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Properties

Label 23.19.b.a.22.1
Level $23$
Weight $19$
Character 23.22
Self dual yes
Analytic conductor $47.239$
Analytic rank $0$
Dimension $1$
CM discriminant -23
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] = [23,19,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, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("23.22");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 19 \)
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: \(47.2388116732\)
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)
 
\(f(q)\) \(=\) \(q+1001.00 q^{2} +28234.0 q^{3} +739857. q^{4} +2.82622e7 q^{6} +4.78191e8 q^{8} +4.09738e8 q^{9} +O(q^{10})\) \(q+1001.00 q^{2} +28234.0 q^{3} +739857. q^{4} +2.82622e7 q^{6} +4.78191e8 q^{8} +4.09738e8 q^{9} +2.08891e10 q^{12} -1.44011e10 q^{13} +2.84720e11 q^{16} +4.10148e11 q^{18} -1.80115e12 q^{23} +1.35012e13 q^{24} +3.81470e12 q^{25} -1.44155e13 q^{26} +6.30120e11 q^{27} -2.58006e13 q^{29} +4.63875e13 q^{31} +1.59650e14 q^{32} +3.03148e14 q^{36} -4.06601e14 q^{39} -5.38722e14 q^{41} -1.80295e15 q^{46} -2.01804e15 q^{47} +8.03878e15 q^{48} +1.62841e15 q^{49} +3.81851e15 q^{50} -1.06548e16 q^{52} +6.30750e14 q^{54} -2.58264e16 q^{58} +1.57584e16 q^{59} +4.64339e16 q^{62} +8.51718e16 q^{64} -5.08537e16 q^{69} +4.05587e16 q^{71} +1.95933e17 q^{72} +5.19733e16 q^{73} +1.07704e17 q^{75} -4.07007e17 q^{78} -1.40950e17 q^{81} -5.39261e17 q^{82} -7.28454e17 q^{87} -1.33260e18 q^{92} +1.30971e18 q^{93} -2.02006e18 q^{94} +4.50755e18 q^{96} +1.63004e18 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1001.00 1.95508 0.977539 − 0.210754i \(-0.0675920\pi\)
0.977539 + 0.210754i \(0.0675920\pi\)
\(3\) 28234.0 1.43444 0.717218 − 0.696849i \(-0.245415\pi\)
0.717218 + 0.696849i \(0.245415\pi\)
\(4\) 739857. 2.82233
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 2.82622e7 2.80443
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 4.78191e8 3.56280
\(9\) 4.09738e8 1.05761
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 2.08891e10 4.04845
\(13\) −1.44011e10 −1.35802 −0.679009 − 0.734130i \(-0.737590\pi\)
−0.679009 + 0.734130i \(0.737590\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.84720e11 4.14322
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 4.10148e11 2.06770
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.80115e12 −1.00000
\(24\) 1.35012e13 5.11061
\(25\) 3.81470e12 1.00000
\(26\) −1.44155e13 −2.65503
\(27\) 6.30120e11 0.0826322
\(28\) 0 0
\(29\) −2.58006e13 −1.77848 −0.889238 − 0.457445i \(-0.848765\pi\)
−0.889238 + 0.457445i \(0.848765\pi\)
\(30\) 0 0
\(31\) 4.63875e13 1.75447 0.877235 − 0.480060i \(-0.159385\pi\)
0.877235 + 0.480060i \(0.159385\pi\)
\(32\) 1.59650e14 4.53752
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 3.03148e14 2.98491
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −4.06601e14 −1.94799
\(40\) 0 0
\(41\) −5.38722e14 −1.64555 −0.822773 − 0.568370i \(-0.807574\pi\)
−0.822773 + 0.568370i \(0.807574\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.80295e15 −1.95508
\(47\) −2.01804e15 −1.80322 −0.901610 − 0.432551i \(-0.857614\pi\)
−0.901610 + 0.432551i \(0.857614\pi\)
\(48\) 8.03878e15 5.94318
\(49\) 1.62841e15 1.00000
\(50\) 3.81851e15 1.95508
\(51\) 0 0
\(52\) −1.06548e16 −3.83277
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 6.30750e14 0.161552
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −2.58264e16 −3.47706
\(59\) 1.57584e16 1.81905 0.909523 − 0.415654i \(-0.136447\pi\)
0.909523 + 0.415654i \(0.136447\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 4.64339e16 3.43013
\(63\) 0 0
\(64\) 8.51718e16 4.72798
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −5.08537e16 −1.43444
