↑

Properties

Label 23.7.b.a.22.1
Level $23$
Weight $7$
Character 23.22
Self dual yes
Analytic conductor $5.291$
Analytic rank $0$
Dimension $1$
CM discriminant -23
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [23,7,Mod(22,23)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(23, 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("23.22");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 23.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: \(5.29124392326\)
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 22.1
Character \(\chi\) \(=\) 23.22

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-7.00000 q^{2} -38.0000 q^{3} -15.0000 q^{4} +266.000 q^{6} +553.000 q^{8} +715.000 q^{9} +O(q^{10})\) \(q-7.00000 q^{2} -38.0000 q^{3} -15.0000 q^{4} +266.000 q^{6} +553.000 q^{8} +715.000 q^{9} +570.000 q^{12} +1082.00 q^{13} -2911.00 q^{16} -5005.00 q^{18} -12167.0 q^{23} -21014.0 q^{24} +15625.0 q^{25} -7574.00 q^{26} +532.000 q^{27} +30746.0 q^{29} +58754.0 q^{31} -15015.0 q^{32} -10725.0 q^{36} -41116.0 q^{39} +43634.0 q^{41} +85169.0 q^{46} -205342. q^{47} +110618. q^{48} +117649. q^{49} -109375. q^{50} -16230.0 q^{52} -3724.00 q^{54} -215222. q^{58} -253942. q^{59} -411278. q^{62} +291409. q^{64} +462346. q^{69} +667154. q^{71} +395395. q^{72} +725042. q^{73} -593750. q^{75} +287812. q^{78} -541451. q^{81} -305438. q^{82} -1.16835e6 q^{87} +182505. q^{92} -2.23265e6 q^{93} +1.43739e6 q^{94} +570570. q^{96} -823543. q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.00000 −0.875000 −0.437500 − 0.899218i \(-0.644136\pi\)
−0.437500 + 0.899218i \(0.644136\pi\)
\(3\) −38.0000 −1.40741 −0.703704 − 0.710494i \(-0.748472\pi\)
−0.703704 + 0.710494i \(0.748472\pi\)
\(4\) −15.0000 −0.234375
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 266.000 1.23148
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 553.000 1.08008
\(9\) 715.000 0.980796
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 570.000 0.329861
\(13\) 1082.00 0.492490 0.246245 − 0.969208i \(-0.420803\pi\)
0.246245 + 0.969208i \(0.420803\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2911.00 −0.710693
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −5005.00 −0.858196
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −12167.0 −1.00000
\(24\) −21014.0 −1.52011
\(25\) 15625.0 1.00000
\(26\) −7574.00 −0.430929
\(27\) 532.000 0.0270284
\(28\) 0 0
\(29\) 30746.0 1.26065 0.630325 − 0.776331i \(-0.282922\pi\)
0.630325 + 0.776331i \(0.282922\pi\)
\(30\) 0 0
\(31\) 58754.0 1.97221 0.986103 − 0.166134i \(-0.0531284\pi\)
0.986103 + 0.166134i \(0.0531284\pi\)
\(32\) −15015.0 −0.458221
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −10725.0 −0.229874
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −41116.0 −0.693134
\(40\) 0 0
\(41\) 43634.0 0.633102 0.316551 − 0.948576i \(-0.397475\pi\)
0.316551 + 0.948576i \(0.397475\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 85169.0 0.875000
\(47\) −205342. −1.97781 −0.988904 − 0.148555i \(-0.952538\pi\)
−0.988904 + 0.148555i \(0.952538\pi\)
\(48\) 110618. 1.00024
\(49\) 117649. 1.00000
\(50\) −109375. −0.875000
\(51\) 0 0
\(52\) −16230.0 −0.115427
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −3724.00 −0.0236499
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −215222. −1.10307
\(59\) −253942. −1.23646 −0.618228 − 0.785999i \(-0.712149\pi\)
−0.618228 + 0.785999i \(0.712149\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −411278. −1.72568
\(63\) 0 0
\(64\) 291409. 1.11164
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 462346. 1.40741
\(70\) 0 0
\(71\) 667154. 1.86402 0.932011 − 0.362430i \(-0.118053\pi\)
0.932011 + 0.362430i \(0.118053\pi\)
