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Properties

Label 7.37.b.a.6.1
Level $7$
Weight $37$
Character 7.6
Self dual yes
Analytic conductor $57.464$
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] = [7,37,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, 37); 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, 37, names="a")
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 37 \)
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: \(57.4638810615\)
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)
 
\(f(q)\) \(=\) \(q+473713. q^{2} +1.55685e11 q^{4} +1.62841e15 q^{7} +4.11965e16 q^{8} +1.50095e17 q^{9} -2.85537e18 q^{11} +7.71401e20 q^{14} +8.81675e21 q^{16} +7.11018e22 q^{18} -1.35262e24 q^{22} -3.75464e24 q^{23} +1.45519e25 q^{25} +2.53519e26 q^{28} +3.80666e26 q^{29} +1.34561e27 q^{32} +2.33674e28 q^{36} +3.24443e28 q^{37} -3.67293e29 q^{43} -4.44536e29 q^{44} -1.77862e30 q^{46} +2.65173e30 q^{49} +6.89343e30 q^{50} -1.14082e31 q^{53} +6.70849e31 q^{56} +1.80326e32 q^{58} +2.44416e32 q^{63} +3.15495e31 q^{64} -1.27080e33 q^{67} +2.02544e33 q^{71} +6.18337e33 q^{72} +1.53693e34 q^{74} -4.64972e33 q^{77} -1.31079e34 q^{79} +2.25284e34 q^{81} -1.73991e35 q^{86} -1.17631e35 q^{88} -5.84540e35 q^{92} +1.25616e36 q^{98} -4.28575e35 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 473713. 1.80707 0.903536 − 0.428513i \(-0.140962\pi\)
0.903536 + 0.428513i \(0.140962\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1.55685e11 2.26551
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 1.62841e15 1.00000
\(8\) 4.11965e16 2.28686
\(9\) 1.50095e17 1.00000
\(10\) 0 0
\(11\) −2.85537e18 −0.513563 −0.256781 − 0.966470i \(-0.582662\pi\)
−0.256781 + 0.966470i \(0.582662\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 7.71401e20 1.80707
\(15\) 0 0
\(16\) 8.81675e21 1.86702
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 7.11018e22 1.80707
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.35262e24 −0.928045
\(23\) −3.75464e24 −1.15736 −0.578679 − 0.815555i \(-0.696432\pi\)
−0.578679 + 0.815555i \(0.696432\pi\)
\(24\) 0 0
\(25\) 1.45519e25 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 2.53519e26 2.26551
\(29\) 3.80666e26 1.80876 0.904378 − 0.426731i \(-0.140335\pi\)
0.904378 + 0.426731i \(0.140335\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1.34561e27 1.08697
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 2.33674e28 2.26551
\(37\) 3.24443e28 1.92091 0.960456 − 0.278432i \(-0.0898148\pi\)
0.960456 + 0.278432i \(0.0898148\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −3.67293e29 −1.45405 −0.727026 − 0.686610i \(-0.759098\pi\)
−0.727026 + 0.686610i \(0.759098\pi\)
\(44\) −4.44536e29 −1.16348
\(45\) 0 0
\(46\) −1.77862e30 −2.09143
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 2.65173e30 1.00000
\(50\) 6.89343e30 1.80707
\(51\) 0 0
\(52\) 0 0
\(53\) −1.14082e31 −1.04773 −0.523866 − 0.851801i \(-0.675511\pi\)
−0.523866 + 0.851801i \(0.675511\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 6.70849e31 2.28686
\(57\) 0 0
\(58\) 1.80326e32 3.26855
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 2.44416e32 1.00000
\(64\) 3.15495e31 0.0972193
\(65\) 0 0
\(66\) 0 0
\(67\) −1.27080e33 −1.71684 −0.858420 − 0.512948i \(-0.828553\pi\)
−0.858420 + 0.512948i \(0.828553\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.02544e33 0.963538 0.481769 − 0.876298i \(-0.339994\pi\)
0.481769 + 0.876298i \(0.339994\pi\)
\(72\) 6.18337e33 2.28686