\(70\) 0 0
\(71\) 4.05587e16 0.884624 0.442312 − 0.896861i \(-0.354158\pi\)
0.442312 + 0.896861i \(0.354158\pi\)
\(72\) 1.95933e17 3.76804
\(73\) 5.19733e16 0.882826 0.441413 − 0.897304i \(-0.354477\pi\)
0.441413 + 0.897304i \(0.354477\pi\)
\(74\) 0 0
\(75\) 1.07704e17 1.43444
\(76\) 0 0
\(77\) 0 0
\(78\) −4.07007e17 −3.80847
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −1.40950e17 −0.939075
\(82\) −5.39261e17 −3.21717
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −7.28454e17 −2.55111
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.33260e18 −2.82233
\(93\) 1.30971e18 2.51668
\(94\) −2.02006e18 −3.52543
\(95\) 0 0
\(96\) 4.50755e18 6.50878
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 1.63004e18 1.95508
\(99\) 0 0
\(100\) 2.82233e18 2.82233
\(101\) −1.48135e18 −1.35446 −0.677231 − 0.735771i \(-0.736820\pi\)
−0.677231 + 0.735771i \(0.736820\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −6.88647e18 −4.83834
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 4.66199e17 0.233215
\(109\) 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
\(116\) −1.90888e19 −5.01945
\(117\) −5.90068e18 −1.43625
\(118\) 1.57741e19 3.55638
\(119\) 0 0
\(120\) 0 0
\(121\) 5.55992e18 1.00000
\(122\) 0 0
\(123\) −1.52103e19 −2.36043
\(124\) 3.43201e19 4.95170
\(125\) 0 0
\(126\) 0 0
\(127\) −1.42579e19 −1.65891 −0.829454 − 0.558575i \(-0.811348\pi\)
−0.829454 + 0.558575i \(0.811348\pi\)
\(128\) 4.34057e19 4.70606
\(129\) 0 0
\(130\) 0 0
\(131\) −1.37169e19 −1.20730 −0.603649 − 0.797250i \(-0.706287\pi\)
−0.603649 + 0.797250i \(0.706287\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) −5.09046e19 −2.80443
\(139\) −2.81811e19 −1.45487 −0.727437 − 0.686175i \(-0.759289\pi\)
−0.727437 + 0.686175i \(0.759289\pi\)
\(140\) 0 0
\(141\) −5.69773e19 −2.58660
\(142\) 4.05993e19 1.72951
\(143\) 0 0
\(144\) 1.16661e20 4.38189
\(145\) 0 0
\(146\) 5.20253e19 1.72599
\(147\) 4.59766e19 1.43444
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 1.07812e20 2.80443
\(151\) 1.70141e19 0.416886 0.208443 − 0.978035i \(-0.433160\pi\)
0.208443 + 0.978035i \(0.433160\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −3.00826e20 −5.49787
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −1.41091e20 −1.83597
\(163\) 1.22369e20 1.50654 0.753272 − 0.657709i \(-0.228474\pi\)
0.753272 + 0.657709i \(0.228474\pi\)
\(164\) −3.98577e20 −4.64428
\(165\) 0 0
\(166\) 0 0
\(167\) 1.71138e20 1.69394 0.846969 − 0.531642i \(-0.178425\pi\)
0.846969 + 0.531642i \(0.178425\pi\)
\(168\) 0 0
\(169\) 9.49362e19 0.844212
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.50427e20 −1.08371 −0.541853 − 0.840473i \(-0.682277\pi\)
−0.541853 + 0.840473i \(0.682277\pi\)
\(174\) −7.29183e20 −4.98762
\(175\) 0 0
\(176\) 0 0
\(177\) 4.44922e20 2.60930
\(178\) 0 0
\(179\) 3.26555e20 1.73093 0.865464 − 0.500971i \(-0.167024\pi\)
0.865464 + 0.500971i \(0.167024\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −8.61294e20 −3.56280
\(185\) 0 0
\(186\) 1.31102e21 4.92030
\(187\) 0 0
\(188\) −1.49306e21 −5.08928
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 2.40474e21 6.78199
\(193\) −5.55299e20 −1.49455 −0.747276 − 0.664513i \(-0.768639\pi\)
−0.747276 + 0.664513i \(0.768639\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.20479e21 2.82233
\(197\) 7.35324e20 1.64544 0.822721 − 0.568446i \(-0.192455\pi\)
0.822721 + 0.568446i \(0.192455\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 1.82415e21 3.56280
\(201\) 0 0
\(202\) −1.48284e21 −2.64808
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −7.38001e20 −1.05761
\(208\) −4.10028e21 −5.62656
\(209\) 0 0
\(210\) 0 0
\(211\) 1.31448e21 1.58567 0.792833 − 0.609439i \(-0.208605\pi\)
