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Properties

Label 7.87.b.a.6.1
Level $7$
Weight $87$
Character 7.6
Self dual yes
Analytic conductor $327.865$
Analytic rank $0$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7,87,Mod(6,7)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7.6"); S:= CuspForms(chi, 87); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 87, names="a")
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 87 \)
Character orbit: \([\chi]\) \(=\) 7.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(327.864794911\)
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 6.1
Character \(\chi\) \(=\) 7.6

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.65948e13 q^{2} +1.98014e26 q^{4} -2.18381e36 q^{7} -2.00204e39 q^{8} +1.07753e41 q^{9} -7.65623e44 q^{11} +3.62399e49 q^{14} +1.79028e52 q^{16} -1.78813e54 q^{18} +1.27053e58 q^{22} +7.16652e58 q^{23} +1.29247e60 q^{25} -4.32427e62 q^{28} +1.43275e63 q^{29} -1.42192e65 q^{32} +2.13366e67 q^{36} -5.04360e67 q^{37} -1.38803e70 q^{43} -1.51604e71 q^{44} -1.18927e72 q^{46} +4.76905e72 q^{49} -2.14482e73 q^{50} +1.80787e74 q^{53} +4.37209e75 q^{56} -2.37762e76 q^{58} -2.35312e77 q^{63} +9.74475e77 q^{64} +4.79961e78 q^{67} -5.56357e79 q^{71} -2.15726e80 q^{72} +8.36972e80 q^{74} +1.67198e81 q^{77} +6.08423e81 q^{79} +1.16106e82 q^{81} +2.30341e83 q^{86} +1.53281e84 q^{88} +1.41907e85 q^{92} -7.91411e85 q^{98} -8.24979e85 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.65948e13 −1.88660 −0.943302 − 0.331935i \(-0.892299\pi\)
−0.943302 + 0.331935i \(0.892299\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1.98014e26 2.55928
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −2.18381e36 −1.00000
\(8\) −2.00204e39 −2.94174
\(9\) 1.07753e41 1.00000
\(10\) 0 0
\(11\) −7.65623e44 −1.27095 −0.635476 − 0.772120i \(-0.719196\pi\)
−0.635476 + 0.772120i \(0.719196\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 3.62399e49 1.88660
\(15\) 0 0
\(16\) 1.79028e52 2.99062
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −1.78813e54 −1.88660
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.27053e58 2.39779
\(23\) 7.16652e58 1.99991 0.999957 − 0.00932719i \(-0.00296898\pi\)
0.999957 + 0.00932719i \(0.00296898\pi\)
\(24\) 0 0
\(25\) 1.29247e60 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −4.32427e62 −2.55928
\(29\) 1.43275e63 1.87524 0.937622 − 0.347656i \(-0.113022\pi\)
0.937622 + 0.347656i \(0.113022\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.42192e65 −2.70038
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 2.13366e67 2.55928
\(37\) −5.04360e67 −1.86237 −0.931185 − 0.364546i \(-0.881224\pi\)
−0.931185 + 0.364546i \(0.881224\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −1.38803e70 −0.800307 −0.400153 − 0.916448i \(-0.631043\pi\)
−0.400153 + 0.916448i \(0.631043\pi\)
\(44\) −1.51604e71 −3.25272
\(45\) 0 0
\(46\) −1.18927e72 −3.77305
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 4.76905e72 1.00000
\(50\) −2.14482e73 −1.88660
\(51\) 0 0
\(52\) 0 0
\(53\) 1.80787e74 1.29809 0.649045 − 0.760750i \(-0.275169\pi\)
0.649045 + 0.760750i \(0.275169\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.37209e75 2.94174
\(57\) 0 0
\(58\) −2.37762e76 −3.53784
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −2.35312e77 −1.00000
\(64\) 9.74475e77 2.10393
\(65\) 0 0
\(66\) 0 0
\(67\) 4.79961e78 1.44540 0.722702 − 0.691160i \(-0.242900\pi\)
0.722702 + 0.691160i \(0.242900\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.56357e79 −1.38435 −0.692173 − 0.721732i \(-0.743347\pi\)
