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Properties

Label 4.37.b.a.3.1
Level $4$
Weight $37$
Character 4.3
Self dual yes
Analytic conductor $32.837$
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] = [4,37,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, 37, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 37);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 37 \)
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: \(32.8365034637\)
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)
 
\(f(q)\) \(=\) \(q-262144. q^{2} +6.87195e10 q^{4} -4.22849e12 q^{5} -1.80144e16 q^{8} +1.50095e17 q^{9} +O(q^{10})\) \(q-262144. q^{2} +6.87195e10 q^{4} -4.22849e12 q^{5} -1.80144e16 q^{8} +1.50095e17 q^{9} +1.10847e18 q^{10} -1.52936e20 q^{13} +4.72237e21 q^{16} -2.31251e22 q^{17} -3.93464e22 q^{18} -2.90580e23 q^{20} +3.32822e24 q^{25} +4.00912e25 q^{26} +1.78879e26 q^{29} -1.23794e27 q^{32} +6.06210e27 q^{34} +1.03144e28 q^{36} +3.18707e28 q^{37} +7.61737e28 q^{40} +1.42746e29 q^{41} -6.34674e29 q^{45} +2.65173e30 q^{49} -8.72472e29 q^{50} -1.05097e31 q^{52} -1.80466e31 q^{53} -4.68921e31 q^{58} +2.71468e32 q^{61} +3.24519e32 q^{64} +6.46688e32 q^{65} -1.58914e33 q^{68} -2.70386e33 q^{72} +6.51296e33 q^{73} -8.35471e33 q^{74} -1.99685e34 q^{80} +2.25284e34 q^{81} -3.74199e34 q^{82} +9.77842e34 q^{85} +7.50733e34 q^{89} +1.66376e35 q^{90} -9.18873e35 q^{97} -6.95135e35 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −262144. −1.00000
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 6.87195e10 1.00000
\(5\) −4.22849e12 −1.10847 −0.554237 − 0.832359i \(-0.686990\pi\)
−0.554237 + 0.832359i \(0.686990\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −1.80144e16 −1.00000
\(9\) 1.50095e17 1.00000
\(10\) 1.10847e18 1.10847
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −1.52936e20 −1.35997 −0.679984 − 0.733226i \(-0.738014\pi\)
−0.679984 + 0.733226i \(0.738014\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.72237e21 1.00000
\(17\) −2.31251e22 −1.64438 −0.822191 − 0.569211i \(-0.807249\pi\)
−0.822191 + 0.569211i \(0.807249\pi\)
\(18\) −3.93464e22 −1.00000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −2.90580e23 −1.10847
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 3.32822e24 0.228713
\(26\) 4.00912e25 1.35997
\(27\) 0 0
\(28\) 0 0
\(29\) 1.78879e26 0.849956 0.424978 − 0.905204i \(-0.360282\pi\)
0.424978 + 0.905204i \(0.360282\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.23794e27 −1.00000
\(33\) 0 0
\(34\) 6.06210e27 1.64438
\(35\) 0 0
\(36\) 1.03144e28 1.00000
\(37\) 3.18707e28 1.88695 0.943475 − 0.331443i \(-0.107536\pi\)
0.943475 + 0.331443i \(0.107536\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 7.61737e28 1.10847
\(41\) 1.42746e29 1.33184 0.665922 − 0.746021i \(-0.268038\pi\)
0.665922 + 0.746021i \(0.268038\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −6.34674e29 −1.10847
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 2.65173e30 1.00000
\(50\) −8.72472e29 −0.228713
\(51\) 0 0
\(52\) −1.05097e31 −1.35997
\(53\) −1.80466e31 −1.65741 −0.828703 − 0.559688i \(-0.810921\pi\)
−0.828703 + 0.559688i \(0.810921\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −4.68921e31 −0.849956
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 2.71468e32 1.98509 0.992547 − 0.121860i \(-0.0388858\pi\)
0.992547 + 0.121860i \(0.0388858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 3.24519e32 1.00000
\(65\) 6.46688e32 1.50749
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −1.58914e33 −1.64438
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −2.70386e33 −1.00000
