Properties

Label 68.5.d.a.67.1
Level $68$
Weight $5$
Character 68.67
Self dual yes
Analytic conductor $7.029$
Analytic rank $0$
Dimension $1$
CM discriminant -68
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] = [68,5,Mod(67,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("68.67");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 68.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.02915748970\)
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 67.1
Character \(\chi\) \(=\) 68.67

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -16.0000 q^{3} +16.0000 q^{4} -64.0000 q^{6} +64.0000 q^{7} +64.0000 q^{8} +175.000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -16.0000 q^{3} +16.0000 q^{4} -64.0000 q^{6} +64.0000 q^{7} +64.0000 q^{8} +175.000 q^{9} +208.000 q^{11} -256.000 q^{12} -274.000 q^{13} +256.000 q^{14} +256.000 q^{16} +289.000 q^{17} +700.000 q^{18} -1024.00 q^{21} +832.000 q^{22} -608.000 q^{23} -1024.00 q^{24} +625.000 q^{25} -1096.00 q^{26} -1504.00 q^{27} +1024.00 q^{28} +256.000 q^{31} +1024.00 q^{32} -3328.00 q^{33} +1156.00 q^{34} +2800.00 q^{36} +4384.00 q^{39} -4096.00 q^{42} +3328.00 q^{44} -2432.00 q^{46} -4096.00 q^{48} +1695.00 q^{49} +2500.00 q^{50} -4624.00 q^{51} -4384.00 q^{52} -4174.00 q^{53} -6016.00 q^{54} +4096.00 q^{56} +1024.00 q^{62} +11200.0 q^{63} +4096.00 q^{64} -13312.0 q^{66} +4624.00 q^{68} +9728.00 q^{69} -7904.00 q^{71} +11200.0 q^{72} -10000.0 q^{75} +13312.0 q^{77} +17536.0 q^{78} +2656.00 q^{79} +9889.00 q^{81} -16384.0 q^{84} +13312.0 q^{88} +542.000 q^{89} -17536.0 q^{91} -9728.00 q^{92} -4096.00 q^{93} -16384.0 q^{96} +6780.00 q^{98} +36400.0 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(-1\) \(-1\)

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
\(3\) −16.0000 −1.77778 −0.888889 0.458123i \(-0.848522\pi\)
−0.888889 + 0.458123i \(0.848522\pi\)
\(4\) 16.0000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −64.0000 −1.77778
\(7\) 64.0000 1.30612 0.653061 0.757305i \(-0.273484\pi\)
0.653061 + 0.757305i \(0.273484\pi\)
\(8\) 64.0000 1.00000
\(9\) 175.000 2.16049
\(10\) 0 0
\(11\) 208.000 1.71901 0.859504 0.511129i \(-0.170772\pi\)
0.859504 + 0.511129i \(0.170772\pi\)
\(12\) −256.000 −1.77778
\(13\) −274.000 −1.62130 −0.810651 0.585530i \(-0.800887\pi\)
−0.810651 + 0.585530i \(0.800887\pi\)
\(14\) 256.000 1.30612
\(15\) 0 0
\(16\) 256.000 1.00000
\(17\) 289.000 1.00000
\(18\) 700.000 2.16049
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −1024.00 −2.32200
\(22\) 832.000 1.71901
\(23\) −608.000 −1.14934 −0.574669 0.818386i \(-0.694869\pi\)
−0.574669 + 0.818386i \(0.694869\pi\)
\(24\) −1024.00 −1.77778
\(25\) 625.000 1.00000
\(26\) −1096.00 −1.62130
\(27\) −1504.00 −2.06310
\(28\) 1024.00 1.30612
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 256.000 0.266389 0.133195 0.991090i \(-0.457476\pi\)
0.133195 + 0.991090i \(0.457476\pi\)
\(32\) 1024.00 1.00000
\(33\) −3328.00 −3.05601
\(34\) 1156.00 1.00000
\(35\) 0 0
\(36\) 2800.00 2.16049
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 4384.00 2.88231
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) −4096.00 −2.32200
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 3328.00 1.71901
\(45\) 0 0
\(46\) −2432.00 −1.14934
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −4096.00 −1.77778
\(49\) 1695.00 0.705956
\(50\) 2500.00 1.00000
\(51\) −4624.00 −1.77778
\(52\) −4384.00 −1.62130
\(53\) −4174.00 −1.48594 −0.742969 0.669326i \(-0.766583\pi\)
