Properties

Label 72.8.d.a.37.2
Level $72$
Weight $8$
Character 72.37
Analytic conductor $22.492$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,8,Mod(37,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.37");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 72.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: no
Analytic conductor: \(22.4917218349\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 37.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 72.37
Dual form 72.8.d.a.37.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+11.3137i q^{2} -128.000 q^{4} +557.200i q^{5} -754.000 q^{7} -1448.15i q^{8} +O(q^{10})\) \(q+11.3137i q^{2} -128.000 q^{4} +557.200i q^{5} -754.000 q^{7} -1448.15i q^{8} -6304.00 q^{10} +6726.00i q^{11} -8530.54i q^{14} +16384.0 q^{16} -71321.6i q^{20} -76096.0 q^{22} -232347. q^{25} +96512.0 q^{28} -251761. i q^{29} +331370. q^{31} +185364. i q^{32} -420129. i q^{35} +806912. q^{40} -860928. i q^{44} -255027. q^{49} -2.62871e6i q^{50} +239554. i q^{53} -3.74773e6 q^{55} +1.09191e6i q^{56} +2.84835e6 q^{58} +1.48050e6i q^{59} +3.74902e6i q^{62} -2.09715e6 q^{64} +4.75322e6 q^{70} -2.81936e6 q^{73} -5.07140e6i q^{77} -7.56127e6 q^{79} +9.12917e6i q^{80} +8.36317e6i q^{83} +9.74029e6 q^{88} -1.17452e7 q^{97} -2.88530e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 256 q^{4} - 1508 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 256 q^{4} - 1508 q^{7} - 12608 q^{10} + 32768 q^{16} - 152192 q^{22} - 464694 q^{25} + 193024 q^{28} + 662740 q^{31} + 1613824 q^{40} - 510054 q^{49} - 7495456 q^{55} + 5696704 q^{58} - 4194304 q^{64} + 9506432 q^{70} - 5638724 q^{73} - 15122540 q^{79} + 19480576 q^{88} - 23490428 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 11.3137i 1.00000i
\(3\) 0 0
\(4\) −128.000 −1.00000
\(5\) 557.200i 1.99350i 0.0805581 + 0.996750i \(0.474330\pi\)
−0.0805581 + 0.996750i \(0.525670\pi\)
\(6\) 0 0
\(7\) −754.000 −0.830861 −0.415430 0.909625i \(-0.636369\pi\)
−0.415430 + 0.909625i \(0.636369\pi\)
\(8\) − 1448.15i − 1.00000i
\(9\) 0 0
\(10\) −6304.00 −1.99350
\(11\) 6726.00i 1.52364i 0.647789 + 0.761820i \(0.275694\pi\)
−0.647789 + 0.761820i \(0.724306\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) − 8530.54i − 0.830861i
\(15\) 0 0
\(16\) 16384.0 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) − 71321.6i − 1.99350i
\(21\) 0 0
\(22\) −76096.0 −1.52364
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −232347. −2.97404
\(26\) 0 0
\(27\) 0 0
\(28\) 96512.0 0.830861
\(29\) − 251761.i − 1.91688i −0.285289 0.958442i \(-0.592089\pi\)
0.285289 0.958442i \(-0.407911\pi\)
\(30\) 0 0
\(31\) 331370. 1.99778 0.998889 0.0471234i \(-0.0150054\pi\)
0.998889 + 0.0471234i \(0.0150054\pi\)
\(32\) 185364.i 1.00000i
\(33\) 0 0
\(34\) 0 0
\(35\) − 420129.i − 1.65632i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 806912. 1.99350
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) − 860928.i − 1.52364i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −255027. −0.309671
\(50\) − 2.62871e6i − 2.97404i
\(51\) 0 0
\(52\) 0 0
\(53\) 239554.i 0.221023i 0.993875 + 0.110511i \(0.0352489\pi\)
−0.993875 + 0.110511i \(0.964751\pi\)
\(54\) 0 0
\(55\) −3.74773e6 −3.03738
\(56\) 1.09191e6i 0.830861i
\(57\) 0 0
\(58\) 2.84835e6 1.91688
\(59\) 1.48050e6i 0.938483i 0.883070 + 0.469242i \(0.155473\pi\)
−0.883070 + 0.469242i \(0.844527\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 3.74902e6i 1.99778i
\(63\) 0 0