\(73\) 6.51296e33 1.87917 0.939587 − 0.342309i \(-0.111209\pi\)
0.939587 + 0.342309i \(0.111209\pi\)
\(74\) −8.35471e33 −1.88695
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −1.99685e34 −1.10847
\(81\) 2.25284e34 1.00000
\(82\) −3.74199e34 −1.33184
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 9.77842e34 1.82275
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.50733e34 0.611597 0.305798 − 0.952096i \(-0.401077\pi\)
0.305798 + 0.952096i \(0.401077\pi\)
\(90\) 1.66376e35 1.10847
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.18873e35 −1.58988 −0.794940 − 0.606688i \(-0.792498\pi\)
−0.794940 + 0.606688i \(0.792498\pi\)
\(98\) −6.95135e35 −1.00000
\(99\) 0 0
\(100\) 2.28713e35 0.228713
\(101\) −2.15779e36 −1.80395 −0.901973 − 0.431792i \(-0.857881\pi\)
−0.901973 + 0.431792i \(0.857881\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 2.75505e36 1.35997
\(105\) 0 0
\(106\) 4.73080e36 1.65741
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −4.54891e36 −0.964339 −0.482170 − 0.876078i \(-0.660151\pi\)
−0.482170 + 0.876078i \(0.660151\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.32656e37 1.46999 0.734994 − 0.678074i \(-0.237185\pi\)
0.734994 + 0.678074i \(0.237185\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.22925e37 0.849956
\(117\) −2.29549e37 −1.35997
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.09127e37 1.00000
\(122\) −7.11636e37 −1.98509
\(123\) 0 0
\(124\) 0 0
\(125\) 4.74593e37 0.854951
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −8.50706e37 −1.00000
\(129\) 0 0
\(130\) −1.69525e38 −1.50749
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 4.16585e38 1.64438
\(137\) 5.78064e38 1.99989 0.999944 − 0.0106143i \(-0.00337870\pi\)
0.999944 + 0.0106143i \(0.00337870\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 7.08802e38 1.00000
\(145\) −7.56390e38 −0.942153
\(146\) −1.70733e39 −1.87917
\(147\) 0 0
\(148\) 2.19014e39 1.88695
\(149\) −2.62046e39 −1.99997 −0.999987 − 0.00501384i \(-0.998404\pi\)
−0.999987 + 0.00501384i \(0.998404\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −3.47095e39 −1.64438
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.31598e39 1.28495 0.642474 − 0.766307i \(-0.277908\pi\)
0.642474 + 0.766307i \(0.277908\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 5.23462e39 1.10847
\(161\) 0 0
\(162\) −5.90568e39 −1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 9.80940e39 1.33184
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.07432e40 0.849516
\(170\) −2.56336e40 −1.82275
\(171\) 0 0
\(172\) 0 0
\(173\) 2.71829e40 1.41080 0.705400 − 0.708810i \(-0.250768\pi\)
0.705400 + 0.708810i \(0.250768\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −1.96800e40 −0.611597
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −4.36144e40 −1.10847
\(181\) 2.75310e40 0.633297 0.316648 − 0.948543i \(-0.397442\pi\)
0.316648 + 0.948543i \(0.397442\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.34765e41 −2.09163
\(186\) 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
\(193\) 2.72521e41 1.97410 0.987048 − 0.160428i \(-0.0512874\pi\)
0.987048 + 0.160428i \(0.0512874\pi\)
\(194\) 2.40877e41 1.58988
\(195\) 0 0
\(196\) 1.82226e41 1.00000
\(197\) −3.35288e41 −1.67890 −0.839451 − 0.543435i \(-0.817123\pi\)
−0.839451 + 0.543435i \(0.817123\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −5.99558e40 −0.228713
\(201\) 0 0
\(202\) 5.65651e41 1.80395
\(203\) 0 0
\(204\) 0 0
\(205\) −6.03598e41 −1.47631
\(206\) 0 0
\(207\) 0 0
\(208\) −7.22219e41 −1.35997
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −1.24015e42 −1.65741
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 1.19247e42 0.964339