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 1.53693e34 3.47123
\(75\) 0 0
\(76\) 0 0
\(77\) −4.64972e33 −0.513563
\(78\) 0 0
\(79\) −1.31079e34 −0.912524 −0.456262 − 0.889845i \(-0.650812\pi\)
−0.456262 + 0.889845i \(0.650812\pi\)
\(80\) 0 0
\(81\) 2.25284e34 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.73991e35 −2.62758
\(87\) 0 0
\(88\) −1.17631e35 −1.17445
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −5.84540e35 −2.62200
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 1.25616e36 1.80707
\(99\) −4.28575e35 −0.513563
\(100\) 2.26551e36 2.26551
\(101\) 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
\(106\) −5.40420e36 −1.89333
\(107\) −6.71643e36 −1.98715 −0.993574 − 0.113182i \(-0.963896\pi\)
−0.993574 + 0.113182i \(0.963896\pi\)
\(108\) 0 0
\(109\) 9.05437e36 1.91947 0.959735 − 0.280906i \(-0.0906350\pi\)
0.959735 + 0.280906i \(0.0906350\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.43573e37 1.86702
\(113\) −1.74874e37 −1.93782 −0.968908 − 0.247421i \(-0.920417\pi\)
−0.968908 + 0.247421i \(0.920417\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.92638e37 4.09775
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.27596e37 −0.736253
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 1.15783e38 1.80707
\(127\) −1.46258e38 −1.97995 −0.989973 − 0.141260i \(-0.954885\pi\)
−0.989973 + 0.141260i \(0.954885\pi\)
\(128\) −7.75240e37 −0.911291
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −6.01993e38 −3.10245
\(135\) 0 0
\(136\) 0 0
\(137\) −5.68614e38 −1.96720 −0.983598 − 0.180374i \(-0.942269\pi\)
−0.983598 + 0.180374i \(0.942269\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 9.59476e38 1.74118
\(143\) 0 0
\(144\) 1.32335e39 1.86702
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 5.05108e39 4.35184
\(149\) −2.37828e39 −1.81514 −0.907569 − 0.419904i \(-0.862064\pi\)
−0.907569 + 0.419904i \(0.862064\pi\)
\(150\) 0 0
\(151\) 1.38925e38 0.0834056 0.0417028 − 0.999130i \(-0.486722\pi\)
0.0417028 + 0.999130i \(0.486722\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −2.20263e39 −0.928045
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) −6.20937e39 −1.64900
\(159\) 0 0
\(160\) 0 0
\(161\) −6.11411e39 −1.15736
\(162\) 1.06720e40 1.80707
\(163\) 7.34741e39 1.11367 0.556836 − 0.830623i \(-0.312015\pi\)
0.556836 + 0.830623i \(0.312015\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.26462e40 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −5.71818e40 −3.29417
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 2.36965e40 1.00000
\(176\) −2.51750e40 −0.958831
\(177\) 0 0
\(178\) 0 0
\(179\) 4.52780e40 1.27213 0.636067 − 0.771634i \(-0.280560\pi\)
0.636067 + 0.771634i \(0.280560\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.54678e41 −2.64672
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.43464e40 −0.824375 −0.412187 − 0.911099i \(-0.635235\pi\)
−0.412187 + 0.911099i \(0.635235\pi\)
\(192\) 0 0
\(193\) 2.68326e41 1.94371 0.971853 − 0.235587i \(-0.0757013\pi\)
0.971853 + 0.235587i \(0.0757013\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 4.12833e41 2.26551
\(197\) 8.75642e40 0.438465 0.219232 − 0.975673i \(-0.429645\pi\)
0.219232 + 0.975673i \(0.429645\pi\)
\(198\) −2.03022e41 −0.928045
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 5.99488e41 2.28686
\(201\) 0 0
\(202\) 0 0
\(203\) 6.19882e41 1.80876
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −5.63552e41 −1.15736
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.27694e42 −1.85817 −0.929085 − 0.369866i \(-0.879404\pi\)
