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Properties

Label 4.33.b.a.3.1
Level $4$
Weight $33$
Character 4.3
Self dual yes
Analytic conductor $25.947$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4,33,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, 33, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 33);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 33 \)
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: \(25.9466620569\)
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)
 
\(f(q)\) \(=\) \(q+65536.0 q^{2} +4.29497e9 q^{4} -1.96496e11 q^{5} +2.81475e14 q^{8} +1.85302e15 q^{9} +O(q^{10})\) \(q+65536.0 q^{2} +4.29497e9 q^{4} -1.96496e11 q^{5} +2.81475e14 q^{8} +1.85302e15 q^{9} -1.28776e16 q^{10} +1.33009e18 q^{13} +1.84467e19 q^{16} +1.42712e18 q^{17} +1.21440e20 q^{18} -8.43944e20 q^{20} +1.53277e22 q^{25} +8.71686e22 q^{26} +4.62879e23 q^{29} +1.20893e24 q^{32} +9.35280e22 q^{34} +7.95866e24 q^{36} +1.33646e25 q^{37} -5.53087e25 q^{40} -1.17480e26 q^{41} -3.64111e26 q^{45} +1.10443e27 q^{49} +1.00451e27 q^{50} +5.71268e27 q^{52} -6.73092e27 q^{53} +3.03352e28 q^{58} -7.13883e28 q^{61} +7.92282e28 q^{64} -2.61357e29 q^{65} +6.12945e27 q^{68} +5.21579e29 q^{72} +6.06524e29 q^{73} +8.75863e29 q^{74} -3.62471e30 q^{80} +3.43368e30 q^{81} -7.69915e30 q^{82} -2.80424e29 q^{85} +1.74085e31 q^{89} -2.38624e31 q^{90} +8.40977e31 q^{97} +7.23798e31 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 65536.0 1.00000
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 4.29497e9 1.00000
\(5\) −1.96496e11 −1.28776 −0.643878 − 0.765128i \(-0.722676\pi\)
−0.643878 + 0.765128i \(0.722676\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 2.81475e14 1.00000
\(9\) 1.85302e15 1.00000
\(10\) −1.28776e16 −1.28776
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 1.33009e18 1.99888 0.999440 − 0.0334664i \(-0.0106547\pi\)
0.999440 + 0.0334664i \(0.0106547\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.84467e19 1.00000
\(17\) 1.42712e18 0.0293278 0.0146639 − 0.999892i \(-0.495332\pi\)
0.0146639 + 0.999892i \(0.495332\pi\)
\(18\) 1.21440e20 1.00000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −8.43944e20 −1.28776
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 1.53277e22 0.658318
\(26\) 8.71686e22 1.99888
\(27\) 0 0
\(28\) 0 0
\(29\) 4.62879e23 1.84969 0.924846 − 0.380342i \(-0.124194\pi\)
0.924846 + 0.380342i \(0.124194\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1.20893e24 1.00000
\(33\) 0 0
\(34\) 9.35280e22 0.0293278
\(35\) 0 0
\(36\) 7.95866e24 1.00000
\(37\) 1.33646e25 1.08325 0.541625 − 0.840620i \(-0.317809\pi\)
0.541625 + 0.840620i \(0.317809\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −5.53087e25 −1.28776
\(41\) −1.17480e26 −1.84256 −0.921280 − 0.388901i \(-0.872855\pi\)
−0.921280 + 0.388901i \(0.872855\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −3.64111e26 −1.28776
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.10443e27 1.00000
\(50\) 1.00451e27 0.658318
\(51\) 0 0
\(52\) 5.71268e27 1.99888
\(53\) −6.73092e27 −1.73644 −0.868222 − 0.496176i \(-0.834737\pi\)
−0.868222 + 0.496176i \(0.834737\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 3.03352e28 1.84969
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −7.13883e28 −1.94245 −0.971225 − 0.238165i \(-0.923454\pi\)
−0.971225 + 0.238165i \(0.923454\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 7.92282e28 1.00000
\(65\) −2.61357e29 −2.57407
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 6.12945e27 0.0293278
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 5.21579e29 1.00000
\(73\) 6.06524e29 0.932571 0.466285 − 0.884634i \(-0.345592\pi\)
