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Properties

Label 7.91.b.a.6.1
Level $7$
Weight $91$
Character 7.6
Self dual yes
Analytic conductor $359.072$
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] = [7,91,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, 91); 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, 91, names="a")
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 91 \)
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: \(359.071719566\)
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)
 
\(f(q)\) \(=\) \(q-3.14720e13 q^{2} -2.47453e26 q^{4} -1.07007e38 q^{7} +4.67483e40 q^{8} +8.72796e42 q^{9} +1.97900e46 q^{11} +3.36772e51 q^{14} -1.16493e54 q^{16} -2.74687e56 q^{18} -6.22832e59 q^{22} -2.60464e61 q^{23} +8.07794e62 q^{25} +2.64792e64 q^{28} +5.80413e65 q^{29} -2.12089e67 q^{32} -2.15976e69 q^{36} +5.64536e70 q^{37} +6.08563e73 q^{43} -4.89710e72 q^{44} +8.19734e74 q^{46} +1.14505e76 q^{49} -2.54229e76 q^{50} -4.37300e77 q^{53} -5.00239e78 q^{56} -1.82668e79 q^{58} -9.33952e80 q^{63} +2.10960e81 q^{64} +2.90658e82 q^{67} -3.61016e83 q^{71} +4.08017e83 q^{72} -1.77671e84 q^{74} -2.11767e84 q^{77} +1.92183e85 q^{79} +7.61773e85 q^{81} -1.91527e87 q^{86} +9.25150e86 q^{88} +6.44527e87 q^{92} -3.60369e89 q^{98} +1.72727e89 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.14720e13 −0.894488 −0.447244 − 0.894412i \(-0.647594\pi\)
−0.447244 + 0.894412i \(0.647594\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −2.47453e26 −0.199891
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −1.07007e38 −1.00000
\(8\) 4.67483e40 1.07329
\(9\) 8.72796e42 1.00000
\(10\) 0 0
\(11\) 1.97900e46 0.271504 0.135752 − 0.990743i \(-0.456655\pi\)
0.135752 + 0.990743i \(0.456655\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 3.36772e51 0.894488
\(15\) 0 0
\(16\) −1.16493e54 −0.760153
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −2.74687e56 −0.894488
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.22832e59 −0.242857
\(23\) −2.60464e61 −1.37403 −0.687014 − 0.726644i \(-0.741079\pi\)
−0.687014 + 0.726644i \(0.741079\pi\)
\(24\) 0 0
\(25\) 8.07794e62 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 2.64792e64 0.199891
\(29\) 5.80413e65 0.903290 0.451645 − 0.892198i \(-0.350837\pi\)
0.451645 + 0.892198i \(0.350837\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −2.12089e67 −0.393341
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −2.15976e69 −0.199891
\(37\) 5.64536e70 1.52270 0.761350 − 0.648341i \(-0.224537\pi\)
0.761350 + 0.648341i \(0.224537\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 6.08563e73 1.89769 0.948844 − 0.315745i \(-0.102254\pi\)
0.948844 + 0.315745i \(0.102254\pi\)
\(44\) −4.89710e72 −0.0542711
\(45\) 0 0
\(46\) 8.19734e74 1.22905
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.14505e76 1.00000
\(50\) −2.54229e76 −0.894488
\(51\) 0 0
\(52\) 0 0
\(53\) −4.37300e77 −1.11780 −0.558902 − 0.829234i \(-0.688777\pi\)
−0.558902 + 0.829234i \(0.688777\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −5.00239e78 −1.07329
\(57\) 0 0
\(58\) −1.82668e79 −0.807982
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −9.33952e80 −1.00000
\(64\) 2.10960e81 1.11199
\(65\) 0 0
\(66\) 0 0
\(67\) 2.90658e82 1.94992 0.974959 − 0.222384i \(-0.0713837\pi\)
0.974959 + 0.222384i \(0.0713837\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.61016e83 −1.78197 −0.890987 − 0.454029i \(-0.849986\pi\)
−0.890987 + 0.454029i \(0.849986\pi\)
\(72\) 4.08017e83 1.07329
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −1.77671e84 −1.36204