\(219\) 0 0
\(220\) 0 0
\(221\) 3.53666e42 2.23631
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 4.99548e41 0.228713
\(226\) −3.47749e42 −1.46999
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 3.56519e41 0.118862 0.0594311 − 0.998232i \(-0.481071\pi\)
0.0594311 + 0.998232i \(0.481071\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.22240e42 −0.849956
\(233\) 4.35731e42 1.06368 0.531841 − 0.846844i \(-0.321500\pi\)
0.531841 + 0.846844i \(0.321500\pi\)
\(234\) 6.01748e42 1.35997
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.50291e43 −1.99814 −0.999072 − 0.0430785i \(-0.986283\pi\)
−0.999072 + 0.0430785i \(0.986283\pi\)
\(242\) −8.10357e42 −1.00000
\(243\) 0 0
\(244\) 1.86551e43 1.98509
\(245\) −1.12128e43 −1.10847
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −1.24412e43 −0.854951
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 2.23007e43 1.00000
\(257\) −2.99453e43 −1.25179 −0.625897 − 0.779906i \(-0.715267\pi\)
−0.625897 + 0.779906i \(0.715267\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 4.44401e43 1.50749
\(261\) 2.68488e43 0.849956
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 7.63097e43 1.83719
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.68695e42 0.0861686 0.0430843 − 0.999071i \(-0.486282\pi\)
0.0430843 + 0.999071i \(0.486282\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −1.09205e44 −1.64438
\(273\) 0 0
\(274\) −1.51536e44 −1.99989
\(275\) 0 0
\(276\) 0 0
\(277\) −2.71487e43 −0.294519 −0.147260 − 0.989098i \(-0.547045\pi\)
−0.147260 + 0.989098i \(0.547045\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.18572e43 −0.183181 −0.0915905 − 0.995797i \(-0.529195\pi\)
−0.0915905 + 0.995797i \(0.529195\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.85808e44 −1.00000
\(289\) 3.37000e44 1.70399
\(290\) 1.98283e44 0.942153
\(291\) 0 0
\(292\) 4.47567e44 1.87917
\(293\) −2.41298e44 −0.952657 −0.476328 − 0.879267i \(-0.658033\pi\)
−0.476328 + 0.879267i \(0.658033\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −5.74131e44 −1.88695
\(297\) 0 0
\(298\) 6.86938e44 1.99997
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.14790e45 −2.20042
\(306\) 9.09889e44 1.64438
\(307\) 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
\(313\) −2.18127e44 −0.262369 −0.131184 − 0.991358i \(-0.541878\pi\)
−0.131184 + 0.991358i \(0.541878\pi\)
\(314\) −1.13141e45 −1.28495
\(315\) 0 0
\(316\) 0 0
\(317\) 8.38644e44 0.802622 0.401311 − 0.915942i \(-0.368555\pi\)
0.401311 + 0.915942i \(0.368555\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.37222e45 −1.10847
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.54814e45 1.00000
\(325\) −5.09004e44 −0.311043
\(326\) 0 0
\(327\) 0 0
\(328\) −2.57148e45 −1.33184
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 4.78362e45 1.88695
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 5.78102e45 1.83936 0.919679 − 0.392672i \(-0.128449\pi\)
0.919679 + 0.392672i \(0.128449\pi\)
\(338\) −2.81625e45 −0.849516
\(339\) 0 0
\(340\) 6.71968e45 1.82275
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −7.12582e45 −1.41080
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −1.11661e46 −1.89253 −0.946266 − 0.323389i \(-0.895178\pi\)
−0.946266 + 0.323389i \(0.895178\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.43714e46 −1.98405 −0.992026 − 0.126034i \(-0.959775\pi\)
−0.992026 + 0.126034i \(0.959775\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 5.15900e45 0.611597
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 1.14333e46 1.10847
\(361\) 1.08425e46 1.00000
\(362\) −7.21708e45 −0.633297
\(363\) 0 0