\(64\) −2.09715e6 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 4.75322e6 1.65632
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −2.81936e6 −0.848244 −0.424122 0.905605i \(-0.639417\pi\)
−0.424122 + 0.905605i \(0.639417\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 5.07140e6i − 1.26593i
\(78\) 0 0
\(79\) −7.56127e6 −1.72544 −0.862720 0.505682i \(-0.831241\pi\)
−0.862720 + 0.505682i \(0.831241\pi\)
\(80\) 9.12917e6i 1.99350i
\(81\) 0 0
\(82\) 0 0
\(83\) 8.36317e6i 1.60545i 0.596348 + 0.802726i \(0.296618\pi\)
−0.596348 + 0.802726i \(0.703382\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 9.74029e6 1.52364
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.17452e7 −1.30665 −0.653326 0.757077i \(-0.726627\pi\)
−0.653326 + 0.757077i \(0.726627\pi\)
\(98\) − 2.88530e6i − 0.309671i
\(99\) 0 0
\(100\) 2.97404e7 2.97404
\(101\) 7.26269e6i 0.701411i 0.936486 + 0.350705i \(0.114058\pi\)
−0.936486 + 0.350705i \(0.885942\pi\)
\(102\) 0 0
\(103\) 1.81105e7 1.63305 0.816526 0.577308i \(-0.195897\pi\)
0.816526 + 0.577308i \(0.195897\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.71024e6 −0.221023
\(107\) 1.99895e7i 1.57746i 0.614737 + 0.788732i \(0.289262\pi\)
−0.614737 + 0.788732i \(0.710738\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) − 4.24007e7i − 3.03738i
\(111\) 0 0
\(112\) −1.23535e7 −0.830861
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.22254e7i 1.91688i
\(117\) 0 0
\(118\) −1.67500e7 −0.938483
\(119\) 0 0
\(120\) 0 0
\(121\) −2.57519e7 −1.32148
\(122\) 0 0
\(123\) 0 0
\(124\) −4.24154e7 −1.99778
\(125\) − 8.59325e7i − 3.93525i
\(126\) 0 0
\(127\) −1.71035e6 −0.0740919 −0.0370460 0.999314i \(-0.511795\pi\)
−0.0370460 + 0.999314i \(0.511795\pi\)
\(128\) − 2.37266e7i − 1.00000i
\(129\) 0 0
\(130\) 0 0
\(131\) − 2.50054e7i − 0.971815i −0.874010 0.485907i \(-0.838489\pi\)
0.874010 0.485907i \(-0.161511\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 5.37765e7i 1.65632i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 1.40281e8 3.82131
\(146\) − 3.18974e7i − 0.848244i
\(147\) 0 0
\(148\) 0 0
\(149\) − 5.86485e7i − 1.45246i −0.687451 0.726231i \(-0.741270\pi\)
0.687451 0.726231i \(-0.258730\pi\)
\(150\) 0 0
\(151\) −4.56813e7 −1.07974 −0.539870 0.841748i \(-0.681527\pi\)
−0.539870 + 0.841748i \(0.681527\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 5.73764e7 1.26593
\(155\) 1.84639e8i 3.98257i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) − 8.55460e7i − 1.72544i
\(159\) 0 0
\(160\) −1.03285e8 −1.99350
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −9.46184e7 −1.60545
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 6.27485e7 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.26771e7i 0.626662i 0.949644 + 0.313331i \(0.101445\pi\)
−0.949644 + 0.313331i \(0.898555\pi\)
\(174\) 0 0
\(175\) 1.75190e8 2.47101
\(176\) 1.10199e8i 1.52364i
\(177\) 0 0
\(178\) 0 0
\(179\) − 1.31454e7i − 0.171312i −0.996325 0.0856561i \(-0.972701\pi\)
0.996325 0.0856561i \(-0.0272986\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −1.62699e8 −1.62905 −0.814526 0.580127i \(-0.803003\pi\)
−0.814526 + 0.580127i \(0.803003\pi\)
\(194\) − 1.32882e8i − 1.30665i
\(195\) 0 0
\(196\) 3.26435e7 0.309671
\(197\) − 1.12962e8i − 1.05269i −0.850272 0.526343i \(-0.823563\pi\)
0.850272 0.526343i \(-0.176437\pi\)
\(198\) 0 0
\(199\) 1.05238e8 0.946644 0.473322 0.880889i \(-0.343055\pi\)
0.473322 + 0.880889i \(0.343055\pi\)
\(200\) 3.36474e8i 2.97404i
\(201\) 0 0
\(202\) −8.21679e7 −0.701411
