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Properties

Label 24.5.h.a.5.1
Level $24$
Weight $5$
Character 24.5
Self dual yes
Analytic conductor $2.481$
Analytic rank $0$
Dimension $1$
CM discriminant -24
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] = [24,5,Mod(5,24)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(24, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.5");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 24.h (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: \(2.48087911401\)
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 5.1
Character \(\chi\) \(=\) 24.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +46.0000 q^{5} +36.0000 q^{6} +2.00000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +46.0000 q^{5} +36.0000 q^{6} +2.00000 q^{7} -64.0000 q^{8} +81.0000 q^{9} -184.000 q^{10} +142.000 q^{11} -144.000 q^{12} -8.00000 q^{14} -414.000 q^{15} +256.000 q^{16} -324.000 q^{18} +736.000 q^{20} -18.0000 q^{21} -568.000 q^{22} +576.000 q^{24} +1491.00 q^{25} -729.000 q^{27} +32.0000 q^{28} -818.000 q^{29} +1656.00 q^{30} -478.000 q^{31} -1024.00 q^{32} -1278.00 q^{33} +92.0000 q^{35} +1296.00 q^{36} -2944.00 q^{40} +72.0000 q^{42} +2272.00 q^{44} +3726.00 q^{45} -2304.00 q^{48} -2397.00 q^{49} -5964.00 q^{50} -3218.00 q^{53} +2916.00 q^{54} +6532.00 q^{55} -128.000 q^{56} +3272.00 q^{58} +6862.00 q^{59} -6624.00 q^{60} +1912.00 q^{62} +162.000 q^{63} +4096.00 q^{64} +5112.00 q^{66} -368.000 q^{70} -5184.00 q^{72} -8158.00 q^{73} -13419.0 q^{75} +284.000 q^{77} -9118.00 q^{79} +11776.0 q^{80} +6561.00 q^{81} -4178.00 q^{83} -288.000 q^{84} +7362.00 q^{87} -9088.00 q^{88} -14904.0 q^{90} +4302.00 q^{93} +9216.00 q^{96} +17282.0 q^{97} +9588.00 q^{98} +11502.0 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −1.00000
\(3\) −9.00000 −1.00000
\(4\) 16.0000 1.00000
\(5\) 46.0000 1.84000 0.920000 − 0.391918i \(-0.128188\pi\)
0.920000 + 0.391918i \(0.128188\pi\)
\(6\) 36.0000 1.00000
\(7\) 2.00000 0.0408163 0.0204082 − 0.999792i \(-0.493503\pi\)
0.0204082 + 0.999792i \(0.493503\pi\)
\(8\) −64.0000 −1.00000
\(9\) 81.0000 1.00000
\(10\) −184.000 −1.84000
\(11\) 142.000 1.17355 0.586777 − 0.809749i \(-0.300397\pi\)
0.586777 + 0.809749i \(0.300397\pi\)
\(12\) −144.000 −1.00000
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −8.00000 −0.0408163
\(15\) −414.000 −1.84000
\(16\) 256.000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −324.000 −1.00000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 736.000 1.84000
\(21\) −18.0000 −0.0408163
\(22\) −568.000 −1.17355
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 576.000 1.00000
\(25\) 1491.00 2.38560
\(26\) 0 0
\(27\) −729.000 −1.00000
\(28\) 32.0000 0.0408163
\(29\) −818.000 −0.972652 −0.486326 − 0.873778i \(-0.661663\pi\)
−0.486326 + 0.873778i \(0.661663\pi\)
\(30\) 1656.00 1.84000
\(31\) −478.000 −0.497399 −0.248699 − 0.968581i \(-0.580003\pi\)
−0.248699 + 0.968581i \(0.580003\pi\)
\(32\) −1024.00 −1.00000
\(33\) −1278.00 −1.17355
\(34\) 0 0
\(35\) 92.0000 0.0751020
\(36\) 1296.00 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −2944.00 −1.84000
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 72.0000 0.0408163
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 2272.00 1.17355
\(45\) 3726.00 1.84000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −2304.00 −1.00000
\(49\) −2397.00 −0.998334
\(50\) −5964.00 −2.38560
\(51\) 0 0
\(52\) 0 0
\(53\) −3218.00 −1.14560 −0.572802 − 0.819694i \(-0.694143\pi\)
−0.572802 + 0.819694i \(0.694143\pi\)
\(54\) 2916.00 1.00000
\(55\) 6532.00 2.15934
