18 −0.334867
\(206\) 0 0
\(207\) 0 0
\(208\) −7.00609e18 −1.99972
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −2.09471e18 −0.513377
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −5.00512e17 −0.0981211
\(219\) 0 0
\(220\) 0 0
\(221\) 1.62100e19 2.84869
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −1.89010e18 −0.287757
\(226\) 6.59362e18 0.968850
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −7.90634e18 −1.04542 −0.522711 − 0.852510i \(-0.675079\pi\)
−0.522711 + 0.852510i \(0.675079\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.64671e19 1.96206
\(233\) −1.43553e19 −1.65259 −0.826295 − 0.563237i \(-0.809556\pi\)
−0.826295 + 0.563237i \(0.809556\pi\)
\(234\) −1.79761e19 −1.99972
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.17552e19 −1.03299 −0.516493 − 0.856291i \(-0.672763\pi\)
−0.516493 + 0.856291i \(0.672763\pi\)
\(242\) 1.17631e19 1.00000
\(243\) 0 0
\(244\) 3.01400e18 0.239897
\(245\) 1.09558e19 0.843945
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −1.65832e19 −1.08680
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.84467e19 1.00000
\(257\) 2.06067e19 1.08279 0.541395 − 0.840768i \(-0.317896\pi\)
0.541395 + 0.840768i \(0.317896\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −3.52428e19 −1.68765
\(261\) 4.22510e19 1.96206
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −1.05370e19 −0.433262
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.65023e19 −0.966642 −0.483321 − 0.875443i \(-0.660570\pi\)
−0.483321 + 0.875443i \(0.660570\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −4.26803e19 −1.42454
\(273\) 0 0
\(274\) 4.84798e19 1.52601
\(275\) 0 0
\(276\) 0 0
\(277\) −6.65567e19 −1.92023 −0.960116 − 0.279601i \(-0.909798\pi\)
−0.960116 + 0.279601i \(0.909798\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.03691e19 0.266741 0.133370 − 0.991066i \(-0.457420\pi\)
0.133370 + 0.991066i \(0.457420\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 4.73304e19 1.00000
\(289\) 5.00883e19 1.02933
\(290\) 8.28345e19 1.65587
\(291\) 0 0
\(292\) 9.05080e19 1.71248
\(293\) −3.24093e19 −0.596662 −0.298331 − 0.954462i \(-0.596430\pi\)
−0.298331 + 0.954462i \(0.596430\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.03476e20 −1.75592
\(297\) 0 0
\(298\) −1.16976e20 −1.88090
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.51614e19 0.202460
\(306\) −1.09508e20 −1.42454
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 1.47816e20 1.60460 0.802299 − 0.596922i \(-0.203610\pi\)
0.802299 + 0.596922i \(0.203610\pi\)
\(314\) −2.55439e19 −0.270302
\(315\) 0 0
\(316\) 0 0
\(317\) −6.80244e19 −0.667100 −0.333550 − 0.942732i \(-0.608247\pi\)
−0.333550 + 0.942732i \(0.608247\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 9.27927e19 0.843945
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.21440e20 1.00000
\(325\) 7.16245e19 0.575433
\(326\) 0 0
\(327\) 0 0
\(328\) −5.31557e19 −0.396788
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) −2.65496e20 −1.75592
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.12287e20 −0.674981 −0.337491 − 0.941329i \(-0.609578\pi\)
−0.337491 + 0.941329i \(0.609578\pi\)
\(338\) 5.10849e20 2.99888
\(339\) 0 0
\(340\) −2.14695e20 −1.20224
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −3.14414e20 −1.53071
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −1.62144e20 −0.736711 −0.368355 − 0.929685i \(-0.620079\pi\)
−0.368355 + 0.929685i \(0.620079\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.51405e20 1.45751 0.728754 − 0.684776i \(-0.240100\pi\)
0.728754 + 0.684776i \(0.240100\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −4.55944e20 −1.76730