\(203\) 1.89828e8i 1.59266i
\(204\) 0 0
\(205\) 0 0
\(206\) 2.04897e8i 1.63305i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) − 3.06629e7i − 0.221023i
\(213\) 0 0
\(214\) −2.26156e8 −1.57746
\(215\) 0 0
\(216\) 0 0
\(217\) −2.49853e8 −1.65988
\(218\) 0 0
\(219\) 0 0
\(220\) 4.79709e8 3.03738
\(221\) 0 0
\(222\) 0 0
\(223\) −2.96802e8 −1.79226 −0.896128 0.443795i \(-0.853632\pi\)
−0.896128 + 0.443795i \(0.853632\pi\)
\(224\) − 1.39764e8i − 0.830861i
\(225\) 0 0
\(226\) 0 0
\(227\) − 2.74776e8i − 1.55915i −0.626310 0.779574i \(-0.715435\pi\)
0.626310 0.779574i \(-0.284565\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.64589e8 −1.91688
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) − 1.89504e8i − 0.938483i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 3.25392e8 1.49743 0.748717 0.662889i \(-0.230670\pi\)
0.748717 + 0.662889i \(0.230670\pi\)
\(242\) − 2.91350e8i − 1.32148i
\(243\) 0 0
\(244\) 0 0
\(245\) − 1.42101e8i − 0.617328i
\(246\) 0 0
\(247\) 0 0
\(248\) − 4.79875e8i − 1.99778i
\(249\) 0 0
\(250\) 9.72215e8 3.93525
\(251\) 4.76503e8i 1.90199i 0.309208 + 0.950995i \(0.399936\pi\)
−0.309208 + 0.950995i \(0.600064\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) − 1.93504e7i − 0.0740919i
\(255\) 0 0
\(256\) 2.68435e8 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 2.82903e8 0.971815
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −1.33479e8 −0.440609
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.04355e8i 1.26657i 0.773918 + 0.633285i \(0.218294\pi\)
−0.773918 + 0.633285i \(0.781706\pi\)
\(270\) 0 0
\(271\) −6.00928e8 −1.83413 −0.917065 0.398739i \(-0.869448\pi\)
−0.917065 + 0.398739i \(0.869448\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 1.56277e9i − 4.53137i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −6.08412e8 −1.65632
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4.10339e8 −1.00000
\(290\) 1.58710e9i 3.82131i
\(291\) 0 0
\(292\) 3.60878e8 0.848244
\(293\) 1.37732e8i 0.319887i 0.987126 + 0.159944i \(0.0511313\pi\)
−0.987126 + 0.159944i \(0.948869\pi\)
\(294\) 0 0
\(295\) −8.24935e8 −1.87087
\(296\) 0 0
\(297\) 0 0
\(298\) 6.63532e8 1.45246
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) − 5.16825e8i − 1.07974i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 6.49140e8i 1.26593i
\(309\) 0 0
\(310\) −2.08896e9 −3.98257
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 4.15736e8 0.766325 0.383162 0.923681i \(-0.374835\pi\)
0.383162 + 0.923681i \(0.374835\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 9.67843e8 1.72544
\(317\) 1.03839e9i 1.83086i 0.402483 + 0.915428i \(0.368147\pi\)
−0.402483 + 0.915428i \(0.631853\pi\)
\(318\) 0 0
\(319\) 1.69335e9 2.92064
\(320\) − 1.16853e9i − 1.99350i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) − 1.07049e9i − 1.60545i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.37289e9 −1.95404 −0.977018 0.213159i \(-0.931625\pi\)
−0.977018 + 0.213159i \(0.931625\pi\)
\(338\) 7.09918e8i 1.00000i
\(339\) 0 0
\(340\) 0 0
\(341\) 2.22879e9i 3.04390i
\(342\) 0 0
\(343\) 8.13242e8 1.08815
\(344\) 0 0
\(345\) 0 0
\(346\) −4.82836e8 −0.626662
\(347\) 3.83177e8i 0.492318i 0.969229 + 0.246159i \(0.0791686\pi\)
−0.969229 + 0.246159i \(0.920831\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 1.98204e9i 2.47101i
\(351\) 0 0
\(352\) −1.24676e9 −1.52364
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 1.48723e8 0.171312