\(56\) −128.000 −0.0408163
\(57\) 0 0
\(58\) 3272.00 0.972652
\(59\) 6862.00 1.97127 0.985636 − 0.168882i \(-0.0540156\pi\)
0.985636 + 0.168882i \(0.0540156\pi\)
\(60\) −6624.00 −1.84000
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 1912.00 0.497399
\(63\) 162.000 0.0408163
\(64\) 4096.00 1.00000
\(65\) 0 0
\(66\) 5112.00 1.17355
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −368.000 −0.0751020
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −5184.00 −1.00000
\(73\) −8158.00 −1.53087 −0.765434 − 0.643514i \(-0.777476\pi\)
−0.765434 + 0.643514i \(0.777476\pi\)
\(74\) 0 0
\(75\) −13419.0 −2.38560
\(76\) 0 0
\(77\) 284.000 0.0479002
\(78\) 0 0
\(79\) −9118.00 −1.46098 −0.730492 − 0.682921i \(-0.760709\pi\)
−0.730492 + 0.682921i \(0.760709\pi\)
\(80\) 11776.0 1.84000
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) −4178.00 −0.606474 −0.303237 − 0.952915i \(-0.598067\pi\)
−0.303237 + 0.952915i \(0.598067\pi\)
\(84\) −288.000 −0.0408163
\(85\) 0 0
\(86\) 0 0
\(87\) 7362.00 0.972652
\(88\) −9088.00 −1.17355
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) −14904.0 −1.84000
\(91\) 0 0
\(92\) 0 0
\(93\) 4302.00 0.497399
\(94\) 0 0
\(95\) 0 0
\(96\) 9216.00 1.00000
\(97\) 17282.0 1.83675 0.918376 − 0.395709i \(-0.129501\pi\)
0.918376 + 0.395709i \(0.129501\pi\)
\(98\) 9588.00 0.998334
\(99\) 11502.0 1.17355
\(100\) 23856.0 2.38560
\(101\) −15698.0 −1.53887 −0.769434 − 0.638726i \(-0.779462\pi\)
−0.769434 + 0.638726i \(0.779462\pi\)
\(102\) 0 0
\(103\) −21118.0 −1.99057 −0.995287 − 0.0969729i \(-0.969084\pi\)
−0.995287 + 0.0969729i \(0.969084\pi\)
\(104\) 0 0
\(105\) −828.000 −0.0751020
\(106\) 12872.0 1.14560
\(107\) 15502.0 1.35400 0.677002 − 0.735981i \(-0.263279\pi\)
0.677002 + 0.735981i \(0.263279\pi\)
\(108\) −11664.0 −1.00000
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) −26128.0 −2.15934
\(111\) 0 0
\(112\) 512.000 0.0408163
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −13088.0 −0.972652
\(117\) 0 0
\(118\) −27448.0 −1.97127
\(119\) 0 0
\(120\) 26496.0 1.84000
\(121\) 5523.00 0.377228
\(122\) 0 0
\(123\) 0 0
\(124\) −7648.00 −0.497399
\(125\) 39836.0 2.54950
\(126\) −648.000 −0.0408163
\(127\) 20642.0 1.27981 0.639903 − 0.768455i \(-0.278974\pi\)
0.639903 + 0.768455i \(0.278974\pi\)
\(128\) −16384.0 −1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) −28178.0 −1.64198 −0.820989 − 0.570943i \(-0.806578\pi\)
−0.820989 + 0.570943i \(0.806578\pi\)
\(132\) −20448.0 −1.17355
\(133\) 0 0
\(134\) 0 0
\(135\) −33534.0 −1.84000
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 1472.00 0.0751020
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 20736.0 1.00000
\(145\) −37628.0 −1.78968
\(146\) 32632.0 1.53087
\(147\) 21573.0 0.998334
\(148\) 0 0
\(149\) −39698.0 −1.78812 −0.894059 − 0.447950i \(-0.852154\pi\)
−0.894059 + 0.447950i \(0.852154\pi\)
\(150\) 53676.0 2.38560
\(151\) 43202.0 1.89474 0.947371 − 0.320139i \(-0.103729\pi\)
0.947371 + 0.320139i \(0.103729\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −1136.00 −0.0479002
\(155\) −21988.0 −0.915213
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 36472.0 1.46098
\(159\) 28962.0 1.14560
\(160\) −47104.0 −1.84000
\(161\) 0 0
\(162\) −26244.0 −1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) −58788.0 −2.15934
\(166\) 16712.0 0.606474
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 1152.00 0.0408163
\(169\) 28561.0 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 57742.0 1.92930 0.964650 − 0.263536i \(-0.0848886\pi\)
0.964650 + 0.263536i \(0.0848886\pi\)
\(174\) −29448.0 −0.972652
