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Properties

Label 31.7.b.a.30.1
Level $31$
Weight $7$
Character 31.30
Self dual yes
Analytic conductor $7.132$
Analytic rank $0$
Dimension $1$
CM discriminant -31
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [31,7,Mod(30,31)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(31, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.30");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 31.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: \(7.13167659222\)
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 30.1
Character \(\chi\) \(=\) 31.30

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-15.0000 q^{2} +161.000 q^{4} -246.000 q^{5} -430.000 q^{7} -1455.00 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-15.0000 q^{2} +161.000 q^{4} -246.000 q^{5} -430.000 q^{7} -1455.00 q^{8} +729.000 q^{9} +3690.00 q^{10} +6450.00 q^{14} +11521.0 q^{16} -10935.0 q^{18} +10618.0 q^{19} -39606.0 q^{20} +44891.0 q^{25} -69230.0 q^{28} -29791.0 q^{31} -79695.0 q^{32} +105780. q^{35} +117369. q^{36} -159270. q^{38} +357930. q^{40} -60558.0 q^{41} -179334. q^{45} +171810. q^{47} +67251.0 q^{49} -673365. q^{50} +625650. q^{56} +136842. q^{59} +446865. q^{62} -313470. q^{63} +458081. q^{64} -133670. q^{67} -1.58670e6 q^{70} +284178. q^{71} -1.06070e6 q^{72} +1.70950e6 q^{76} -2.83417e6 q^{80} +531441. q^{81} +908370. q^{82} +2.69001e6 q^{90} -2.57715e6 q^{94} -2.61203e6 q^{95} +1.80749e6 q^{97} -1.00876e6 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −15.0000 −1.87500 −0.937500 − 0.347985i \(-0.886866\pi\)
−0.937500 + 0.347985i \(0.886866\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 161.000 2.51562
\(5\) −246.000 −1.96800 −0.984000 − 0.178168i \(-0.942983\pi\)
−0.984000 + 0.178168i \(0.942983\pi\)
\(6\) 0 0
\(7\) −430.000 −1.25364 −0.626822 − 0.779162i \(-0.715645\pi\)
−0.626822 + 0.779162i \(0.715645\pi\)
\(8\) −1455.00 −2.84180
\(9\) 729.000 1.00000
\(10\) 3690.00 3.69000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 6450.00 2.35058
\(15\) 0 0
\(16\) 11521.0 2.81274
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −10935.0 −1.87500
\(19\) 10618.0 1.54804 0.774020 − 0.633162i \(-0.218243\pi\)
0.774020 + 0.633162i \(0.218243\pi\)
\(20\) −39606.0 −4.95075
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 44891.0 2.87302
\(26\) 0 0
\(27\) 0 0
\(28\) −69230.0 −3.15370
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −29791.0 −1.00000
\(32\) −79695.0 −2.43210
\(33\) 0 0
\(34\) 0 0
\(35\) 105780. 2.46717
\(36\) 117369. 2.51562
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −159270. −2.90257
\(39\) 0 0
\(40\) 357930. 5.59266
\(41\) −60558.0 −0.878658 −0.439329 − 0.898326i \(-0.644784\pi\)
−0.439329 + 0.898326i \(0.644784\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −179334. −1.96800
\(46\) 0 0
\(47\) 171810. 1.65484 0.827418 − 0.561587i \(-0.189809\pi\)
0.827418 + 0.561587i \(0.189809\pi\)
\(48\) 0 0
\(49\) 67251.0 0.571624
\(50\) −673365. −5.38692
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 625650. 3.56260
\(57\) 0 0
\(58\) 0 0
\(59\) 136842. 0.666290 0.333145 − 0.942876i \(-0.391890\pi\)
0.333145 + 0.942876i \(0.391890\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 446865. 1.87500
\(63\) −313470. −1.25364
\(64\) 458081. 1.74744
\(65\) 0 0
\(66\) 0 0
\(67\) −133670. −0.444436 −0.222218 − 0.974997i \(-0.571330\pi\)
−0.222218 + 0.974997i \(0.571330\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −1.58670e6 −4.62595