0.466285 + 0.884634i \(0.345592\pi\)
\(74\) 8.75863e29 1.08325
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −3.62471e30 −1.28776
\(81\) 3.43368e30 1.00000
\(82\) −7.69915e30 −1.84256
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −2.80424e29 −0.0377670
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.74085e31 1.12337 0.561683 − 0.827353i \(-0.310154\pi\)
0.561683 + 0.827353i \(0.310154\pi\)
\(90\) −2.38624e31 −1.28776
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.40977e31 1.36910 0.684552 − 0.728964i \(-0.259998\pi\)
0.684552 + 0.728964i \(0.259998\pi\)
\(98\) 7.23798e31 1.00000
\(99\) 0 0
\(100\) 6.58318e31 0.658318
\(101\) −2.34248e32 −1.99772 −0.998858 − 0.0477860i \(-0.984783\pi\)
−0.998858 + 0.0477860i \(0.984783\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 3.74386e32 1.99888
\(105\) 0 0
\(106\) −4.41118e32 −1.73644
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −7.90239e32 −1.99037 −0.995186 − 0.0980030i \(-0.968755\pi\)
−0.995186 + 0.0980030i \(0.968755\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.50076e32 −1.06133 −0.530665 − 0.847582i \(-0.678058\pi\)
−0.530665 + 0.847582i \(0.678058\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.98805e33 1.84969
\(117\) 2.46468e33 1.99888
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.11138e33 1.00000
\(122\) −4.67850e33 −1.94245
\(123\) 0 0
\(124\) 0 0
\(125\) 1.56321e33 0.440004
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 5.19230e33 1.00000
\(129\) 0 0
\(130\) −1.71283e34 −2.57407
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 4.01700e32 0.0293278
\(137\) 5.06204e33 0.328698 0.164349 − 0.986402i \(-0.447448\pi\)
0.164349 + 0.986402i \(0.447448\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 3.41822e34 1.00000
\(145\) −9.09539e34 −2.38195
\(146\) 3.97491e34 0.932571
\(147\) 0 0
\(148\) 5.74005e34 1.08325
\(149\) 9.07572e34 1.53780 0.768902 − 0.639367i \(-0.220804\pi\)
0.768902 + 0.639367i \(0.220804\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 2.64449e33 0.0293278
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.62580e35 −1.92694 −0.963468 − 0.267822i \(-0.913696\pi\)
−0.963468 + 0.267822i \(0.913696\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −2.37549e35 −1.28776
\(161\) 0 0
\(162\) 2.25030e35 1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −5.04572e35 −1.84256
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.32635e36 2.99552
\(170\) −1.83779e34 −0.0377670
\(171\) 0 0
\(172\) 0 0
\(173\) 2.20864e35 0.343074 0.171537 − 0.985178i \(-0.445127\pi\)
0.171537 + 0.985178i \(0.445127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 1.14088e36 1.12337
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −1.56385e36 −1.28776
\(181\) −2.96253e35 −0.223257 −0.111629 − 0.993750i \(-0.535607\pi\)
−0.111629 + 0.993750i \(0.535607\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.62609e36 −1.39496
\(186\) 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
\(193\) −2.75193e36 −0.742542 −0.371271 − 0.928525i \(-0.621078\pi\)
−0.371271 + 0.928525i \(0.621078\pi\)
\(194\) 5.51142e36 1.36910
\(195\) 0 0
\(196\) 4.74348e36 1.00000
\(197\) −6.71665e36 −1.30525 −0.652624 − 0.757682i \(-0.726332\pi\)
−0.652624 + 0.757682i \(0.726332\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 4.31435e36 0.658318
\(201\) 0 0
\(202\) −1.53517e37 −1.99772
\(203\) 0 0
\(204\) 0 0
\(205\) 2.30843e37 2.37277
\(206\) 0 0
\(207\) 0 0
\(208\) 2.45358e37 1.99888
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −2.89091e37 −1.73644
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −5.17891e37 −1.99037
\(219\) 0 0
\(220\) 0 0