−0.929085 + 0.369866i \(0.879404\pi\)
\(212\) −1.77608e42 −2.37365
\(213\) 0 0
\(214\) −3.18166e42 −3.59092
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 4.28917e42 3.46862
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 2.19121e42 1.08697
\(225\) 2.18416e42 1.00000
\(226\) −8.28400e42 −3.50177
\(227\) 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
\(232\) 1.56821e43 4.13638
\(233\) −6.29020e42 −1.53553 −0.767764 − 0.640733i \(-0.778631\pi\)
−0.767764 + 0.640733i \(0.778631\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.64105e42 1.48923 0.744616 − 0.667494i \(-0.232633\pi\)
0.744616 + 0.667494i \(0.232633\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −1.07815e43 −1.33046
\(243\) 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 \(\pi\)
\(252\) 3.80518e43 2.26551
\(253\) 1.07209e43 0.594376
\(254\) −6.92844e43 −3.57790
\(255\) 0 0
\(256\) −3.88922e43 −1.74399
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 5.28327e43 1.92091
\(260\) 0 0
\(261\) 5.71359e43 1.80876
\(262\) 0 0
\(263\) 6.72139e43 1.85462 0.927312 − 0.374289i \(-0.122113\pi\)
0.927312 + 0.374289i \(0.122113\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.97843e44 −3.88951
\(269\) 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
\(274\) −2.69360e44 −3.55486
\(275\) −4.15510e43 −0.513563
\(276\) 0 0
\(277\) −1.98260e42 −0.0215080 −0.0107540 − 0.999942i \(-0.503423\pi\)
−0.0107540 + 0.999942i \(0.503423\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.05274e44 1.72037 0.860183 − 0.509985i \(-0.170349\pi\)
0.860183 + 0.509985i \(0.170349\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 3.15329e44 2.18290
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 2.01968e44 1.08697
\(289\) 1.97770e44 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
\(296\) 1.33659e45 4.39286
\(297\) 0 0
\(298\) −1.12662e45 −3.28008
\(299\) 0 0
\(300\) 0 0
\(301\) −5.98105e44 −1.45405
\(302\) 6.58106e43 0.150720
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) −7.23889e44 −1.16348
\(309\) 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
\(316\) −2.04069e45 −2.06733
\(317\) −2.75550e44 −0.263714 −0.131857 − 0.991269i \(-0.542094\pi\)
−0.131857 + 0.991269i \(0.542094\pi\)
\(318\) 0 0
\(319\) −1.08694e45 −0.928910
\(320\) 0 0
\(321\) 0 0
\(322\) −2.89634e45 −2.09143
\(323\) 0 0
\(324\) 3.50732e45 2.26551
\(325\) 0 0
\(326\) 3.48056e45 2.01248
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.34472e45 1.03083 0.515416 − 0.856940i \(-0.327638\pi\)
0.515416 + 0.856940i \(0.327638\pi\)
\(332\) 0 0
\(333\) 4.86972e45 1.92091
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −6.23421e45 −1.98355 −0.991776 − 0.127987i \(-0.959149\pi\)
−0.991776 + 0.127987i \(0.959149\pi\)
\(338\) 5.99068e45 1.80707
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 4.31811e45 1.00000
\(344\) −1.51312e46 −3.32522
\(345\) 0 0
\(346\) 0 0
\(347\) −7.92166e45 −1.48897 −0.744484 − 0.667640i \(-0.767304\pi\)
−0.744484 + 0.667640i \(0.767304\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 1.12254e46 1.80707
\(351\) 0 0
\(352\) −3.84220e45 −0.558229
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 2.14488e46 2.29884
\(359\) 3.82768e45 0.390154 0.195077 − 0.980788i \(-0.437504\pi\)
0.195077 + 0.980788i \(0.437504\pi\)
\(360\) 0 0
\(361\) 1.08425e46 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 \(\pi\)
\(368\) −3.31037e46 −2.16081
\(369\) 0 0
\(370\) 0 0
\(371\) −1.85772e46 −1.04773
\(372\) 0 0
\(373\) −3.84597e46 −1.96900 −0.984502 − 0.175371i \(-0.943888\pi\)