\(72\) 395395. 1.05934
\(73\) 725042. 1.86378 0.931890 − 0.362741i \(-0.118159\pi\)
0.931890 + 0.362741i \(0.118159\pi\)
\(74\) 0 0
\(75\) −593750. −1.40741
\(76\) 0 0
\(77\) 0 0
\(78\) 287812. 0.606492
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −541451. −1.01884
\(82\) −305438. −0.553964
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.16835e6 −1.77425
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 182505. 0.234375
\(93\) −2.23265e6 −2.77570
\(94\) 1.43739e6 1.73058
\(95\) 0 0
\(96\) 570570. 0.644904
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −823543. −0.875000
\(99\) 0 0
\(100\) −234375. −0.234375
\(101\) 505802. 0.490926 0.245463 − 0.969406i \(-0.421060\pi\)
0.245463 + 0.969406i \(0.421060\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 598346. 0.531927
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −7980.00 −0.00633478
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −461190. −0.295465
\(117\) 773630. 0.483032
\(118\) 1.77759e6 1.08190
\(119\) 0 0
\(120\) 0 0
\(121\) 1.77156e6 1.00000
\(122\) 0 0
\(123\) −1.65809e6 −0.891032
\(124\) −881310. −0.462236
\(125\) 0 0
\(126\) 0 0
\(127\) 2.70490e6 1.32050 0.660252 − 0.751044i \(-0.270449\pi\)
0.660252 + 0.751044i \(0.270449\pi\)
\(128\) −1.07890e6 −0.514461
\(129\) 0 0
\(130\) 0 0
\(131\) 3.32143e6 1.47745 0.738723 − 0.674009i \(-0.235429\pi\)
0.738723 + 0.674009i \(0.235429\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) −3.23642e6 −1.23148
\(139\) −5.20149e6 −1.93680 −0.968398 − 0.249411i \(-0.919763\pi\)
−0.968398 + 0.249411i \(0.919763\pi\)
\(140\) 0 0
\(141\) 7.80300e6 2.78358
\(142\) −4.67008e6 −1.63102
\(143\) 0 0
\(144\) −2.08136e6 −0.697045
\(145\) 0 0
\(146\) −5.07529e6 −1.63081
\(147\) −4.47066e6 −1.40741
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 4.15625e6 1.23148
\(151\) 6.18955e6 1.79775 0.898873 − 0.438208i \(-0.144387\pi\)
0.898873 + 0.438208i \(0.144387\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 616740. 0.162453
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 3.79016e6 0.891481
\(163\) −5.98500e6 −1.38198 −0.690989 − 0.722865i \(-0.742825\pi\)
−0.690989 + 0.722865i \(0.742825\pi\)
\(164\) −654510. −0.148383
\(165\) 0 0
\(166\) 0 0
\(167\) −6.07493e6 −1.30434 −0.652171 − 0.758072i \(-0.726142\pi\)
−0.652171 + 0.758072i \(0.726142\pi\)
\(168\) 0 0
\(169\) −3.65608e6 −0.757454
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.96467e6 0.379446 0.189723 − 0.981838i \(-0.439241\pi\)
0.189723 + 0.981838i \(0.439241\pi\)
\(174\) 8.17844e6 1.55247
\(175\) 0 0
\(176\) 0 0
\(177\) 9.64980e6 1.74020
\(178\) 0 0
\(179\) −3.91917e6 −0.683338 −0.341669 − 0.939820i \(-0.610992\pi\)
−0.341669 + 0.939820i \(0.610992\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −6.72835e6 −1.08008
\(185\) 0 0
\(186\) 1.56286e7 2.42874
\(187\) 0 0
\(188\) 3.08013e6 0.463549
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −1.10735e7 −1.56453
\(193\) 3.99168e6 0.555244 0.277622 − 0.960690i \(-0.410454\pi\)
0.277622 + 0.960690i \(0.410454\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.76474e6 −0.234375
\(197\) 1.49813e7 1.95952 0.979760 − 0.200177i \(-0.0641517\pi\)
0.979760 + 0.200177i \(0.0641517\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 8.64062e6 1.08008
\(201\) 0 0
\(202\) −3.54061e6 −0.429561
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −8.69940e6 −0.980796
\(208\) −3.14970e6 −0.350009
\(209\) 0 0
\(210\) 0 0
\(211\) −1.26968e7 −1.35160 −0.675800 − 0.737085i \(-0.736202\pi\)
−0.675800 + 0.737085i \(0.736202\pi\)
\(212\) 0 0
\(213\) −2.53519e7 −2.62344
\(214\) 0 0
\(215\) 0 0
\(216\) 294196. 0.0291928
\(217\) 0 0
\(218\) 0 0
\(219\) −2.75516e7 −2.62310