−0.742969 + 0.669326i \(0.766583\pi\)
\(54\) −6016.00 −2.06310
\(55\) 0 0
\(56\) 4096.00 1.30612
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 1024.00 0.266389
\(63\) 11200.0 2.82187
\(64\) 4096.00 1.00000
\(65\) 0 0
\(66\) −13312.0 −3.05601
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 4624.00 1.00000
\(69\) 9728.00 2.04327
\(70\) 0 0
\(71\) −7904.00 −1.56794 −0.783971 0.620797i \(-0.786809\pi\)
−0.783971 + 0.620797i \(0.786809\pi\)
\(72\) 11200.0 2.16049
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −10000.0 −1.77778
\(76\) 0 0
\(77\) 13312.0 2.24524
\(78\) 17536.0 2.88231
\(79\) 2656.00 0.425573 0.212786 0.977099i \(-0.431746\pi\)
0.212786 + 0.977099i \(0.431746\pi\)
\(80\) 0 0
\(81\) 9889.00 1.50724
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −16384.0 −2.32200
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 13312.0 1.71901
\(89\) 542.000 0.0684257 0.0342129 0.999415i \(-0.489108\pi\)
0.0342129 + 0.999415i \(0.489108\pi\)
\(90\) 0 0
\(91\) −17536.0 −2.11762
\(92\) −9728.00 −1.14934
\(93\) −4096.00 −0.473581
\(94\) 0 0
\(95\) 0 0
\(96\) −16384.0 −1.77778
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 6780.00 0.705956
\(99\) 36400.0 3.71391
\(100\) 10000.0 1.00000
\(101\) −9586.00 −0.939712 −0.469856 0.882743i \(-0.655694\pi\)
−0.469856 + 0.882743i \(0.655694\pi\)
\(102\) −18496.0 −1.77778
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −17536.0 −1.62130
\(105\) 0 0
\(106\) −16696.0 −1.48594
\(107\) −9776.00 −0.853874 −0.426937 0.904281i \(-0.640407\pi\)
−0.426937 + 0.904281i \(0.640407\pi\)
\(108\) −24064.0 −2.06310
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 16384.0 1.30612
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −47950.0 −3.50281
\(118\) 0 0
\(119\) 18496.0 1.30612
\(120\) 0 0
\(121\) 28623.0 1.95499
\(122\) 0 0
\(123\) 0 0
\(124\) 4096.00 0.266389
\(125\) 0 0
\(126\) 44800.0 2.82187
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 16384.0 1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) 32656.0 1.90292 0.951460 0.307773i \(-0.0995838\pi\)
0.951460 + 0.307773i \(0.0995838\pi\)
\(132\) −53248.0 −3.05601
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 18496.0 1.00000
\(137\) −36514.0 −1.94544 −0.972721 0.231978i \(-0.925480\pi\)
−0.972721 + 0.231978i \(0.925480\pi\)
\(138\) 38912.0 2.04327
\(139\) −36464.0 −1.88727 −0.943636 0.330984i \(-0.892619\pi\)
−0.943636 + 0.330984i \(0.892619\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −31616.0 −1.56794
\(143\) −56992.0 −2.78703
\(144\) 44800.0 2.16049
\(145\) 0 0
\(146\) 0 0
\(147\) −27120.0 −1.25503
\(148\) 0 0
\(149\) −43726.0 −1.96955 −0.984775 0.173831i \(-0.944385\pi\)
−0.984775 + 0.173831i \(0.944385\pi\)
\(150\) −40000.0 −1.77778
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 50575.0 2.16049
\(154\) 53248.0 2.24524
\(155\) 0 0
\(156\) 70144.0 2.88231
\(157\) 39506.0 1.60274 0.801371 0.598167i \(-0.204104\pi\)
0.801371 + 0.598167i \(0.204104\pi\)
\(158\) 10624.0 0.425573
\(159\) 66784.0 2.64167
\(160\) 0 0
\(161\) −38912.0 −1.50118
\(162\) 39556.0 1.50724
\(163\) −28496.0 −1.07253 −0.536264 0.844050i \(-0.680165\pi\)
−0.536264 + 0.844050i \(0.680165\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19328.0 −0.693033 −0.346517 0.938044i \(-0.612636\pi\)
−0.346517 + 0.938044i \(0.612636\pi\)
\(168\) −65536.0 −2.32200
\(169\) 46515.0 1.62862
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 40000.0 1.30612
\(176\) 53248.0 1.71901
\(177\) 0 0
\(178\) 2168.00 0.0684257