\(75\) 0 0
\(76\) 0 0
\(77\) −2.11767e84 −0.271504
\(78\) 0 0
\(79\) 1.92183e85 0.777136 0.388568 − 0.921420i \(-0.372970\pi\)
0.388568 + 0.921420i \(0.372970\pi\)
\(80\) 0 0
\(81\) 7.61773e85 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.91527e87 −1.69746
\(87\) 0 0
\(88\) 9.25150e86 0.291402
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.44527e87 0.274656
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −3.60369e89 −0.894488
\(99\) 1.72727e89 0.271504
\(100\) −1.99891e89 −0.199891
\(101\) 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
\(106\) 1.37627e91 0.999862
\(107\) −1.17520e91 −0.559554 −0.279777 − 0.960065i \(-0.590261\pi\)
−0.279777 + 0.960065i \(0.590261\pi\)
\(108\) 0 0
\(109\) −7.31827e91 −1.51431 −0.757157 − 0.653233i \(-0.773412\pi\)
−0.757157 + 0.653233i \(0.773412\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.24656e92 0.760153
\(113\) 2.86295e92 1.17027 0.585133 − 0.810938i \(-0.301042\pi\)
0.585133 + 0.810938i \(0.301042\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.43625e92 −0.180559
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.92138e93 −0.926286
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 2.93934e94 0.894488
\(127\) 3.25450e94 0.693932 0.346966 − 0.937878i \(-0.387212\pi\)
0.346966 + 0.937878i \(0.387212\pi\)
\(128\) −4.01380e94 −0.601322
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −9.14760e95 −1.74418
\(135\) 0 0
\(136\) 0 0
\(137\) 1.24443e96 0.876080 0.438040 − 0.898956i \(-0.355673\pi\)
0.438040 + 0.898956i \(0.355673\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.13619e97 1.59395
\(143\) 0 0
\(144\) −1.01675e97 −0.760153
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −1.39696e97 −0.304374
\(149\) −1.09808e98 −1.76706 −0.883531 − 0.468373i \(-0.844840\pi\)
−0.883531 + 0.468373i \(0.844840\pi\)
\(150\) 0 0
\(151\) 1.75931e98 1.55375 0.776874 − 0.629656i \(-0.216804\pi\)
0.776874 + 0.629656i \(0.216804\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 6.66473e97 0.242857
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) −6.04840e98 −0.695139
\(159\) 0 0
\(160\) 0 0
\(161\) 2.78715e99 1.37403
\(162\) −2.39745e99 −0.894488
\(163\) −5.44873e99 −1.54118 −0.770590 − 0.637332i \(-0.780038\pi\)
−0.770590 + 0.637332i \(0.780038\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.79846e100 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −1.50591e100 −0.379331
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −8.64395e100 −1.00000
\(176\) −2.30540e100 −0.206384
\(177\) 0 0
\(178\) 0 0
\(179\) −2.82652e101 −1.18268 −0.591338 − 0.806424i \(-0.701400\pi\)
−0.591338 + 0.806424i \(0.701400\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.21763e102 −1.47473
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.42125e102 −0.546434 −0.273217 − 0.961952i \(-0.588088\pi\)
−0.273217 + 0.961952i \(0.588088\pi\)
\(192\) 0 0
\(193\) −1.17313e103 −1.65679 −0.828397 − 0.560141i \(-0.810747\pi\)
−0.828397 + 0.560141i \(0.810747\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.83345e102 −0.199891
\(197\) −3.46841e103 −1.94604 −0.973018 − 0.230731i \(-0.925888\pi\)
−0.973018 + 0.230731i \(0.925888\pi\)
\(198\) −5.43605e102 −0.242857
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 3.77630e103 1.07329
\(201\) 0 0
\(202\) 0 0
\(203\) −6.21082e103 −0.903290
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.27332e104 −1.37403
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −6.35578e104 −1.62352 −0.811761 − 0.583989i \(-0.801491\pi\)
−0.811761 + 0.583989i \(0.801491\pi\)