−0.692173 + 0.721732i \(0.743347\pi\)
\(72\) −2.15726e80 −2.94174
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 8.36972e80 3.51356
\(75\) 0 0
\(76\) 0 0
\(77\) 1.67198e81 1.27095
\(78\) 0 0
\(79\) 6.08423e81 1.53547 0.767734 − 0.640769i \(-0.221384\pi\)
0.767734 + 0.640769i \(0.221384\pi\)
\(80\) 0 0
\(81\) 1.16106e82 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.30341e83 1.50986
\(87\) 0 0
\(88\) 1.53281e84 3.73881
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.41907e85 5.11833
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −7.91411e85 −1.88660
\(99\) −8.24979e85 −1.27095
\(100\) 2.55928e86 2.55928
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −3.00012e87 −2.44898
\(107\) −3.55241e87 −1.93651 −0.968256 − 0.249961i \(-0.919582\pi\)
−0.968256 + 0.249961i \(0.919582\pi\)
\(108\) 0 0
\(109\) −1.26885e87 −0.311940 −0.155970 − 0.987762i \(-0.549850\pi\)
−0.155970 + 0.987762i \(0.549850\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.90964e88 −2.99062
\(113\) 9.41311e87 0.491315 0.245658 − 0.969357i \(-0.420996\pi\)
0.245658 + 0.969357i \(0.420996\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.83706e89 4.79927
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.23292e89 0.615322
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 3.90494e90 1.88660
\(127\) −1.93165e90 −0.664308 −0.332154 − 0.943225i \(-0.607775\pi\)
−0.332154 + 0.943225i \(0.607775\pi\)
\(128\) −5.16962e90 −1.26891
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −7.96483e91 −2.72691
\(135\) 0 0
\(136\) 0 0
\(137\) −1.46323e92 −1.93343 −0.966715 − 0.255855i \(-0.917643\pi\)
−0.966715 + 0.255855i \(0.917643\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 9.23261e92 2.61171
\(143\) 0 0
\(144\) 1.92907e93 2.99062
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −9.98705e93 −4.76632
\(149\) 3.64957e93 1.30387 0.651933 − 0.758277i \(-0.273958\pi\)
0.651933 + 0.758277i \(0.273958\pi\)
\(150\) 0 0
\(151\) 9.75904e93 1.96516 0.982581 − 0.185832i \(-0.0594981\pi\)
0.982581 + 0.185832i \(0.0594981\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −2.77461e94 −2.39779
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) −1.00966e95 −2.89682
\(159\) 0 0
\(160\) 0 0
\(161\) −1.56503e95 −1.99991
\(162\) −1.92676e95 −1.88660
\(163\) 1.08952e95 0.818781 0.409390 − 0.912359i \(-0.365741\pi\)
0.409390 + 0.912359i \(0.365741\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 6.29692e95 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −2.74851e96 −2.04821
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −2.82251e96 −1.00000
\(176\) −1.37068e97 −3.80094
\(177\) 0 0
\(178\) 0 0
\(179\) −1.40745e97 −1.88692 −0.943459 − 0.331489i \(-0.892449\pi\)
−0.943459 + 0.331489i \(0.892449\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.43477e98 −5.88322
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.41187e98 −1.98573 −0.992864 − 0.119252i \(-0.961950\pi\)
−0.992864 + 0.119252i \(0.961950\pi\)
\(192\) 0 0
\(193\) 2.31170e98 1.21609 0.608047 − 0.793901i \(-0.291953\pi\)
0.608047 + 0.793901i \(0.291953\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 9.44340e98 2.55928
\(197\) −2.33808e98 −0.509110 −0.254555 − 0.967058i \(-0.581929\pi\)
−0.254555 + 0.967058i \(0.581929\pi\)
\(198\) 1.36903e99 2.39779
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −2.58758e99 −2.94174
\(201\) 0 0
\(202\) 0 0
\(203\) −3.12887e99 −1.87524
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 7.72211e99 1.99991
