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Properties

Label 7.39.b.a.6.1
Level $7$
Weight $39$
Character 7.6
Self dual yes
Analytic conductor $64.024$
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] = [7,39,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, 39); 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, 39, names="a")
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 39 \)
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: \(64.0244283235\)
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)
 
\(f(q)\) \(=\) \(q-413367. q^{2} -1.04006e11 q^{4} -1.13989e16 q^{7} +1.56618e17 q^{8} +1.35085e18 q^{9} -1.05169e20 q^{11} +4.71193e21 q^{14} -3.61518e22 q^{16} -5.58398e23 q^{18} +4.34735e25 q^{22} -1.45793e26 q^{23} +3.63798e26 q^{25} +1.18555e27 q^{28} -8.37709e27 q^{29} -2.81068e28 q^{32} -1.40496e29 q^{36} -9.15056e29 q^{37} -2.16644e31 q^{43} +1.09382e31 q^{44} +6.02661e31 q^{46} +1.29935e32 q^{49} -1.50382e32 q^{50} +1.01577e33 q^{53} -1.78527e33 q^{56} +3.46281e33 q^{58} -1.53982e34 q^{63} +2.15558e34 q^{64} +9.90861e34 q^{67} +2.71406e35 q^{71} +2.11568e35 q^{72} +3.78254e35 q^{74} +1.19881e36 q^{77} -1.00721e36 q^{79} +1.82480e36 q^{81} +8.95535e36 q^{86} -1.64714e37 q^{88} +1.51633e37 q^{92} -5.37108e37 q^{98} -1.42068e38 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −413367. −0.788435 −0.394217 − 0.919017i \(-0.628984\pi\)
−0.394217 + 0.919017i \(0.628984\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −1.04006e11 −0.378370
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −1.13989e16 −1.00000
\(8\) 1.56618e17 1.08676
\(9\) 1.35085e18 1.00000
\(10\) 0 0
\(11\) −1.05169e20 −1.71960 −0.859801 − 0.510630i \(-0.829412\pi\)
−0.859801 + 0.510630i \(0.829412\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 4.71193e21 0.788435
\(15\) 0 0
\(16\) −3.61518e22 −0.478466
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −5.58398e23 −0.788435
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.34735e25 1.35579
\(23\) −1.45793e26 −1.95393 −0.976964 − 0.213406i \(-0.931544\pi\)
−0.976964 + 0.213406i \(0.931544\pi\)
\(24\) 0 0
\(25\) 3.63798e26 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 1.18555e27 0.378370
\(29\) −8.37709e27 −1.37256 −0.686280 − 0.727337i \(-0.740758\pi\)
−0.686280 + 0.727337i \(0.740758\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −2.81068e28 −0.709516
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −1.40496e29 −0.378370
\(37\) −9.15056e29 −1.46425 −0.732125 − 0.681171i \(-0.761471\pi\)
−0.732125 + 0.681171i \(0.761471\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −2.16644e31 −1.99456 −0.997278 − 0.0737371i \(-0.976507\pi\)
−0.997278 + 0.0737371i \(0.976507\pi\)
\(44\) 1.09382e31 0.650646
\(45\) 0 0
\(46\) 6.02661e31 1.54054
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.29935e32 1.00000
\(50\) −1.50382e32 −0.788435
\(51\) 0 0
\(52\) 0 0
\(53\) 1.01577e33 1.76018 0.880088 − 0.474811i \(-0.157484\pi\)
0.880088 + 0.474811i \(0.157484\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.78527e33 −1.08676
\(57\) 0 0
\(58\) 3.46281e33 1.08217
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −1.53982e34 −1.00000
\(64\) 2.15558e34 1.03787
\(65\) 0 0
\(66\) 0 0
\(67\) 9.90861e34 1.99798 0.998990 − 0.0449242i \(-0.0143046\pi\)
0.998990 + 0.0449242i \(0.0143046\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.71406e35 1.81849 0.909244 − 0.416264i \(-0.136661\pi\)
0.909244 + 0.416264i \(0.136661\pi\)
\(72\) 2.11568e35 1.08676