\(364\) 0 0
\(365\) −2.75400e46 −2.08302
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 2.14254e46 1.33184
\(370\) 3.53278e46 2.09163
\(371\) 0 0
\(372\) 0 0
\(373\) 2.76144e46 1.41376 0.706882 − 0.707331i \(-0.250101\pi\)
0.706882 + 0.707331i \(0.250101\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.73571e46 −1.15591
\(378\) 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
\(386\) −7.14397e46 −1.97410
\(387\) 0 0
\(388\) −6.31445e46 −1.58988
\(389\) 7.96658e46 1.91505 0.957523 − 0.288357i \(-0.0931090\pi\)
0.957523 + 0.288357i \(0.0931090\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −4.77693e46 −1.00000
\(393\) 0 0
\(394\) 8.78936e46 1.67890
\(395\) 0 0
\(396\) 0 0
\(397\) 1.92807e45 0.0321288 0.0160644 − 0.999871i \(-0.494886\pi\)
0.0160644 + 0.999871i \(0.494886\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.57171e46 0.228713
\(401\) −3.24522e46 −0.451487 −0.225743 − 0.974187i \(-0.572481\pi\)
−0.225743 + 0.974187i \(0.572481\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −1.48282e47 −1.80395
\(405\) −9.52611e46 −1.10847
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.23763e47 1.20663 0.603313 − 0.797504i \(-0.293847\pi\)
0.603313 + 0.797504i \(0.293847\pi\)
\(410\) 1.58230e47 1.47631
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.89325e47 1.35997
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −1.11839e47 −0.647913 −0.323957 − 0.946072i \(-0.605013\pi\)
−0.323957 + 0.946072i \(0.605013\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 3.25098e47 1.65741
\(425\) −7.69653e46 −0.376092
\(426\) 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
\(433\) −5.67206e47 −1.98134 −0.990669 − 0.136292i \(-0.956481\pi\)
−0.990669 + 0.136292i \(0.956481\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.12598e47 −0.964339
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 3.98011e47 1.00000
\(442\) −9.27113e47 −2.23631
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −3.17447e47 −0.677939
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.92524e47 1.80429 0.902146 − 0.431430i \(-0.141991\pi\)
0.902146 + 0.431430i \(0.141991\pi\)
\(450\) −1.30953e47 −0.228713
\(451\) 0 0
\(452\) 9.11603e47 1.46999
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.33200e48 1.76203 0.881017 − 0.473085i \(-0.156860\pi\)
0.881017 + 0.473085i \(0.156860\pi\)
\(458\) −9.34593e46 −0.118862
\(459\) 0 0
\(460\) 0 0
\(461\) 2.86051e47 0.323465 0.161732 − 0.986835i \(-0.448292\pi\)
0.161732 + 0.986835i \(0.448292\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 8.44734e47 0.849956
\(465\) 0 0
\(466\) −1.14224e48 −1.06368
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −1.57745e48 −1.35997
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2.70869e48 −1.65741
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −4.87417e48 −2.56619
\(482\) 3.93979e48 1.99814
\(483\) 0 0
\(484\) 2.12430e48 1.00000
\(485\) 3.88545e48 1.76234
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) −4.89033e48 −1.98509
\(489\) 0 0
\(490\) 2.93937e48 1.10847
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −4.13660e48 −1.39765
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 3.26138e48 0.854951
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 9.12418e48 1.99963
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.99864e48 −0.380027 −0.190014 − 0.981781i \(-0.560853\pi\)
−0.190014 + 0.981781i \(0.560853\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −5.84601e48 −1.00000
\(513\) 0 0
\(514\) 7.84999e48 1.25179
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −1.16497e49 −1.50749
\(521\) 1.17372e49 1.46719 0.733594 − 0.679588i \(-0.237841\pi\)