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 8.93872e8 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 1.57095e9i − 1.69097i
\(366\) 0 0
\(367\) 1.18099e9 1.24714 0.623572 0.781766i \(-0.285681\pi\)
0.623572 + 0.781766i \(0.285681\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 1.80623e8i − 0.183639i
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 2.82579e9 2.52364
\(386\) − 1.84073e9i − 1.62905i
\(387\) 0 0
\(388\) 1.50339e9 1.30665
\(389\) 1.81138e8i 0.156022i 0.996952 + 0.0780112i \(0.0248570\pi\)
−0.996952 + 0.0780112i \(0.975143\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 3.69319e8i 0.309671i
\(393\) 0 0
\(394\) 1.27801e9 1.05269
\(395\) − 4.21314e9i − 3.43966i
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 1.19063e9i 0.946644i
\(399\) 0 0
\(400\) −3.80677e9 −2.97404
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) − 9.29624e8i − 0.701411i
\(405\) 0 0
\(406\) −2.14766e9 −1.59266
\(407\) 0 0
\(408\) 0 0
\(409\) 2.72362e9 1.96841 0.984204 0.177038i \(-0.0566517\pi\)
0.984204 + 0.177038i \(0.0566517\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.31814e9 −1.63305
\(413\) − 1.11630e9i − 0.779749i
\(414\) 0 0
\(415\) −4.65996e9 −3.20047
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.74239e8i 0.447780i 0.974614 + 0.223890i \(0.0718756\pi\)
−0.974614 + 0.223890i \(0.928124\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 3.46911e8 0.221023
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) − 2.55866e9i − 1.57746i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −2.27806e9 −1.34852 −0.674261 0.738493i \(-0.735538\pi\)
−0.674261 + 0.738493i \(0.735538\pi\)
\(434\) − 2.82676e9i − 1.65988i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 1.18970e9 0.671136 0.335568 0.942016i \(-0.391072\pi\)
0.335568 + 0.942016i \(0.391072\pi\)
\(440\) 5.42729e9i 3.03738i
\(441\) 0 0
\(442\) 0 0
\(443\) 3.30306e9i 1.80511i 0.430578 + 0.902553i \(0.358310\pi\)
−0.430578 + 0.902553i \(0.641690\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) − 3.35793e9i − 1.79226i
\(447\) 0 0
\(448\) 1.58125e9 0.830861
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 3.10873e9 1.55915
\(455\) 0 0
\(456\) 0 0
\(457\) −4.00635e9 −1.96355 −0.981775 0.190045i \(-0.939137\pi\)
−0.981775 + 0.190045i \(0.939137\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.88431e8i 0.422348i 0.977448 + 0.211174i \(0.0677287\pi\)
−0.977448 + 0.211174i \(0.932271\pi\)
\(462\) 0 0
\(463\) 4.20875e9 1.97069 0.985347 0.170561i \(-0.0545580\pi\)
0.985347 + 0.170561i \(0.0545580\pi\)
\(464\) − 4.12485e9i − 1.91688i
\(465\) 0 0
\(466\) 0 0
\(467\) 4.25926e9i 1.93520i 0.252492 + 0.967599i \(0.418750\pi\)
−0.252492 + 0.967599i \(0.581250\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 2.14399e9 0.938483
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 3.68140e9i 1.49743i
\(483\) 0 0
\(484\) 3.29624e9 1.32148
\(485\) − 6.54443e9i − 2.60481i
\(486\) 0 0
\(487\) −1.59056e9 −0.624022 −0.312011 0.950078i \(-0.601003\pi\)
−0.312011 + 0.950078i \(0.601003\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 1.60769e9 0.617328
\(491\) − 3.00270e9i − 1.14479i −0.819977 0.572397i \(-0.806014\pi\)
0.819977 0.572397i \(-0.193986\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 5.42917e9 1.99778
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 1.09994e10i 3.93525i
\(501\) 0 0
\(502\) −5.39102e9 −1.90199
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −4.04677e9 −1.39826
\(506\) 0 0
\(507\) 0 0
\(508\) 2.18924e8 0.0740919