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.24647e6 0.112400 0.0561999 − 0.998420i \(-0.482102\pi\)
0.0561999 + 0.998420i \(0.482102\pi\)
\(224\) 0 0
\(225\) 1.11719e7 0.980796
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.70025e7 1.36160
\(233\) 7.80984e6 0.617411 0.308706 − 0.951158i \(-0.400104\pi\)
0.308706 + 0.951158i \(0.400104\pi\)
\(234\) −5.41541e6 −0.422653
\(235\) 0 0
\(236\) 3.80913e6 0.289794
\(237\) 0 0
\(238\) 0 0
\(239\) 2.69836e7 1.97654 0.988271 − 0.152712i \(-0.0488005\pi\)
0.988271 + 0.152712i \(0.0488005\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −1.24009e7 −0.875000
\(243\) 2.01873e7 1.40689
\(244\) 0 0
\(245\) 0 0
\(246\) 1.16066e7 0.779653
\(247\) 0 0
\(248\) 3.24910e7 2.13014
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.89343e7 −1.15544
\(255\) 0 0
\(256\) −1.10979e7 −0.661484
\(257\) −2.31777e7 −1.36543 −0.682716 − 0.730684i \(-0.739201\pi\)
−0.682716 + 0.730684i \(0.739201\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2.19834e7 1.23644
\(262\) −2.32500e7 −1.29277
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.79262e7 −1.94842 −0.974210 − 0.225643i \(-0.927552\pi\)
−0.974210 + 0.225643i \(0.927552\pi\)
\(270\) 0 0
\(271\) 3.96187e7 1.99064 0.995320 − 0.0966371i \(-0.0308086\pi\)
0.995320 + 0.0966371i \(0.0308086\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −6.93519e6 −0.329861
\(277\) −3.91296e7 −1.84105 −0.920527 − 0.390680i \(-0.872240\pi\)
−0.920527 + 0.390680i \(0.872240\pi\)
\(278\) 3.64105e7 1.69470
\(279\) 4.20091e7 1.93433
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) −5.46210e7 −2.43563
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −1.00073e7 −0.436880
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.07357e7 −0.449422
\(289\) 2.41376e7 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −1.08756e7 −0.436823
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 3.12946e7 1.23148
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.31647e7 −0.492490
\(300\) 8.90625e6 0.329861
\(301\) 0 0
\(302\) −4.33269e7 −1.57303
\(303\) −1.92205e7 −0.690934
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5.07075e7 1.75250 0.876248 − 0.481860i \(-0.160039\pi\)
0.876248 + 0.481860i \(0.160039\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2.58677e7 −0.859958 −0.429979 − 0.902839i \(-0.641479\pi\)
−0.429979 + 0.902839i \(0.641479\pi\)
\(312\) −2.27371e7 −0.748639
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.13691e7 −1.92651 −0.963257 − 0.268582i \(-0.913445\pi\)
−0.963257 + 0.268582i \(0.913445\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 8.12176e6 0.238790
\(325\) 1.69062e7 0.492490
\(326\) 4.18950e7 1.20923
\(327\) 0 0
\(328\) 2.41296e7 0.683799
\(329\) 0 0
\(330\) 0 0
\(331\) 7.25286e7 1.99998 0.999989 − 0.00477828i \(-0.00152098\pi\)
0.999989 + 0.00477828i \(0.00152098\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 4.25245e7 1.14130
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 2.55926e7 0.662772
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −1.37527e7 −0.332016
\(347\) 708554. 0.0169584 0.00847919 − 0.999964i \(-0.497301\pi\)
0.00847919 + 0.999964i \(0.497301\pi\)
\(348\) 1.75252e7 0.415840
\(349\) 9.50019e6 0.223489 0.111744 − 0.993737i \(-0.464356\pi\)
0.111744 + 0.993737i \(0.464356\pi\)
\(350\) 0 0
\(351\) 575624. 0.0133112
\(352\) 0 0
\(353\) −7.62365e7 −1.73316 −0.866580 − 0.499038i \(-0.833687\pi\)
−0.866580 + 0.499038i \(0.833687\pi\)
\(354\) −6.75486e7 −1.52267
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 2.74342e7 0.597921
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 4.70459e7 1.00000
\(362\) 0 0
\(363\) −6.73193e7 −1.40741