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 2.38086e20 0.843945
\(361\) 2.88441e20 1.00000
\(362\) 3.93080e20 1.33295
\(363\) 0 0
\(364\) 0 0
\(365\) 4.55283e20 1.44523
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −1.36386e20 −0.396788
\(370\) −5.20514e20 −1.48190
\(371\) 0 0
\(372\) 0 0
\(373\) 7.04221e20 1.87948 0.939741 − 0.341887i \(-0.111066\pi\)
0.939741 + 0.341887i \(0.111066\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.60108e21 −3.92358
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5.52644e20 −1.12136
\(387\) 0 0
\(388\) 9.42781e20 1.83551
\(389\) −6.37627e20 −1.21610 −0.608051 − 0.793898i \(-0.708048\pi\)
−0.608051 + 0.793898i \(0.708048\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 5.57556e20 1.00000
\(393\) 0 0
\(394\) 4.84044e20 0.833518
\(395\) 0 0
\(396\) 0 0
\(397\) 2.22973e20 0.361351 0.180675 − 0.983543i \(-0.442172\pi\)
0.180675 + 0.983543i \(0.442172\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.88584e20 −0.287757
\(401\) 9.32250e20 1.39437 0.697184 − 0.716892i \(-0.254436\pi\)
0.697184 + 0.716892i \(0.254436\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −3.39216e19 −0.0477997
\(405\) 6.10878e20 0.843945
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.13586e21 −1.45056 −0.725281 − 0.688453i \(-0.758290\pi\)
−0.725281 + 0.688453i \(0.758290\pi\)
\(410\) −2.67389e20 −0.334867
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −1.79356e21 −1.99972
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 1.68109e21 1.70347 0.851737 − 0.523970i \(-0.175550\pi\)
0.851737 + 0.523970i \(0.175550\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −5.36245e20 −0.513377
\(425\) 4.36328e20 0.409923
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −2.26669e21 −1.83438 −0.917189 − 0.398453i \(-0.869547\pi\)
−0.917189 + 0.398453i \(0.869547\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.28131e20 −0.0981211
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.43057e21 1.00000
\(442\) 4.14976e21 2.84869
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −2.29354e21 −1.49151
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.06295e21 1.24887 0.624435 − 0.781077i \(-0.285329\pi\)
0.624435 + 0.781077i \(0.285329\pi\)
\(450\) −4.83867e20 −0.287757
\(451\) 0 0
\(452\) 1.68797e21 0.968850
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.77291e21 −1.98311 −0.991556 − 0.129682i \(-0.958604\pi\)
−0.991556 + 0.129682i \(0.958604\pi\)
\(458\) −2.02402e21 −1.04542
\(459\) 0 0
\(460\) 0 0
\(461\) 4.16776e20 0.204312 0.102156 − 0.994768i \(-0.467426\pi\)
0.102156 + 0.994768i \(0.467426\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 4.21557e21 1.96206
\(465\) 0 0
\(466\) −3.67496e21 −1.65259
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −4.60189e21 −1.99972
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.37589e21 −0.513377
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 1.00608e22 3.51135
\(482\) −3.00934e21 −1.03299
\(483\) 0 0
\(484\) 3.01136e21 1.00000
\(485\) 4.74248e21 1.54907
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 7.71585e20 0.239897
\(489\) 0 0
\(490\) 2.80468e21 0.843945
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −9.75359e21 −2.79505
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −4.24530e21 −1.08680
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −1.70636e20 −0.0403403
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −8.17546e21 −1.81455 −0.907277 − 0.420533i \(-0.861843\pi\)
−0.907277 + 0.420533i \(0.861843\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 4.72237e21 1.00000
\(513\) 0 0
\(514\) 5.27532e21 1.08279
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −9.02215e21 −1.68765