\(509\) − 2.52951e9i − 0.850205i −0.905145 0.425103i \(-0.860238\pi\)
0.905145 0.425103i \(-0.139762\pi\)
\(510\) 0 0
\(511\) 2.12580e9 0.704773
\(512\) 3.03700e9i 1.00000i
\(513\) 0 0
\(514\) 0 0
\(515\) 1.00912e10i 3.25549i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 3.20069e9i 0.971815i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −3.40483e9 −1.00000
\(530\) − 1.51015e9i − 0.440609i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −1.11382e10 −3.14468
\(536\) 0 0
\(537\) 0 0
\(538\) −4.57475e9 −1.26657
\(539\) − 1.71531e9i − 0.471826i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) − 6.79872e9i − 1.83413i
\(543\) 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
\(550\) 1.76807e10 4.53137
\(551\) 0 0
\(552\) 0 0
\(553\) 5.70120e9 1.43360
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.91442e9i 1.69536i 0.530505 + 0.847682i \(0.322002\pi\)
−0.530505 + 0.847682i \(0.677998\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) − 6.88339e9i − 1.65632i
\(561\) 0 0
\(562\) 0 0
\(563\) 8.22048e9i 1.94141i 0.240268 + 0.970707i \(0.422765\pi\)
−0.240268 + 0.970707i \(0.577235\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −9.55149e8 −0.206993 −0.103496 0.994630i \(-0.533003\pi\)
−0.103496 + 0.994630i \(0.533003\pi\)
\(578\) − 4.64245e9i − 1.00000i
\(579\) 0 0
\(580\) −1.79560e10 −3.82131
\(581\) − 6.30583e9i − 1.33391i
\(582\) 0 0
\(583\) −1.61124e9 −0.336759
\(584\) 4.08287e9i 0.848244i
\(585\) 0 0
\(586\) −1.55826e9 −0.319887
\(587\) − 8.18263e9i − 1.66978i −0.550417 0.834890i \(-0.685531\pi\)
0.550417 0.834890i \(-0.314469\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) − 9.33308e9i − 1.87087i
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.50701e9i 1.45246i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −2.99883e9 −0.563496 −0.281748 0.959489i \(-0.590914\pi\)
−0.281748 + 0.959489i \(0.590914\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 5.84721e9 1.07974
\(605\) − 1.43490e10i − 2.63437i
\(606\) 0 0
\(607\) −1.08013e9 −0.196027 −0.0980134 0.995185i \(-0.531249\pi\)
−0.0980134 + 0.995185i \(0.531249\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −7.34418e9 −1.26593
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) − 2.36338e10i − 3.98257i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.97295e10 4.87088
\(626\) 4.70352e9i 0.766325i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 9.78539e9 1.55051 0.775257 0.631646i \(-0.217620\pi\)
0.775257 + 0.631646i \(0.217620\pi\)
\(632\) 1.09499e10i 1.72544i
\(633\) 0 0
\(634\) −1.17481e10 −1.83086
\(635\) − 9.53005e8i − 0.147702i
\(636\) 0 0
\(637\) 0 0
\(638\) 1.91580e10i 2.92064i
\(639\) 0 0
\(640\) 1.32204e10 1.99350
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −9.95785e9 −1.42991
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.06861e10i 1.50183i 0.660396 + 0.750917i \(0.270388\pi\)
−0.660396 + 0.750917i \(0.729612\pi\)
\(654\) 0 0
\(655\) 1.39330e10 1.93731
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1.40300e10i − 1.90968i −0.297123 0.954839i \(-0.596027\pi\)
0.297123 0.954839i \(-0.403973\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 1.21112e10 1.60545
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.51565e10 −1.91667 −0.958334 0.285651i \(-0.907790\pi\)
−0.958334 + 0.285651i \(0.907790\pi\)
\(674\) − 1.55325e10i − 1.95404i
\(675\) 0 0
\(676\) −8.03181e9 −1.00000
\(677\) − 5.99907e9i − 0.743059i −0.928421 0.371530i \(-0.878833\pi\)
0.928421 0.371530i \(-0.121167\pi\)