0.792833 + 0.609439i \(0.208605\pi\)
\(212\) 0 0
\(213\) 1.14513e21 1.26894
\(214\) 0 0
\(215\) 0 0
\(216\) 3.01318e20 0.294402
\(217\) 0 0
\(218\) 0 0
\(219\) 1.46742e21 1.26636
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −4.57929e20 −0.335780 −0.167890 − 0.985806i \(-0.553695\pi\)
−0.167890 + 0.985806i \(0.553695\pi\)
\(224\) 0 0
\(225\) 1.56303e21 1.05761
\(226\) 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
\(232\) −1.23376e22 −6.33635
\(233\) −3.27251e21 −1.61688 −0.808439 − 0.588580i \(-0.799687\pi\)
−0.808439 + 0.588580i \(0.799687\pi\)
\(234\) −5.90658e21 −2.80798
\(235\) 0 0
\(236\) 1.16590e22 5.13395
\(237\) 0 0
\(238\) 0 0
\(239\) 4.55994e21 1.79216 0.896082 − 0.443889i \(-0.146402\pi\)
0.896082 + 0.443889i \(0.146402\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 5.56548e21 1.95508
\(243\) −4.22371e21 −1.42968
\(244\) 0 0
\(245\) 0 0
\(246\) −1.52255e22 −4.61483
\(247\) 0 0
\(248\) 2.21821e22 6.25083
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.42722e22 −3.24330
\(255\) 0 0
\(256\) 2.11219e22 4.47273
\(257\) 7.58388e21 1.55058 0.775288 − 0.631607i \(-0.217604\pi\)
0.775288 + 0.631607i \(0.217604\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.05715e22 −1.88093
\(262\) −1.37306e22 −2.36036
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.14433e22 −1.55161 −0.775804 − 0.630974i \(-0.782656\pi\)
−0.775804 + 0.630974i \(0.782656\pi\)
\(270\) 0 0
\(271\) 1.51071e22 1.91628 0.958140 − 0.286301i \(-0.0924258\pi\)
0.958140 + 0.286301i \(0.0924258\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −3.76245e22 −4.04845
\(277\) −6.88440e21 −0.717048 −0.358524 − 0.933521i \(-0.616720\pi\)
−0.358524 + 0.933521i \(0.616720\pi\)
\(278\) −2.82093e22 −2.84439
\(279\) 1.90068e22 1.85554
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) −5.70343e22 −5.05701
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 3.00076e22 2.49670
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 6.54146e22 4.79891
\(289\) 1.40631e22 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 3.84528e22 2.49163
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 4.60226e22 2.80443
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.59386e22 1.35802
\(300\) 7.96857e22 4.04845
\(301\) 0 0
\(302\) 1.70311e22 0.815044
\(303\) −4.18246e22 −1.94289
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3.02450e21 0.124856 0.0624278 − 0.998049i \(-0.480116\pi\)
0.0624278 + 0.998049i \(0.480116\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.29078e22 1.94391 0.971955 − 0.235165i \(-0.0755630\pi\)
0.971955 + 0.235165i \(0.0755630\pi\)
\(312\) −1.94433e23 −6.94029
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.43051e22 −1.37063 −0.685315 − 0.728247i \(-0.740335\pi\)
−0.685315 + 0.728247i \(0.740335\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.04283e23 −2.65038
\(325\) −5.49358e22 −1.35802
\(326\) 1.22491e23 2.94541
\(327\) 0 0
\(328\) −2.57612e23 −5.86275
\(329\) 0 0
\(330\) 0 0
\(331\) 9.53756e22 1.99979 0.999897 − 0.0143344i \(-0.00456294\pi\)
0.999897 + 0.0143344i \(0.00456294\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 1.71309e23 3.31178
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 9.50312e22 1.65050
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −1.50578e23 −2.11873
\(347\) −3.71047e21 −0.0508703 −0.0254351 − 0.999676i \(-0.508097\pi\)
−0.0254351 + 0.999676i \(0.508097\pi\)
\(348\) −5.38952e23 −7.20007
\(349\) −5.06424e22 −0.659304 −0.329652 − 0.944103i \(-0.606931\pi\)
−0.329652 + 0.944103i \(0.606931\pi\)
\(350\) 0 0
\(351\) −9.07442e21 −0.112216
\(352\) 0 0
\(353\) −5.67058e20 −0.00666277 −0.00333139 − 0.999994i \(-0.501060\pi\)
−0.00333139 + 0.999994i \(0.501060\pi\)