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 3.78254e35 1.15447
\(75\) 0 0
\(76\) 0 0
\(77\) 1.19881e36 1.71960
\(78\) 0 0
\(79\) −1.00721e36 −0.887573 −0.443786 − 0.896133i \(-0.646365\pi\)
−0.443786 + 0.896133i \(0.646365\pi\)
\(80\) 0 0
\(81\) 1.82480e36 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.95535e36 1.57258
\(87\) 0 0
\(88\) −1.64714e37 −1.86879
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.51633e37 0.739308
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −5.37108e37 −0.788435
\(99\) −1.42068e38 −1.71960
\(100\) −3.78370e37 −0.378370
\(101\) 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
\(106\) −4.19888e38 −1.38778
\(107\) −6.65113e38 −1.83909 −0.919546 − 0.392982i \(-0.871443\pi\)
−0.919546 + 0.392982i \(0.871443\pi\)
\(108\) 0 0
\(109\) 2.27464e38 0.442394 0.221197 − 0.975229i \(-0.429004\pi\)
0.221197 + 0.975229i \(0.429004\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.12091e38 0.478466
\(113\) 1.84658e39 1.81083 0.905415 − 0.424528i \(-0.139560\pi\)
0.905415 + 0.424528i \(0.139560\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 8.71265e38 0.519336
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.32014e39 1.95703
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 6.36511e39 0.788435
\(127\) −2.79662e39 −0.298101 −0.149050 − 0.988830i \(-0.547622\pi\)
−0.149050 + 0.988830i \(0.547622\pi\)
\(128\) −1.18450e39 −0.108779
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −4.09589e40 −1.57528
\(135\) 0 0
\(136\) 0 0
\(137\) 3.84342e40 0.970570 0.485285 − 0.874356i \(-0.338716\pi\)
0.485285 + 0.874356i \(0.338716\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.12190e41 −1.43376
\(143\) 0 0
\(144\) −4.88358e40 −0.478466
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 9.51710e40 0.554028
\(149\) −9.30842e40 −0.476800 −0.238400 − 0.971167i \(-0.576623\pi\)
−0.238400 + 0.971167i \(0.576623\pi\)
\(150\) 0 0
\(151\) −1.92315e41 −0.764629 −0.382315 − 0.924032i \(-0.624873\pi\)
−0.382315 + 0.924032i \(0.624873\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −4.95550e41 −1.35579
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 4.16346e41 0.699793
\(159\) 0 0
\(160\) 0 0
\(161\) 1.66188e42 1.95393
\(162\) −7.54312e41 −0.788435
\(163\) 2.01170e42 1.87068 0.935338 − 0.353754i \(-0.115095\pi\)
0.935338 + 0.353754i \(0.115095\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 2.13721e42 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 2.25322e42 0.754680
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −4.14689e42 −1.00000
\(176\) 3.80206e42 0.822770
\(177\) 0 0
\(178\) 0 0
\(179\) −1.06491e43 −1.67149 −0.835746 − 0.549116i \(-0.814964\pi\)
−0.835746 + 0.549116i \(0.814964\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2.28338e43 −2.12344
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.06907e42 −0.323391 −0.161696 − 0.986841i \(-0.551696\pi\)
−0.161696 + 0.986841i \(0.551696\pi\)
\(192\) 0 0
\(193\) 4.07256e42 0.152855 0.0764274 − 0.997075i \(-0.475649\pi\)
0.0764274 + 0.997075i \(0.475649\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.35140e43 −0.378370
\(197\) −4.12675e43 −1.04894 −0.524469 − 0.851430i \(-0.675736\pi\)
−0.524469 + 0.851430i \(0.675736\pi\)
\(198\) 5.87263e43 1.35579
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 5.69773e43 1.08676
\(201\) 0 0
\(202\) 0 0
\(203\) 9.54896e43 1.37256
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.96945e44 −1.95393
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 2.58192e44 1.78064 0.890319 − 0.455338i \(-0.150482\pi\)