\(175\) 2982.00 0.0973714
\(176\) 36352.0 1.17355
\(177\) −61758.0 −1.97127
\(178\) 0 0
\(179\) 11182.0 0.348990 0.174495 − 0.984658i \(-0.444171\pi\)
0.174495 + 0.984658i \(0.444171\pi\)
\(180\) 59616.0 1.84000
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) −17208.0 −0.497399
\(187\) 0 0
\(188\) 0 0
\(189\) −1458.00 −0.0408163
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −36864.0 −1.00000
\(193\) 9602.00 0.257779 0.128889 − 0.991659i \(-0.458859\pi\)
0.128889 + 0.991659i \(0.458859\pi\)
\(194\) −69128.0 −1.83675
\(195\) 0 0
\(196\) −38352.0 −0.998334
\(197\) 39982.0 1.03022 0.515112 − 0.857123i \(-0.327750\pi\)
0.515112 + 0.857123i \(0.327750\pi\)
\(198\) −46008.0 −1.17355
\(199\) −38398.0 −0.969622 −0.484811 − 0.874619i \(-0.661112\pi\)
−0.484811 + 0.874619i \(0.661112\pi\)
\(200\) −95424.0 −2.38560
\(201\) 0 0
\(202\) 62792.0 1.53887
\(203\) −1636.00 −0.0397001
\(204\) 0 0
\(205\) 0 0
\(206\) 84472.0 1.99057
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 3312.00 0.0751020
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −51488.0 −1.14560
\(213\) 0 0
\(214\) −62008.0 −1.35400
\(215\) 0 0
\(216\) 46656.0 1.00000
\(217\) −956.000 −0.0203020
\(218\) 0 0
\(219\) 73422.0 1.53087
\(220\) 104512. 2.15934
\(221\) 0 0
\(222\) 0 0
\(223\) −46558.0 −0.936234 −0.468117 − 0.883666i \(-0.655067\pi\)
−0.468117 + 0.883666i \(0.655067\pi\)
\(224\) −2048.00 −0.0408163
\(225\) 120771. 2.38560
\(226\) 0 0
\(227\) −16658.0 −0.323274 −0.161637 − 0.986850i \(-0.551677\pi\)
−0.161637 + 0.986850i \(0.551677\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) −2556.00 −0.0479002
\(232\) 52352.0 0.972652
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 109792. 1.97127
\(237\) 82062.0 1.46098
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −105984. −1.84000
\(241\) 29762.0 0.512422 0.256211 − 0.966621i \(-0.417526\pi\)
0.256211 + 0.966621i \(0.417526\pi\)
\(242\) −22092.0 −0.377228
\(243\) −59049.0 −1.00000
\(244\) 0 0
\(245\) −110262. −1.83693
\(246\) 0 0
\(247\) 0 0
\(248\) 30592.0 0.497399
\(249\) 37602.0 0.606474
\(250\) −159344. −2.54950
\(251\) −94898.0 −1.50629 −0.753147 − 0.657853i \(-0.771465\pi\)
−0.753147 + 0.657853i \(0.771465\pi\)
\(252\) 2592.00 0.0408163
\(253\) 0 0
\(254\) −82568.0 −1.27981
\(255\) 0 0
\(256\) 65536.0 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −66258.0 −0.972652
\(262\) 112712. 1.64198
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 81792.0 1.17355
\(265\) −148028. −2.10791
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −40178.0 −0.555244 −0.277622 − 0.960690i \(-0.589546\pi\)
−0.277622 + 0.960690i \(0.589546\pi\)
\(270\) 134136. 1.84000
\(271\) −143518. −1.95419 −0.977097 − 0.212793i \(-0.931744\pi\)
−0.977097 + 0.212793i \(0.931744\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 211722. 2.79963
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −38718.0 −0.497399
\(280\) −5888.00 −0.0751020
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −82944.0 −1.00000
\(289\) 83521.0 1.00000
\(290\) 150512. 1.78968
\(291\) −155538. −1.83675
\(292\) −130528. −1.53087
\(293\) 22702.0 0.264441 0.132221 − 0.991220i \(-0.457789\pi\)
0.132221 + 0.991220i \(0.457789\pi\)
\(294\) −86292.0 −0.998334
\(295\) 315652. 3.62714
\(296\) 0 0
\(297\) −103518. −1.17355
\(298\) 158792. 1.78812
\(299\) 0 0
\(300\) −214704. −2.38560
\(301\) 0 0
\(302\) −172808. −1.89474
\(303\) 141282. 1.53887
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 4544.00 0.0479002
\(309\) 190062. 1.99057
\(310\) 87952.0 0.915213
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 84962.0 0.867234 0.433617 − 0.901097i \(-0.357237\pi\)