\(221\) 1.89820e36 0.0586227
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 2.84025e37 0.658318
\(226\) −4.91570e37 −1.06133
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −5.18823e37 −0.907092 −0.453546 − 0.891233i \(-0.649841\pi\)
−0.453546 + 0.891233i \(0.649841\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.30289e38 1.84969
\(233\) 5.51627e37 0.731056 0.365528 − 0.930800i \(-0.380888\pi\)
0.365528 + 0.930800i \(0.380888\pi\)
\(234\) 1.61525e38 1.99888
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.20816e38 −0.932938 −0.466469 − 0.884537i \(-0.654474\pi\)
−0.466469 + 0.884537i \(0.654474\pi\)
\(242\) 1.38371e38 1.00000
\(243\) 0 0
\(244\) −3.06610e38 −1.94245
\(245\) −2.17016e38 −1.28776
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 1.02446e38 0.440004
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 3.40282e38 1.00000
\(257\) −2.99732e38 −0.827566 −0.413783 − 0.910376i \(-0.635793\pi\)
−0.413783 + 0.910376i \(0.635793\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.12252e39 −2.57407
\(261\) 8.57724e38 1.84969
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 1.32260e39 2.23612
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −8.00999e38 −1.06560 −0.532802 − 0.846240i \(-0.678861\pi\)
−0.532802 + 0.846240i \(0.678861\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 2.63258e37 0.0293278
\(273\) 0 0
\(274\) 3.31746e38 0.328698
\(275\) 0 0
\(276\) 0 0
\(277\) 2.02706e39 1.68729 0.843647 − 0.536899i \(-0.180404\pi\)
0.843647 + 0.536899i \(0.180404\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.91474e39 −1.92885 −0.964425 − 0.264358i \(-0.914840\pi\)
−0.964425 + 0.264358i \(0.914840\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 2.24016e39 1.00000
\(289\) −2.36587e39 −0.999140
\(290\) −5.96075e39 −2.38195
\(291\) 0 0
\(292\) 2.60500e39 0.932571
\(293\) −4.85045e39 −1.64399 −0.821997 − 0.569492i \(-0.807140\pi\)
−0.821997 + 0.569492i \(0.807140\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.76180e39 1.08325
\(297\) 0 0
\(298\) 5.94786e39 1.53780
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.40275e40 2.50140
\(306\) 1.73309e38 0.0293278
\(307\) 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
\(313\) 4.87729e39 0.574737 0.287369 − 0.957820i \(-0.407220\pi\)
0.287369 + 0.957820i \(0.407220\pi\)
\(314\) −1.72085e40 −1.92694
\(315\) 0 0
\(316\) 0 0
\(317\) −1.61686e40 −1.55498 −0.777489 − 0.628896i \(-0.783507\pi\)
−0.777489 + 0.628896i \(0.783507\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.55680e40 −1.28776
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.47476e40 1.00000
\(325\) 2.03871e40 1.31590
\(326\) 0 0
\(327\) 0 0
\(328\) −3.30676e40 −1.84256
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 2.47649e40 1.08325
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.27404e40 −1.54440 −0.772200 − 0.635379i \(-0.780844\pi\)
−0.772200 + 0.635379i \(0.780844\pi\)
\(338\) 8.69239e40 2.99552
\(339\) 0 0
\(340\) −1.20441e39 −0.0377670
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 1.44746e40 0.343074
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −7.05900e40 −1.45726 −0.728629 − 0.684909i \(-0.759842\pi\)
−0.728629 + 0.684909i \(0.759842\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.22712e39 0.124328 0.0621642 − 0.998066i \(-0.480200\pi\)
0.0621642 + 0.998066i \(0.480200\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 7.47689e40 1.12337
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −1.02488e41 −1.28776
\(361\) 8.31984e40 1.00000
\(362\) −1.94152e40 −0.223257
\(363\) 0 0
\(364\) 0 0
\(365\) −1.19180e41 −1.20092