\(71\) 284178. 0.793991 0.396995 − 0.917821i \(-0.370053\pi\)
0.396995 + 0.917821i \(0.370053\pi\)
\(72\) −1.06070e6 −2.84180
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 1.70950e6 3.89429
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −2.83417e6 −5.53548
\(81\) 531441. 1.00000
\(82\) 908370. 1.64748
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 2.69001e6 3.69000
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −2.57715e6 −3.10282
\(95\) −2.61203e6 −3.04654
\(96\) 0 0
\(97\) 1.80749e6 1.98044 0.990218 − 0.139531i \(-0.0445594\pi\)
0.990218 + 0.139531i \(0.0445594\pi\)
\(98\) −1.00876e6 −1.07180
\(99\) 0 0
\(100\) 7.22745e6 7.22745
\(101\) 1.14350e6 1.10987 0.554934 − 0.831894i \(-0.312744\pi\)
0.554934 + 0.831894i \(0.312744\pi\)
\(102\) 0 0
\(103\) −567470. −0.519315 −0.259658 − 0.965701i \(-0.583610\pi\)
−0.259658 + 0.965701i \(0.583610\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 122730. 0.100184 0.0500921 − 0.998745i \(-0.484049\pi\)
0.0500921 + 0.998745i \(0.484049\pi\)
\(108\) 0 0
\(109\) −199942. −0.154392 −0.0771960 − 0.997016i \(-0.524597\pi\)
−0.0771960 + 0.997016i \(0.524597\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.95403e6 −3.52618
\(113\) 2.88381e6 1.99862 0.999312 − 0.0370748i \(-0.0118040\pi\)
0.999312 + 0.0370748i \(0.0118040\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −2.05263e6 −1.24929
\(119\) 0 0
\(120\) 0 0
\(121\) 1.77156e6 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −4.79635e6 −2.51562
\(125\) −7.19944e6 −3.68611
\(126\) 4.70205e6 2.35058
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −1.77074e6 −0.844352
\(129\) 0 0
\(130\) 0 0
\(131\) −4.47658e6 −1.99128 −0.995641 − 0.0932711i \(-0.970268\pi\)
−0.995641 + 0.0932711i \(0.970268\pi\)
\(132\) 0 0
\(133\) −4.56574e6 −1.94069
\(134\) 2.00505e6 0.833318
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 1.70306e7 6.20648
\(141\) 0 0
\(142\) −4.26267e6 −1.48873
\(143\) 0 0
\(144\) 8.39881e6 2.81274
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.42500e6 1.63999 0.819995 − 0.572371i \(-0.193976\pi\)
0.819995 + 0.572371i \(0.193976\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) −1.54492e7 −4.39921
\(153\) 0 0
\(154\) 0 0
\(155\) 7.32859e6 1.96800
\(156\) 0 0
\(157\) −6.42647e6 −1.66063 −0.830316 − 0.557293i \(-0.811840\pi\)
−0.830316 + 0.557293i \(0.811840\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 1.96050e7 4.78637
\(161\) 0 0
\(162\) −7.97162e6 −1.87500
\(163\) −1.83895e6 −0.424627 −0.212313 − 0.977202i \(-0.568100\pi\)
−0.212313 + 0.977202i \(0.568100\pi\)
\(164\) −9.74984e6 −2.21037
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 4.82681e6 1.00000
\(170\) 0 0
\(171\) 7.74052e6 1.54804
\(172\) 0 0
\(173\) 1.00930e7 1.94932 0.974662 − 0.223682i \(-0.0718077\pi\)
0.974662 + 0.223682i \(0.0718077\pi\)
\(174\) 0 0
\(175\) −1.93031e7 −3.60175
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −2.88728e7 −4.95075
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 2.76614e7 4.16295
\(189\) 0 0
\(190\) 3.91804e7 5.71226
\(191\) 1.38582e7 1.98888 0.994439 − 0.105316i \(-0.0335855\pi\)
0.994439 + 0.105316i \(0.0335855\pi\)
\(192\) 0 0
\(193\) 6.25165e6 0.869606 0.434803 − 0.900525i \(-0.356818\pi\)
0.434803 + 0.900525i \(0.356818\pi\)
\(194\) −2.71124e7 −3.71332
\(195\) 0 0
\(196\) 1.08274e7 1.43799
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −6.53164e7 −8.16455
\(201\) 0 0
\(202\) −1.71525e7 −2.08100
\(203\) 0 0
\(204\) 0 0
\(205\) 1.48973e7 1.72920
\(206\) 8.51205e6 0.973716