0.433617 + 0.901097i \(0.357237\pi\)
\(314\) 0 0
\(315\) 7452.00 0.0751020
\(316\) −145888. −1.46098
\(317\) 89422.0 0.889869 0.444934 − 0.895563i \(-0.353227\pi\)
0.444934 + 0.895563i \(0.353227\pi\)
\(318\) −115848. −1.14560
\(319\) −116156. −1.14146
\(320\) 188416. 1.84000
\(321\) −139518. −1.35400
\(322\) 0 0
\(323\) 0 0
\(324\) 104976. 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 235152. 2.15934
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −66848.0 −0.606474
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −4608.00 −0.0408163
\(337\) −191038. −1.68213 −0.841066 − 0.540933i \(-0.818071\pi\)
−0.841066 + 0.540933i \(0.818071\pi\)
\(338\) −114244. −1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) −67876.0 −0.583724
\(342\) 0 0
\(343\) −9596.00 −0.0815647
\(344\) 0 0
\(345\) 0 0
\(346\) −230968. −1.92930
\(347\) 229582. 1.90668 0.953342 − 0.301891i \(-0.0976180\pi\)
0.953342 + 0.301891i \(0.0976180\pi\)
\(348\) 117792. 0.972652
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) −11928.0 −0.0973714
\(351\) 0 0
\(352\) −145408. −1.17355
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 247032. 1.97127
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −44728.0 −0.348990
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −238464. −1.84000
\(361\) 130321. 1.00000
\(362\) 0 0
\(363\) −49707.0 −0.377228
\(364\) 0 0
\(365\) −375268. −2.81680
\(366\) 0 0
\(367\) 234722. 1.74270 0.871348 − 0.490665i \(-0.163246\pi\)
0.871348 + 0.490665i \(0.163246\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6436.00 −0.0467593
\(372\) 68832.0 0.497399
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −358524. −2.54950
\(376\) 0 0
\(377\) 0 0
\(378\) 5832.00 0.0408163
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −185778. −1.27981
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 147456. 1.00000
\(385\) 13064.0 0.0881363
\(386\) −38408.0 −0.257779
\(387\) 0 0
\(388\) 276512. 1.83675
\(389\) 266542. 1.76143 0.880717 − 0.473643i \(-0.157061\pi\)
0.880717 + 0.473643i \(0.157061\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 153408. 0.998334
\(393\) 253602. 1.64198
\(394\) −159928. −1.03022
\(395\) −419428. −2.68821
\(396\) 184032. 1.17355
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 153592. 0.969622
\(399\) 0 0
\(400\) 381696. 2.38560
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −251168. −1.53887
\(405\) 301806. 1.84000
\(406\) 6544.00 0.0397001
\(407\) 0 0
\(408\) 0 0
\(409\) 180962. 1.08178 0.540892 − 0.841092i \(-0.318087\pi\)
0.540892 + 0.841092i \(0.318087\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −337888. −1.99057
\(413\) 13724.0 0.0804601
\(414\) 0 0
\(415\) −192188. −1.11591
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −181778. −1.03541 −0.517706 − 0.855559i \(-0.673214\pi\)
−0.517706 + 0.855559i \(0.673214\pi\)
\(420\) −13248.0 −0.0751020
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 205952. 1.14560
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 248032. 1.35400
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −186624. −1.00000
\(433\) 73922.0 0.394274 0.197137 − 0.980376i \(-0.436836\pi\)
0.197137 + 0.980376i \(0.436836\pi\)
\(434\) 3824.00 0.0203020
\(435\) 338652. 1.78968
\(436\) 0 0
\(437\) 0 0
\(438\) −293688. −1.53087
\(439\) −308158. −1.59899 −0.799493 − 0.600676i \(-0.794898\pi\)
−0.799493 + 0.600676i \(0.794898\pi\)
\(440\) −418048. −2.15934
\(441\) −194157. −0.998334
\(442\) 0 0
\(443\) 385102. 1.96231 0.981157 − 0.193214i \(-0.0618912\pi\)