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −2.17692e41 −1.84256
\(370\) −1.72104e41 −1.39496
\(371\) 0 0
\(372\) 0 0
\(373\) 2.15143e41 1.53245 0.766226 − 0.642571i \(-0.222132\pi\)
0.766226 + 0.642571i \(0.222132\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.15669e41 3.69731
\(378\) 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
\(386\) −1.80351e41 −0.742542
\(387\) 0 0
\(388\) 3.61197e41 1.36910
\(389\) −1.43255e41 −0.521095 −0.260548 − 0.965461i \(-0.583903\pi\)
−0.260548 + 0.965461i \(0.583903\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 3.10869e41 1.00000
\(393\) 0 0
\(394\) −4.40182e41 −1.30525
\(395\) 0 0
\(396\) 0 0
\(397\) −7.11797e41 −1.86943 −0.934713 − 0.355404i \(-0.884343\pi\)
−0.934713 + 0.355404i \(0.884343\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 2.82745e41 0.658318
\(401\) −2.49148e40 −0.0557375 −0.0278687 − 0.999612i \(-0.508872\pi\)
−0.0278687 + 0.999612i \(0.508872\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −1.00609e42 −1.99772
\(405\) −6.74706e41 −1.28776
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.38492e40 0.104132 0.0520658 − 0.998644i \(-0.483419\pi\)
0.0520658 + 0.998644i \(0.483419\pi\)
\(410\) 1.51285e42 2.37277
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.60798e42 1.99888
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 8.78282e41 0.901821 0.450910 − 0.892569i \(-0.351099\pi\)
0.450910 + 0.892569i \(0.351099\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −1.89459e42 −1.73644
\(425\) 2.18745e40 0.0193070
\(426\) 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
\(433\) 2.08411e42 1.36494 0.682471 − 0.730913i \(-0.260905\pi\)
0.682471 + 0.730913i \(0.260905\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.39405e42 −1.99037
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 2.04653e42 1.00000
\(442\) 1.24400e41 0.0586227
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −3.42070e42 −1.44662
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.20147e42 −0.440323 −0.220162 − 0.975463i \(-0.570658\pi\)
−0.220162 + 0.975463i \(0.570658\pi\)
\(450\) 1.86138e42 0.658318
\(451\) 0 0
\(452\) −3.22155e42 −1.06133
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.99569e42 1.93273 0.966365 − 0.257174i \(-0.0827915\pi\)
0.966365 + 0.257174i \(0.0827915\pi\)
\(458\) −3.40016e42 −0.907092
\(459\) 0 0
\(460\) 0 0
\(461\) −8.14863e42 −1.95826 −0.979128 − 0.203244i \(-0.934852\pi\)
−0.979128 + 0.203244i \(0.934852\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 8.53861e42 1.84969
\(465\) 0 0
\(466\) 3.61514e42 0.731056
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 1.05857e43 1.99888
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.24725e43 −1.73644
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 1.77761e43 2.16529
\(482\) −7.91782e42 −0.932938
\(483\) 0 0
\(484\) 9.06830e42 1.00000
\(485\) −1.65249e43 −1.76307
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) −2.00940e43 −1.94245
\(489\) 0 0
\(490\) −1.42223e43 −1.28776
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 6.60586e41 0.0542473
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 6.71392e42 0.440004
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 4.60288e43 2.57257
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.62392e43 1.29261 0.646304 − 0.763080i \(-0.276314\pi\)
0.646304 + 0.763080i \(0.276314\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 2.23007e43 1.00000
\(513\) 0 0
\(514\) −1.96432e43 −0.827566
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −7.35655e43 −2.57407
\(521\) −5.46722e43 −1.85508 −0.927540 − 0.373723i \(-0.878081\pi\)
−0.927540 + 0.373723i \(0.878081\pi\)