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) −70144.0 −2.11762
\(183\) 0 0
\(184\) −38912.0 −1.14934
\(185\) 0 0
\(186\) −16384.0 −0.473581
\(187\) 60112.0 1.71901
\(188\) 0 0
\(189\) −96256.0 −2.69466
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −65536.0 −1.77778
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 27120.0 0.705956
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 145600. 3.71391
\(199\) 22048.0 0.556754 0.278377 0.960472i \(-0.410204\pi\)
0.278377 + 0.960472i \(0.410204\pi\)
\(200\) 40000.0 1.00000
\(201\) 0 0
\(202\) −38344.0 −0.939712
\(203\) 0 0
\(204\) −73984.0 −1.77778
\(205\) 0 0
\(206\) 0 0
\(207\) −106400. −2.48314
\(208\) −70144.0 −1.62130
\(209\) 0 0
\(210\) 0 0
\(211\) 87376.0 1.96258 0.981290 0.192537i \(-0.0616715\pi\)
0.981290 + 0.192537i \(0.0616715\pi\)
\(212\) −66784.0 −1.48594
\(213\) 126464. 2.78745
\(214\) −39104.0 −0.853874
\(215\) 0 0
\(216\) −96256.0 −2.06310
\(217\) 16384.0 0.347937
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −79186.0 −1.62130
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 65536.0 1.30612
\(225\) 109375. 2.16049
\(226\) 0 0
\(227\) 103024. 1.99934 0.999670 0.0256849i \(-0.00817665\pi\)
0.999670 + 0.0256849i \(0.00817665\pi\)
\(228\) 0 0
\(229\) 74894.0 1.42816 0.714079 0.700065i \(-0.246846\pi\)
0.714079 + 0.700065i \(0.246846\pi\)
\(230\) 0 0
\(231\) −212992. −3.99153
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) −191800. −3.50281
\(235\) 0 0
\(236\) 0 0
\(237\) −42496.0 −0.756574
\(238\) 73984.0 1.30612
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 114492. 1.95499
\(243\) −36400.0 −0.616437
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 16384.0 0.266389
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 179200. 2.82187
\(253\) −126464. −1.97572
\(254\) 0 0
\(255\) 0 0
\(256\) 65536.0 1.00000
\(257\) −88834.0 −1.34497 −0.672486 0.740110i \(-0.734773\pi\)
−0.672486 + 0.740110i \(0.734773\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 130624. 1.90292
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) −212992. −3.05601
\(265\) 0 0
\(266\) 0 0
\(267\) −8672.00 −0.121646
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 73984.0 1.00000
\(273\) 280576. 3.76466
\(274\) −146056. −1.94544
\(275\) 130000. 1.71901
\(276\) 155648. 2.04327
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −145856. −1.88727
\(279\) 44800.0 0.575532
\(280\) 0 0
\(281\) 118754. 1.50396 0.751979 0.659187i \(-0.229100\pi\)
0.751979 + 0.659187i \(0.229100\pi\)
\(282\) 0 0
\(283\) −159728. −1.99438 −0.997191 0.0749057i \(-0.976134\pi\)
−0.997191 + 0.0749057i \(0.976134\pi\)
\(284\) −126464. −1.56794
\(285\) 0 0
\(286\) −227968. −2.78703
\(287\) 0 0
\(288\) 179200. 2.16049
\(289\) 83521.0 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −73102.0 −0.851518 −0.425759 0.904837i \(-0.639993\pi\)
−0.425759 + 0.904837i \(0.639993\pi\)
\(294\) −108480. −1.25503
\(295\) 0 0
\(296\) 0 0
\(297\) −312832. −3.54649
\(298\) −174904. −1.96955
\(299\) 166592. 1.86342
\(300\) −160000. −1.77778
\(301\) 0 0
\(302\) 0 0
\(303\) 153376. 1.67060
\(304\) 0 0
\(305\) 0 0
\(306\) 202300. 2.16049
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 212992. 2.24524
\(309\) 0 0
\(310\) 0 0
\(311\) −126464. −1.30751 −0.653757 0.756705i \(-0.726808\pi\)
−0.653757 + 0.756705i \(0.726808\pi\)
\(312\) 280576. 2.88231
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 158024. 1.60274
\(315\) 0 0
\(316\) 42496.0 0.425573