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.23363e7 −1.31322 −0.656608 − 0.754232i \(-0.728009\pi\)
−0.656608 + 0.754232i \(0.728009\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −1.84095e6 −0.187845
\(215\) 0 0
\(216\) 0 0
\(217\) 1.28101e7 1.25364
\(218\) 2.99913e6 0.289485
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 3.42688e7 3.04899
\(225\) 3.27255e7 2.87302
\(226\) −4.32572e7 −3.74742
\(227\) −1.50767e7 −1.28893 −0.644465 − 0.764634i \(-0.722920\pi\)
−0.644465 + 0.764634i \(0.722920\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.83973e6 0.303552 0.151776 − 0.988415i \(-0.451501\pi\)
0.151776 + 0.988415i \(0.451501\pi\)
\(234\) 0 0
\(235\) −4.22653e7 −3.25672
\(236\) 2.20316e7 1.67614
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −2.65734e7 −1.87500
\(243\) 0 0
\(244\) 0 0
\(245\) −1.65437e7 −1.12496
\(246\) 0 0
\(247\) 0 0
\(248\) 4.33459e7 2.84180
\(249\) 0 0
\(250\) 1.07992e8 6.91146
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −5.04687e7 −3.15370
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −2.75616e6 −0.164280
\(257\) −1.55794e7 −0.917806 −0.458903 − 0.888486i \(-0.651758\pi\)
−0.458903 + 0.888486i \(0.651758\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 6.71487e7 3.73365
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 6.84861e7 3.63879
\(267\) 0 0
\(268\) −2.15209e7 −1.11804
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −2.17176e7 −1.00000
\(280\) −1.53910e8 −7.01120
\(281\) −4.39536e7 −1.98096 −0.990479 − 0.137663i \(-0.956041\pi\)
−0.990479 + 0.137663i \(0.956041\pi\)
\(282\) 0 0
\(283\) −3.42282e7 −1.51016 −0.755082 − 0.655630i \(-0.772403\pi\)
−0.755082 + 0.655630i \(0.772403\pi\)
\(284\) 4.57527e7 1.99738
\(285\) 0 0
\(286\) 0 0
\(287\) 2.60399e7 1.10152
\(288\) −5.80977e7 −2.43210
\(289\) 2.41376e7 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.24502e7 −0.892520 −0.446260 − 0.894903i \(-0.647244\pi\)
−0.446260 + 0.894903i \(0.647244\pi\)
\(294\) 0 0
\(295\) −3.36631e7 −1.31126
\(296\) 0 0
\(297\) 0 0
\(298\) −8.13750e7 −3.07498
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 1.22330e8 4.35424
\(305\) 0 0
\(306\) 0 0
\(307\) 4.06169e7 1.40376 0.701878 − 0.712297i \(-0.252345\pi\)
0.701878 + 0.712297i \(0.252345\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.09929e8 −3.69000
\(311\) 4.45334e7 1.48049 0.740243 − 0.672339i \(-0.234710\pi\)
0.740243 + 0.672339i \(0.234710\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 9.63970e7 3.11369
\(315\) 7.71136e7 2.46717
\(316\) 0 0
\(317\) 6.18644e7 1.94206 0.971031 − 0.238954i \(-0.0768044\pi\)
0.971031 + 0.238954i \(0.0768044\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.12688e8 −3.43896
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 8.55620e7 2.51562
\(325\) 0 0
\(326\) 2.75842e7 0.796175
\(327\) 0 0
\(328\) 8.81119e7 2.49697
\(329\) −7.38783e7 −2.07458
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.28828e7 0.874651
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −7.24021e7 −1.87500
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) −1.16108e8 −2.90257
\(343\) 2.16711e7 0.537031
\(344\) 0 0
\(345\) 0 0
\(346\) −1.51396e8 −3.65498
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −6.55690e7 −1.54249 −0.771245 − 0.636539i \(-0.780366\pi\)
−0.771245 + 0.636539i \(0.780366\pi\)
\(350\) 2.89547e8 6.75328
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −6.99078e7 −1.56257
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7.57593e7 −1.63739 −0.818696 − 0.574227i \(-0.805303\pi\)
−0.818696 + 0.574227i \(0.805303\pi\)