\(354\) 4.45367e23 5.10139
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 3.26882e23 3.38410
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.04127e23 1.00000
\(362\) 0 0
\(363\) 1.56979e23 1.43444
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −5.12824e23 −4.14322
\(369\) −2.20735e23 −1.74034
\(370\) 0 0
\(371\) 0 0
\(372\) 9.68995e23 7.10289
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.65007e23 −6.42451
\(377\) 3.71557e23 2.41520
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −4.02558e23 −2.37960
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 1.22552e24 6.75054
\(385\) 0 0
\(386\) −5.55855e23 −2.92197
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 7.78692e23 3.56280
\(393\) −3.87283e23 −1.73179
\(394\) 7.36059e23 3.21697
\(395\) 0 0
\(396\) 0 0
\(397\) −1.30451e23 −0.532514 −0.266257 − 0.963902i \(-0.585787\pi\)
−0.266257 + 0.963902i \(0.585787\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.08612e24 4.14322
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −6.68032e23 −2.38260
\(404\) −1.09599e24 −3.82274
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.72225e23 0.537757 0.268878 − 0.963174i \(-0.413347\pi\)
0.268878 + 0.963174i \(0.413347\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −7.38739e23 −2.06770
\(415\) 0 0
\(416\) −2.29913e24 −6.16203
\(417\) −7.95666e23 −2.08692
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 1.31579e24 3.10010
\(423\) −8.26867e23 −1.90710
\(424\) 0 0
\(425\) 0 0
\(426\) 1.14628e24 2.48087
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 1.79408e23 0.342363
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 1.46888e24 2.47583
\(439\) 9.28281e23 1.53285 0.766423 − 0.642336i \(-0.222035\pi\)
0.766423 + 0.642336i \(0.222035\pi\)
\(440\) 0 0
\(441\) 6.67223e23 1.05761
\(442\) 0 0
\(443\) −3.25199e23 −0.494898 −0.247449 − 0.968901i \(-0.579592\pi\)
−0.247449 + 0.968901i \(0.579592\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4.58387e23 −0.656475
\(447\) 0 0
\(448\) 0 0
\(449\) 6.24135e22 0.0841514 0.0420757 − 0.999114i \(-0.486603\pi\)
0.0420757 + 0.999114i \(0.486603\pi\)
\(450\) 1.56459e24 2.06770
\(451\) 0 0
\(452\) 0 0
\(453\) 4.80377e23 0.597996
\(454\) 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
\(461\) −1.09182e24 −1.16103 −0.580513 − 0.814251i \(-0.697148\pi\)
−0.580513 + 0.814251i \(0.697148\pi\)
\(462\) 0 0
\(463\) 1.59548e24 1.63178 0.815892 − 0.578204i \(-0.196246\pi\)
0.815892 + 0.578204i \(0.196246\pi\)
\(464\) −7.34595e24 −7.36862
\(465\) 0 0
\(466\) −3.27578e24 −3.16112
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −4.36566e24 −4.05357
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 7.53551e24 6.48089
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 4.56450e24 3.50382
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 4.11354e24 2.82233
\(485\) 0 0
\(486\) −4.22793e24 −2.79513
\(487\) −2.44746e24 −1.58838 −0.794190 − 0.607669i \(-0.792105\pi\)
−0.794190 + 0.607669i \(0.792105\pi\)
\(488\) 0 0
\(489\) 3.45496e24 2.16104
\(490\) 0 0
\(491\) −1.07725e24 −0.649506 −0.324753 − 0.945799i \(-0.605281\pi\)
−0.324753 + 0.945799i \(0.605281\pi\)
\(492\) −1.12534e25 −6.66192
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.32075e25 7.26916
\(497\) 0 0
\(498\) 0 0
\(499\) −3.51041e22 −0.0183001 −0.00915003 − 0.999958i \(-0.502913\pi\)
−0.00915003 + 0.999958i \(0.502913\pi\)
\(500\) 0 0
\(501\) 4.83190e24 2.42985
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.68043e24 1.21097
\(508\) −1.05488e25 −4.68199
\(509\) 2.41029e24 1.05101 0.525507 − 0.850789i \(-0.323875\pi\)
0.525507 + 0.850789i \(0.323875\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 9.76444e24 4.03848
\(513\) 0 0