\(212\) 1.08211e104 0.223439
\(213\) 0 0
\(214\) 3.69859e104 0.500515
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 2.30321e105 1.35454
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 2.26950e105 0.393341
\(225\) 7.05039e105 1.00000
\(226\) −9.01029e105 −1.04679
\(227\) 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
\(232\) 2.71333e106 0.969490
\(233\) 6.69732e106 1.97190 0.985949 − 0.167047i \(-0.0534231\pi\)
0.985949 + 0.167047i \(0.0534231\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −5.40648e106 −0.507002 −0.253501 − 0.967335i \(-0.581582\pi\)
−0.253501 + 0.967335i \(0.581582\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 1.54886e107 0.828552
\(243\) 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 \(\pi\)
\(252\) 2.31109e107 0.199891
\(253\) −5.15460e107 −0.373054
\(254\) −1.02426e108 −0.620714
\(255\) 0 0
\(256\) −1.34833e108 −0.574115
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −6.04093e108 −1.52270
\(260\) 0 0
\(261\) 5.06582e108 0.903290
\(262\) 0 0
\(263\) −9.08155e108 −1.14855 −0.574277 − 0.818661i \(-0.694717\pi\)
−0.574277 + 0.818661i \(0.694717\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −7.19242e108 −0.389771
\(269\) 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
\(274\) −3.91646e109 −0.783643
\(275\) 1.59863e109 0.271504
\(276\) 0 0
\(277\) 1.12230e110 1.37569 0.687843 − 0.725860i \(-0.258558\pi\)
0.687843 + 0.725860i \(0.258558\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.17808e109 0.461555 0.230778 − 0.973007i \(-0.425873\pi\)
0.230778 + 0.973007i \(0.425873\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 8.93346e109 0.356200
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.85110e110 −0.393341
\(289\) 5.50051e110 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
\(296\) 2.63911e111 1.63430
\(297\) 0 0
\(298\) 3.45588e111 1.58062
\(299\) 0 0
\(300\) 0 0
\(301\) −6.51204e111 −1.89769
\(302\) −5.53691e111 −1.38981
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 5.24023e110 0.0542711
\(309\) 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
\(316\) −4.75564e111 −0.155342
\(317\) −3.10052e112 −0.878552 −0.439276 − 0.898352i \(-0.644765\pi\)
−0.439276 + 0.898352i \(0.644765\pi\)
\(318\) 0 0
\(319\) 1.14864e112 0.245246
\(320\) 0 0
\(321\) 0 0
\(322\) −8.77172e112 −1.22905
\(323\) 0 0
\(324\) −1.88503e112 −0.199891
\(325\) 0 0
\(326\) 1.71482e113 1.37857
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.15924e113 1.68560 0.842799 − 0.538228i \(-0.180906\pi\)
0.842799 + 0.538228i \(0.180906\pi\)
\(332\) 0 0
\(333\) 4.92725e113 1.52270
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.49298e113 0.630740 0.315370 − 0.948969i \(-0.397871\pi\)
0.315370 + 0.948969i \(0.397871\pi\)
\(338\) −5.66013e113 −0.894488
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.22528e114 −1.00000
\(344\) 2.84493e114 2.03677
\(345\) 0 0
\(346\) 0 0
\(347\) 3.99372e114 1.93442 0.967210 − 0.253977i \(-0.0817388\pi\)
0.967210 + 0.253977i \(0.0817388\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 2.72042e114 0.894488
\(351\) 0 0
\(352\) −4.19724e113 −0.106793
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 8.89563e114 1.05789
\(359\) −1.82457e115 −1.91386 −0.956930 − 0.290318i \(-0.906239\pi\)
−0.956930 + 0.290318i \(0.906239\pi\)
\(360\) 0 0
\(361\) 1.22412e115 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 \(\pi\)
\(368\) 3.03423e115 1.04447
\(369\) 0 0
\(370\) 0 0
\(371\) 4.67941e115 1.11780
\(372\) 0 0
\(373\) 4.54910e115 0.853155 0.426578 − 0.904451i \(-0.359719\pi\)