0.890319 + 0.455338i \(0.150482\pi\)
\(212\) −1.05646e44 −0.665998
\(213\) 0 0
\(214\) 2.74936e44 1.45000
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −9.40261e43 −0.348799
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 3.20387e44 0.709516
\(225\) 4.91437e44 1.00000
\(226\) −7.63315e44 −1.42772
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.31200e45 −1.49164
\(233\) −1.27882e45 −1.33982 −0.669910 − 0.742442i \(-0.733668\pi\)
−0.669910 + 0.742442i \(0.733668\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.59115e44 0.490623 0.245311 − 0.969444i \(-0.421110\pi\)
0.245311 + 0.969444i \(0.421110\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −3.02590e45 −1.54299
\(243\) 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 \(\pi\)
\(252\) 1.60150e45 0.378370
\(253\) 1.53330e46 3.35998
\(254\) 1.15603e45 0.235033
\(255\) 0 0
\(256\) −5.43557e45 −0.952108
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 1.04306e46 1.46425
\(260\) 0 0
\(261\) −1.13162e46 −1.37256
\(262\) 0 0
\(263\) 1.90332e46 1.99688 0.998439 − 0.0558481i \(-0.0177863\pi\)
0.998439 + 0.0558481i \(0.0177863\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.03055e46 −0.755977
\(269\) 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
\(274\) −1.58874e46 −0.765231
\(275\) −3.82604e46 −1.71960
\(276\) 0 0
\(277\) −2.88144e46 −1.12848 −0.564241 − 0.825610i \(-0.690831\pi\)
−0.564241 + 0.825610i \(0.690831\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.51882e46 1.34774 0.673869 − 0.738851i \(-0.264631\pi\)
0.673869 + 0.738851i \(0.264631\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −2.82277e46 −0.688062
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −3.79682e46 −0.709516
\(289\) 5.71556e46 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
\(296\) −1.43314e47 −1.59128
\(297\) 0 0
\(298\) 3.84779e46 0.375926
\(299\) 0 0
\(300\) 0 0
\(301\) 2.46950e47 1.99456
\(302\) 7.94966e46 0.602860
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) −1.24683e47 −0.650646
\(309\) 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
\(316\) 1.04755e47 0.335831
\(317\) 3.00770e47 0.908047 0.454024 − 0.890990i \(-0.349988\pi\)
0.454024 + 0.890990i \(0.349988\pi\)
\(318\) 0 0
\(319\) 8.81013e47 2.36026
\(320\) 0 0
\(321\) 0 0
\(322\) −6.86967e47 −1.54054
\(323\) 0 0
\(324\) −1.89790e47 −0.378370
\(325\) 0 0
\(326\) −8.31571e47 −1.47491
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.11459e48 1.48041 0.740205 − 0.672381i \(-0.234729\pi\)
0.740205 + 0.672381i \(0.234729\pi\)
\(332\) 0 0
\(333\) −1.23611e48 −1.46425
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.49042e47 −0.423954 −0.211977 − 0.977275i \(-0.567990\pi\)
−0.211977 + 0.977275i \(0.567990\pi\)
\(338\) −8.83452e47 −0.788435
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.48111e48 −1.00000
\(344\) −3.39303e48 −2.16759
\(345\) 0 0
\(346\) 0 0
\(347\) 2.15917e48 1.16957 0.584786 − 0.811188i \(-0.301179\pi\)
0.584786 + 0.811188i \(0.301179\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 1.71419e48 0.788435
\(351\) 0 0
\(352\) 2.95598e48 1.22009
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 4.40198e48 1.31786
\(359\) 1.59964e48 0.454181 0.227090 − 0.973874i \(-0.427079\pi\)
0.227090 + 0.973874i \(0.427079\pi\)
\(360\) 0 0
\(361\) 3.91414e48 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 \(\pi\)
\(368\) 5.27069e48 0.934887
\(369\) 0 0
\(370\) 0 0
\(371\) −1.15787e49 −1.76018
\(372\) 0 0
\(373\) 7.43081e48 1.01993 0.509963 − 0.860196i \(-0.329659\pi\)