\(514\) 7.59146e24 3.03150
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −4.24716e24 −1.55451
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) −1.05821e25 −3.67736
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −1.01486e25 −3.40740
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.24415e24 1.00000
\(530\) 0 0
\(531\) 6.45681e24 1.92383
\(532\) 0 0
\(533\) 7.75819e24 2.23468
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 9.21996e24 2.48291
\(538\) −1.14548e25 −3.03352
\(539\) 0 0
\(540\) 0 0
\(541\) −7.74360e24 −1.95060 −0.975299 − 0.220890i \(-0.929104\pi\)
−0.975299 + 0.220890i \(0.929104\pi\)
\(542\) 1.51223e25 3.74648
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −8.18027e24 −1.86587 −0.932936 − 0.360043i \(-0.882762\pi\)
−0.932936 + 0.360043i \(0.882762\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) −2.43178e25 −5.11061
\(553\) 0 0
\(554\) −6.89128e24 −1.40188
\(555\) 0 0
\(556\) −2.08500e25 −4.10613
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 1.90258e25 3.62772
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) −4.21550e25 −7.30025
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 1.93948e25 3.15174
\(569\) 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
\(575\) −6.87085e24 −1.00000
\(576\) 3.48981e25 5.00034
\(577\) −9.24167e24 −1.30367 −0.651836 − 0.758360i \(-0.726001\pi\)
−0.651836 + 0.758360i \(0.726001\pi\)
\(578\) 1.40771e25 1.95508
\(579\) −1.56783e25 −2.14384
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 2.48532e25 3.14533
\(585\) 0 0
\(586\) 0 0
\(587\) −1.48760e25 −1.79781 −0.898903 − 0.438147i \(-0.855635\pi\)
−0.898903 + 0.438147i \(0.855635\pi\)
\(588\) 3.40161e25 4.04845
\(589\) 0 0
\(590\) 0 0
\(591\) 2.07611e25 2.36028
\(592\) 0 0
\(593\) −1.47479e25 −1.62644 −0.813222 − 0.581954i \(-0.802288\pi\)
−0.813222 + 0.581954i \(0.802288\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 2.59645e25 2.65503
\(599\) −1.82212e25 −1.83542 −0.917709 − 0.397253i \(-0.869964\pi\)
−0.917709 + 0.397253i \(0.869964\pi\)
\(600\) 5.15031e25 5.11061
\(601\) −2.04147e25 −1.99560 −0.997798 − 0.0663311i \(-0.978871\pi\)
−0.997798 + 0.0663311i \(0.978871\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.25880e25 1.17659
\(605\) 0 0
\(606\) −4.18664e25 −3.79850
\(607\) 5.08809e24 0.454837 0.227419 − 0.973797i \(-0.426971\pi\)
0.227419 + 0.973797i \(0.426971\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.90620e25 2.44880
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 3.02752e24 0.244103
\(615\) 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
\(621\) −1.13494e24 −0.0826322
\(622\) 5.29607e25 3.80050
\(623\) 0 0
\(624\) −1.15767e26 −8.07095
\(625\) 1.45519e25 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
\(633\) 3.71130e25 2.27454
\(634\) −4.43494e25 −2.67969
\(635\) 0 0
\(636\) 0 0
\(637\) −2.34509e25 −1.35802
\(638\) 0 0
\(639\) 1.66185e25 0.935584
\(640\) 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
\(647\) −3.97061e25 −1.99857 −0.999284 − 0.0378456i \(-0.987950\pi\)
−0.999284 + 0.0378456i \(0.987950\pi\)
\(648\) −6.74011e25 −3.34574
\(649\) 0 0
\(650\) −5.49908e25 −2.65503
\(651\) 0 0
\(652\) 9.05354e25 4.25197
\(653\) 1.63183e25 0.755888 0.377944 − 0.925829i \(-0.376631\pi\)
0.377944 + 0.925829i \(0.376631\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.53385e26 −6.81786
\(657\) 2.12955e25 0.933682
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 9.54710e25 3.90975
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.64708e25 1.77848
\(668\) 1.26617e26 4.78085
\(669\) −1.29292e25 −0.481654
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −5.20100e24 −0.183633 −0.0918167 − 0.995776i \(-0.529267\pi\)