0.981157 + 0.193214i \(0.0618912\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 186232. 0.936234
\(447\) 357282. 1.78812
\(448\) 8192.00 0.0408163
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) −483084. −2.38560
\(451\) 0 0
\(452\) 0 0
\(453\) −388818. −1.89474
\(454\) 66632.0 0.323274
\(455\) 0 0
\(456\) 0 0
\(457\) −136798. −0.655009 −0.327505 − 0.944850i \(-0.606208\pi\)
−0.327505 + 0.944850i \(0.606208\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 144142. 0.678248 0.339124 − 0.940742i \(-0.389869\pi\)
0.339124 + 0.940742i \(0.389869\pi\)
\(462\) 10224.0 0.0479002
\(463\) −366238. −1.70845 −0.854223 − 0.519906i \(-0.825967\pi\)
−0.854223 + 0.519906i \(0.825967\pi\)
\(464\) −209408. −0.972652
\(465\) 197892. 0.915213
\(466\) 0 0
\(467\) 34222.0 0.156918 0.0784588 − 0.996917i \(-0.475000\pi\)
0.0784588 + 0.996917i \(0.475000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −439168. −1.97127
\(473\) 0 0
\(474\) −328248. −1.46098
\(475\) 0 0
\(476\) 0 0
\(477\) −260658. −1.14560
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 423936. 1.84000
\(481\) 0 0
\(482\) −119048. −0.512422
\(483\) 0 0
\(484\) 88368.0 0.377228
\(485\) 794972. 3.37962
\(486\) 236196. 1.00000
\(487\) 466562. 1.96721 0.983607 − 0.180327i \(-0.0577158\pi\)
0.983607 + 0.180327i \(0.0577158\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 441048. 1.83693
\(491\) 261262. 1.08371 0.541855 − 0.840472i \(-0.317722\pi\)
0.541855 + 0.840472i \(0.317722\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 529092. 2.15934
\(496\) −122368. −0.497399
\(497\) 0 0
\(498\) −150408. −0.606474
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 637376. 2.54950
\(501\) 0 0
\(502\) 379592. 1.50629
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) −10368.0 −0.0408163
\(505\) −722108. −2.83152
\(506\) 0 0
\(507\) −257049. −1.00000
\(508\) 330272. 1.27981
\(509\) −501938. −1.93738 −0.968689 − 0.248276i \(-0.920136\pi\)
−0.968689 + 0.248276i \(0.920136\pi\)
\(510\) 0 0
\(511\) −16316.0 −0.0624844
\(512\) −262144. −1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) −971428. −3.66266
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −519678. −1.92930
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 265032. 0.972652
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −450848. −1.64198
\(525\) −26838.0 −0.0973714
\(526\) 0 0
\(527\) 0 0
\(528\) −327168. −1.17355
\(529\) 279841. 1.00000
\(530\) 592112. 2.10791
\(531\) 555822. 1.97127
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 713092. 2.49137
\(536\) 0 0
\(537\) −100638. −0.348990
\(538\) 160712. 0.555244
\(539\) −340374. −1.17160
\(540\) −536544. −1.84000
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 574072. 1.95419
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −846888. −2.79963
\(551\) 0 0
\(552\) 0 0
\(553\) −18236.0 −0.0596320
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −214898. −0.692663 −0.346331 − 0.938112i \(-0.612573\pi\)
−0.346331 + 0.938112i \(0.612573\pi\)
\(558\) 154872. 0.497399
\(559\) 0 0
\(560\) 23552.0 0.0751020
\(561\) 0 0
\(562\) 0 0
\(563\) 527662. 1.66471 0.832356 − 0.554242i \(-0.186992\pi\)
0.832356 + 0.554242i \(0.186992\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 13122.0 0.0408163
\(568\) 0 0
\(569\) 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
\(575\) 0 0
\(576\) 331776. 1.00000
\(577\) −581758. −1.74739 −0.873697 − 0.486471i \(-0.838284\pi\)
−0.873697 + 0.486471i \(0.838284\pi\)
\(578\) −334084. −1.00000
\(579\) −86418.0 −0.257779
\(580\) −602048. −1.78968
\(581\) −8356.00 −0.0247540
\(582\) 622152. 1.83675
\(583\) −456956. −1.34443