0.733594 + 0.679588i \(0.237841\pi\)
\(522\) −7.03826e48 −0.849956
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.05245e49 1.00000
\(530\) −2.00041e49 −1.83719
\(531\) 0 0
\(532\) 0 0
\(533\) −2.18309e49 −1.81127
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −1.22866e48 −0.0861686
\(539\) 0 0
\(540\) 0 0
\(541\) −3.01933e49 −1.91584 −0.957921 − 0.287033i \(-0.907331\pi\)
−0.957921 + 0.287033i \(0.907331\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 2.86275e49 1.64438
\(545\) 1.92350e49 1.06894
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 3.97242e49 1.99989
\(549\) 4.07459e49 1.98509
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 7.11687e48 0.294519
\(555\) 0 0
\(556\) 0 0
\(557\) −3.43528e49 −1.28994 −0.644969 − 0.764209i \(-0.723130\pi\)
−0.644969 + 0.764209i \(0.723130\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 5.72973e48 0.183181
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) −5.60933e49 −1.62944
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.52686e49 −0.646488 −0.323244 − 0.946316i \(-0.604774\pi\)
−0.323244 + 0.946316i \(0.604774\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 4.87085e49 1.00000
\(577\) 7.19649e48 0.143204 0.0716022 − 0.997433i \(-0.477189\pi\)
0.0716022 + 0.997433i \(0.477189\pi\)
\(578\) −8.83424e49 −1.70399
\(579\) 0 0
\(580\) −5.19787e49 −0.942153
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −1.17327e50 −1.87917
\(585\) 9.70644e49 1.50749
\(586\) 6.32548e49 0.952657
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.50505e50 1.88695
\(593\) 1.43045e50 1.73975 0.869876 − 0.493270i \(-0.164198\pi\)
0.869876 + 0.493270i \(0.164198\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.80077e50 −1.99997
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 9.28713e49 0.887444 0.443722 − 0.896165i \(-0.353658\pi\)
0.443722 + 0.896165i \(0.353658\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.30714e50 −1.10847
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 3.00915e50 2.20042
\(611\) 0 0
\(612\) −2.38522e50 −1.64438
\(613\) −1.54243e50 −1.03257 −0.516283 − 0.856418i \(-0.672685\pi\)
−0.516283 + 0.856418i \(0.672685\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.35189e50 1.99598 0.997991 − 0.0633596i \(-0.0201815\pi\)
0.997991 + 0.0633596i \(0.0201815\pi\)
\(618\) 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
\(625\) −2.49113e50 −1.17640
\(626\) 5.71808e49 0.262369
\(627\) 0 0
\(628\) 2.96592e50 1.28495
\(629\) −7.37013e50 −3.10287
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −2.19846e50 −0.802622
\(635\) 0 0
\(636\) 0 0
\(637\) −4.05545e50 −1.35997
\(638\) 0 0
\(639\) 0 0
\(640\) 3.59720e50 1.10847
\(641\) 5.61423e50 1.68208 0.841039 − 0.540975i \(-0.181945\pi\)
0.841039 + 0.540975i \(0.181945\pi\)
\(642\) 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 \(\pi\)
\(648\) −4.05836e50 −1.00000
\(649\) 0 0
\(650\) 1.33432e50 0.311043
\(651\) 0 0
\(652\) 0 0
\(653\) 8.71059e50 1.86900 0.934502 − 0.355958i \(-0.115845\pi\)
0.934502 + 0.355958i \(0.115845\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.74097e50 1.33184
\(657\) 9.77561e50 1.87917
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −6.79623e50 −1.17123 −0.585614 − 0.810590i \(-0.699147\pi\)
−0.585614 + 0.810590i \(0.699147\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −1.25400e51 −1.88695
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 5.32268e49 0.0663528 0.0331764 − 0.999450i \(-0.489438\pi\)
0.0331764 + 0.999450i \(0.489438\pi\)
\(674\) −1.51546e51 −1.83936
\(675\) 0 0
\(676\) 7.38264e50 0.849516