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 3.54181e7 0.710693
\(369\) 3.11983e7 0.620943
\(370\) 0 0
\(371\) 0 0
\(372\) 3.34898e7 0.650554
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −1.13554e8 −2.13619
\(377\) 3.32672e7 0.620857
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −1.02786e8 −1.85849
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 4.09983e7 0.724056
\(385\) 0 0
\(386\) −2.79418e7 −0.485839
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 6.50599e7 1.08008
\(393\) −1.26214e8 −2.07937
\(394\) −1.04869e8 −1.71458
\(395\) 0 0
\(396\) 0 0
\(397\) 1.02325e8 1.63535 0.817676 − 0.575679i \(-0.195262\pi\)
0.817676 + 0.575679i \(0.195262\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −4.54844e7 −0.710693
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 6.35718e7 0.971291
\(404\) −7.58703e6 −0.115061
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.11816e8 −1.63431 −0.817153 − 0.576421i \(-0.804449\pi\)
−0.817153 + 0.576421i \(0.804449\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 6.08958e7 0.858196
\(415\) 0 0
\(416\) −1.62462e7 −0.225669
\(417\) 1.97657e8 2.72586
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 8.88779e7 1.18265
\(423\) −1.46820e8 −1.93983
\(424\) 0 0
\(425\) 0 0
\(426\) 1.77463e8 2.29551
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −1.54865e6 −0.0192089
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 1.92861e8 2.29521
\(439\) −1.16094e8 −1.37220 −0.686099 − 0.727509i \(-0.740678\pi\)
−0.686099 + 0.727509i \(0.740678\pi\)
\(440\) 0 0
\(441\) 8.41190e7 0.980796
\(442\) 0 0
\(443\) 1.42821e8 1.64279 0.821393 − 0.570363i \(-0.193197\pi\)
0.821393 + 0.570363i \(0.193197\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −8.72526e6 −0.0983499
\(447\) 0 0
\(448\) 0 0
\(449\) 1.58038e8 1.74591 0.872955 − 0.487801i \(-0.162201\pi\)
0.872955 + 0.487801i \(0.162201\pi\)
\(450\) −7.82031e7 −0.858196
\(451\) 0 0
\(452\) 0 0
\(453\) −2.35203e8 −2.53016
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.86172e8 −1.90026 −0.950129 − 0.311856i \(-0.899049\pi\)
−0.950129 + 0.311856i \(0.899049\pi\)
\(462\) 0 0
\(463\) −6.20833e7 −0.625506 −0.312753 − 0.949834i \(-0.601251\pi\)
−0.312753 + 0.949834i \(0.601251\pi\)
\(464\) −8.95016e7 −0.895936
\(465\) 0 0
\(466\) −5.46689e7 −0.540235
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −1.16044e7 −0.113211
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −1.40430e8 −1.33547
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −1.88885e8 −1.72947
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −2.65734e7 −0.234375
\(485\) 0 0
\(486\) −1.41311e8 −1.23103
\(487\) 1.55985e8 1.35050 0.675252 − 0.737587i \(-0.264035\pi\)
0.675252 + 0.737587i \(0.264035\pi\)
\(488\) 0 0
\(489\) 2.27430e8 1.94501
\(490\) 0 0
\(491\) 1.90755e8 1.61151 0.805754 − 0.592250i \(-0.201760\pi\)
0.805754 + 0.592250i \(0.201760\pi\)
\(492\) 2.48714e7 0.208836
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.71033e8 −1.40163
\(497\) 0 0
\(498\) 0 0
\(499\) 757946. 0.00610010 0.00305005 − 0.999995i \(-0.499029\pi\)
0.00305005 + 0.999995i \(0.499029\pi\)
\(500\) 0 0
\(501\) 2.30847e8 1.83574
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.38931e8 1.06605
\(508\) −4.05735e7 −0.309493
\(509\) −2.00351e8 −1.51928 −0.759641 − 0.650343i \(-0.774625\pi\)
−0.759641 + 0.650343i \(0.774625\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.46735e8 1.09326
\(513\) 0 0
\(514\) 1.62244e8 1.19475
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −7.46573e7 −0.534036
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) −1.53884e8 −1.08189
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −4.98215e7 −0.346277