\(360\) 2.60931e8 5.59266
\(361\) 6.56960e7 1.39642
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −4.41468e7 −0.878658
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 9.95003e7 1.91734 0.958668 − 0.284529i \(-0.0918371\pi\)
0.958668 + 0.284529i \(0.0918371\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −2.49984e8 −4.70271
\(377\) 0 0
\(378\) 0 0
\(379\) 1.08388e8 1.99096 0.995480 − 0.0949732i \(-0.0302766\pi\)
0.995480 + 0.0949732i \(0.0302766\pi\)
\(380\) −4.20537e8 −7.66395
\(381\) 0 0
\(382\) −2.07874e8 −3.72915
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −9.37748e7 −1.63051
\(387\) 0 0
\(388\) 2.91006e8 4.98203
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −9.78502e7 −1.62444
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.19030e8 1.90233 0.951166 − 0.308681i \(-0.0998876\pi\)
0.951166 + 0.308681i \(0.0998876\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 5.17189e8 8.08108
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1.84103e8 2.79201
\(405\) −1.30734e8 −1.96800
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) −2.23459e8 −3.24225
\(411\) 0 0
\(412\) −9.13627e7 −1.30640
\(413\) −5.88421e7 −0.835291
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.38240e8 −1.87928 −0.939638 − 0.342169i \(-0.888838\pi\)
−0.939638 + 0.342169i \(0.888838\pi\)
\(420\) 0 0
\(421\) −9.09868e6 −0.121936 −0.0609680 − 0.998140i \(-0.519419\pi\)
−0.0609680 + 0.998140i \(0.519419\pi\)
\(422\) 1.85044e8 2.46228
\(423\) 1.25249e8 1.65484
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 1.97595e7 0.252026
\(429\) 0 0
\(430\) 0 0
\(431\) −9.32758e6 −0.116503 −0.0582515 − 0.998302i \(-0.518553\pi\)
−0.0582515 + 0.998302i \(0.518553\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) −1.92152e8 −2.35058
\(435\) 0 0
\(436\) −3.21907e7 −0.388392
\(437\) 0 0
\(438\) 0 0
\(439\) −1.65523e8 −1.95643 −0.978214 − 0.207599i \(-0.933435\pi\)
−0.978214 + 0.207599i \(0.933435\pi\)
\(440\) 0 0
\(441\) 4.90260e7 0.571624
\(442\) 0 0
\(443\) 1.22927e8 1.41396 0.706980 − 0.707233i \(-0.250057\pi\)
0.706980 + 0.707233i \(0.250057\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.96975e8 −2.19067
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) −4.90883e8 −5.38692
\(451\) 0 0
\(452\) 4.64293e8 5.02779
\(453\) 0 0
\(454\) 2.26151e8 2.41674
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −5.75960e7 −0.569160
\(467\) −1.31037e8 −1.28659 −0.643297 − 0.765616i \(-0.722434\pi\)
−0.643297 + 0.765616i \(0.722434\pi\)
\(468\) 0 0
\(469\) 5.74781e7 0.557165
\(470\) 6.33979e8 6.10634
\(471\) 0 0
\(472\) −1.99105e8 −1.89346
\(473\) 0 0
\(474\) 0 0
\(475\) 4.76653e8 4.44755
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.32899e7 0.666865 0.333432 − 0.942774i \(-0.391793\pi\)
0.333432 + 0.942774i \(0.391793\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.85221e8 2.51562
\(485\) −4.44643e8 −3.89750
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 2.48156e8 2.10929
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −3.43222e8 −2.81274
\(497\) −1.22197e8 −0.995382
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −1.15911e9 −9.27287
\(501\) 0 0
\(502\) 0 0
\(503\) 2.54066e8 1.99637 0.998187 − 0.0601855i \(-0.0191692\pi\)
0.998187 + 0.0601855i \(0.0191692\pi\)
\(504\) 4.56099e8 3.56260
\(505\) −2.81301e8 −2.18422
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.54669e8 1.15238
\(513\) 0 0
\(514\) 2.33691e8 1.72089
\(515\) 1.39598e8 1.02201
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.88046e7 −0.486524 −0.243262 − 0.969961i \(-0.578218\pi\)