0.509963 + 0.860196i \(0.329659\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.55763e49 −1.57878 −0.789388 − 0.613895i \(-0.789602\pi\)
−0.789388 + 0.613895i \(0.789602\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 2.92212e48 0.254973
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.68346e48 −0.120516
\(387\) −2.92654e49 −1.99456
\(388\) 0 0
\(389\) −2.02023e49 −1.24841 −0.624206 − 0.781259i \(-0.714578\pi\)
−0.624206 + 0.781259i \(0.714578\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 2.03501e49 1.08676
\(393\) 0 0
\(394\) 1.70586e49 0.827019
\(395\) 0 0
\(396\) 1.47759e49 0.650646
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.31520e49 −0.478466
\(401\) −3.88211e49 −1.34687 −0.673434 − 0.739247i \(-0.735181\pi\)
−0.673434 + 0.739247i \(0.735181\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −3.94722e49 −1.08217
\(407\) 9.62358e49 2.51793
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 8.14106e49 1.54054
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −8.55039e49 −1.17659 −0.588297 − 0.808645i \(-0.700201\pi\)
−0.588297 + 0.808645i \(0.700201\pi\)
\(422\) −1.06728e50 −1.40392
\(423\) 0 0
\(424\) 1.59089e50 1.91288
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 6.91755e49 0.695858
\(429\) 0 0
\(430\) 0 0
\(431\) 9.88628e49 0.870893 0.435446 − 0.900215i \(-0.356591\pi\)
0.435446 + 0.900215i \(0.356591\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.36575e49 −0.167389
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.75523e50 1.00000
\(442\) 0 0
\(443\) −3.82313e50 −1.99871 −0.999353 − 0.0359774i \(-0.988546\pi\)
−0.999353 + 0.0359774i \(0.988546\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −2.45712e50 −1.03787
\(449\) 1.09344e50 0.442704 0.221352 − 0.975194i \(-0.428953\pi\)
0.221352 + 0.975194i \(0.428953\pi\)
\(450\) −2.03144e50 −0.788435
\(451\) 0 0
\(452\) −1.92055e50 −0.685164
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.11496e50 −1.19113 −0.595563 − 0.803309i \(-0.703071\pi\)
−0.595563 + 0.803309i \(0.703071\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.92490e50 −0.434879 −0.217440 − 0.976074i \(-0.569771\pi\)
−0.217440 + 0.976074i \(0.569771\pi\)
\(464\) 3.02847e50 0.656723
\(465\) 0 0
\(466\) 5.28622e50 1.05636
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −1.12947e51 −1.99798
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.27843e51 3.42984
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.37216e51 1.76018
\(478\) −3.13793e50 −0.386824
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −7.61336e50 −0.740482
\(485\) 0 0
\(486\) 0 0
\(487\) 2.28856e51 1.97930 0.989650 − 0.143499i \(-0.0458354\pi\)
0.989650 + 0.143499i \(0.0458354\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4.41827e50 −0.327118 −0.163559 − 0.986534i \(-0.552297\pi\)
−0.163559 + 0.986534i \(0.552297\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.09372e51 −1.81849
\(498\) 0 0
\(499\) 1.83094e51 0.997158 0.498579 − 0.866844i \(-0.333855\pi\)
0.498579 + 0.866844i \(0.333855\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) −2.41164e51 −1.08676
\(505\) 0 0
\(506\) −6.33814e51 −2.64912
\(507\) 0 0
\(508\) 2.90864e50 0.112792
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 2.57248e51 0.859454
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −4.31168e51 −1.15447
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 4.67775e51 1.08217
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −7.86768e51 −1.57441
\(527\) 0 0
\(528\) 0 0