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −8.29517e99 −0.943365 −0.471682 − 0.881769i \(-0.656353\pi\)
−0.471682 + 0.881769i \(0.656353\pi\)
\(212\) 3.57985e100 3.32217
\(213\) 0 0
\(214\) 5.89513e100 3.65343
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 2.10563e100 0.588508
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 3.10520e101 2.70038
\(225\) 1.39267e101 1.00000
\(226\) −1.56208e101 −0.926918
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.86844e102 −5.51648
\(233\) −1.24610e102 −1.99180 −0.995902 − 0.0904374i \(-0.971173\pi\)
−0.995902 + 0.0904374i \(0.971173\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.50925e102 1.87977 0.939884 − 0.341493i \(-0.110933\pi\)
0.939884 + 0.341493i \(0.110933\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −3.70547e102 −1.16087
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −4.65951e103 −2.55928
\(253\) −5.48685e103 −2.54180
\(254\) 3.20553e103 1.25329
\(255\) 0 0
\(256\) 1.03921e103 0.289992
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 1.10143e104 1.86237
\(260\) 0 0
\(261\) 1.54383e104 1.87524
\(262\) 0 0
\(263\) 1.00073e103 0.0875430 0.0437715 − 0.999042i \(-0.486063\pi\)
0.0437715 + 0.999042i \(0.486063\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 9.50392e104 3.69919
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 2.42820e105 3.64762
\(275\) −9.89544e104 −1.27095
\(276\) 0 0
\(277\) −9.47897e104 −0.891520 −0.445760 − 0.895153i \(-0.647066\pi\)
−0.445760 + 0.895153i \(0.647066\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.35688e105 −1.19664 −0.598322 − 0.801255i \(-0.704166\pi\)
−0.598322 + 0.801255i \(0.704166\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −1.10167e106 −3.54293
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.53215e106 −2.70038
\(289\) 6.58578e105 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.00975e107 5.47861
\(297\) 0 0
\(298\) −6.05638e106 −2.45988
\(299\) 0 0
\(300\) 0 0
\(301\) 3.03121e106 0.800307
\(302\) −1.61949e107 −3.70749
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 3.31076e107 3.25272
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.20477e108 3.92969
\(317\) 4.79929e107 1.36656 0.683279 − 0.730157i \(-0.260553\pi\)
0.683279 + 0.730157i \(0.260553\pi\)
\(318\) 0 0
\(319\) −1.09695e108 −2.38335
\(320\) 0 0
\(321\) 0 0
\(322\) 2.59713e108 3.77305
\(323\) 0 0
\(324\) 2.29907e108 2.55928
\(325\) 0 0
\(326\) −1.80803e108 −1.54472
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.50383e108 1.99976 0.999881 − 0.0154307i \(-0.00491193\pi\)
0.999881 + 0.0154307i \(0.00491193\pi\)
\(332\) 0 0
\(333\) −5.43461e108 −1.86237
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8.23121e108 −1.68802 −0.844011 − 0.536326i \(-0.819812\pi\)
−0.844011 + 0.536326i \(0.819812\pi\)
\(338\) −1.04496e109 −1.88660
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.04147e109 −1.00000
\(344\) 2.77891e109 2.35429
\(345\) 0 0
\(346\) 0 0
\(347\) 3.37775e109 1.96997 0.984986 − 0.172633i \(-0.0552274\pi\)
0.984986 + 0.172633i \(0.0552274\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 4.68389e109 1.88660
\(351\) 0 0
\(352\) 1.08865e110 3.43206
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 2.33563e110 3.55987
\(359\) −1.25635e110 −1.69844 −0.849220 − 0.528040i \(-0.822927\pi\)
−0.849220 + 0.528040i \(0.822927\pi\)
\(360\) 0 0
\(361\) 9.39312e109 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 1.28301e111 5.98099
\(369\) 0 0
\(370\) 0 0
\(371\) −3.94805e110 −1.29809
\(372\) 0 0