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.48036e8 1.00000
\(530\) 0 0
\(531\) −1.81569e8 −1.21271
\(532\) 0 0
\(533\) 4.72120e7 0.311796
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.48929e8 0.961735
\(538\) 2.65483e8 1.70487
\(539\) 0 0
\(540\) 0 0
\(541\) −3.15808e8 −1.99449 −0.997245 − 0.0741740i \(-0.976368\pi\)
−0.997245 + 0.0741740i \(0.976368\pi\)
\(542\) −2.77331e8 −1.74181
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.27720e8 0.780361 0.390180 − 0.920738i \(-0.372413\pi\)
0.390180 + 0.920738i \(0.372413\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 2.55677e8 1.52011
\(553\) 0 0
\(554\) 2.73907e8 1.61092
\(555\) 0 0
\(556\) 7.80224e7 0.453936
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) −2.94064e8 −1.69254
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) −1.17045e8 −0.652402
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 3.68936e8 2.01329
\(569\) 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
\(574\) 0 0
\(575\) −1.90109e8 −1.00000
\(576\) 2.08357e8 1.09029
\(577\) 9.00812e7 0.468929 0.234464 − 0.972125i \(-0.424666\pi\)
0.234464 + 0.972125i \(0.424666\pi\)
\(578\) −1.68963e8 −0.875000
\(579\) −1.51684e8 −0.781455
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 4.00948e8 2.01303
\(585\) 0 0
\(586\) 0 0
\(587\) −3.99910e8 −1.97719 −0.988594 − 0.150604i \(-0.951878\pi\)
−0.988594 + 0.150604i \(0.951878\pi\)
\(588\) 6.70599e7 0.329861
\(589\) 0 0
\(590\) 0 0
\(591\) −5.69288e8 −2.75784
\(592\) 0 0
\(593\) 2.78321e8 1.33470 0.667348 − 0.744746i \(-0.267429\pi\)
0.667348 + 0.744746i \(0.267429\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 9.21529e7 0.430929
\(599\) 1.62397e8 0.755611 0.377806 − 0.925885i \(-0.376679\pi\)
0.377806 + 0.925885i \(0.376679\pi\)
\(600\) −3.28344e8 −1.52011
\(601\) −4.34057e8 −1.99951 −0.999755 − 0.0221248i \(-0.992957\pi\)
−0.999755 + 0.0221248i \(0.992957\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −9.28433e7 −0.421347
\(605\) 0 0
\(606\) 1.34543e8 0.604567
\(607\) −3.69151e8 −1.65059 −0.825294 − 0.564704i \(-0.808990\pi\)
−0.825294 + 0.564704i \(0.808990\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.22180e8 −0.974050
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −3.54953e8 −1.53343
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) −6.47284e6 −0.0270284
\(622\) 1.81074e8 0.752463
\(623\) 0 0
\(624\) 1.19689e8 0.492606
\(625\) 2.44141e8 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 4.82480e8 1.90225
\(634\) 4.29584e8 1.68570
\(635\) 0 0
\(636\) 0 0
\(637\) 1.27296e8 0.492490
\(638\) 0 0
\(639\) 4.77015e8 1.82822
\(640\) 0 0
\(641\) 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
\(647\) 2.76738e8 1.02178 0.510888 − 0.859648i \(-0.329317\pi\)
0.510888 + 0.859648i \(0.329317\pi\)
\(648\) −2.99422e8 −1.10042
\(649\) 0 0
\(650\) −1.18344e8 −0.430929
\(651\) 0 0
\(652\) 8.97750e7 0.323901
\(653\) −7.17455e7 −0.257665 −0.128832 − 0.991666i \(-0.541123\pi\)
−0.128832 + 0.991666i \(0.541123\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.27019e8 −0.449941
\(657\) 5.18405e8 1.82799
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −5.07700e8 −1.74998
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.74087e8 −1.26065
\(668\) 9.11239e7 0.305705
\(669\) −4.73657e7 −0.158192
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 5.18377e8 1.70059 0.850297 − 0.526304i \(-0.176422\pi\)
0.850297 + 0.526304i \(0.176422\pi\)
\(674\) 0 0
\(675\) 8.31250e6 0.0270284
\(676\) 5.48413e7 0.177528
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.13389e8 1.92519 0.962595 − 0.270945i \(-0.0873361\pi\)