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 267136. 2.64167
\(319\) 0 0
\(320\) 0 0
\(321\) 156416. 1.51800
\(322\) −155648. −1.50118
\(323\) 0 0
\(324\) 158224. 1.50724
\(325\) −171250. −1.62130
\(326\) −113984. −1.07253
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −77312.0 −0.693033
\(335\) 0 0
\(336\) −262144. −2.32200
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 186060. 1.62862
\(339\) 0 0
\(340\) 0 0
\(341\) 53248.0 0.457925
\(342\) 0 0
\(343\) −45184.0 −0.384058
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −240656. −1.99865 −0.999327 0.0366737i \(-0.988324\pi\)
−0.999327 + 0.0366737i \(0.988324\pi\)
\(348\) 0 0
\(349\) −236206. −1.93928 −0.969639 0.244541i \(-0.921363\pi\)
−0.969639 + 0.244541i \(0.921363\pi\)
\(350\) 160000. 1.30612
\(351\) 412096. 3.34491
\(352\) 212992. 1.71901
\(353\) −103294. −0.828945 −0.414472 0.910062i \(-0.636034\pi\)
−0.414472 + 0.910062i \(0.636034\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 8672.00 0.0684257
\(357\) −295936. −2.32200
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 130321. 1.00000
\(362\) 0 0
\(363\) −457968. −3.47554
\(364\) −280576. −2.11762
\(365\) 0 0
\(366\) 0 0
\(367\) 269344. 1.99975 0.999874 0.0158877i \(-0.00505741\pi\)
0.999874 + 0.0158877i \(0.00505741\pi\)
\(368\) −155648. −1.14934
\(369\) 0 0
\(370\) 0 0
\(371\) −267136. −1.94082
\(372\) −65536.0 −0.473581
\(373\) 228686. 1.64370 0.821849 0.569706i \(-0.192943\pi\)
0.821849 + 0.569706i \(0.192943\pi\)
\(374\) 240448. 1.71901
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) −385024. −2.69466
\(379\) −73424.0 −0.511163 −0.255582 0.966787i \(-0.582267\pi\)
−0.255582 + 0.966787i \(0.582267\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −262144. −1.77778
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −79858.0 −0.527739 −0.263870 0.964558i \(-0.584999\pi\)
−0.263870 + 0.964558i \(0.584999\pi\)
\(390\) 0 0
\(391\) −175712. −1.14934
\(392\) 108480. 0.705956
\(393\) −522496. −3.38297
\(394\) 0 0
\(395\) 0 0
\(396\) 582400. 3.71391
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 88192.0 0.556754
\(399\) 0 0
\(400\) 160000. 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −70144.0 −0.431897
\(404\) −153376. −0.939712
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −295936. −1.77778
\(409\) −292126. −1.74632 −0.873160 0.487435i \(-0.837933\pi\)
−0.873160 + 0.487435i \(0.837933\pi\)
\(410\) 0 0
\(411\) 584224. 3.45856
\(412\) 0 0
\(413\) 0 0
\(414\) −425600. −2.48314
\(415\) 0 0
\(416\) −280576. −1.62130
\(417\) 583424. 3.35515
\(418\) 0 0
\(419\) 138928. 0.791337 0.395669 0.918393i \(-0.370513\pi\)
0.395669 + 0.918393i \(0.370513\pi\)
\(420\) 0 0
\(421\) 324494. 1.83081 0.915403 0.402538i \(-0.131872\pi\)
0.915403 + 0.402538i \(0.131872\pi\)
\(422\) 349504. 1.96258
\(423\) 0 0
\(424\) −267136. −1.48594
\(425\) 180625. 1.00000
\(426\) 505856. 2.78745
\(427\) 0 0
\(428\) −156416. −0.853874
\(429\) 911872. 4.95472
\(430\) 0 0
\(431\) −62624.0 −0.337121 −0.168561 0.985691i \(-0.553912\pi\)
−0.168561 + 0.985691i \(0.553912\pi\)
\(432\) −385024. −2.06310
\(433\) −374722. −1.99863 −0.999317 0.0369452i \(-0.988237\pi\)
−0.999317 + 0.0369452i \(0.988237\pi\)
\(434\) 65536.0 0.347937
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 204256. 1.05985 0.529927 0.848043i \(-0.322219\pi\)
0.529927 + 0.848043i \(0.322219\pi\)
\(440\) 0 0
\(441\) 296625. 1.52521
\(442\) −316744. −1.62130
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 699616. 3.50142