\(521\) −2.06664e21 −0.380683 −0.190341 − 0.981718i \(-0.560959\pi\)
−0.190341 + 0.981718i \(0.560959\pi\)
\(522\) 1.08163e22 1.96206
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 6.13261e21 1.00000
\(530\) −2.69748e21 −0.433262
\(531\) 0 0
\(532\) 0 0
\(533\) 5.16828e21 0.793465
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −6.78459e21 −0.966642
\(539\) 0 0
\(540\) 0 0
\(541\) 9.93129e21 1.35340 0.676702 − 0.736257i \(-0.263408\pi\)
0.676702 + 0.736257i \(0.263408\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −1.09262e22 −1.42454
\(545\) −6.44538e20 −0.0828088
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 1.24108e22 1.52601
\(549\) 1.97972e21 0.239897
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −1.70385e22 −1.92023
\(555\) 0 0
\(556\) 0 0
\(557\) −1.37379e22 −1.48279 −0.741395 − 0.671069i \(-0.765835\pi\)
−0.741395 + 0.671069i \(0.765835\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 2.65448e21 0.266741
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 8.49099e21 0.817656
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.15247e22 −1.95902 −0.979512 − 0.201388i \(-0.935455\pi\)
−0.979512 + 0.201388i \(0.935455\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.21166e22 1.00000
\(577\) 1.93165e22 1.57225 0.786126 − 0.618067i \(-0.212084\pi\)
0.786126 + 0.618067i \(0.212084\pi\)
\(578\) 1.28226e22 1.02933
\(579\) 0 0
\(580\) 2.12056e22 1.65587
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 2.31701e22 1.71248
\(585\) −2.31489e22 −1.68765
\(586\) −8.29678e21 −0.596662
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −2.64898e22 −1.75592
\(593\) 1.83468e22 1.19984 0.599919 − 0.800060i \(-0.295199\pi\)
0.599919 + 0.800060i \(0.295199\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2.99459e22 −1.88090
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −3.36869e22 −1.97909 −0.989544 − 0.144229i \(-0.953930\pi\)
−0.989544 + 0.144229i \(0.953930\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.51481e22 0.843945
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 3.88131e21 0.202460
\(611\) 0 0
\(612\) −2.80342e22 −1.42454
\(613\) 3.57838e22 1.79474 0.897372 − 0.441275i \(-0.145474\pi\)
0.897372 + 0.441275i \(0.145474\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.45691e21 0.402652 0.201326 − 0.979524i \(-0.435475\pi\)
0.201326 + 0.979524i \(0.435475\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.46553e22 −0.629439
\(626\) 3.78409e22 1.60460
\(627\) 0 0
\(628\) −6.53923e21 −0.270302
\(629\) 6.12894e22 2.50138
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −1.74142e22 −0.667100
\(635\) 0 0
\(636\) 0 0
\(637\) −5.42107e22 −1.99972
\(638\) 0 0
\(639\) 0 0
\(640\) 2.37549e22 0.843945
\(641\) −4.69460e22 −1.64715 −0.823576 − 0.567205i \(-0.808025\pi\)
−0.823576 + 0.567205i \(0.808025\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 3.10885e22 1.00000
\(649\) 0 0
\(650\) 1.83359e22 0.575433
\(651\) 0 0
\(652\) 0 0
\(653\) −4.18505e22 −1.26589 −0.632944 − 0.774198i \(-0.718153\pi\)
−0.632944 + 0.774198i \(0.718153\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.36079e22 −0.396788
\(657\) 5.94494e22 1.71248
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −5.90167e22 −1.61943 −0.809717 − 0.586821i \(-0.800379\pi\)
−0.809717 + 0.586821i \(0.800379\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −6.79670e22 −1.75592
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.33918e22 0.318212 0.159106 − 0.987261i \(-0.449139\pi\)
0.159106 + 0.987261i \(0.449139\pi\)
\(674\) −2.87456e22 −0.674981
\(675\) 0 0
\(676\) 1.30777e23 2.99888
\(677\) 7.20802e22 1.63345 0.816726 − 0.577025i \(-0.195787\pi\)