0.426578 + 0.904451i \(0.359719\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.18018e116 1.99402 0.997009 − 0.0772919i \(-0.0246274\pi\)
0.997009 + 0.0772919i \(0.0246274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 7.62015e115 0.488779
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.69209e116 1.48198
\(387\) 5.31151e116 1.89769
\(388\) 0 0
\(389\) 5.45711e116 1.54608 0.773038 − 0.634360i \(-0.218736\pi\)
0.773038 + 0.634360i \(0.218736\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 5.35290e116 1.07329
\(393\) 0 0
\(394\) 1.09158e117 1.74071
\(395\) 0 0
\(396\) −4.27417e115 −0.0542711
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −9.41024e116 −0.760153
\(401\) 2.73853e117 1.97706 0.988531 − 0.151016i \(-0.0482545\pi\)
0.988531 + 0.151016i \(0.0482545\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 1.95467e117 0.807982
\(407\) 1.11722e117 0.413418
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 7.15460e117 1.22905
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −1.17511e118 −0.949260 −0.474630 − 0.880185i \(-0.657418\pi\)
−0.474630 + 0.880185i \(0.657418\pi\)
\(422\) 2.00029e118 1.45222
\(423\) 0 0
\(424\) −2.04430e118 −1.19973
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 2.90807e117 0.111850
\(429\) 0 0
\(430\) 0 0
\(431\) 5.65255e118 1.58769 0.793847 − 0.608117i \(-0.208075\pi\)
0.793847 + 0.608117i \(0.208075\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.81093e118 0.302698
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 9.99394e118 1.00000
\(442\) 0 0
\(443\) −2.05462e119 −1.67712 −0.838561 − 0.544808i \(-0.816603\pi\)
−0.838561 + 0.544808i \(0.816603\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −2.25742e119 −1.11199
\(449\) 2.00658e119 0.894070 0.447035 − 0.894516i \(-0.352480\pi\)
0.447035 + 0.894516i \(0.352480\pi\)
\(450\) −2.21890e119 −0.894488
\(451\) 0 0
\(452\) −7.08446e118 −0.233925
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −9.88181e119 −1.98888 −0.994438 − 0.105328i \(-0.966411\pi\)
−0.994438 + 0.105328i \(0.966411\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.51836e120 −1.69915 −0.849574 − 0.527470i \(-0.823141\pi\)
−0.849574 + 0.527470i \(0.823141\pi\)
\(464\) −6.76141e119 −0.686638
\(465\) 0 0
\(466\) −2.10778e120 −1.76384
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −3.11024e120 −1.94992
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.20435e120 0.515229
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −3.81674e120 −1.11780
\(478\) 1.70153e120 0.453507
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.21781e120 0.185156
\(485\) 0 0
\(486\) 0 0
\(487\) −1.61360e120 −0.185777 −0.0928887 − 0.995676i \(-0.529610\pi\)
−0.0928887 + 0.995676i \(0.529610\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −5.57036e120 −0.443829 −0.221915 − 0.975066i \(-0.571231\pi\)
−0.221915 + 0.975066i \(0.571231\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.86313e121 1.78197
\(498\) 0 0
\(499\) 4.57811e121 1.76263 0.881316 − 0.472528i \(-0.156658\pi\)
0.881316 + 0.472528i \(0.156658\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) −4.36607e121 −1.07329
\(505\) 0 0
\(506\) 1.62225e121 0.333692
\(507\) 0 0
\(508\) −8.05336e120 −0.138711
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 9.21233e121 1.11486
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 1.90120e122 1.36204
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) −1.59432e122 −0.807982
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 2.85815e122 1.02737
\(527\) 0 0
\(528\) 0 0
\(529\) 3.19077e122 0.887954