\(373\) 2.09624e110 0.546968 0.273484 − 0.961877i \(-0.411824\pi\)
0.273484 + 0.961877i \(0.411824\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 3.62206e110 0.475850 0.237925 − 0.971284i \(-0.423533\pi\)
0.237925 + 0.971284i \(0.423533\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 4.00244e111 3.74628
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.83621e111 −2.29429
\(387\) −1.49564e111 −0.800307
\(388\) 0 0
\(389\) −9.39860e110 −0.402930 −0.201465 − 0.979496i \(-0.564570\pi\)
−0.201465 + 0.979496i \(0.564570\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −9.54784e111 −2.94174
\(393\) 0 0
\(394\) 3.87999e111 0.960490
\(395\) 0 0
\(396\) −1.63358e112 −3.25272
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 2.31388e112 2.99062
\(401\) −1.24553e112 −1.44593 −0.722964 − 0.690885i \(-0.757221\pi\)
−0.722964 + 0.690885i \(0.757221\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 5.19228e112 3.53784
\(407\) 3.86149e112 2.36699
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −1.28146e113 −3.77305
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 1.08675e113 1.55596 0.777979 − 0.628290i \(-0.216245\pi\)
0.777979 + 0.628290i \(0.216245\pi\)
\(422\) 1.37656e113 1.77976
\(423\) 0 0
\(424\) −3.61944e113 −3.81864
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −7.03428e113 −4.95607
\(429\) 0 0
\(430\) 0 0
\(431\) −3.68850e113 −1.92454 −0.962271 − 0.272094i \(-0.912284\pi\)
−0.962271 + 0.272094i \(0.912284\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.51251e113 −0.798342
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 5.13877e113 1.00000
\(442\) 0 0
\(443\) −2.07016e113 −0.331622 −0.165811 − 0.986158i \(-0.553024\pi\)
−0.165811 + 0.986158i \(0.553024\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −2.12807e114 −2.10393
\(449\) 1.35150e114 1.21402 0.607008 − 0.794696i \(-0.292370\pi\)
0.607008 + 0.794696i \(0.292370\pi\)
\(450\) −2.31110e114 −1.88660
\(451\) 0 0
\(452\) 1.86393e114 1.25741
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.26835e114 −1.79417 −0.897084 − 0.441860i \(-0.854319\pi\)
−0.897084 + 0.441860i \(0.854319\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 3.96784e114 0.951856 0.475928 − 0.879484i \(-0.342112\pi\)
0.475928 + 0.879484i \(0.342112\pi\)
\(464\) 2.56503e115 5.60815
\(465\) 0 0
\(466\) 2.06786e115 3.75775
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −1.04815e115 −1.44540
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.06271e115 1.01715
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.94803e115 1.29809
\(478\) −5.82351e115 −3.54638
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 4.42150e115 1.57478
\(485\) 0 0
\(486\) 0 0
\(487\) −4.98657e115 −1.36162 −0.680811 − 0.732459i \(-0.738373\pi\)
−0.680811 + 0.732459i \(0.738373\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.72767e115 −1.29229 −0.646146 − 0.763214i \(-0.723620\pi\)
−0.646146 + 0.763214i \(0.723620\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.21498e116 1.38435
\(498\) 0 0
\(499\) −9.48886e115 −0.909684 −0.454842 − 0.890572i \(-0.650304\pi\)
−0.454842 + 0.890572i \(0.650304\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 4.71105e116 2.94174
\(505\) 0 0
\(506\) 9.10529e116 4.79536
\(507\) 0 0
\(508\) −3.82495e116 −1.70015
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 2.27525e116 0.721806
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −1.82779e117 −3.51356
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) −2.56195e117 −3.53784