−0.0918167 + 0.995776i \(0.529267\pi\)
\(674\) 0 0
\(675\) 2.40372e24 0.0826322
\(676\) 7.02392e25 2.38265
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.39828e25 1.35987 0.679934 − 0.733273i \(-0.262008\pi\)
0.679934 + 0.733273i \(0.262008\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −3.87997e25 −1.08025 −0.540126 − 0.841584i \(-0.681624\pi\)
−0.540126 + 0.841584i \(0.681624\pi\)
\(692\) −1.11295e26 −3.05858
\(693\) 0 0
\(694\) −3.71418e24 −0.0994554
\(695\) 0 0
\(696\) −3.48340e26 −9.08909
\(697\) 0 0
\(698\) −5.06931e25 −1.28899
\(699\) −9.23960e25 −2.31931
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) −9.08350e24 −0.219391
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −5.67625e23 −0.0130262
\(707\) 0 0
\(708\) 3.29179e26 7.36432
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8.35510e25 −1.75447
\(714\) 0 0
\(715\) 0 0
\(716\) 2.41604e26 4.88525
\(717\) 1.28745e26 2.57074
\(718\) 0 0
\(719\) 4.90528e25 0.955220 0.477610 − 0.878572i \(-0.341503\pi\)
0.477610 + 0.878572i \(0.341503\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.04231e26 1.95508
\(723\) 0 0
\(724\) 0 0
\(725\) −9.84215e25 −1.77848
\(726\) 1.57136e26 2.80443
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −6.46452e25 −1.11170
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −2.87554e26 −4.53752
\(737\) 0 0
\(738\) −2.20956e26 −3.40250
\(739\) −5.95740e25 −0.906269 −0.453134 − 0.891442i \(-0.649694\pi\)
−0.453134 + 0.891442i \(0.649694\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 6.26289e26 8.96641
\(745\) 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 \(\pi\)
\(752\) −5.74575e26 −7.47113
\(753\) 0 0
\(754\) 3.71929e26 4.72191
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.04512e26 −1.22096 −0.610481 − 0.792031i \(-0.709024\pi\)
−0.610481 + 0.792031i \(0.709024\pi\)
\(762\) −4.02960e26 −4.65230
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.26938e26 −2.47030
\(768\) 5.96355e26 6.41584
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 2.14123e26 2.22420
\(772\) −4.10842e26 −4.21812
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 1.76954e26 1.75447
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1.62575e25 −0.146959
\(784\) 4.63642e26 4.14322
\(785\) 0 0
\(786\) −3.87671e26 −3.38579
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 5.44034e26 4.64398
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −1.30581e26 −1.04111
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 6.09015e26 4.53752
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −6.68700e26 −4.65817
\(807\) −3.23091e26 −2.22568
\(808\) −7.08370e26 −4.82567
\(809\) 2.15677e26 1.45301 0.726504 − 0.687163i \(-0.241144\pi\)
0.726504 + 0.687163i \(0.241144\pi\)
\(810\) 0 0
\(811\) −2.74971e26 −1.81175 −0.905877 − 0.423542i \(-0.860787\pi\)
−0.905877 + 0.423542i \(0.860787\pi\)
\(812\) 0 0
\(813\) 4.26535e26 2.74878
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 1.72397e26 1.05136
\(819\) 0 0
\(820\) 0 0
\(821\) 3.35666e26 1.98070 0.990349 − 0.138594i \(-0.0442582\pi\)
0.990349 + 0.138594i \(0.0442582\pi\)
\(822\) 0 0
\(823\) 2.28084e26 1.31673 0.658366 − 0.752698i \(-0.271248\pi\)
0.658366 + 0.752698i \(0.271248\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −5.46015e26 −2.98491
\(829\) −2.99868e26 −1.62158 −0.810792 − 0.585334i \(-0.800964\pi\)
−0.810792 + 0.585334i \(0.800964\pi\)
\(830\) 0 0
\(831\) −1.94374e26 −1.02856
\(832\) −1.22657e27 −6.42069
\(833\) 0 0
\(834\) −7.96462e26 −4.08010
\(835\) 0 0
\(836\) 0 0
\(837\) 2.92297e25 0.144976
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 4.55214e26 2.16298
\(842\) 0 0
\(843\) 0 0
\(844\) 9.72527e26 4.47527
\(845\) 0 0
\(846\) −8.27694e26 −3.72852