\(678\) 0 0
\(679\) 8.85589e9 1.08565
\(680\) 0 0
\(681\) 0 0
\(682\) −2.52159e10 −3.04390
\(683\) 1.14617e10i 1.37650i 0.725474 + 0.688250i \(0.241621\pi\)
−0.725474 + 0.688250i \(0.758379\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 9.20078e9i 1.08815i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) − 5.46266e9i − 0.626662i
\(693\) 0 0
\(694\) −4.33515e9 −0.492318
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −2.24243e10 −2.47101
\(701\) 1.44039e10i 1.57931i 0.613551 + 0.789655i \(0.289740\pi\)
−0.613551 + 0.789655i \(0.710260\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) − 1.41054e10i − 1.52364i
\(705\) 0 0
\(706\) 0 0
\(707\) − 5.47606e9i − 0.582775i
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.68261e9i 0.171312i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −1.36553e10 −1.35684
\(722\) 1.01130e10i 1.00000i
\(723\) 0 0
\(724\) 0 0
\(725\) 5.84959e10i 5.70089i
\(726\) 0 0
\(727\) −5.32034e9 −0.513533 −0.256767 0.966473i \(-0.582657\pi\)
−0.256767 + 0.966473i \(0.582657\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1.77733e10 1.69097
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 1.33614e10i 1.24714i
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.04352e9 0.183639
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 3.26789e10 2.89548
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 1.50721e10i − 1.31065i
\(750\) 0 0
\(751\) 2.22692e10 1.91851 0.959257 0.282533i \(-0.0911748\pi\)
0.959257 + 0.282533i \(0.0911748\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 2.54536e10i − 2.15246i
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.82131e9 0.540911 0.270456 0.962732i \(-0.412826\pi\)
0.270456 + 0.962732i \(0.412826\pi\)
\(770\) 3.19701e10i 2.52364i
\(771\) 0 0
\(772\) 2.08255e10 1.62905
\(773\) − 1.43588e10i − 1.11813i −0.829125 0.559063i \(-0.811161\pi\)
0.829125 0.559063i \(-0.188839\pi\)
\(774\) 0 0
\(775\) −7.69928e10 −5.94148
\(776\) 1.70089e10i 1.30665i
\(777\) 0 0
\(778\) −2.04935e9 −0.156022
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −4.17836e9 −0.309671
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 1.44591e10i 1.05269i
\(789\) 0 0
\(790\) 4.76662e10 3.43966
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −1.34705e10 −0.946644
\(797\) − 1.64157e10i − 1.14856i −0.818657 0.574282i \(-0.805281\pi\)
0.818657 0.574282i \(-0.194719\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) − 4.30687e10i − 2.97404i
\(801\) 0 0
\(802\) 0 0
\(803\) − 1.89630e10i − 1.29242i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 1.05175e10 0.701411
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) − 2.42980e10i − 1.59266i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 3.08143e10i 1.96841i
\(819\) 0 0
\(820\) 0 0
\(821\) 2.00801e10i 1.26638i 0.773995 + 0.633191i \(0.218255\pi\)
−0.773995 + 0.633191i \(0.781745\pi\)
\(822\) 0 0
\(823\) 7.23236e9 0.452252 0.226126 0.974098i \(-0.427394\pi\)
0.226126 + 0.974098i \(0.427394\pi\)
\(824\) − 2.62268e10i − 1.63305i
\(825\) 0 0
\(826\) 1.26295e10 0.779749
\(827\) − 9.58238e9i − 0.589121i −0.955633 0.294560i \(-0.904827\pi\)
0.955633 0.294560i \(-0.0951732\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) − 5.27214e10i − 3.20047i
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −7.62814e9 −0.447780
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −4.61338e10 −2.67444
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.49635e10i 1.99350i
\(846\) 0 0