−0.984502 + 0.175371i \(0.943888\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.45487e46 0.558882 0.279441 − 0.960163i \(-0.409851\pi\)
0.279441 + 0.960163i \(0.409851\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4.46931e46 −1.48970
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.27109e47 3.51242
\(387\) −5.51287e46 −1.45405
\(388\) 0 0
\(389\) −7.80569e46 −1.87637 −0.938185 − 0.346133i \(-0.887495\pi\)
−0.938185 + 0.346133i \(0.887495\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.09242e47 2.28686
\(393\) 0 0
\(394\) 4.14803e46 0.792337
\(395\) 0 0
\(396\) −6.67225e46 −1.16348
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.28301e47 1.86702
\(401\) 5.26271e46 0.732168 0.366084 − 0.930582i \(-0.380698\pi\)
0.366084 + 0.930582i \(0.380698\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 2.93646e47 3.26855
\(407\) −9.26404e46 −0.986508
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −2.66962e47 −2.09143
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 3.13730e47 1.81751 0.908757 − 0.417326i \(-0.137033\pi\)
0.908757 + 0.417326i \(0.137033\pi\)
\(422\) −6.04902e47 −3.35785
\(423\) 0 0
\(424\) −4.69976e47 −2.39602
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.04564e48 −4.50190
\(429\) 0 0
\(430\) 0 0
\(431\) 5.08865e47 1.93202 0.966010 − 0.258505i \(-0.0832300\pi\)
0.966010 + 0.258505i \(0.0832300\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.40963e48 4.34858
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 3.98011e47 1.00000
\(442\) 0 0
\(443\) 4.47412e47 1.03619 0.518096 − 0.855323i \(-0.326641\pi\)
0.518096 + 0.855323i \(0.326641\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 5.13756e46 0.0972193
\(449\) −1.41290e47 −0.256849 −0.128424 − 0.991719i \(-0.540992\pi\)
−0.128424 + 0.991719i \(0.540992\pi\)
\(450\) 1.03467e48 1.80707
\(451\) 0 0
\(452\) −2.72251e48 −4.39014
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.27460e47 0.697747 0.348873 − 0.937170i \(-0.386564\pi\)
0.348873 + 0.937170i \(0.386564\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 9.77130e47 1.02210 0.511051 − 0.859550i \(-0.329256\pi\)
0.511051 + 0.859550i \(0.329256\pi\)
\(464\) 3.35624e48 3.37698
\(465\) 0 0
\(466\) −2.97975e48 −2.77481
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −2.06938e48 −1.71684
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.04876e48 0.746747
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.71231e48 −1.04773
\(478\) 4.56709e48 2.69115
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −3.54331e48 −1.66799
\(485\) 0 0
\(486\) 0 0
\(487\) −1.29834e48 −0.546847 −0.273423 − 0.961894i \(-0.588156\pi\)
−0.273423 + 0.961894i \(0.588156\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −2.16105e48 −0.785593 −0.392796 − 0.919626i \(-0.628492\pi\)
−0.392796 + 0.919626i \(0.628492\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.29825e48 0.963538
\(498\) 0 0
\(499\) −4.99276e48 −1.35685 −0.678424 − 0.734671i \(-0.737337\pi\)
−0.678424 + 0.734671i \(0.737337\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 1.00691e49 2.28686
\(505\) 0 0
\(506\) 5.07862e48 1.07408
\(507\) 0 0
\(508\) −2.27701e49 −4.48558
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.30963e49 −2.24022
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 2.50276e49 3.47123
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 2.70660e49 3.26855
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 3.18401e49 3.35144
\(527\) 0 0
\(528\) 0 0
\(529\) 3.57284e48 0.339478