−0.243262 + 0.969961i \(0.578218\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −7.20730e8 −5.00932
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.48036e8 1.00000
\(530\) 0 0
\(531\) 9.97578e7 0.666290
\(532\) −7.35084e8 −4.88205
\(533\) 0 0
\(534\) 0 0
\(535\) −3.01916e7 −0.197163
\(536\) 1.94490e8 1.26300
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 3.01836e8 1.90625 0.953124 − 0.302580i \(-0.0978481\pi\)
0.953124 + 0.302580i \(0.0978481\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.91857e7 0.303843
\(546\) 0 0
\(547\) −2.93182e8 −1.79133 −0.895665 − 0.444729i \(-0.853300\pi\)
−0.895665 + 0.444729i \(0.853300\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 3.25765e8 1.87500
\(559\) 0 0
\(560\) 1.21869e9 6.93952
\(561\) 0 0
\(562\) 6.59304e8 3.71430
\(563\) −1.44564e8 −0.810093 −0.405046 − 0.914296i \(-0.632745\pi\)
−0.405046 + 0.914296i \(0.632745\pi\)
\(564\) 0 0
\(565\) −7.09417e8 −3.93329
\(566\) 5.13422e8 2.83156
\(567\) −2.28520e8 −1.25364
\(568\) −4.13479e8 −2.25636
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −3.90599e8 −2.06536
\(575\) 0 0
\(576\) 3.33941e8 1.74744
\(577\) 2.57206e8 1.33892 0.669459 − 0.742849i \(-0.266526\pi\)
0.669459 + 0.742849i \(0.266526\pi\)
\(578\) −3.62064e8 −1.87500
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 3.36753e8 1.67347
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) −3.16321e8 −1.54804
\(590\) 5.04947e8 2.45861
\(591\) 0 0
\(592\) 0 0
\(593\) 2.25161e8 1.07977 0.539883 − 0.841740i \(-0.318469\pi\)
0.539883 + 0.841740i \(0.318469\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8.73425e8 4.12560
\(597\) 0 0
\(598\) 0 0
\(599\) −2.87376e8 −1.33712 −0.668558 − 0.743660i \(-0.733088\pi\)
−0.668558 + 0.743660i \(0.733088\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −9.74454e7 −0.444436
\(604\) 0 0
\(605\) −4.35804e8 −1.96800
\(606\) 0 0
\(607\) 9.02008e7 0.403315 0.201657 − 0.979456i \(-0.435367\pi\)
0.201657 + 0.979456i \(0.435367\pi\)
\(608\) −8.46202e8 −3.76498
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −6.09253e8 −2.63204
\(615\) 0 0
\(616\) 0 0
\(617\) 4.34675e8 1.85059 0.925293 − 0.379252i \(-0.123819\pi\)
0.925293 + 0.379252i \(0.123819\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 1.17990e9 4.95075
\(621\) 0 0
\(622\) −6.68000e8 −2.77591
\(623\) 0 0
\(624\) 0 0
\(625\) 1.06964e9 4.38124
\(626\) 0 0
\(627\) 0 0
\(628\) −1.03466e9 −4.17753
\(629\) 0 0
\(630\) −1.15670e9 −4.62595
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −9.27966e8 −3.64137
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2.07166e8 0.793991
\(640\) 4.35601e8 1.66169
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 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 \(\pi\)
\(648\) −7.73247e8 −2.84180
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −2.96071e8 −1.06820
\(653\) −5.04733e8 −1.81268 −0.906342 − 0.422545i \(-0.861137\pi\)
−0.906342 + 0.422545i \(0.861137\pi\)
\(654\) 0 0
\(655\) 1.10124e9 3.91884
\(656\) −6.97689e8 −2.47144
\(657\) 0 0
\(658\) 1.10817e9 3.88983
\(659\) −5.69810e8 −1.99101 −0.995505 − 0.0947082i \(-0.969808\pi\)
−0.995505 + 0.0947082i \(0.969808\pi\)
\(660\) 0 0
\(661\) 3.43513e8 1.18943 0.594715 − 0.803937i \(-0.297265\pi\)
0.594715 + 0.803937i \(0.297265\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.12317e9 3.81928
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) −4.93242e8 −1.63997
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 7.77116e8 2.51562