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −1.66069e116 −0.165159
\(527\) 0 0
\(528\) 0 0
\(529\) 3.85181e117 2.99965
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −9.60903e117 −4.25200
\(537\) 0 0
\(538\) 0 0
\(539\) −3.65129e117 −1.27095
\(540\) 0 0
\(541\) −1.32831e117 −0.394290 −0.197145 − 0.980374i \(-0.563167\pi\)
−0.197145 + 0.980374i \(0.563167\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −7.31917e116 −0.135209 −0.0676047 − 0.997712i \(-0.521536\pi\)
−0.0676047 + 0.997712i \(0.521536\pi\)
\(548\) −2.89741e118 −4.94818
\(549\) 0 0
\(550\) 1.64212e118 2.39779
\(551\) 0 0
\(552\) 0 0
\(553\) −1.32868e118 −1.53547
\(554\) 1.57301e118 1.68195
\(555\) 0 0
\(556\) 0 0
\(557\) 1.95041e118 1.65331 0.826654 − 0.562711i \(-0.190242\pi\)
0.826654 + 0.562711i \(0.190242\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 3.91118e118 2.25760
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.53555e118 −1.00000
\(568\) 1.11385e119 4.07239
\(569\) 5.48416e118 1.85901 0.929507 − 0.368804i \(-0.120232\pi\)
0.929507 + 0.368804i \(0.120232\pi\)
\(570\) 0 0
\(571\) −4.13888e118 −1.20651 −0.603254 − 0.797549i \(-0.706130\pi\)
−0.603254 + 0.797549i \(0.706130\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9.26250e118 1.99991
\(576\) 1.05002e119 2.10393
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −1.09289e119 −1.88660
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.38415e119 −1.64981
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −9.02945e119 −5.56965
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.22669e119 3.33695
\(597\) 0 0
\(598\) 0 0
\(599\) −8.38857e117 −0.0312129 −0.0156065 − 0.999878i \(-0.504968\pi\)
−0.0156065 + 0.999878i \(0.504968\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) −5.03021e119 −1.50986
\(603\) 5.17170e119 1.44540
\(604\) 1.93243e120 5.02940
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.43133e120 −1.97214 −0.986071 − 0.166326i \(-0.946810\pi\)
−0.986071 + 0.166326i \(0.946810\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −3.34738e120 −3.73881
\(617\) 1.86308e120 1.94075 0.970374 − 0.241609i \(-0.0776750\pi\)
0.970374 + 0.241609i \(0.0776750\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.67048e120 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 4.64069e120 1.84213 0.921064 − 0.389411i \(-0.127321\pi\)
0.921064 + 0.389411i \(0.127321\pi\)
\(632\) −1.21809e121 −4.51695
\(633\) 0 0
\(634\) −7.96429e120 −2.57816
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 1.82036e121 4.49643
\(639\) −5.99489e120 −1.38435
\(640\) 0 0
\(641\) 8.10910e120 1.63711 0.818555 − 0.574429i \(-0.194776\pi\)
0.818555 + 0.574429i \(0.194776\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −3.09899e121 −5.11833
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −2.32450e121 −2.94174
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 2.15741e121 2.09549
\(653\) 1.34718e121 1.22507 0.612534 − 0.790444i \(-0.290150\pi\)
0.612534 + 0.790444i \(0.290150\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −2.75227e121 −1.68896 −0.844479 − 0.535589i \(-0.820090\pi\)
−0.844479 + 0.535589i \(0.820090\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −7.47399e121 −3.77276
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 9.01860e121 3.51356
\(667\) 1.02679e122 3.75032
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 7.45720e121 1.85322 0.926612 − 0.376018i \(-0.122707\pi\)
0.926612 + 0.376018i \(0.122707\pi\)
\(674\) 1.36595e122 3.18463
\(675\) 0 0
\(676\) 1.24688e122 2.55928