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −5.23524e49 −3.92618
\(537\) 0 0
\(538\) 0 0
\(539\) −7.57166e48 −0.513563
\(540\) 0 0
\(541\) −1.04752e49 −0.664675 −0.332338 − 0.943160i \(-0.607837\pi\)
−0.332338 + 0.943160i \(0.607837\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.82443e49 1.98973 0.994865 − 0.101208i \(-0.0322707\pi\)
0.994865 + 0.101208i \(0.0322707\pi\)
\(548\) −8.85245e49 −4.45670
\(549\) 0 0
\(550\) −1.96833e49 −0.928045
\(551\) 0 0
\(552\) 0 0
\(553\) −2.13450e49 −0.912524
\(554\) −9.39185e47 −0.0388665
\(555\) 0 0
\(556\) 0 0
\(557\) −1.50628e49 −0.565604 −0.282802 − 0.959178i \(-0.591264\pi\)
−0.282802 + 0.959178i \(0.591264\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 9.72411e49 3.10882
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.66856e49 1.00000
\(568\) 8.34409e49 2.20348
\(569\) −1.91774e48 −0.0490646 −0.0245323 − 0.999699i \(-0.507810\pi\)
−0.0245323 + 0.999699i \(0.507810\pi\)
\(570\) 0 0
\(571\) −8.26191e49 −1.98441 −0.992203 − 0.124632i \(-0.960225\pi\)
−0.992203 + 0.124632i \(0.960225\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.46373e49 −1.15736
\(576\) 4.73540e48 0.0972193
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 9.36864e49 1.80707
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 3.25745e49 0.538076
\(584\) 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
\(592\) 2.86053e50 3.58638
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.70261e50 −4.11221
\(597\) 0 0
\(598\) 0 0
\(599\) 1.71257e50 1.73767 0.868833 − 0.495105i \(-0.164870\pi\)
0.868833 + 0.495105i \(0.164870\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) −2.83330e50 −2.62758
\(603\) −1.90740e50 −1.71684
\(604\) 2.16285e49 0.188956
\(605\) 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
\(613\) 2.97891e50 1.99420 0.997100 − 0.0760971i \(-0.0242459\pi\)
0.997100 + 0.0760971i \(0.0242459\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −1.91552e50 −1.17445
\(617\) 1.87398e50 1.11592 0.557958 − 0.829869i \(-0.311585\pi\)
0.557958 + 0.829869i \(0.311585\pi\)
\(618\) 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
\(625\) 2.11758e50 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 3.13434e50 1.24629 0.623143 − 0.782108i \(-0.285855\pi\)
0.623143 + 0.782108i \(0.285855\pi\)
\(632\) −5.39998e50 −2.08682
\(633\) 0 0
\(634\) −1.30532e50 −0.476551
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −5.14898e50 −1.67861
\(639\) 3.04007e50 0.963538
\(640\) 0 0
\(641\) −6.56379e50 −1.96657 −0.983286 − 0.182065i \(-0.941722\pi\)
−0.983286 + 0.182065i \(0.941722\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −9.51873e50 −2.62200
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 9.28091e50 2.28686
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 1.14388e51 2.52303
\(653\) 3.15274e50 0.676473 0.338236 − 0.941061i \(-0.390170\pi\)
0.338236 + 0.941061i \(0.390170\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 8.81359e50 1.60404 0.802020 − 0.597297i \(-0.203759\pi\)
0.802020 + 0.597297i \(0.203759\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 1.11073e51 1.86279
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 2.30685e51 3.47123
\(667\) −1.42927e51 −2.09338
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.57111e51 −1.95856 −0.979278 − 0.202521i \(-0.935087\pi\)
−0.979278 + 0.202521i \(0.935087\pi\)
\(674\) −2.95323e51 −3.58442
\(675\) 0 0
\(676\) 1.96882e51 2.26551
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.82678e51 −1.74628 −0.873140 − 0.487469i \(-0.837920\pi\)