\(529\) 1.56882e52 2.81783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 1.55187e52 2.17132
\(537\) 0 0
\(538\) 0 0
\(539\) −1.36651e52 −1.71960
\(540\) 0 0
\(541\) −1.64328e52 −1.92736 −0.963681 − 0.267055i \(-0.913949\pi\)
−0.963681 + 0.267055i \(0.913949\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.74596e52 1.66064 0.830318 − 0.557290i \(-0.188159\pi\)
0.830318 + 0.557290i \(0.188159\pi\)
\(548\) −3.99737e51 −0.367235
\(549\) 0 0
\(550\) 1.58156e52 1.35579
\(551\) 0 0
\(552\) 0 0
\(553\) 1.14810e52 0.887573
\(554\) 1.19109e52 0.889735
\(555\) 0 0
\(556\) 0 0
\(557\) −1.99819e52 −1.34706 −0.673531 − 0.739159i \(-0.735223\pi\)
−0.673531 + 0.739159i \(0.735223\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −1.86793e52 −1.06260
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.08007e52 −1.00000
\(568\) 4.25070e52 1.97625
\(569\) −3.57629e52 −1.60805 −0.804025 − 0.594596i \(-0.797312\pi\)
−0.804025 + 0.594596i \(0.797312\pi\)
\(570\) 0 0
\(571\) 4.75030e52 1.99818 0.999091 − 0.0426206i \(-0.0135707\pi\)
0.999091 + 0.0426206i \(0.0135707\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.30393e52 −1.95393
\(576\) 2.91187e52 1.03787
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −2.36263e52 −0.788435
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.06828e53 −3.02680
\(584\) 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
\(592\) 3.30810e52 0.700593
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 9.68128e51 0.180407
\(597\) 0 0
\(598\) 0 0
\(599\) 4.29378e52 0.727327 0.363663 − 0.931530i \(-0.381526\pi\)
0.363663 + 0.931530i \(0.381526\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) −1.02081e53 −1.57258
\(603\) 1.33851e53 1.99798
\(604\) 2.00018e52 0.289313
\(605\) 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
\(613\) 4.61843e52 0.504366 0.252183 − 0.967680i \(-0.418852\pi\)
0.252183 + 0.967680i \(0.418852\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 1.87756e53 1.86879
\(617\) 1.01102e53 0.975754 0.487877 − 0.872912i \(-0.337772\pi\)
0.487877 + 0.872912i \(0.337772\pi\)
\(618\) 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
\(625\) 1.32349e53 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −3.07535e53 −1.93792 −0.968962 − 0.247211i \(-0.920486\pi\)
−0.968962 + 0.247211i \(0.920486\pi\)
\(632\) −1.57747e53 −0.964574
\(633\) 0 0
\(634\) −1.24328e53 −0.715936
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −3.64182e53 −1.86091
\(639\) 3.66629e53 1.81849
\(640\) 0 0
\(641\) −3.32875e53 −1.55589 −0.777946 − 0.628332i \(-0.783738\pi\)
−0.777946 + 0.628332i \(0.783738\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −1.72845e53 −0.739308
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 2.85796e53 1.08676
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −2.09228e53 −0.707809
\(653\) −4.38597e52 −0.144117 −0.0720585 − 0.997400i \(-0.522957\pi\)
−0.0720585 + 0.997400i \(0.522957\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.09718e53 −0.303009 −0.151504 − 0.988457i \(-0.548412\pi\)
−0.151504 + 0.988457i \(0.548412\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −4.60733e53 −1.16721
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 5.10965e53 1.15447
\(667\) 1.22132e54 2.68188
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.44043e53 0.266812 0.133406 − 0.991061i \(-0.457409\pi\)
0.133406 + 0.991061i \(0.457409\pi\)
\(674\) 1.85619e53 0.334260
\(675\) 0 0
\(676\) −2.22282e53 −0.378370
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.08180e54 −1.51410 −0.757050 − 0.653357i \(-0.773360\pi\)