\(584\) 522112. 1.53087
\(585\) 0 0
\(586\) −90808.0 −0.264441
\(587\) −74738.0 −0.216903 −0.108451 − 0.994102i \(-0.534589\pi\)
−0.108451 + 0.994102i \(0.534589\pi\)
\(588\) 345168. 0.998334
\(589\) 0 0
\(590\) −1.26261e6 −3.62714
\(591\) −359838. −1.03022
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 414072. 1.17355
\(595\) 0 0
\(596\) −635168. −1.78812
\(597\) 345582. 0.969622
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 858816. 2.38560
\(601\) 712802. 1.97342 0.986711 − 0.162485i \(-0.0519509\pi\)
0.986711 + 0.162485i \(0.0519509\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 691232. 1.89474
\(605\) 254058. 0.694100
\(606\) −565128. −1.53887
\(607\) −203998. −0.553667 −0.276833 − 0.960918i \(-0.589285\pi\)
−0.276833 + 0.960918i \(0.589285\pi\)
\(608\) 0 0
\(609\) 14724.0 0.0397001
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −18176.0 −0.0479002
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) −760248. −1.99057
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) −351808. −0.915213
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 900581. 2.30549
\(626\) −339848. −0.867234
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) −29808.0 −0.0751020
\(631\) 736322. 1.84931 0.924654 − 0.380809i \(-0.124355\pi\)
0.924654 + 0.380809i \(0.124355\pi\)
\(632\) 583552. 1.46098
\(633\) 0 0
\(634\) −357688. −0.889869
\(635\) 949532. 2.35484
\(636\) 463392. 1.14560
\(637\) 0 0
\(638\) 464624. 1.14146
\(639\) 0 0
\(640\) −753664. −1.84000
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 558072. 1.35400
\(643\) 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 \(\pi\)
\(648\) −419904. −1.00000
\(649\) 974404. 2.31339
\(650\) 0 0
\(651\) 8604.00 0.0203020
\(652\) 0 0
\(653\) −159218. −0.373393 −0.186696 − 0.982418i \(-0.559778\pi\)
−0.186696 + 0.982418i \(0.559778\pi\)
\(654\) 0 0
\(655\) −1.29619e6 −3.02124
\(656\) 0 0
\(657\) −660798. −1.53087
\(658\) 0 0
\(659\) 335662. 0.772914 0.386457 − 0.922307i \(-0.373699\pi\)
0.386457 + 0.922307i \(0.373699\pi\)
\(660\) −940608. −2.15934
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 267392. 0.606474
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 419022. 0.936234
\(670\) 0 0
\(671\) 0 0
\(672\) 18432.0 0.0408163
\(673\) −869758. −1.92030 −0.960148 − 0.279491i \(-0.909834\pi\)
−0.960148 + 0.279491i \(0.909834\pi\)
\(674\) 764152. 1.68213
\(675\) −1.08694e6 −2.38560
\(676\) 456976. 1.00000
\(677\) −895058. −1.95287 −0.976436 − 0.215807i \(-0.930762\pi\)
−0.976436 + 0.215807i \(0.930762\pi\)
\(678\) 0 0
\(679\) 34564.0 0.0749695
\(680\) 0 0
\(681\) 149922. 0.323274
\(682\) 271504. 0.583724
\(683\) −846578. −1.81479 −0.907393 − 0.420282i \(-0.861931\pi\)
−0.907393 + 0.420282i \(0.861931\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 38384.0 0.0815647
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 923872. 1.92930
\(693\) 23004.0 0.0479002
\(694\) −918328. −1.90668
\(695\) 0 0
\(696\) −471168. −0.972652
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 47712.0 0.0973714
\(701\) 980302. 1.99491 0.997456 − 0.0712813i \(-0.0227088\pi\)
0.997456 + 0.0712813i \(0.0227088\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 581632. 1.17355
\(705\) 0 0
\(706\) 0 0
\(707\) −31396.0 −0.0628110
\(708\) −988128. −1.97127
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) −738558. −1.46098
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 178912. 0.348990
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 953856. 1.84000
\(721\) −42236.0 −0.0812479
\(722\) −521284. −1.00000
\(723\) −267858. −0.512422