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.15723e122 −1.52521 −0.762604 − 0.646866i \(-0.776079\pi\)
−0.762604 + 0.646866i \(0.776079\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.72830e122 1.88660
\(687\) 0 0
\(688\) −2.48497e122 −2.39342
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 1.80160e122 1.27095
\(694\) −5.60529e122 −3.71656
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −5.58899e122 −2.55928
\(701\) 4.61740e122 1.98849 0.994243 − 0.107149i \(-0.0341722\pi\)
0.994243 + 0.107149i \(0.0341722\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −7.46081e122 −2.67400
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 7.25950e122 1.91919 0.959596 − 0.281381i \(-0.0907924\pi\)
0.959596 + 0.281381i \(0.0907924\pi\)
\(710\) 0 0
\(711\) 6.55592e122 1.53547
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −2.78696e123 −4.82915
\(717\) 0 0
\(718\) 2.08488e123 3.20428
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.55876e123 −1.88660
\(723\) 0 0
\(724\) 0 0
\(725\) 1.85179e123 1.87524
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.25108e123 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −1.01902e124 −5.40053
\(737\) −3.67469e123 −1.83704
\(738\) 0 0
\(739\) 4.22498e123 1.87981 0.939905 − 0.341435i \(-0.110913\pi\)
0.939905 + 0.341435i \(0.110913\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 6.55170e123 2.44898
\(743\) 9.05728e122 0.319506 0.159753 − 0.987157i \(-0.448930\pi\)
0.159753 + 0.987157i \(0.448930\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −3.47866e123 −1.03191
\(747\) 0 0
\(748\) 0 0
\(749\) 7.75780e123 1.93651
\(750\) 0 0
\(751\) 7.10567e123 1.58157 0.790784 − 0.612096i \(-0.209673\pi\)
0.790784 + 0.612096i \(0.209673\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −7.50772e123 −1.18682 −0.593410 − 0.804900i \(-0.702219\pi\)
−0.593410 + 0.804900i \(0.702219\pi\)
\(758\) −6.01072e123 −0.897740
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 2.77094e123 0.311940
\(764\) −4.77586e124 −5.08203
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.57750e124 3.11232
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 2.48198e124 1.50986
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 1.55967e124 0.760171
\(779\) 0 0
\(780\) 0 0
\(781\) 4.25960e124 1.75944
\(782\) 0 0
\(783\) 0 0
\(784\) 8.53793e124 2.99062
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −4.62974e124 −1.30295
\(789\) 0 0
\(790\) 0 0
\(791\) −2.05565e124 −0.491315
\(792\) 1.65164e125 3.73881
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.83779e125 −2.70038
\(801\) 0 0
\(802\) 2.06693e125 2.72790
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2.19509e125 −1.99373 −0.996865 − 0.0791189i \(-0.974789\pi\)
−0.996865 + 0.0791189i \(0.974789\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −6.19561e125 −4.79927
\(813\) 0 0
\(814\) −6.40805e125 −4.46557
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.24606e125 1.56531 0.782656 − 0.622455i \(-0.213865\pi\)
0.782656 + 0.622455i \(0.213865\pi\)
\(822\) 0 0
\(823\) −3.75917e125 −1.63267 −0.816337 − 0.577576i \(-0.803999\pi\)
−0.816337 + 0.577576i \(0.803999\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −5.01380e125 −1.76779 −0.883894 − 0.467687i \(-0.845087\pi\)
−0.883894 + 0.467687i \(0.845087\pi\)
\(828\) 1.52909e126 5.11833
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.46903e126 2.51654
\(842\) −1.80343e126 −2.93548
\(843\) 0 0
\(844\) −1.64256e126 −2.41433
\(845\) 0 0
\(846\) 0 0
\(847\) −4.87628e125 −0.615322
\(848\) 3.23659e126 3.88210
\(849\) 0 0