\(522\) 5.62118e43 1.84969
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.76089e43 1.00000
\(530\) 8.66779e43 2.23612
\(531\) 0 0
\(532\) 0 0
\(533\) −1.56258e44 −3.68305
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −5.24943e43 −1.06560
\(539\) 0 0
\(540\) 0 0
\(541\) −9.06214e42 −0.168296 −0.0841482 − 0.996453i \(-0.526817\pi\)
−0.0841482 + 0.996453i \(0.526817\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.72529e42 0.0293278
\(545\) 1.55279e44 2.56312
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 2.17413e43 0.328698
\(549\) −1.32284e44 −1.94245
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 1.32846e44 1.68729
\(555\) 0 0
\(556\) 0 0
\(557\) 1.70532e43 0.198665 0.0993326 − 0.995054i \(-0.468329\pi\)
0.0993326 + 0.995054i \(0.468329\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −1.91020e44 −1.92885
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 1.47387e44 1.36673
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.21865e44 1.83777 0.918886 − 0.394523i \(-0.129090\pi\)
0.918886 + 0.394523i \(0.129090\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.46811e44 1.00000
\(577\) 7.12411e43 0.471973 0.235987 − 0.971756i \(-0.424168\pi\)
0.235987 + 0.971756i \(0.424168\pi\)
\(578\) −1.55050e44 −0.999140
\(579\) 0 0
\(580\) −3.90644e44 −2.38195
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 1.70721e44 0.932571
\(585\) −4.84300e44 −2.57407
\(586\) −3.17879e44 −1.64399
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 2.46533e44 1.08325
\(593\) −1.31028e44 −0.560387 −0.280193 − 0.959944i \(-0.590399\pi\)
−0.280193 + 0.959944i \(0.590399\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.89799e44 1.53780
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 5.55350e44 1.91679 0.958396 − 0.285442i \(-0.0921403\pi\)
0.958396 + 0.285442i \(0.0921403\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.14877e44 −1.28776
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 9.19308e44 2.50140
\(611\) 0 0
\(612\) 1.13580e43 0.0293278
\(613\) 4.85424e44 1.22111 0.610553 − 0.791975i \(-0.290947\pi\)
0.610553 + 0.791975i \(0.290947\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −8.10733e44 −1.83787 −0.918936 − 0.394408i \(-0.870950\pi\)
−0.918936 + 0.394408i \(0.870950\pi\)
\(618\) 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
\(625\) −6.64039e44 −1.22494
\(626\) 3.19638e44 0.574737
\(627\) 0 0
\(628\) −1.12777e45 −1.92694
\(629\) 1.90730e43 0.0317693
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −1.05963e45 −1.55498
\(635\) 0 0
\(636\) 0 0
\(637\) 1.46899e45 1.99888
\(638\) 0 0
\(639\) 0 0
\(640\) −1.02027e45 −1.28776
\(641\) 5.79278e44 0.713113 0.356556 − 0.934274i \(-0.383951\pi\)
0.356556 + 0.934274i \(0.383951\pi\)
\(642\) 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 \(\pi\)
\(648\) 9.66496e44 1.00000
\(649\) 0 0
\(650\) 1.33609e45 1.31590
\(651\) 0 0
\(652\) 0 0
\(653\) −4.34491e44 −0.397530 −0.198765 − 0.980047i \(-0.563693\pi\)
−0.198765 + 0.980047i \(0.563693\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.16712e45 −1.84256
\(657\) 1.12390e45 0.932571
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 8.26814e44 0.622565 0.311282 − 0.950317i \(-0.399241\pi\)
0.311282 + 0.950317i \(0.399241\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 1.62299e45 1.08325
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −3.36284e45 −1.89874 −0.949370 − 0.314159i \(-0.898278\pi\)
−0.949370 + 0.314159i \(0.898278\pi\)
\(674\) −2.80103e45 −1.54440
\(675\) 0 0
\(676\) 5.69665e45 2.99552
\(677\) 1.30108e45 0.668168 0.334084 − 0.942543i \(-0.391573\pi\)
0.334084 + 0.942543i \(0.391573\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −7.89325e43 −0.0377670