\(724\) 0 0
\(725\) −1.21964e6 −2.32036
\(726\) 198828. 0.377228
\(727\) 1.04544e6 1.97802 0.989011 − 0.147842i \(-0.0472327\pi\)
0.989011 + 0.147842i \(0.0472327\pi\)
\(728\) 0 0
\(729\) 531441. 1.00000
\(730\) 1.50107e6 2.81680
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) −938888. −1.74270
\(735\) 992358. 1.83693
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 25744.0 0.0467593
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) −275328. −0.497399
\(745\) −1.82611e6 −3.29014
\(746\) 0 0
\(747\) −338418. −0.606474
\(748\) 0 0
\(749\) 31004.0 0.0552655
\(750\) 1.43410e6 2.54950
\(751\) 837602. 1.48511 0.742554 − 0.669787i \(-0.233614\pi\)
0.742554 + 0.669787i \(0.233614\pi\)
\(752\) 0 0
\(753\) 854082. 1.50629
\(754\) 0 0
\(755\) 1.98729e6 3.48632
\(756\) −23328.0 −0.0408163
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 743112. 1.27981
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −589824. −1.00000
\(769\) −439678. −0.743502 −0.371751 − 0.928333i \(-0.621242\pi\)
−0.371751 + 0.928333i \(0.621242\pi\)
\(770\) −52256.0 −0.0881363
\(771\) 0 0
\(772\) 153632. 0.257779
\(773\) −136658. −0.228705 −0.114353 − 0.993440i \(-0.536479\pi\)
−0.114353 + 0.993440i \(0.536479\pi\)
\(774\) 0 0
\(775\) −712698. −1.18659
\(776\) −1.10605e6 −1.83675
\(777\) 0 0
\(778\) −1.06617e6 −1.76143
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 596322. 0.972652
\(784\) −613632. −0.998334
\(785\) 0 0
\(786\) −1.01441e6 −1.64198
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 639712. 1.03022
\(789\) 0 0
\(790\) 1.67771e6 2.68821
\(791\) 0 0
\(792\) −736128. −1.17355
\(793\) 0 0
\(794\) 0 0
\(795\) 1.33225e6 2.10791
\(796\) −614368. −0.969622
\(797\) −404018. −0.636039 −0.318020 − 0.948084i \(-0.603018\pi\)
−0.318020 + 0.948084i \(0.603018\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.52678e6 −2.38560
\(801\) 0 0
\(802\) 0 0
\(803\) −1.15844e6 −1.79656
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 361602. 0.555244
\(808\) 1.00467e6 1.53887
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) −1.20722e6 −1.84000
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −26176.0 −0.0397001
\(813\) 1.29166e6 1.95419
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −723848. −1.08178
\(819\) 0 0
\(820\) 0 0
\(821\) 899182. 1.33402 0.667008 − 0.745050i \(-0.267575\pi\)
0.667008 + 0.745050i \(0.267575\pi\)
\(822\) 0 0
\(823\) −1.13376e6 −1.67387 −0.836933 − 0.547305i \(-0.815654\pi\)
−0.836933 + 0.547305i \(0.815654\pi\)
\(824\) 1.35155e6 1.99057
\(825\) −1.90550e6 −2.79963
\(826\) −54896.0 −0.0804601
\(827\) −1.02226e6 −1.49468 −0.747342 − 0.664439i \(-0.768670\pi\)
−0.747342 + 0.664439i \(0.768670\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 768752. 1.11591
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 348462. 0.497399
\(838\) 727112. 1.03541
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 52992.0 0.0751020
\(841\) −38157.0 −0.0539489
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.31381e6 1.84000
\(846\) 0 0
\(847\) 11046.0 0.0153971
\(848\) −823808. −1.14560
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −992128. −1.35400
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 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 \(\pi\)
\(864\) 746496. 1.00000
\(865\) 2.65613e6 3.54991
\(866\) −295688. −0.394274
\(867\) −751689. −1.00000
\(868\) −15296.0 −0.0203020
\(869\) −1.29476e6 −1.71454
\(870\) −1.35461e6 −1.78968
\(871\) 0 0
\(872\) 0 0
\(873\) 1.39984e6 1.83675
\(874\) 0 0
\(875\) 79672.0 0.104061
\(876\) 1.17475e6 1.53087