0.962595 + 0.270945i \(0.0873361\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −6.23542e8 −1.88987 −0.944934 − 0.327261i \(-0.893875\pi\)
−0.944934 + 0.327261i \(0.893875\pi\)
\(692\) −2.94700e7 −0.0889327
\(693\) 0 0
\(694\) −4.95988e6 −0.0148386
\(695\) 0 0
\(696\) −6.46096e8 −1.91633
\(697\) 0 0
\(698\) −6.65013e7 −0.195553
\(699\) −2.96774e8 −0.868949
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) −4.02937e6 −0.0116473
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 5.33655e8 1.51652
\(707\) 0 0
\(708\) −1.44747e8 −0.407859
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −7.14860e8 −1.97221
\(714\) 0 0
\(715\) 0 0
\(716\) 5.87876e7 0.160157
\(717\) −1.02538e9 −2.78180
\(718\) 0 0
\(719\) 6.96357e8 1.87346 0.936732 − 0.350047i \(-0.113834\pi\)
0.936732 + 0.350047i \(0.113834\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −3.29321e8 −0.875000
\(723\) 0 0
\(724\) 0 0
\(725\) 4.80406e8 1.26065
\(726\) 4.71235e8 1.23148
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −3.72400e8 −0.961229
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 1.82688e8 0.458221
\(737\) 0 0
\(738\) −2.18388e8 −0.543325
\(739\) 1.26013e8 0.312236 0.156118 − 0.987738i \(-0.450102\pi\)
0.156118 + 0.987738i \(0.450102\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) −1.23466e9 −2.99797
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 5.97751e8 1.40562
\(753\) 0 0
\(754\) −2.32870e8 −0.543250
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.49422e8 1.47358 0.736789 − 0.676123i \(-0.236341\pi\)
0.736789 + 0.676123i \(0.236341\pi\)
\(762\) 7.19503e8 1.62618
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.74765e8 −0.608942
\(768\) 4.21718e8 0.930977
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 8.80751e8 1.92172
\(772\) −5.98752e7 −0.130135
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 9.18031e8 1.97221
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 1.63569e7 0.0340734
\(784\) −3.42476e8 −0.710693
\(785\) 0 0
\(786\) 8.83501e8 1.81945
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −2.24719e8 −0.459262
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −7.16276e8 −1.43093
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −2.34609e8 −0.458221
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −4.45003e8 −0.849880
\(807\) 1.44120e9 2.74222
\(808\) 2.79709e8 0.530239
\(809\) −2.83551e8 −0.535531 −0.267766 − 0.963484i \(-0.586285\pi\)
−0.267766 + 0.963484i \(0.586285\pi\)
\(810\) 0 0
\(811\) −1.05551e9 −1.97878 −0.989392 − 0.145268i \(-0.953596\pi\)
−0.989392 + 0.145268i \(0.953596\pi\)
\(812\) 0 0
\(813\) −1.50551e9 −2.80164
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 7.82711e8 1.43002
\(819\) 0 0
\(820\) 0 0
\(821\) −5.08386e8 −0.918679 −0.459340 − 0.888261i \(-0.651914\pi\)
−0.459340 + 0.888261i \(0.651914\pi\)
\(822\) 0 0
\(823\) −8.05686e8 −1.44533 −0.722664 − 0.691199i \(-0.757083\pi\)
−0.722664 + 0.691199i \(0.757083\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 1.30491e8 0.229874
\(829\) −1.11478e9 −1.95671 −0.978357 − 0.206925i \(-0.933655\pi\)
−0.978357 + 0.206925i \(0.933655\pi\)
\(830\) 0 0
\(831\) 1.48693e9 2.59111
\(832\) 3.15305e8 0.547470
\(833\) 0 0
\(834\) −1.38360e9 −2.38513
\(835\) 0 0
\(836\) 0 0
\(837\) 3.12571e7 0.0533056
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 3.50493e8 0.589239
\(842\) 0 0
\(843\) 0 0
\(844\) 1.90453e8 0.316781
\(845\) 0 0
\(846\) 1.02774e9 1.69735
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 3.80278e8 0.614868
\(853\) −6.36633e8 −1.02575 −0.512876 − 0.858463i \(-0.671420\pi\)
−0.512876 + 0.858463i \(0.671420\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.12572e8 −0.814353 −0.407177 − 0.913349i \(-0.633487\pi\)