0.816726 + 0.577025i \(0.195787\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −5.49619e22 −1.20224
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 6.24303e22 1.28787
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 5.21386e22 1.02661
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) −8.04900e22 −1.53071
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3.14845e22 0.565243
\(698\) −4.15088e22 −0.736711
\(699\) 0 0
\(700\) 0 0
\(701\) −1.16017e23 −1.98965 −0.994825 − 0.101604i \(-0.967602\pi\)
−0.994825 + 0.101604i \(0.967602\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 8.99598e22 1.45751
\(707\) 0 0
\(708\) 0 0
\(709\) −1.26270e23 −1.97756 −0.988782 − 0.149363i \(-0.952278\pi\)
−0.988782 + 0.149363i \(0.952278\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.16722e23 −1.76730
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 6.09501e22 0.843945
\(721\) 0 0
\(722\) 7.38410e22 1.00000
\(723\) 0 0
\(724\) 1.00628e23 1.33295
\(725\) −4.30966e22 −0.564597
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 7.97664e22 1.00000
\(730\) 1.16552e23 1.44523
\(731\) 0 0
\(732\) 0 0
\(733\) 6.30230e22 0.756253 0.378127 − 0.925754i \(-0.376568\pi\)
0.378127 + 0.925754i \(0.376568\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −3.49148e22 −0.396788
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) −1.33252e23 −1.48190
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −1.50637e23 −1.58738
\(746\) 1.80280e23 1.87948
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −4.09876e23 −3.92358
\(755\) 0 0
\(756\) 0 0
\(757\) 1.25230e23 1.16129 0.580644 − 0.814157i \(-0.302801\pi\)
0.580644 + 0.814157i \(0.302801\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.06301e23 1.83411 0.917054 − 0.398762i \(-0.130560\pi\)
0.917054 + 0.398762i \(0.130560\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.41020e23 −1.20224
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.56381e23 1.27871 0.639357 − 0.768910i \(-0.279201\pi\)
0.639357 + 0.768910i \(0.279201\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.41477e23 −1.12136
\(773\) −1.15320e23 −0.904625 −0.452313 − 0.891859i \(-0.649401\pi\)
−0.452313 + 0.891859i \(0.649401\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2.41352e23 1.83551
\(777\) 0 0
\(778\) −1.63233e23 −1.21610
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.42734e23 1.00000
\(785\) −3.28943e22 −0.228120
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 1.23915e23 0.833518
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −7.50205e22 −0.479727
\(794\) 5.70811e22 0.361351
\(795\) 0 0
\(796\) 0 0
\(797\) 3.23242e23 1.98546 0.992729 − 0.120373i \(-0.0384089\pi\)
0.992729 + 0.120373i \(0.0384089\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −4.82776e22 −0.287757
\(801\) −2.99483e23 −1.76730
\(802\) 2.38656e23 1.39437
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −8.68392e21 −0.0477997
\(809\) −3.48989e23 −1.90206 −0.951028 − 0.309104i \(-0.899971\pi\)
−0.951028 + 0.309104i \(0.899971\pi\)
\(810\) 1.56385e23 0.843945
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −2.90779e23 −1.45056
\(819\) 0 0
\(820\) −6.84516e22 −0.334867
\(821\) −1.27444e23 −0.617411 −0.308705 − 0.951158i \(-0.599896\pi\)
−0.308705 + 0.951158i \(0.599896\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 3.66211e23 1.64170 0.820852 − 0.571141i \(-0.193499\pi\)
0.820852 + 0.571141i \(0.193499\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.59151e23 −1.99972
\(833\) −3.30245e23 −1.42454
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 7.13125e23 2.84969
\(842\) 4.30360e23 1.70347
\(843\) 0 0