\(448\) 262144. 1.30612
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 437500. 2.16049
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 412096. 1.99934
\(455\) 0 0
\(456\) 0 0
\(457\) 147806. 0.707717 0.353859 0.935299i \(-0.384869\pi\)
0.353859 + 0.935299i \(0.384869\pi\)
\(458\) 299576. 1.42816
\(459\) −434656. −2.06310
\(460\) 0 0
\(461\) 180242. 0.848114 0.424057 0.905636i \(-0.360606\pi\)
0.424057 + 0.905636i \(0.360606\pi\)
\(462\) −851968. −3.99153
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −767200. −3.50281
\(469\) 0 0
\(470\) 0 0
\(471\) −632096. −2.84932
\(472\) 0 0
\(473\) 0 0
\(474\) −169984. −0.756574
\(475\) 0 0
\(476\) 295936. 1.30612
\(477\) −730450. −3.21036
\(478\) 0 0
\(479\) −422432. −1.84114 −0.920568 0.390583i \(-0.872274\pi\)
−0.920568 + 0.390583i \(0.872274\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 622592. 2.66876
\(484\) 457968. 1.95499
\(485\) 0 0
\(486\) −145600. −0.616437
\(487\) −473888. −1.99810 −0.999051 0.0435486i \(-0.986134\pi\)
−0.999051 + 0.0435486i \(0.986134\pi\)
\(488\) 0 0
\(489\) 455936. 1.90672
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 65536.0 0.266389
\(497\) −505856. −2.04793
\(498\) 0 0
\(499\) −140144. −0.562825 −0.281413 0.959587i \(-0.590803\pi\)
−0.281413 + 0.959587i \(0.590803\pi\)
\(500\) 0 0
\(501\) 309248. 1.23206
\(502\) 0 0
\(503\) 488032. 1.92891 0.964456 0.264244i \(-0.0851225\pi\)
0.964456 + 0.264244i \(0.0851225\pi\)
\(504\) 716800. 2.82187
\(505\) 0 0
\(506\) −505856. −1.97572
\(507\) −744240. −2.89532
\(508\) 0 0
\(509\) 38354.0 0.148039 0.0740193 0.997257i \(-0.476417\pi\)
0.0740193 + 0.997257i \(0.476417\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 262144. 1.00000
\(513\) 0 0
\(514\) −355336. −1.34497
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 522496. 1.90292
\(525\) −640000. −2.32200
\(526\) 0 0
\(527\) 73984.0 0.266389
\(528\) −851968. −3.05601
\(529\) 89823.0 0.320979
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) −34688.0 −0.121646
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 352560. 1.21354
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 295936. 1.00000
\(545\) 0 0
\(546\) 1.12230e6 3.76466
\(547\) 329104. 1.09991 0.549957 0.835193i \(-0.314644\pi\)
0.549957 + 0.835193i \(0.314644\pi\)
\(548\) −584224. −1.94544
\(549\) 0 0
\(550\) 520000. 1.71901
\(551\) 0 0
\(552\) 622592. 2.04327
\(553\) 169984. 0.555850
\(554\) 0 0
\(555\) 0 0
\(556\) −583424. −1.88727
\(557\) 32366.0 0.104323 0.0521613 0.998639i \(-0.483389\pi\)
0.0521613 + 0.998639i \(0.483389\pi\)
\(558\) 179200. 0.575532
\(559\) 0 0
\(560\) 0 0
\(561\) −961792. −3.05601
\(562\) 475016. 1.50396
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −638912. −1.99438
\(567\) 632896. 1.96864
\(568\) −505856. −1.56794
\(569\) −331678. −1.02445 −0.512227 0.858850i \(-0.671179\pi\)
−0.512227 + 0.858850i \(0.671179\pi\)
\(570\) 0 0
\(571\) 650416. 1.99489 0.997445 0.0714371i \(-0.0227585\pi\)
0.997445 + 0.0714371i \(0.0227585\pi\)
\(572\) −911872. −2.78703
\(573\) 0 0
\(574\) 0 0
\(575\) −380000. −1.14934
\(576\) 716800. 2.16049
\(577\) −573442. −1.72242 −0.861208 0.508253i \(-0.830291\pi\)
−0.861208 + 0.508253i \(0.830291\pi\)
\(578\) 334084. 1.00000
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −868192. −2.55434
\(584\) 0 0
\(585\) 0 0
\(586\) −292408. −0.851518
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −433920. −1.25503
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 546626. 1.55446 0.777232 0.629214i \(-0.216623\pi\)