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 1.23263e6 1.59899
\(879\) −204318. −0.264441
\(880\) 1.67219e6 2.15934
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 776628. 0.998334
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) −2.84087e6 −3.62714
\(886\) −1.54041e6 −1.96231
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 41284.0 0.0522370
\(890\) 0 0
\(891\) 931662. 1.17355
\(892\) −744928. −0.936234
\(893\) 0 0
\(894\) −1.42913e6 −1.78812
\(895\) 514372. 0.642142
\(896\) −32768.0 −0.0408163
\(897\) 0 0
\(898\) 0 0
\(899\) 391004. 0.483795
\(900\) 1.93234e6 2.38560
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 1.55527e6 1.89474
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) −266528. −0.323274
\(909\) −1.27154e6 −1.53887
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −593276. −0.711730
\(914\) 547192. 0.655009
\(915\) 0 0
\(916\) 0 0
\(917\) −56356.0 −0.0670195
\(918\) 0 0
\(919\) −1.25088e6 −1.48110 −0.740549 − 0.672002i \(-0.765435\pi\)
−0.740549 + 0.672002i \(0.765435\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −576568. −0.678248
\(923\) 0 0
\(924\) −40896.0 −0.0479002
\(925\) 0 0
\(926\) 1.46495e6 1.70845
\(927\) −1.71056e6 −1.99057
\(928\) 837632. 0.972652
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) −791568. −0.915213
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) −136888. −0.156918
\(935\) 0 0
\(936\) 0 0
\(937\) 464162. 0.528677 0.264338 − 0.964430i \(-0.414846\pi\)
0.264338 + 0.964430i \(0.414846\pi\)
\(938\) 0 0
\(939\) −764658. −0.867234
\(940\) 0 0
\(941\) 1.58606e6 1.79119 0.895593 − 0.444873i \(-0.146751\pi\)
0.895593 + 0.444873i \(0.146751\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.75667e6 1.97127
\(945\) −67068.0 −0.0751020
\(946\) 0 0
\(947\) 87982.0 0.0981056 0.0490528 − 0.998796i \(-0.484380\pi\)
0.0490528 + 0.998796i \(0.484380\pi\)
\(948\) 1.31299e6 1.46098
\(949\) 0 0
\(950\) 0 0
\(951\) −804798. −0.889869
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 1.04263e6 1.14560
\(955\) 0 0
\(956\) 0 0
\(957\) 1.04540e6 1.14146
\(958\) 0 0
\(959\) 0 0
\(960\) −1.69574e6 −1.84000
\(961\) −695037. −0.752595
\(962\) 0 0
\(963\) 1.25566e6 1.35400
\(964\) 476192. 0.512422
\(965\) 441692. 0.474313
\(966\) 0 0
\(967\) 1.77792e6 1.90134 0.950670 − 0.310204i \(-0.100397\pi\)
0.950670 + 0.310204i \(0.100397\pi\)
\(968\) −353472. −0.377228
\(969\) 0 0
\(970\) −3.17989e6 −3.37962
\(971\) −1.83922e6 −1.95072 −0.975360 − 0.220621i \(-0.929192\pi\)
−0.975360 + 0.220621i \(0.929192\pi\)
\(972\) −944784. −1.00000
\(973\) 0 0
\(974\) −1.86625e6 −1.96721
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.76419e6 −1.83693
\(981\) 0 0
\(982\) −1.04505e6 −1.08371
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 1.83917e6 1.89561
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) −2.11637e6 −2.15934
\(991\) −54238.0 −0.0552276 −0.0276138 − 0.999619i \(-0.508791\pi\)
−0.0276138 + 0.999619i \(0.508791\pi\)
\(992\) 489472. 0.497399
\(993\) 0 0
\(994\) 0 0
\(995\) −1.76631e6 −1.78410
\(996\) 601632. 0.606474
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 24.5.h.a.5.1 ✓ 1
3.2 odd 2 24.5.h.b.5.1 yes 1
4.3 odd 2 96.5.h.b.17.1 1
8.3 odd 2 96.5.h.a.17.1 1
8.5 even 2 24.5.h.b.5.1 yes 1
12.11 even 2 96.5.h.a.17.1 1
24.5 odd 2 CM 24.5.h.a.5.1 ✓ 1
24.11 even 2 96.5.h.b.17.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.5.h.a.5.1 ✓ 1 1.1 even 1 trivial
24.5.h.a.5.1 ✓ 1 24.5 odd 2 CM
24.5.h.b.5.1 yes 1 3.2 odd 2
24.5.h.b.5.1 yes 1 8.5 even 2
96.5.h.a.17.1 1 8.3 odd 2
96.5.h.a.17.1 1 12.11 even 2
96.5.h.b.17.1 1 4.3 odd 2
96.5.h.b.17.1 1 24.11 even 2