\(677\) 3.31604e50 0.371555 0.185778 − 0.982592i \(-0.440520\pi\)
0.185778 + 0.982592i \(0.440520\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −1.76152e51 −1.82275
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −2.44434e51 −2.21682
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.75997e51 2.25402
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 1.86799e51 1.41080
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −3.30101e51 −2.19006
\(698\) 2.92712e51 1.89253
\(699\) 0 0
\(700\) 0 0
\(701\) 2.83757e51 1.69833 0.849166 − 0.528127i \(-0.177105\pi\)
0.849166 + 0.528127i \(0.177105\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 3.76736e51 1.98405
\(707\) 0 0
\(708\) 0 0
\(709\) −3.69424e51 −1.80257 −0.901286 − 0.433225i \(-0.857375\pi\)
−0.901286 + 0.433225i \(0.857375\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.35240e51 −0.611597
\(713\) 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 \(\pi\)
\(720\) −2.99716e51 −1.10847
\(721\) 0 0
\(722\) −2.84230e51 −1.00000
\(723\) 0 0
\(724\) 1.89191e51 0.633297
\(725\) 5.95349e50 0.194396
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 3.38139e51 1.00000
\(730\) 7.21945e51 2.08302
\(731\) 0 0
\(732\) 0 0
\(733\) −6.62019e51 −1.77419 −0.887093 − 0.461590i \(-0.847279\pi\)
−0.887093 + 0.461590i \(0.847279\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −5.61653e51 −1.33184
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) −9.26097e51 −2.09163
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 1.10806e52 2.21692
\(746\) −7.23896e51 −1.41376
\(747\) 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
\(754\) 7.17149e51 1.15591
\(755\) 0 0
\(756\) 0 0
\(757\) 1.12237e52 1.68426 0.842130 − 0.539275i \(-0.181302\pi\)
0.842130 + 0.539275i \(0.181302\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −8.84409e51 −1.20706 −0.603531 − 0.797340i \(-0.706240\pi\)
−0.603531 + 0.797340i \(0.706240\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1.46769e52 1.82275
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1.62736e52 −1.83997 −0.919984 − 0.391955i \(-0.871799\pi\)
−0.919984 + 0.391955i \(0.871799\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.87275e52 1.97410
\(773\) 2.36430e51 0.243484 0.121742 − 0.992562i \(-0.461152\pi\)
0.121742 + 0.992562i \(0.461152\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.65530e52 1.58988
\(777\) 0 0
\(778\) −2.08839e52 −1.91505
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.25224e52 1.00000
\(785\) −1.82501e52 −1.42433
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −2.30408e52 −1.67890
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4.15172e52 −2.69967
\(794\) −5.05433e50 −0.0321288
\(795\) 0 0
\(796\) 0 0
\(797\) −9.03017e51 −0.536347 −0.268174 − 0.963371i \(-0.586420\pi\)
−0.268174 + 0.963371i \(0.586420\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −4.12013e51 −0.228713
\(801\) 1.12681e52 0.611597
\(802\) 8.50714e51 0.451487
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 3.88712e52 1.80395
\(809\) 4.39308e52 1.99387 0.996933 − 0.0782534i \(-0.0249343\pi\)
0.996933 + 0.0782534i \(0.0249343\pi\)
\(810\) 2.49721e52 1.10847
\(811\) 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
\(818\) −3.24438e52 −1.20663
\(819\) 0 0
\(820\) −4.14790e52 −1.47631
\(821\) 5.11593e52 1.78134 0.890671 − 0.454647i \(-0.150235\pi\)
0.890671 + 0.454647i \(0.150235\pi\)
\(822\) 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
\(829\) 6.69882e52 1.95892 0.979460 − 0.201641i \(-0.0646274\pi\)
0.979460 + 0.201641i \(0.0646274\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.96305e52 −1.35997
\(833\) −6.13215e52 −1.64438