0.777232 + 0.629214i \(0.216623\pi\)
\(594\) −1.25133e6 −3.54649
\(595\) 0 0
\(596\) −699616. −1.96955
\(597\) −352768. −0.989784
\(598\) 666368. 1.86342
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −640000. −1.77778
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 613504. 1.67060
\(607\) −38336.0 −0.104047 −0.0520235 0.998646i \(-0.516567\pi\)
−0.0520235 + 0.998646i \(0.516567\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 809200. 2.16049
\(613\) 741746. 1.97394 0.986971 0.160900i \(-0.0514397\pi\)
0.986971 + 0.160900i \(0.0514397\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 851968. 2.24524
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 594928. 1.55268 0.776342 0.630312i \(-0.217073\pi\)
0.776342 + 0.630312i \(0.217073\pi\)
\(620\) 0 0
\(621\) 914432. 2.37120
\(622\) −505856. −1.30751
\(623\) 34688.0 0.0893723
\(624\) 1.12230e6 2.88231
\(625\) 390625. 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 632096. 1.60274
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 169984. 0.425573
\(633\) −1.39802e6 −3.48903
\(634\) 0 0
\(635\) 0 0
\(636\) 1.06854e6 2.64167
\(637\) −464430. −1.14457
\(638\) 0 0
\(639\) −1.38320e6 −3.38753
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 625664. 1.51800
\(643\) 614704. 1.48677 0.743386 0.668863i \(-0.233219\pi\)
0.743386 + 0.668863i \(0.233219\pi\)
\(644\) −622592. −1.50118
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 632896. 1.50724
\(649\) 0 0
\(650\) −685000. −1.62130
\(651\) −262144. −0.618554
\(652\) −455936. −1.07253
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 394034. 0.901843 0.450921 0.892564i \(-0.351095\pi\)
0.450921 + 0.892564i \(0.351095\pi\)
\(662\) 0 0
\(663\) 1.26698e6 2.88231
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −309248. −0.693033
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) −1.04858e6 −2.32200
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −940000. −2.06310
\(676\) 744240. 1.62862
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −1.64838e6 −3.55438
\(682\) 212992. 0.457925
\(683\) 875824. 1.87748 0.938740 0.344626i \(-0.111994\pi\)
0.938740 + 0.344626i \(0.111994\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −180736. −0.384058
\(687\) −1.19830e6 −2.53895
\(688\) 0 0
\(689\) 1.14368e6 2.40915
\(690\) 0 0
\(691\) −646064. −1.35307 −0.676534 0.736412i \(-0.736519\pi\)
−0.676534 + 0.736412i \(0.736519\pi\)
\(692\) 0 0
\(693\) 2.32960e6 4.85082
\(694\) −962624. −1.99865
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −944824. −1.93928
\(699\) 0 0
\(700\) 640000. 1.30612
\(701\) 845102. 1.71978 0.859890 0.510479i \(-0.170532\pi\)
0.859890 + 0.510479i \(0.170532\pi\)
\(702\) 1.64838e6 3.34491
\(703\) 0 0
\(704\) 851968. 1.71901
\(705\) 0 0
\(706\) −413176. −0.828945
\(707\) −613504. −1.22738
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 464800. 0.919447
\(712\) 34688.0 0.0684257
\(713\) −155648. −0.306171
\(714\) −1.18374e6 −2.32200
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −232544. −0.449829 −0.224914 0.974379i \(-0.572210\pi\)
−0.224914 + 0.974379i \(0.572210\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 521284. 1.00000
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) −1.83187e6 −3.47554
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) −1.12230e6 −2.11762
\(729\) −218609. −0.411351
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −110254. −0.205204 −0.102602 0.994722i \(-0.532717\pi\)