−0.407177 + 0.913349i \(0.633487\pi\)
\(858\) 0 0
\(859\) 1.26724e9 1.99931 0.999654 − 0.0262855i \(-0.00836789\pi\)
0.999654 + 0.0262855i \(0.00836789\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.00781e8 0.623555 0.311777 − 0.950155i \(-0.399076\pi\)
0.311777 + 0.950155i \(0.399076\pi\)
\(864\) −7.98798e6 −0.0123850
\(865\) 0 0
\(866\) 0 0
\(867\) −9.17228e8 −1.40741
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 4.13274e8 0.614788
\(877\) −1.86924e8 −0.277119 −0.138560 − 0.990354i \(-0.544247\pi\)
−0.138560 + 0.990354i \(0.544247\pi\)
\(878\) 8.12659e8 1.20067
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −5.88833e8 −0.858196
\(883\) −1.36875e9 −1.98812 −0.994060 − 0.108838i \(-0.965287\pi\)
−0.994060 + 0.108838i \(0.965287\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −9.99747e8 −1.43744
\(887\) −1.39031e9 −1.99223 −0.996117 − 0.0880391i \(-0.971940\pi\)
−0.996117 + 0.0880391i \(0.971940\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −1.86970e7 −0.0263437
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 5.00258e8 0.693134
\(898\) −1.10626e9 −1.52767
\(899\) 1.80645e9 2.48626
\(900\) −1.67578e8 −0.229874
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 1.64642e9 2.21389
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 3.61648e8 0.481498
\(910\) 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
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −1.92689e9 −2.46648
\(922\) 1.30321e9 1.66273
\(923\) 7.21861e8 0.918012
\(924\) 0 0
\(925\) 0 0
\(926\) 4.34583e8 0.547318
\(927\) 0 0
\(928\) −4.61651e8 −0.577657
\(929\) 8.42564e8 1.05089 0.525443 − 0.850829i \(-0.323900\pi\)
0.525443 + 0.850829i \(0.323900\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.17148e8 −0.144706
\(933\) 9.82974e8 1.21031
\(934\) 0 0
\(935\) 0 0
\(936\) 4.27817e8 0.521712
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) −5.30895e8 −0.633102
\(944\) 7.39225e8 0.878741
\(945\) 0 0
\(946\) 0 0
\(947\) −1.32946e9 −1.56539 −0.782697 − 0.622403i \(-0.786157\pi\)
−0.782697 + 0.622403i \(0.786157\pi\)
\(948\) 0 0
\(949\) 7.84495e8 0.917892
\(950\) 0 0
\(951\) 2.33203e9 2.71139
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −4.04754e8 −0.463252
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.56453e9 2.88960
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.73601e9 −1.91987 −0.959936 − 0.280219i \(-0.909593\pi\)
−0.959936 + 0.280219i \(0.909593\pi\)
\(968\) 9.79673e8 1.08008
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −3.02810e8 −0.329739
\(973\) 0 0
\(974\) −1.09189e9 −1.18169
\(975\) −6.42438e8 −0.693134
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) −1.59201e9 −1.70188
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −1.33529e9 −1.41007
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −9.16925e8 −0.962384
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.86316e9 −1.91439 −0.957194 − 0.289446i \(-0.906529\pi\)
−0.957194 + 0.289446i \(0.906529\pi\)
\(992\) −8.82191e8 −0.903707
\(993\) −2.75609e9 −2.81478
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.64078e9 1.65564 0.827819 − 0.560995i \(-0.189581\pi\)
0.827819 + 0.560995i \(0.189581\pi\)
\(998\) −5.30562e6 −0.00533758
\(999\) 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 23.7.b.a.22.1 ✓ 1
3.2 odd 2 207.7.d.a.91.1 1
4.3 odd 2 368.7.f.a.321.1 1
23.22 odd 2 CM 23.7.b.a.22.1 ✓ 1
69.68 even 2 207.7.d.a.91.1 1
92.91 even 2 368.7.f.a.321.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.7.b.a.22.1 ✓ 1 1.1 even 1 trivial
23.7.b.a.22.1 ✓ 1 23.22 odd 2 CM
207.7.d.a.91.1 1 3.2 odd 2
207.7.d.a.91.1 1 69.68 even 2
368.7.f.a.321.1 1 4.3 odd 2
368.7.f.a.321.1 1 92.91 even 2