−0.757050 + 0.653357i \(0.773360\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 6.12243e53 0.788435
\(687\) 0 0
\(688\) 7.83208e53 0.954326
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 1.61942e54 1.71960
\(694\) −8.92529e53 −0.922131
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 4.31300e53 0.378370
\(701\) −3.15771e53 −0.269606 −0.134803 − 0.990872i \(-0.543040\pi\)
−0.134803 + 0.990872i \(0.543040\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −2.26701e54 −1.78473
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.19008e54 0.819024 0.409512 − 0.912305i \(-0.365699\pi\)
0.409512 + 0.912305i \(0.365699\pi\)
\(710\) 0 0
\(711\) −1.36059e54 −0.887573
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.10756e54 0.632443
\(717\) 0 0
\(718\) −6.61240e53 −0.358092
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.61798e54 −0.788435
\(723\) 0 0
\(724\) 0 0
\(725\) −3.04757e54 −1.37256
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2.46503e54 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
\(736\) 4.09779e54 1.38634
\(737\) −1.04208e55 −3.43573
\(738\) 0 0
\(739\) 1.66381e54 0.521027 0.260513 − 0.965470i \(-0.416108\pi\)
0.260513 + 0.965470i \(0.416108\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.78626e54 1.38778
\(743\) −7.03234e54 −1.98752 −0.993762 − 0.111526i \(-0.964426\pi\)
−0.993762 + 0.111526i \(0.964426\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −3.07165e54 −0.804145
\(747\) 0 0
\(748\) 0 0
\(749\) 7.58155e54 1.83909
\(750\) 0 0
\(751\) 8.30268e54 1.91452 0.957259 − 0.289233i \(-0.0934001\pi\)
0.957259 + 0.289233i \(0.0934001\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 4.83599e54 0.958658 0.479329 − 0.877635i \(-0.340880\pi\)
0.479329 + 0.877635i \(0.340880\pi\)
\(758\) 6.43874e54 1.24476
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −2.59284e54 −0.442394
\(764\) 7.35223e53 0.122362
\(765\) 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
\(772\) −4.23570e53 −0.0578357
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 1.20974e55 1.57258
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 8.35096e54 0.984292
\(779\) 0 0
\(780\) 0 0
\(781\) −2.85435e55 −3.12707
\(782\) 0 0
\(783\) 0 0
\(784\) −4.69738e54 −0.478466
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 4.29205e54 0.396887
\(789\) 0 0
\(790\) 0 0
\(791\) −2.10490e55 −1.81083
\(792\) −2.22504e55 −1.86879
\(793\) 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
\(800\) −1.02252e55 −0.709516
\(801\) 0 0
\(802\) 1.60474e55 1.06192
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.16794e55 1.77728 0.888640 − 0.458606i \(-0.151651\pi\)
0.888640 + 0.458606i \(0.151651\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −9.93146e54 −0.519336
\(813\) 0 0
\(814\) −3.97807e55 −1.98522
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.49753e55 1.48334 0.741672 − 0.670763i \(-0.234033\pi\)
0.741672 + 0.670763i \(0.234033\pi\)
\(822\) 0 0
\(823\) 4.35479e55 1.76348 0.881739 − 0.471737i \(-0.156373\pi\)
0.881739 + 0.471737i \(0.156373\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.14574e54 0.226970 0.113485 − 0.993540i \(-0.463799\pi\)
0.113485 + 0.993540i \(0.463799\pi\)
\(828\) 2.04834e55 0.739308
\(829\) 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
\(841\) 3.29259e55 0.883921
\(842\) 3.53445e55 0.927667
\(843\) 0 0
\(844\) −2.68534e55 −0.673740
\(845\) 0 0
\(846\) 0 0
\(847\) −8.34415e55 −1.95703
\(848\) −3.67221e55 −0.842183
\(849\) 0 0