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 1.35878e123 2.09282
\(537\) 0 0
\(538\) 0 0
\(539\) 2.26605e122 0.271504
\(540\) 0 0
\(541\) 1.21309e122 0.123031 0.0615155 − 0.998106i \(-0.480407\pi\)
0.0615155 + 0.998106i \(0.480407\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.13587e123 1.93610 0.968052 − 0.250749i \(-0.0806769\pi\)
0.968052 + 0.250749i \(0.0806769\pi\)
\(548\) −3.07937e122 −0.175120
\(549\) 0 0
\(550\) −5.03120e122 −0.242857
\(551\) 0 0
\(552\) 0 0
\(553\) −2.05650e123 −0.777136
\(554\) −3.53210e123 −1.23053
\(555\) 0 0
\(556\) 0 0
\(557\) 5.01856e122 0.137118 0.0685592 − 0.997647i \(-0.478160\pi\)
0.0685592 + 0.997647i \(0.478160\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −2.25909e123 −0.412856
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −8.15150e123 −1.00000
\(568\) −1.68769e124 −1.91257
\(569\) −1.26539e124 −1.32486 −0.662432 − 0.749122i \(-0.730476\pi\)
−0.662432 + 0.749122i \(0.730476\pi\)
\(570\) 0 0
\(571\) −6.87495e123 −0.614674 −0.307337 − 0.951601i \(-0.599438\pi\)
−0.307337 + 0.951601i \(0.599438\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.10401e124 −1.37403
\(576\) 1.84125e124 1.11199
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −1.73112e124 −0.894488
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −8.65418e123 −0.303488
\(584\) 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
\(592\) −6.57646e124 −1.15748
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.71723e124 0.353219
\(597\) 0 0
\(598\) 0 0
\(599\) −5.25385e124 −0.544842 −0.272421 − 0.962178i \(-0.587824\pi\)
−0.272421 + 0.962178i \(0.587824\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 2.04947e125 1.69746
\(603\) 2.53685e125 1.94992
\(604\) −4.35347e124 −0.310580
\(605\) 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
\(613\) 5.35586e125 1.96385 0.981924 − 0.189278i \(-0.0606148\pi\)
0.981924 + 0.189278i \(0.0606148\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −9.89974e124 −0.291402
\(617\) −5.61652e125 −1.53686 −0.768431 − 0.639933i \(-0.778962\pi\)
−0.768431 + 0.639933i \(0.778962\pi\)
\(618\) 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
\(625\) 6.52530e125 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.25252e126 1.24871 0.624356 − 0.781140i \(-0.285361\pi\)
0.624356 + 0.781140i \(0.285361\pi\)
\(632\) 8.98425e125 0.834091
\(633\) 0 0
\(634\) 9.75795e125 0.785854
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −3.61500e125 −0.219370
\(639\) −3.15094e126 −1.78197
\(640\) 0 0
\(641\) 1.79873e126 0.883801 0.441900 − 0.897064i \(-0.354304\pi\)
0.441900 + 0.897064i \(0.354304\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −6.89688e125 −0.274656
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 3.56116e126 1.07329
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 1.34830e126 0.308068
\(653\) 9.35034e126 1.99404 0.997021 − 0.0771309i \(-0.0245759\pi\)
0.997021 + 0.0771309i \(0.0245759\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.17751e125 0.0590301 0.0295151 − 0.999564i \(-0.490604\pi\)
0.0295151 + 0.999564i \(0.490604\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −1.30900e127 −1.50775
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −1.55071e127 −1.36204
\(667\) −1.51177e127 −1.24115
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.77890e127 −0.976055 −0.488027 − 0.872828i \(-0.662283\pi\)
−0.488027 + 0.872828i \(0.662283\pi\)
\(674\) −1.09931e127 −0.564189
\(675\) 0 0
\(676\) −4.45035e126 −0.199891
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −6.76720e127 −1.91195 −0.955977 − 0.293441i \(-0.905199\pi\)