−0.873140 + 0.487469i \(0.837920\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 2.04555e51 1.80707
\(687\) 0 0
\(688\) −3.23833e51 −2.71474
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) −6.97898e50 −0.513563
\(694\) −3.75259e51 −2.69067
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 3.68918e51 2.26551
\(701\) 6.99111e50 0.418429 0.209214 − 0.977870i \(-0.432909\pi\)
0.209214 + 0.977870i \(0.432909\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −9.00852e49 −0.0499282
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.47317e51 −1.20676 −0.603380 − 0.797454i \(-0.706180\pi\)
−0.603380 + 0.797454i \(0.706180\pi\)
\(710\) 0 0
\(711\) −1.96742e51 −0.912524
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 7.04909e51 2.88203
\(717\) 0 0
\(718\) 1.81322e51 0.705036
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 5.13624e51 1.80707
\(723\) 0 0
\(724\) 0 0
\(725\) 5.53942e51 1.80876
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 3.38139e51 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
\(736\) −5.05228e51 −1.25802
\(737\) 3.62859e51 0.881705
\(738\) 0 0
\(739\) −2.79256e51 −0.646254 −0.323127 − 0.946356i \(-0.604734\pi\)
−0.323127 + 0.946356i \(0.604734\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −8.80027e51 −1.89333
\(743\) −1.78517e51 −0.374869 −0.187435 − 0.982277i \(-0.560017\pi\)
−0.187435 + 0.982277i \(0.560017\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.82189e52 −3.55813
\(747\) 0 0
\(748\) 0 0
\(749\) −1.09371e52 −1.98715
\(750\) 0 0
\(751\) 8.72747e51 1.51137 0.755683 − 0.654938i \(-0.227305\pi\)
0.755683 + 0.654938i \(0.227305\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.03309e52 1.55029 0.775147 − 0.631781i \(-0.217676\pi\)
0.775147 + 0.631781i \(0.217676\pi\)
\(758\) 6.89193e51 1.00994
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.47443e52 1.91947
\(764\) −1.46883e52 −1.86763
\(765\) 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
\(772\) 4.17742e52 4.40348
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −2.61152e52 −2.62758
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −3.69766e52 −3.39074
\(779\) 0 0
\(780\) 0 0
\(781\) −5.78337e51 −0.494837
\(782\) 0 0
\(783\) 0 0
\(784\) 2.33796e52 1.86702
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 1.36324e52 0.993346
\(789\) 0 0
\(790\) 0 0
\(791\) −2.84767e52 −1.93782
\(792\) −1.76558e52 −1.17445
\(793\) 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
\(800\) 1.95812e52 1.08697
\(801\) 0 0
\(802\) 2.49302e52 1.32308
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.58806e52 0.720764 0.360382 − 0.932805i \(-0.382646\pi\)
0.360382 + 0.932805i \(0.382646\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 9.65060e52 4.09775
\(813\) 0 0
\(814\) −4.38849e52 −1.78269
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.72854e51 −0.0950064 −0.0475032 − 0.998871i \(-0.515126\pi\)
−0.0475032 + 0.998871i \(0.515126\pi\)
\(822\) 0 0
\(823\) −4.62054e52 −1.53991 −0.769955 − 0.638098i \(-0.779722\pi\)
−0.769955 + 0.638098i \(0.779722\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.53745e52 1.99668 0.998341 − 0.0575707i \(-0.0183355\pi\)
0.998341 + 0.0575707i \(0.0183355\pi\)
\(828\) −8.77363e52 −2.62200
\(829\) 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
\(841\) 1.00614e53 2.27160
\(842\) 1.48618e53 3.28438
\(843\) 0 0
\(844\) −1.98799e53 −4.20970
\(845\) 0 0
\(846\) 0 0
\(847\) −3.70620e52 −0.736253
\(848\) −1.00583e53 −1.95614
\(849\) 0 0