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −7.77221e8 −2.48276
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −6.81231e6 −0.0213812 −0.0106906 − 0.999943i \(-0.503403\pi\)
−0.0106906 + 0.999943i \(0.503403\pi\)
\(684\) 1.24622e9 3.89429
\(685\) 0 0
\(686\) −3.25067e8 −1.00693
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 3.40452e8 1.03186 0.515931 − 0.856630i \(-0.327446\pi\)
0.515931 + 0.856630i \(0.327446\pi\)
\(692\) 1.62498e9 4.90377
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 9.83535e8 2.89217
\(699\) 0 0
\(700\) −3.10780e9 −9.06065
\(701\) −3.75583e8 −1.09032 −0.545158 − 0.838333i \(-0.683530\pi\)
−0.545158 + 0.838333i \(0.683530\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.91704e8 −1.39138
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 1.04862e9 2.92983
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 1.13639e9 3.07011
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −2.06611e9 −5.53548
\(721\) 2.44012e8 0.651037
\(722\) −9.85441e8 −2.61830
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 2.38693e8 0.621206 0.310603 − 0.950540i \(-0.399469\pi\)
0.310603 + 0.950540i \(0.399469\pi\)
\(728\) 0 0
\(729\) 3.87420e8 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −5.44476e7 −0.138251 −0.0691253 − 0.997608i \(-0.522021\pi\)
−0.0691253 + 0.997608i \(0.522021\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 6.62202e8 1.64748
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −1.33455e9 −3.22750
\(746\) −1.49250e9 −3.59500
\(747\) 0 0
\(748\) 0 0
\(749\) −5.27739e7 −0.125595
\(750\) 0 0
\(751\) 7.88264e8 1.86102 0.930512 − 0.366262i \(-0.119362\pi\)
0.930512 + 0.366262i \(0.119362\pi\)
\(752\) 1.97942e9 4.65463
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −1.62582e9 −3.73305
\(759\) 0 0
\(760\) 3.80050e9 8.65765
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 8.59751e7 0.193553
\(764\) 2.23118e9 5.00327
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −2.75984e8 −0.606884 −0.303442 − 0.952850i \(-0.598136\pi\)
−0.303442 + 0.952850i \(0.598136\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.00652e9 2.18760
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −1.33735e9 −2.87302
\(776\) −2.62990e9 −5.62800
\(777\) 0 0
\(778\) 0 0
\(779\) −6.43005e8 −1.36020
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 7.74799e8 1.60783
\(785\) 1.58091e9 3.26812
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.24004e9 −2.50556
\(792\) 0 0
\(793\) 0 0
\(794\) −1.78545e9 −3.56687
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −3.57759e9 −6.98748
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −1.66379e9 −3.15402
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 1.96102e9 3.69000
\(811\) −3.12332e8 −0.585537 −0.292769 − 0.956183i \(-0.594577\pi\)
−0.292769 + 0.956183i \(0.594577\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.52382e8 0.835665
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 2.39846e9 4.35002
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 8.25669e8 1.47579
\(825\) 0 0
\(826\) 8.82631e8 1.56617
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 2.07360e9 3.52364
\(839\) 1.27524e8 0.215926 0.107963 − 0.994155i \(-0.465567\pi\)
0.107963 + 0.994155i \(0.465567\pi\)
\(840\) 0 0
\(841\) 5.94823e8 1.00000
\(842\) 1.36480e8 0.228630
\(843\) 0 0
\(844\) −1.98614e9 −3.30356
\(845\) −1.18740e9 −1.96800
\(846\) −1.87874e9 −3.10282
\(847\) −7.61771e8 −1.25364
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −8.69660e8 −1.40121 −0.700603 − 0.713551i \(-0.747086\pi\)