\(850\) 0 0
\(851\) 1.33409e56 2.86104
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.04169e56 −1.99864
\(857\) 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
\(862\) −4.08666e55 −0.686642
\(863\) −5.59618e55 −0.919785 −0.459893 − 0.887975i \(-0.652112\pi\)
−0.459893 + 0.887975i \(0.652112\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.05927e56 1.52627
\(870\) 0 0
\(871\) 0 0
\(872\) 3.56249e55 0.480774
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.45541e56 −1.76195 −0.880975 − 0.473162i \(-0.843113\pi\)
−0.880975 + 0.473162i \(0.843113\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −7.25553e55 −0.788435
\(883\) −2.40152e55 −0.255408 −0.127704 − 0.991812i \(-0.540761\pi\)
−0.127704 + 0.991812i \(0.540761\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.58036e56 1.57585
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 3.18784e55 0.298101
\(890\) 0 0
\(891\) −1.91913e56 −1.71960
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.35020e55 0.108779
\(897\) 0 0
\(898\) −4.51991e55 −0.349044
\(899\) 0 0
\(900\) −5.11122e55 −0.378370
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 2.89208e56 1.96793
\(905\) 0 0
\(906\) 0 0
\(907\) −3.06713e56 −1.95972 −0.979859 − 0.199693i \(-0.936006\pi\)
−0.979859 + 0.199693i \(0.936006\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 2.44548e56 1.43719 0.718597 − 0.695427i \(-0.244785\pi\)
0.718597 + 0.695427i \(0.244785\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1.70099e56 0.939126
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 3.96811e56 1.97510 0.987549 − 0.157309i \(-0.0502818\pi\)
0.987549 + 0.157309i \(0.0502818\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −3.32896e56 −1.46425
\(926\) 7.95689e55 0.342874
\(927\) 0 0
\(928\) 2.35454e56 0.973854
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.33004e56 0.506948
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 4.66886e56 1.57528
\(939\) 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
\(946\) −9.41828e56 −2.70421
\(947\) −7.10651e56 −1.99989 −0.999947 − 0.0103204i \(-0.996715\pi\)
−0.999947 + 0.0103204i \(0.996715\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 5.96912e56 1.48986 0.744929 − 0.667143i \(-0.232483\pi\)
0.744929 + 0.667143i \(0.232483\pi\)
\(954\) −5.67206e56 −1.38778
\(955\) 0 0
\(956\) −7.89523e55 −0.185637
\(957\) 0 0
\(958\) 0 0
\(959\) −4.38107e56 −0.970570
\(960\) 0 0
\(961\) 4.69618e56 1.00000
\(962\) 0 0
\(963\) −8.98469e56 −1.83909
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.64479e56 −0.311177 −0.155589 − 0.987822i \(-0.549727\pi\)
−0.155589 + 0.987822i \(0.549727\pi\)
\(968\) 1.14647e57 2.12681
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −9.46015e56 −1.56055
\(975\) 0 0
\(976\) 0 0
\(977\) −1.28534e57 −1.99996 −0.999981 − 0.00624410i \(-0.998012\pi\)
−0.999981 + 0.00624410i \(0.998012\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 3.07270e56 0.442394
\(982\) 1.82637e56 0.257911
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.15852e57 3.89722
\(990\) 0 0
\(991\) 1.11581e57 1.32492 0.662460 − 0.749097i \(-0.269513\pi\)
0.662460 + 0.749097i \(0.269513\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 1.27884e57 1.43376
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −7.56851e56 −0.786194
\(999\) 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 7.39.b.a.6.1 ✓ 1
7.6 odd 2 CM 7.39.b.a.6.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.39.b.a.6.1 ✓ 1 1.1 even 1 trivial
7.39.b.a.6.1 ✓ 1 7.6 odd 2 CM