\(844\) 0 0
\(845\) 6.57850e23 2.53089
\(846\) 0 0
\(847\) 0 0
\(848\) −1.37279e23 −0.513377
\(849\) 0 0
\(850\) 1.11700e23 0.409923
\(851\) 0 0
\(852\) 0 0
\(853\) −2.03144e23 −0.724789 −0.362394 − 0.932025i \(-0.618041\pi\)
−0.362394 + 0.932025i \(0.618041\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.39197e23 −1.16575 −0.582875 − 0.812562i \(-0.698072\pi\)
−0.582875 + 0.812562i \(0.698072\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −4.04889e23 −1.29184
\(866\) −5.80272e23 −1.83438
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −3.28016e22 −0.0981211
\(873\) 6.19257e23 1.83551
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6.94624e23 −1.98496 −0.992482 − 0.122387i \(-0.960945\pi\)
−0.992482 + 0.122387i \(0.960945\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −6.89948e23 −1.90112 −0.950559 − 0.310546i \(-0.899488\pi\)
−0.950559 + 0.310546i \(0.899488\pi\)
\(882\) 3.66226e23 1.00000
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 1.06234e24 2.84869
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −5.87145e23 −1.49151
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 5.28114e23 1.24887
\(899\) 0 0
\(900\) −1.23870e23 −0.287757
\(901\) 3.17622e23 0.731329
\(902\) 0 0
\(903\) 0 0
\(904\) 4.32120e23 0.968850
\(905\) 5.06192e23 1.12493
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −2.22811e22 −0.0477997
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −9.65865e23 −1.98311
\(915\) 0 0
\(916\) −5.18150e23 −1.04542
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.06695e23 0.204312
\(923\) 0 0
\(924\) 0 0
\(925\) 2.70810e23 0.505278
\(926\) 0 0
\(927\) 0 0
\(928\) 1.07919e24 1.96206
\(929\) 4.88934e23 0.881301 0.440650 − 0.897679i \(-0.354748\pi\)
0.440650 + 0.897679i \(0.354748\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −9.40790e23 −1.65259
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −1.17808e24 −1.99972
\(937\) −3.29952e23 −0.555308 −0.277654 − 0.960681i \(-0.589557\pi\)
−0.277654 + 0.960681i \(0.589557\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 6.94828e23 1.13021 0.565107 − 0.825018i \(-0.308835\pi\)
0.565107 + 0.825018i \(0.308835\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −2.25280e24 −3.42447
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.06527e24 1.56573 0.782866 − 0.622191i \(-0.213757\pi\)
0.782866 + 0.622191i \(0.213757\pi\)
\(954\) −3.52228e23 −0.513377
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.27423e23 1.00000
\(962\) 2.57557e24 3.51135
\(963\) 0 0
\(964\) −7.70391e23 −1.03299
\(965\) −7.11671e23 −0.946370
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 7.70909e23 1.00000
\(969\) 0 0
\(970\) 1.21407e24 1.54907
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.97526e23 0.239897
\(977\) −7.70188e23 −0.927770 −0.463885 − 0.885896i \(-0.653545\pi\)
−0.463885 + 0.885896i \(0.653545\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 7.17997e23 0.843945
\(981\) −8.41617e22 −0.0981211
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 6.23331e23 0.703443
\(986\) −2.49692e24 −2.79505
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.94584e24 −1.99317 −0.996587 − 0.0825473i \(-0.973694\pi\)
−0.996587 + 0.0825473i \(0.973694\pi\)
\(998\) 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 4.17.b.a.3.1 ✓ 1
3.2 odd 2 36.17.d.a.19.1 1
4.3 odd 2 CM 4.17.b.a.3.1 ✓ 1
8.3 odd 2 64.17.c.a.63.1 1
8.5 even 2 64.17.c.a.63.1 1
12.11 even 2 36.17.d.a.19.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.17.b.a.3.1 ✓ 1 1.1 even 1 trivial
4.17.b.a.3.1 ✓ 1 4.3 odd 2 CM
36.17.d.a.19.1 1 3.2 odd 2
36.17.d.a.19.1 1 12.11 even 2
64.17.c.a.63.1 1 8.3 odd 2
64.17.c.a.63.1 1 8.5 even 2