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 8.47236e26 3.58136
\(853\) 4.77678e26 1.99799 0.998997 − 0.0447818i \(-0.0142593\pi\)
0.998997 + 0.0447818i \(0.0142593\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.74534e26 1.90300 0.951502 − 0.307642i \(-0.0995401\pi\)
0.951502 + 0.307642i \(0.0995401\pi\)
\(858\) 0 0
\(859\) 5.07711e26 1.99378 0.996892 − 0.0787838i \(-0.0251037\pi\)
0.996892 + 0.0787838i \(0.0251037\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −4.32323e26 −1.62821 −0.814107 − 0.580715i \(-0.802773\pi\)
−0.814107 + 0.580715i \(0.802773\pi\)
\(864\) 1.00599e26 0.374945
\(865\) 0 0
\(866\) 0 0
\(867\) 3.97057e26 1.43444
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 1.08568e27 3.57408
\(877\) 2.48612e26 0.810076 0.405038 − 0.914300i \(-0.367258\pi\)
0.405038 + 0.914300i \(0.367258\pi\)
\(878\) 9.29209e26 2.99683
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 6.67891e26 2.06770
\(883\) −6.18027e26 −1.89392 −0.946959 − 0.321356i \(-0.895861\pi\)
−0.946959 + 0.321356i \(0.895861\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −3.25524e26 −0.967564
\(887\) −6.56108e26 −1.93047 −0.965234 − 0.261388i \(-0.915820\pi\)
−0.965234 + 0.261388i \(0.915820\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −3.38802e26 −0.947681
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 7.32350e26 1.94799
\(898\) 6.24759e25 0.164523
\(899\) −1.19683e27 −3.12028
\(900\) 1.15642e27 2.98491
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 4.80857e26 1.16913
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −6.06968e26 −1.43249
\(910\) 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
\(921\) 8.53937e25 0.179097
\(922\) −1.09291e27 −2.26990
\(923\) −5.84090e26 −1.20134
\(924\) 0 0
\(925\) 0 0
\(926\) 1.59708e27 3.19027
\(927\) 0 0
\(928\) −4.11906e27 −8.06987
\(929\) −1.02672e27 −1.99210 −0.996050 − 0.0887934i \(-0.971699\pi\)
−0.996050 + 0.0887934i \(0.971699\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.42119e27 −4.56337
\(933\) 1.49380e27 2.78842
\(934\) 0 0
\(935\) 0 0
\(936\) −2.82165e27 −5.11706
\(937\) 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
\(943\) 9.70321e26 1.64555
\(944\) 4.48673e27 7.53670
\(945\) 0 0
\(946\) 0 0
\(947\) 5.26954e26 0.860247 0.430123 − 0.902770i \(-0.358470\pi\)
0.430123 + 0.902770i \(0.358470\pi\)
\(948\) 0 0
\(949\) −7.48473e26 −1.19889
\(950\) 0 0
\(951\) −1.25091e27 −1.96608
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 3.37370e27 5.05808
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.45275e27 2.07817
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −9.73592e26 −1.31686 −0.658429 − 0.752643i \(-0.728779\pi\)
−0.658429 + 0.752643i \(0.728779\pi\)
\(968\) 2.65870e27 3.56280
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −3.12494e27 −4.03502
\(973\) 0 0
\(974\) −2.44990e27 −3.10541
\(975\) −1.55106e27 −1.94799
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 3.45841e27 4.22500
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −1.07833e27 −1.26983
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −7.27342e27 −8.40974
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.17338e27 −1.27285 −0.636424 − 0.771339i \(-0.719587\pi\)
−0.636424 + 0.771339i \(0.719587\pi\)
\(992\) 7.40576e27 7.96094
\(993\) 2.69283e27 2.86858
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −4.17148e26 −0.428582 −0.214291 − 0.976770i \(-0.568744\pi\)
−0.214291 + 0.976770i \(0.568744\pi\)
\(998\) −3.51392e25 −0.0357780
\(999\) 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 23.19.b.a.22.1 ✓ 1
23.22 odd 2 CM 23.19.b.a.22.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.19.b.a.22.1 ✓ 1 1.1 even 1 trivial
23.19.b.a.22.1 ✓ 1 23.22 odd 2 CM