−0.102602 + 0.994722i \(0.532717\pi\)
\(734\) 1.07738e6 1.99975
\(735\) 0 0
\(736\) −622592. −1.14934
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.06854e6 −1.94082
\(743\) 891904. 1.61562 0.807812 0.589440i \(-0.200651\pi\)
0.807812 + 0.589440i \(0.200651\pi\)
\(744\) −262144. −0.473581
\(745\) 0 0
\(746\) 914744. 1.64370
\(747\) 0 0
\(748\) 961792. 1.71901
\(749\) −625664. −1.11526
\(750\) 0 0
\(751\) −846752. −1.50133 −0.750665 0.660683i \(-0.770267\pi\)
−0.750665 + 0.660683i \(0.770267\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −1.54010e6 −2.69466
\(757\) −705202. −1.23061 −0.615307 0.788288i \(-0.710968\pi\)
−0.615307 + 0.788288i \(0.710968\pi\)
\(758\) −293696. −0.511163
\(759\) 2.02342e6 3.51239
\(760\) 0 0
\(761\) 320414. 0.553276 0.276638 0.960974i \(-0.410780\pi\)
0.276638 + 0.960974i \(0.410780\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.04858e6 −1.77778
\(769\) 1.07929e6 1.82510 0.912551 0.408963i \(-0.134110\pi\)
0.912551 + 0.408963i \(0.134110\pi\)
\(770\) 0 0
\(771\) 1.42134e6 2.39106
\(772\) 0 0
\(773\) 264206. 0.442164 0.221082 0.975255i \(-0.429041\pi\)
0.221082 + 0.975255i \(0.429041\pi\)
\(774\) 0 0
\(775\) 160000. 0.266389
\(776\) 0 0
\(777\) 0 0
\(778\) −319432. −0.527739
\(779\) 0 0
\(780\) 0 0
\(781\) −1.64403e6 −2.69531
\(782\) −702848. −1.14934
\(783\) 0 0
\(784\) 433920. 0.705956
\(785\) 0 0
\(786\) −2.08998e6 −3.38297
\(787\) 757264. 1.22264 0.611319 0.791384i \(-0.290639\pi\)
0.611319 + 0.791384i \(0.290639\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 2.32960e6 3.71391
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 352768. 0.556754
\(797\) 477266. 0.751353 0.375676 0.926751i \(-0.377410\pi\)
0.375676 + 0.926751i \(0.377410\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 640000. 1.00000
\(801\) 94850.0 0.147833
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −280576. −0.431897
\(807\) 0 0
\(808\) −613504. −0.939712
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 1.29746e6 1.97265 0.986327 0.164800i \(-0.0526978\pi\)
0.986327 + 0.164800i \(0.0526978\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −1.18374e6 −1.77778
\(817\) 0 0
\(818\) −1.16850e6 −1.74632
\(819\) −3.06880e6 −4.57510
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 2.33690e6 3.45856
\(823\) −1.33002e6 −1.96362 −0.981809 0.189869i \(-0.939194\pi\)
−0.981809 + 0.189869i \(0.939194\pi\)
\(824\) 0 0
\(825\) −2.08000e6 −3.05601
\(826\) 0 0
\(827\) −877808. −1.28348 −0.641739 0.766923i \(-0.721787\pi\)
−0.641739 + 0.766923i \(0.721787\pi\)
\(828\) −1.70240e6 −2.48314
\(829\) −280366. −0.407959 −0.203979 0.978975i \(-0.565388\pi\)
−0.203979 + 0.978975i \(0.565388\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.12230e6 −1.62130
\(833\) 489855. 0.705956
\(834\) 2.33370e6 3.35515
\(835\) 0 0
\(836\) 0 0
\(837\) −385024. −0.549588
\(838\) 555712. 0.791337
\(839\) −1.39270e6 −1.97849 −0.989247 0.146252i \(-0.953279\pi\)
−0.989247 + 0.146252i \(0.953279\pi\)
\(840\) 0 0
\(841\) 707281. 1.00000
\(842\) 1.29798e6 1.83081
\(843\) −1.90006e6 −2.67370
\(844\) 1.39802e6 1.96258
\(845\) 0 0
\(846\) 0 0
\(847\) 1.83187e6 2.55346
\(848\) −1.06854e6 −1.48594
\(849\) 2.55565e6 3.54557
\(850\) 722500. 1.00000
\(851\) 0 0
\(852\) 2.02342e6 2.78745
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −625664. −0.853874
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 3.64749e6 4.95472
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −250496. −0.337121