−0.955977 + 0.293441i \(0.905199\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 3.85620e127 0.894488
\(687\) 0 0
\(688\) −7.08933e127 −1.44253
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) −1.84829e127 −0.271504
\(694\) −1.25690e128 −1.73032
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 2.13897e127 0.199891
\(701\) 2.20633e128 1.93356 0.966782 − 0.255601i \(-0.0822733\pi\)
0.966782 + 0.255601i \(0.0822733\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.17490e127 0.301910
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.81633e128 1.48116 0.740581 − 0.671967i \(-0.234550\pi\)
0.740581 + 0.671967i \(0.234550\pi\)
\(710\) 0 0
\(711\) 1.67737e128 0.777136
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 6.99431e127 0.236406
\(717\) 0 0
\(718\) 5.74229e128 1.71193
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −3.85255e128 −0.894488
\(723\) 0 0
\(724\) 0 0
\(725\) 4.68854e128 0.903290
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 6.64873e128 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
\(736\) 5.52416e128 0.540461
\(737\) 5.75213e128 0.529410
\(738\) 0 0
\(739\) 9.12545e127 0.0743455 0.0371727 − 0.999309i \(-0.488165\pi\)
0.0371727 + 0.999309i \(0.488165\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.47270e129 −0.999862
\(743\) −9.66779e128 −0.617775 −0.308888 − 0.951099i \(-0.599957\pi\)
−0.308888 + 0.951099i \(0.599957\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.43169e129 −0.763138
\(747\) 0 0
\(748\) 0 0
\(749\) 1.25755e129 0.559554
\(750\) 0 0
\(751\) 1.07852e129 0.425628 0.212814 − 0.977093i \(-0.431737\pi\)
0.212814 + 0.977093i \(0.431737\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.00327e129 0.276760 0.138380 − 0.990379i \(-0.455810\pi\)
0.138380 + 0.990379i \(0.455810\pi\)
\(758\) −6.86148e129 −1.78362
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 7.83106e129 1.51431
\(764\) 5.99145e128 0.109227
\(765\) 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
\(772\) 2.90295e129 0.331178
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −1.67164e130 −1.69746
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −1.71746e130 −1.38295
\(779\) 0 0
\(780\) 0 0
\(781\) −7.14452e129 −0.483812
\(782\) 0 0
\(783\) 0 0
\(784\) −1.33390e130 −0.760153
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 8.58269e129 0.388995
\(789\) 0 0
\(790\) 0 0
\(791\) −3.06356e130 −1.17027
\(792\) 8.07468e129 0.291402
\(793\) 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
\(800\) −1.71324e130 −0.393341
\(801\) 0 0
\(802\) −8.61870e130 −1.76846
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.42780e131 1.98145 0.990726 − 0.135874i \(-0.0433842\pi\)
0.990726 + 0.135874i \(0.0433842\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 1.53688e130 0.180559
\(813\) 0 0
\(814\) −3.51611e130 −0.369798
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.72855e131 1.23663 0.618314 − 0.785931i \(-0.287816\pi\)
0.618314 + 0.785931i \(0.287816\pi\)
\(822\) 0 0
\(823\) 3.07958e131 1.97469 0.987344 − 0.158594i \(-0.0506961\pi\)
0.987344 + 0.158594i \(0.0506961\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.83936e131 1.97930 0.989649 − 0.143509i \(-0.0458387\pi\)
0.989649 + 0.143509i \(0.0458387\pi\)
\(828\) 5.62540e130 0.274656
\(829\) 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
\(841\) −7.59971e130 −0.184068
\(842\) 3.69832e131 0.849102
\(843\) 0 0
\(844\) 1.57276e131 0.324527
\(845\) 0 0
\(846\) 0 0
\(847\) 5.26621e131 0.926286
\(848\) 5.09424e131 0.849701
\(849\) 0 0
\(850\) 0 0