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −9.94671e44 −0.423283
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8.95272e45 −3.47094
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 9.48605e44 0.343074
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1.67658e44 −0.0540382
\(698\) −4.62619e45 −1.45726
\(699\) 0 0
\(700\) 0 0
\(701\) 6.65974e45 1.95871 0.979353 − 0.202157i \(-0.0647952\pi\)
0.979353 + 0.202157i \(0.0647952\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 4.73637e44 0.124328
\(707\) 0 0
\(708\) 0 0
\(709\) 7.79016e45 1.91076 0.955381 − 0.295376i \(-0.0954448\pi\)
0.955381 + 0.295376i \(0.0954448\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 4.90006e45 1.12337
\(713\) 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 \(\pi\)
\(720\) −6.71667e45 −1.28776
\(721\) 0 0
\(722\) 5.45249e45 1.00000
\(723\) 0 0
\(724\) −1.27240e45 −0.223257
\(725\) 7.09485e45 1.21768
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 6.36269e45 1.00000
\(730\) −7.81055e45 −1.20092
\(731\) 0 0
\(732\) 0 0
\(733\) −9.91783e45 −1.42808 −0.714041 − 0.700104i \(-0.753137\pi\)
−0.714041 + 0.700104i \(0.753137\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −1.42667e46 −1.84256
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) −1.12790e46 −1.39496
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −1.78334e46 −1.98032
\(746\) 1.40996e46 1.53245
\(747\) 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
\(754\) 4.03485e46 3.69731
\(755\) 0 0
\(756\) 0 0
\(757\) −7.57509e45 −0.651409 −0.325704 − 0.945472i \(-0.605601\pi\)
−0.325704 + 0.945472i \(0.605601\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.72566e46 1.36396 0.681978 − 0.731373i \(-0.261120\pi\)
0.681978 + 0.731373i \(0.261120\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −5.19632e44 −0.0377670
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −5.45736e45 −0.364889 −0.182445 − 0.983216i \(-0.558401\pi\)
−0.182445 + 0.983216i \(0.558401\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.18195e46 −0.742542
\(773\) −1.92027e46 −1.18165 −0.590827 − 0.806798i \(-0.701198\pi\)
−0.590827 + 0.806798i \(0.701198\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2.36714e46 1.36910
\(777\) 0 0
\(778\) −9.38838e45 −0.521095
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 2.03731e46 1.00000
\(785\) 5.15960e46 2.48143
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −2.88478e46 −1.30525
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −9.49527e46 −3.88272
\(794\) −4.66483e46 −1.86943
\(795\) 0 0
\(796\) 0 0
\(797\) 5.14744e46 1.94204 0.971021 − 0.238995i \(-0.0768179\pi\)
0.971021 + 0.238995i \(0.0768179\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.85300e46 0.658318
\(801\) 3.22583e46 1.12337
\(802\) −1.63282e45 −0.0557375
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −6.59349e46 −1.99772
\(809\) 5.44636e46 1.61782 0.808909 − 0.587934i \(-0.200058\pi\)
0.808909 + 0.587934i \(0.200058\pi\)
\(810\) −4.42175e46 −1.28776
\(811\) 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
\(818\) 4.18442e45 0.104132
\(819\) 0 0
\(820\) 9.91464e46 2.37277
\(821\) −6.89739e46 −1.61880 −0.809402 − 0.587255i \(-0.800209\pi\)
−0.809402 + 0.587255i \(0.800209\pi\)
\(822\) 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
\(829\) 3.45922e46 0.695193 0.347597 − 0.937644i \(-0.386998\pi\)
0.347597 + 0.937644i \(0.386998\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.05380e47 1.99888
\(833\) 1.57616e45 0.0293278
\(834\) 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
\(841\) 1.51633e47 2.42136