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) −1.54010e6 −2.06310
\(865\) 0 0
\(866\) −1.49889e6 −1.99863
\(867\) −1.33634e6 −1.77778
\(868\) 262144. 0.347937
\(869\) 552448. 0.731563
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 817024. 1.05985
\(879\) 1.16963e6 1.51381
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 1.18650e6 1.52521
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −1.26698e6 −1.62130
\(885\) 0 0
\(886\) 0 0
\(887\) 798304. 1.01466 0.507331 0.861752i \(-0.330632\pi\)
0.507331 + 0.861752i \(0.330632\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.05691e6 2.59096
\(892\) 0 0
\(893\) 0 0
\(894\) 2.79846e6 3.50142
\(895\) 0 0
\(896\) 1.04858e6 1.30612
\(897\) −2.66547e6 −3.31275
\(898\) 0 0
\(899\) 0 0
\(900\) 1.75000e6 2.16049
\(901\) −1.20629e6 −1.48594
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.15525e6 −1.40430 −0.702151 0.712028i \(-0.747777\pi\)
−0.702151 + 0.712028i \(0.747777\pi\)
\(908\) 1.64838e6 1.99934
\(909\) −1.67755e6 −2.03024
\(910\) 0 0
\(911\) 58816.0 0.0708694 0.0354347 0.999372i \(-0.488718\pi\)
0.0354347 + 0.999372i \(0.488718\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 591224. 0.707717
\(915\) 0 0
\(916\) 1.19830e6 1.42816
\(917\) 2.08998e6 2.48545
\(918\) −1.73862e6 −2.06310
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 720968. 0.848114
\(923\) 2.16570e6 2.54211
\(924\) −3.40787e6 −3.99153
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 2.02342e6 2.32447
\(934\) 0 0
\(935\) 0 0
\(936\) −3.06880e6 −3.50281
\(937\) −163294. −0.185991 −0.0929953 0.995667i \(-0.529644\pi\)
−0.0929953 + 0.995667i \(0.529644\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) −2.52838e6 −2.84932
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.61243e6 1.79797 0.898983 0.437984i \(-0.144307\pi\)
0.898983 + 0.437984i \(0.144307\pi\)
\(948\) −679936. −0.756574
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 1.18374e6 1.30612
\(953\) −1.09731e6 −1.20822 −0.604109 0.796902i \(-0.706471\pi\)
−0.604109 + 0.796902i \(0.706471\pi\)
\(954\) −2.92180e6 −3.21036
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) −1.68973e6 −1.84114
\(959\) −2.33690e6 −2.54099
\(960\) 0 0
\(961\) −857985. −0.929037
\(962\) 0 0
\(963\) −1.71080e6 −1.84479
\(964\) 0 0
\(965\) 0 0
\(966\) 2.49037e6 2.66876
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 1.83187e6 1.95499
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −582400. −0.616437
\(973\) −2.33370e6 −2.46501
\(974\) −1.89555e6 −1.99810
\(975\) 2.74000e6 2.88231
\(976\) 0 0
\(977\) −10174.0 −0.0106587 −0.00532933 0.999986i \(-0.501696\pi\)
−0.00532933 + 0.999986i \(0.501696\pi\)
\(978\) 1.82374e6 1.90672
\(979\) 112736. 0.117624
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.76118e6 1.82263 0.911313 0.411714i \(-0.135070\pi\)
0.911313 + 0.411714i \(0.135070\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.89718e6 −1.93180 −0.965900 0.258916i \(-0.916635\pi\)
−0.965900 + 0.258916i \(0.916635\pi\)
\(992\) 262144. 0.266389
\(993\) 0 0
\(994\) −2.02342e6 −2.04793
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −560576. −0.562825
\(999\) 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 68.5.d.a.67.1 1
4.3 odd 2 68.5.d.b.67.1 yes 1
17.16 even 2 68.5.d.b.67.1 yes 1
68.67 odd 2 CM 68.5.d.a.67.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
68.5.d.a.67.1 1 1.1 even 1 trivial
68.5.d.a.67.1 1 68.67 odd 2 CM
68.5.d.b.67.1 yes 1 4.3 odd 2
68.5.d.b.67.1 yes 1 17.16 even 2