\(850\) 0 0
\(851\) −3.61450e126 −3.72458
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 7.11208e126 5.69671
\(857\) 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
\(862\) 6.12098e126 3.63085
\(863\) −2.41783e126 −1.36446 −0.682232 − 0.731136i \(-0.738990\pi\)
−0.682232 + 0.731136i \(0.738990\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −4.65823e126 −1.95151
\(870\) 0 0
\(871\) 0 0
\(872\) 2.54030e126 0.917647
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2.88077e125 −0.0813816 −0.0406908 − 0.999172i \(-0.512956\pi\)
−0.0406908 + 0.999172i \(0.512956\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −8.52766e126 −1.88660
\(883\) −1.94366e126 −0.409554 −0.204777 − 0.978809i \(-0.565647\pi\)
−0.204777 + 0.978809i \(0.565647\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 3.43538e126 0.625640
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 4.21837e126 0.664308
\(890\) 0 0
\(891\) −8.88937e126 −1.27095
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.12895e127 1.26891
\(897\) 0 0
\(898\) −2.24279e127 −2.29037
\(899\) 0 0
\(900\) 2.75769e127 2.55928
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −1.88455e127 −1.44532
\(905\) 0 0
\(906\) 0 0
\(907\) 8.57188e126 0.570116 0.285058 − 0.958510i \(-0.407987\pi\)
0.285058 + 0.958510i \(0.407987\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.71572e127 0.944401 0.472200 − 0.881491i \(-0.343460\pi\)
0.472200 + 0.881491i \(0.343460\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 7.08321e127 3.38488
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 5.28982e127 1.99928 0.999641 − 0.0268026i \(-0.00853254\pi\)
0.999641 + 0.0268026i \(0.00853254\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −6.51869e127 −1.86237
\(926\) −6.58453e127 −1.79578
\(927\) 0 0
\(928\) −2.03726e128 −5.06388
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.46745e128 −5.09758
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 1.73937e128 2.72691
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −1.76354e128 −1.91896
\(947\) 1.88930e128 1.96450 0.982248 − 0.187586i \(-0.0600664\pi\)
0.982248 + 0.187586i \(0.0600664\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.18208e127 0.489937 0.244968 − 0.969531i \(-0.421222\pi\)
0.244968 + 0.969531i \(0.421222\pi\)
\(954\) −3.23270e128 −2.44898
\(955\) 0 0
\(956\) 6.94882e128 4.81085
\(957\) 0 0
\(958\) 0 0
\(959\) 3.19542e128 1.93343
\(960\) 0 0
\(961\) 1.80761e128 1.00000
\(962\) 0 0
\(963\) −3.82781e128 −1.93651
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 4.36895e128 1.84943 0.924717 − 0.380654i \(-0.124301\pi\)
0.924717 + 0.380654i \(0.124301\pi\)
\(968\) −4.47040e128 −1.81012
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 8.27509e128 2.56884
\(975\) 0 0
\(976\) 0 0
\(977\) 3.04875e128 0.829192 0.414596 − 0.910006i \(-0.363923\pi\)
0.414596 + 0.910006i \(0.363923\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.36722e128 −0.311940
\(982\) 1.11644e129 2.43804
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9.94737e128 −1.60054
\(990\) 0 0
\(991\) 8.16911e128 1.20506 0.602529 − 0.798097i \(-0.294160\pi\)
0.602529 + 0.798097i \(0.294160\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −2.01623e129 −2.61171
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 1.57465e129 1.71621
\(999\) 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 7.87.b.a.6.1 ✓ 1
7.6 odd 2 CM 7.87.b.a.6.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.87.b.a.6.1 ✓ 1 1.1 even 1 trivial
7.87.b.a.6.1 ✓ 1 7.6 odd 2 CM