\(851\) −1.47042e132 −2.09223
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −5.49387e131 −0.600563
\(857\) 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
\(862\) −1.77897e132 −1.42017
\(863\) −2.54126e132 −1.92559 −0.962793 − 0.270240i \(-0.912897\pi\)
−0.962793 + 0.270240i \(0.912897\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 3.80332e131 0.210995
\(870\) 0 0
\(871\) 0 0
\(872\) −3.42117e132 −1.62530
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 4.32994e132 1.59038 0.795189 − 0.606362i \(-0.207372\pi\)
0.795189 + 0.606362i \(0.207372\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −3.14529e132 −0.894488
\(883\) −6.30406e132 −1.70368 −0.851842 − 0.523799i \(-0.824514\pi\)
−0.851842 + 0.523799i \(0.824514\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 6.46631e132 1.50017
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −3.48254e132 −0.693932
\(890\) 0 0
\(891\) 1.50755e132 0.271504
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 4.29505e132 0.601322
\(897\) 0 0
\(898\) −6.31512e132 −0.799735
\(899\) 0 0
\(900\) −1.74464e132 −0.199891
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 1.33838e133 1.25603
\(905\) 0 0
\(906\) 0 0
\(907\) −1.55225e132 −0.125497 −0.0627485 − 0.998029i \(-0.519987\pi\)
−0.0627485 + 0.998029i \(0.519987\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.65570e133 −1.09813 −0.549065 − 0.835780i \(-0.685016\pi\)
−0.549065 + 0.835780i \(0.685016\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 3.11000e133 1.77903
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −3.83446e133 −1.71596 −0.857979 − 0.513685i \(-0.828280\pi\)
−0.857979 + 0.513685i \(0.828280\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 4.56029e133 1.52270
\(926\) 4.77859e133 1.51987
\(927\) 0 0
\(928\) −1.23099e133 −0.355300
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.65727e133 −0.394164
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 9.78856e133 1.74418
\(939\) 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
\(946\) −3.79032e133 −0.460866
\(947\) 1.00387e133 0.116394 0.0581970 − 0.998305i \(-0.481465\pi\)
0.0581970 + 0.998305i \(0.481465\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 2.28530e134 1.99417 0.997087 − 0.0762733i \(-0.0243022\pi\)
0.997087 + 0.0762733i \(0.0243022\pi\)
\(954\) 1.20120e134 0.999862
\(955\) 0 0
\(956\) 1.33785e133 0.101345
\(957\) 0 0
\(958\) 0 0
\(959\) −1.33162e134 −0.876080
\(960\) 0 0
\(961\) 1.66937e134 1.00000
\(962\) 0 0
\(963\) −1.02571e134 −0.559554
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −4.41380e134 −1.99812 −0.999060 − 0.0433515i \(-0.986196\pi\)
−0.999060 + 0.0433515i \(0.986196\pi\)
\(968\) −2.30066e134 −0.994172
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 5.07833e133 0.166176
\(975\) 0 0
\(976\) 0 0
\(977\) −3.92560e134 −1.11854 −0.559269 − 0.828987i \(-0.688918\pi\)
−0.559269 + 0.828987i \(0.688918\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −6.38736e134 −1.51431
\(982\) 1.75310e134 0.397000
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.58509e135 −2.60748
\(990\) 0 0
\(991\) 1.06783e135 1.60394 0.801968 − 0.597366i \(-0.203786\pi\)
0.801968 + 0.597366i \(0.203786\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −1.21580e135 −1.59395
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −1.44082e135 −1.57665
\(999\) 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 7.91.b.a.6.1 ✓ 1
7.6 odd 2 CM 7.91.b.a.6.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.91.b.a.6.1 ✓ 1 1.1 even 1 trivial
7.91.b.a.6.1 ✓ 1 7.6 odd 2 CM