\(842\) 5.75591e46 0.901821
\(843\) 0 0
\(844\) 0 0
\(845\) −2.60623e47 −3.85750
\(846\) 0 0
\(847\) 0 0
\(848\) −1.24164e47 −1.73644
\(849\) 0 0
\(850\) 1.43357e45 0.0193070
\(851\) 0 0
\(852\) 0 0
\(853\) −1.15846e47 −1.47468 −0.737341 − 0.675521i \(-0.763919\pi\)
−0.737341 + 0.675521i \(0.763919\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.42712e46 −0.641027 −0.320513 − 0.947244i \(-0.603855\pi\)
−0.320513 + 0.947244i \(0.603855\pi\)
\(858\) 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
\(865\) −4.33990e46 −0.441796
\(866\) 1.36584e47 1.36494
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −2.22432e47 −1.99037
\(873\) 1.55835e47 1.36910
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.37582e47 1.94009 0.970043 − 0.242935i \(-0.0781101\pi\)
0.970043 + 0.242935i \(0.0781101\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2.12610e47 1.61425 0.807123 − 0.590383i \(-0.201023\pi\)
0.807123 + 0.590383i \(0.201023\pi\)
\(882\) 1.34121e47 1.00000
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 8.15271e45 0.0586227
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −2.24179e47 −1.44662
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −7.87395e46 −0.440323
\(899\) 0 0
\(900\) 1.21988e47 0.658318
\(901\) −9.60587e45 −0.0509260
\(902\) 0 0
\(903\) 0 0
\(904\) −2.11128e47 −1.06133
\(905\) 5.82125e46 0.287501
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −4.34066e47 −1.99772
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 4.58469e47 1.93273
\(915\) 0 0
\(916\) −2.22833e47 −0.907092
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −5.34028e47 −1.95826
\(923\) 0 0
\(924\) 0 0
\(925\) 2.04848e47 0.713122
\(926\) 0 0
\(927\) 0 0
\(928\) 5.59586e47 1.84969
\(929\) −3.76520e47 −1.22331 −0.611654 − 0.791125i \(-0.709496\pi\)
−0.611654 + 0.791125i \(0.709496\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.36922e47 0.731056
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 6.93746e47 1.99888
\(937\) −5.97227e47 −1.69163 −0.845817 − 0.533474i \(-0.820886\pi\)
−0.845817 + 0.533474i \(0.820886\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −2.73113e47 −0.722618 −0.361309 − 0.932446i \(-0.617670\pi\)
−0.361309 + 0.932446i \(0.617670\pi\)
\(942\) 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
\(949\) 8.06729e47 1.86410
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 2.09005e47 0.451514 0.225757 − 0.974184i \(-0.427514\pi\)
0.225757 + 0.974184i \(0.427514\pi\)
\(954\) −8.17400e47 −1.73644
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 5.29144e47 1.00000
\(962\) 1.16497e48 2.16529
\(963\) 0 0
\(964\) −5.18902e47 −0.932938
\(965\) 5.40744e47 0.956213
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 5.94300e47 1.00000
\(969\) 0 0
\(970\) −1.08297e48 −1.76307
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −1.31688e48 −1.94245
\(977\) −7.85108e47 −1.13924 −0.569622 − 0.821907i \(-0.692910\pi\)
−0.569622 + 0.821907i \(0.692910\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −9.32076e47 −1.28776
\(981\) −1.46433e48 −1.99037
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 1.31980e48 1.68084
\(986\) 4.32921e46 0.0542473
\(987\) 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
\(997\) 1.88015e48 1.97274 0.986372 − 0.164531i \(-0.0526111\pi\)
0.986372 + 0.164531i \(0.0526111\pi\)
\(998\) 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 4.33.b.a.3.1 ✓ 1
3.2 odd 2 36.33.d.a.19.1 1
4.3 odd 2 CM 4.33.b.a.3.1 ✓ 1
12.11 even 2 36.33.d.a.19.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.33.b.a.3.1 ✓ 1 1.1 even 1 trivial
4.33.b.a.3.1 ✓ 1 4.3 odd 2 CM
36.33.d.a.19.1 1 3.2 odd 2
36.33.d.a.19.1 1 12.11 even 2