\(850\) 0 0
\(851\) −1.21817e53 −2.22318
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2.76693e53 −4.54434
\(857\) 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
\(862\) 2.41056e53 3.49130
\(863\) 2.86622e52 0.406552 0.203276 − 0.979122i \(-0.434841\pi\)
0.203276 + 0.979122i \(0.434841\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 3.74278e52 0.468638
\(870\) 0 0
\(871\) 0 0
\(872\) 3.73008e53 4.38957
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.00861e52 0.107085 0.0535426 − 0.998566i \(-0.482949\pi\)
0.0535426 + 0.998566i \(0.482949\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 1.88543e53 1.80707
\(883\) 1.08523e53 1.01913 0.509567 − 0.860431i \(-0.329806\pi\)
0.509567 + 0.860431i \(0.329806\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 2.11945e53 1.87247
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −2.38169e53 −1.97995
\(890\) 0 0
\(891\) −6.43268e52 −0.513563
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −1.26241e53 −0.911291
\(897\) 0 0
\(898\) −6.69309e52 −0.464144
\(899\) 0 0
\(900\) 3.40041e53 2.26551
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −7.20418e53 −4.43152
\(905\) 0 0
\(906\) 0 0
\(907\) −3.45004e53 −1.99937 −0.999685 − 0.0251133i \(-0.992005\pi\)
−0.999685 + 0.0251133i \(0.992005\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3.63509e53 −1.94619 −0.973094 − 0.230409i \(-0.925994\pi\)
−0.973094 + 0.230409i \(0.925994\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 2.49865e53 1.26088
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 4.29407e52 0.196422 0.0982109 − 0.995166i \(-0.468688\pi\)
0.0982109 + 0.995166i \(0.468688\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 4.72127e53 1.92091
\(926\) 4.62879e53 1.84701
\(927\) 0 0
\(928\) 5.12227e53 1.96607
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −9.79287e53 −3.47875
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) −9.80294e53 −3.10245
\(939\) 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
\(946\) 4.96809e53 1.34943
\(947\) 5.96700e53 1.59022 0.795108 − 0.606468i \(-0.207414\pi\)
0.795108 + 0.606468i \(0.207414\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.40426e53 1.99907 0.999534 − 0.0305342i \(-0.00972086\pi\)
0.999534 + 0.0305342i \(0.00972086\pi\)
\(954\) −8.11141e53 −1.89333
\(955\) 0 0
\(956\) 1.50096e54 3.37387
\(957\) 0 0
\(958\) 0 0
\(959\) −9.25939e53 −1.96720
\(960\) 0 0
\(961\) 4.88676e53 1.00000
\(962\) 0 0
\(963\) −1.00810e54 −1.98715
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.09305e54 1.99970 0.999850 − 0.0173452i \(-0.00552141\pi\)
0.999850 + 0.0173452i \(0.00552141\pi\)
\(968\) −9.37614e53 −1.68371
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −6.15038e53 −0.988191
\(975\) 0 0
\(976\) 0 0
\(977\) −1.00886e53 −0.153365 −0.0766826 − 0.997056i \(-0.524433\pi\)
−0.0766826 + 0.997056i \(0.524433\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.35901e54 1.91947
\(982\) −1.02372e54 −1.41962
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.37905e54 1.68286
\(990\) 0 0
\(991\) 9.86492e52 0.116083 0.0580414 − 0.998314i \(-0.481514\pi\)
0.0580414 + 0.998314i \(0.481514\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 1.56242e54 1.74118
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −2.36514e54 −2.45192
\(999\) 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 7.37.b.a.6.1 ✓ 1
7.6 odd 2 CM 7.37.b.a.6.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.37.b.a.6.1 ✓ 1 1.1 even 1 trivial
7.37.b.a.6.1 ✓ 1 7.6 odd 2 CM