\(834\) 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
\(841\) −1.22944e52 −0.277575
\(842\) 2.93180e52 0.647913
\(843\) 0 0
\(844\) 0 0
\(845\) −4.54274e52 −0.941665
\(846\) 0 0
\(847\) 0 0
\(848\) −8.52225e52 −1.65741
\(849\) 0 0
\(850\) 2.01760e52 0.376092
\(851\) 0 0
\(852\) 0 0
\(853\) 3.85237e52 0.673978 0.336989 − 0.941509i \(-0.390591\pi\)
0.336989 + 0.941509i \(0.390591\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.73812e52 0.440349 0.220174 − 0.975461i \(-0.429337\pi\)
0.220174 + 0.975461i \(0.429337\pi\)
\(858\) 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
\(865\) −1.14942e53 −1.56383
\(866\) 1.48690e53 1.98134
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 8.19458e52 0.964339
\(873\) −1.37918e53 −1.58988
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 9.18508e52 0.975192 0.487596 − 0.873069i \(-0.337874\pi\)
0.487596 + 0.873069i \(0.337874\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.53039e52 −0.149705 −0.0748526 − 0.997195i \(-0.523849\pi\)
−0.0748526 + 0.997195i \(0.523849\pi\)
\(882\) −1.04336e53 −1.00000
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 2.43037e53 2.23631
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 8.32167e52 0.677939
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −2.60184e53 −1.80429
\(899\) 0 0
\(900\) 3.43286e52 0.228713
\(901\) 4.17329e53 2.72541
\(902\) 0 0
\(903\) 0 0
\(904\) −2.38971e53 −1.46999
\(905\) −1.16414e53 −0.701993
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −3.23872e53 −1.80395
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −3.49177e53 −1.76203
\(915\) 0 0
\(916\) 2.44998e52 0.118862
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −7.49866e52 −0.323465
\(923\) 0 0
\(924\) 0 0
\(925\) 1.06073e53 0.431571
\(926\) 0 0
\(927\) 0 0
\(928\) −2.21442e53 −0.849956
\(929\) 4.28014e53 1.61130 0.805648 − 0.592395i \(-0.201817\pi\)
0.805648 + 0.592395i \(0.201817\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.99432e53 1.06368
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 4.13518e53 1.35997
\(937\) 3.21413e53 1.03693 0.518467 − 0.855098i \(-0.326503\pi\)
0.518467 + 0.855098i \(0.326503\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 5.47762e53 1.63674 0.818369 − 0.574694i \(-0.194879\pi\)
0.818369 + 0.574694i \(0.194879\pi\)
\(942\) 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
\(949\) −9.96066e53 −2.55562
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −8.39314e53 −1.99642 −0.998210 − 0.0597998i \(-0.980954\pi\)
−0.998210 + 0.0597998i \(0.980954\pi\)
\(954\) 7.10068e53 1.65741
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 4.88676e53 1.00000
\(962\) 1.27773e54 2.56619
\(963\) 0 0
\(964\) −1.03279e54 −1.99814
\(965\) −1.15235e54 −2.18823
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −5.56873e53 −1.00000
\(969\) 0 0
\(970\) −1.01855e54 −1.76234
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.28197e54 1.98509
\(977\) −1.21853e53 −0.185240 −0.0926198 − 0.995702i \(-0.529524\pi\)
−0.0926198 + 0.995702i \(0.529524\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −7.70539e53 −1.10847
\(981\) −6.82766e53 −0.964339
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 1.41776e54 1.86102
\(986\) 1.08439e54 1.39765
\(987\) 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
\(997\) 1.56435e54 1.65129 0.825643 − 0.564193i \(-0.190813\pi\)
0.825643 + 0.564193i \(0.190813\pi\)
\(998\) 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 4.37.b.a.3.1 ✓ 1
4.3 odd 2 CM 4.37.b.a.3.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.37.b.a.3.1 ✓ 1 1.1 even 1 trivial
4.37.b.a.3.1 ✓ 1 4.3 odd 2 CM