−0.700603 + 0.713551i \(0.747086\pi\)
\(854\) 0 0
\(855\) −1.90417e9 −3.04654
\(856\) −1.78572e8 −0.284703
\(857\) 5.75221e8 0.913886 0.456943 − 0.889496i \(-0.348944\pi\)
0.456943 + 0.889496i \(0.348944\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.39914e8 0.218443
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −2.48289e9 −3.83627
\(866\) 0 0
\(867\) 0 0
\(868\) 2.06243e9 3.15370
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 2.90916e8 0.438750
\(873\) 1.31766e9 1.98044
\(874\) 0 0
\(875\) 3.09576e9 4.62107
\(876\) 0 0
\(877\) 1.43534e8 0.212792 0.106396 − 0.994324i \(-0.466069\pi\)
0.106396 + 0.994324i \(0.466069\pi\)
\(878\) 2.48284e9 3.66830
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −7.35390e8 −1.07180
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1.84391e9 −2.65118
\(887\) 1.26848e9 1.81767 0.908833 − 0.417161i \(-0.136975\pi\)
0.908833 + 0.417161i \(0.136975\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.82428e9 2.56175
\(894\) 0 0
\(895\) 0 0
\(896\) 7.61416e8 1.05852
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 5.26881e9 7.22745
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −4.19594e9 −5.67969
\(905\) 0 0
\(906\) 0 0
\(907\) −1.09404e9 −1.46627 −0.733133 − 0.680085i \(-0.761943\pi\)
−0.733133 + 0.680085i \(0.761943\pi\)
\(908\) −2.42735e9 −3.24246
\(909\) 8.33610e8 1.10987
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.92493e9 2.49636
\(918\) 0 0
\(919\) 1.18759e9 1.53010 0.765051 − 0.643970i \(-0.222714\pi\)
0.765051 + 0.643970i \(0.222714\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −4.13686e8 −0.519315
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 7.14071e8 0.884896
\(932\) 6.18197e8 0.763623
\(933\) 0 0
\(934\) 1.96555e9 2.41237
\(935\) 0 0
\(936\) 0 0
\(937\) 2.88399e8 0.350570 0.175285 − 0.984518i \(-0.443915\pi\)
0.175285 + 0.984518i \(0.443915\pi\)
\(938\) −8.62172e8 −1.04468
\(939\) 0 0
\(940\) −6.80471e9 −8.19268
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.57656e9 1.87410
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −7.14979e9 −8.33916
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −3.40913e9 −3.91411
\(956\) 0 0
\(957\) 0 0
\(958\) −1.09935e9 −1.25037
\(959\) 0 0
\(960\) 0 0
\(961\) 8.87504e8 1.00000
\(962\) 0 0
\(963\) 8.94702e7 0.100184
\(964\) 0 0
\(965\) −1.53791e9 −1.71139
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −2.57762e9 −2.84180
\(969\) 0 0
\(970\) 6.66964e9 7.30781
\(971\) 1.72259e9 1.88159 0.940795 − 0.338976i \(-0.110080\pi\)
0.940795 + 0.338976i \(0.110080\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.32477e9 −1.42055 −0.710277 − 0.703922i \(-0.751430\pi\)
−0.710277 + 0.703922i \(0.751430\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −2.66354e9 −2.82997
\(981\) −1.45758e8 −0.154392
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 2.37419e9 2.43210
\(993\) 0 0
\(994\) 1.83295e9 1.86634
\(995\) 0 0
\(996\) 0 0
\(997\) 1.00850e9 1.01764 0.508818 − 0.860874i \(-0.330083\pi\)
0.508818 + 0.860874i \(0.330083\pi\)
\(998\) 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 31.7.b.a.30.1 ✓ 1
3.2 odd 2 279.7.d.a.154.1 1
4.3 odd 2 496.7.e.a.433.1 1
31.30 odd 2 CM 31.7.b.a.30.1 ✓ 1
93.92 even 2 279.7.d.a.154.1 1
124.123 even 2 496.7.e.a.433.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.7.b.a.30.1 ✓ 1 1.1 even 1 trivial
31.7.b.a.30.1 ✓ 1 31.30 odd 2 CM
279.7.d.a.154.1 1 3.2 odd 2
279.7.d.a.154.1 1 93.92 even 2
496.7.e.a.433.1 1 4.3 odd 2
496.7.e.a.433.1 1 124.123 even 2