\(847\) 1.94169e10 1.09797
\(848\) 3.92485e9i 0.221023i
\(849\) 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
\(856\) 2.89480e10 1.57746
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −2.37797e10 −1.24925
\(866\) − 2.57733e10i − 1.34852i
\(867\) 0 0
\(868\) 3.19812e10 1.65988
\(869\) − 5.08571e10i − 2.62895i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 6.47931e10i 3.26965i
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 1.34599e10i 0.671136i
\(879\) 0 0
\(880\) −6.14028e10 −3.03738
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −3.73698e10 −1.80511
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 1.28960e9 0.0615601
\(890\) 0 0
\(891\) 0 0
\(892\) 3.79907e10 1.79226
\(893\) 0 0
\(894\) 0 0
\(895\) 7.32462e9 0.341511
\(896\) 1.78898e10i 0.830861i
\(897\) 0 0
\(898\) 0 0
\(899\) − 8.34261e10i − 3.82951i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 3.51713e10i 1.55915i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −5.62507e10 −2.44613
\(914\) − 4.53267e10i − 1.96355i
\(915\) 0 0
\(916\) 0 0
\(917\) 1.88540e10i 0.807443i
\(918\) 0 0
\(919\) −2.27338e10 −0.966201 −0.483101 0.875565i \(-0.660490\pi\)
−0.483101 + 0.875565i \(0.660490\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.00514e10 −0.422348
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 4.76165e10i 1.97069i
\(927\) 0 0
\(928\) 4.66674e10 1.91688
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) −4.81881e10 −1.93520
\(935\) 0 0
\(936\) 0 0
\(937\) 3.28266e10 1.30358 0.651789 0.758400i \(-0.274019\pi\)
0.651789 + 0.758400i \(0.274019\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.09437e9i 0.0428155i 0.999771 + 0.0214078i \(0.00681482\pi\)
−0.999771 + 0.0214078i \(0.993185\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 2.42565e10i 0.938483i
\(945\) 0 0
\(946\) 0 0
\(947\) 4.98292e10i 1.90660i 0.302031 + 0.953298i \(0.402335\pi\)
−0.302031 + 0.953298i \(0.597665\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 8.22935e10 2.99112
\(962\) 0 0
\(963\) 0 0
\(964\) −4.16502e10 −1.49743
\(965\) − 9.06560e10i − 3.24751i
\(966\) 0 0
\(967\) 4.78857e10 1.70299 0.851496 0.524360i \(-0.175696\pi\)
0.851496 + 0.524360i \(0.175696\pi\)
\(968\) 3.72927e10i 1.32148i
\(969\) 0 0
\(970\) 7.40418e10 2.60481
\(971\) 5.27865e10i 1.85036i 0.379531 + 0.925179i \(0.376085\pi\)
−0.379531 + 0.925179i \(0.623915\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 1.79952e10i − 0.624022i
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.81889e10i 0.617328i
\(981\) 0 0
\(982\) 3.39717e10 1.14479
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 6.29422e10 2.09853
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 5.76781e10 1.88258 0.941289 0.337603i \(-0.109616\pi\)
0.941289 + 0.337603i \(0.109616\pi\)
\(992\) 6.14240e10i 1.99778i
\(993\) 0 0
\(994\) 0 0
\(995\) 5.86386e10i 1.88713i
\(996\) 0 0
\(997\) 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 72.8.d.a.37.2 yes 2
3.2 odd 2 inner 72.8.d.a.37.1 2
4.3 odd 2 288.8.d.a.145.2 2
8.3 odd 2 288.8.d.a.145.1 2
8.5 even 2 inner 72.8.d.a.37.1 2
12.11 even 2 288.8.d.a.145.1 2
24.5 odd 2 CM 72.8.d.a.37.2 yes 2
24.11 even 2 288.8.d.a.145.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.8.d.a.37.1 2 3.2 odd 2 inner
72.8.d.a.37.1 2 8.5 even 2 inner
72.8.d.a.37.2 yes 2 1.1 even 1 trivial
72.8.d.a.37.2 yes 2 24.5 odd 2 CM
288.8.d.a.145.1 2 8.3 odd 2
288.8.d.a.145.1 2 12.11 even 2
288.8.d.a.145.2 2 4.3 odd 2
288.8.d.a.145.2 2 24.11 even 2