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Properties

Label 7.67.b.a.6.1
Level $7$
Weight $67$
Character 7.6
Self dual yes
Analytic conductor $193.108$
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] = [7,67,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, 67); 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, 67, names="a")
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 67 \)
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: \(193.107619924\)
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)
 
\(f(q)\) \(=\) \(q-4.88439e9 q^{2} -4.99298e19 q^{4} -7.73099e27 q^{7} +6.04280e29 q^{8} +3.09032e31 q^{9} -1.41391e34 q^{11} +3.77612e37 q^{14} +7.32627e38 q^{16} -1.50943e41 q^{18} +6.90606e43 q^{22} +1.11613e45 q^{23} +1.35525e46 q^{25} +3.86007e47 q^{28} -3.52897e48 q^{29} -4.81664e49 q^{32} -1.54299e51 q^{36} +9.79008e51 q^{37} -1.04245e54 q^{43} +7.05960e53 q^{44} -5.45161e54 q^{46} +5.97683e55 q^{49} -6.61958e55 q^{50} -1.52244e57 q^{53} -4.67169e57 q^{56} +1.72369e58 q^{58} -2.38912e59 q^{63} +1.81205e59 q^{64} +3.04083e60 q^{67} -1.51345e61 q^{71} +1.86742e61 q^{72} -4.78185e61 q^{74} +1.09309e62 q^{77} -6.87567e62 q^{79} +9.55005e62 q^{81} +5.09172e63 q^{86} -8.54396e63 q^{88} -5.57281e64 q^{92} -2.91931e65 q^{98} -4.36942e65 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.88439e9 −0.568617 −0.284309 − 0.958733i \(-0.591764\pi\)
−0.284309 + 0.958733i \(0.591764\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −4.99298e19 −0.676674
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −7.73099e27 −1.00000
\(8\) 6.04280e29 0.953386
\(9\) 3.09032e31 1.00000
\(10\) 0 0
\(11\) −1.41391e34 −0.608782 −0.304391 − 0.952547i \(-0.598453\pi\)
−0.304391 + 0.952547i \(0.598453\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 3.77612e37 0.568617
\(15\) 0 0
\(16\) 7.32627e38 0.134562
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −1.50943e41 −0.568617
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.90606e43 0.346164
\(23\) 1.11613e45 1.29032 0.645159 − 0.764049i \(-0.276791\pi\)
0.645159 + 0.764049i \(0.276791\pi\)
\(24\) 0 0
\(25\) 1.35525e46 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 3.86007e47 0.676674
\(29\) −3.52897e48 −1.94319 −0.971593 − 0.236659i \(-0.923947\pi\)
−0.971593 + 0.236659i \(0.923947\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −4.81664e49 −1.02990
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −1.54299e51 −0.676674
\(37\) 9.79008e51 1.73832 0.869160 − 0.494532i \(-0.164660\pi\)
0.869160 + 0.494532i \(0.164660\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −1.04245e54 −1.29896 −0.649480 − 0.760379i \(-0.725013\pi\)
−0.649480 + 0.760379i \(0.725013\pi\)
\(44\) 7.05960e53 0.411947
\(45\) 0 0
\(46\) −5.45161e54 −0.733697
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 5.97683e55 1.00000
\(50\) −6.61958e55 −0.568617
\(51\) 0 0
\(52\) 0 0
\(53\) −1.52244e57 −1.91178 −0.955888 − 0.293730i \(-0.905103\pi\)
−0.955888 + 0.293730i \(0.905103\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.67169e57 −0.953386
\(57\) 0 0
\(58\) 1.72369e58 1.10493
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −2.38912e59 −1.00000
\(64\) 1.81205e59 0.451057
\(65\) 0 0
\(66\) 0 0
\(67\) 3.04083e60 1.66926 0.834630 − 0.550811i \(-0.185681\pi\)
0.834630 + 0.550811i \(0.185681\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.51345e61 −1.22586 −0.612932 − 0.790136i \(-0.710010\pi\)
−0.612932 + 0.790136i \(0.710010\pi\)
\(72\) 1.86742e61 0.953386
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −4.78185e61 −0.988438
\(75\) 0 0
\(76\) 0 0
\(77\) 1.09309e62 0.608782
\(78\) 0 0
\(79\) −6.87567e62 −1.64294 −0.821468 − 0.570254i \(-0.806845\pi\)
−0.821468 + 0.570254i \(0.806845\pi\)
\(80\) 0 0
\(81\) 9.55005e62 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.09172e63 0.738611
\(87\) 0 0
\(88\) −8.54396e63 −0.580405
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −5.57281e64 −0.873125
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −2.91931e65 −0.568617
\(99\) −4.36942e65 −0.608782
\(100\) −6.76674e65 −0.676674
\(101\) 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
\(106\) 7.43620e66 1.08707
\(107\) −3.85043e66 −0.412900 −0.206450 − 0.978457i \(-0.566191\pi\)
−0.206450 + 0.978457i \(0.566191\pi\)
\(108\) 0 0
\(109\) −2.97872e67 −1.73363 −0.866814 − 0.498632i \(-0.833836\pi\)
−0.866814 + 0.498632i \(0.833836\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −5.66394e66 −0.134562
\(113\) −1.12640e68 −1.99575 −0.997873 − 0.0651872i \(-0.979236\pi\)
−0.997873 + 0.0651872i \(0.979236\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.76201e68 1.31490
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3.39495e68 −0.629384
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 1.16694e69 0.568617
\(127\) 1.36889e69 0.513862 0.256931 − 0.966430i \(-0.417289\pi\)
0.256931 + 0.966430i \(0.417289\pi\)
\(128\) 2.66898e69 0.773422
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.48526e70 −0.949170
\(135\) 0 0
\(136\) 0 0
\(137\) 4.25897e70 1.31077 0.655385 − 0.755295i \(-0.272507\pi\)
0.655385 + 0.755295i \(0.272507\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.39227e70 0.697048
\(143\) 0 0
\(144\) 2.26405e70 0.134562
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −4.88816e71 −1.17628
\(149\) −2.59550e71 −0.500121 −0.250060 − 0.968230i \(-0.580450\pi\)
−0.250060 + 0.968230i \(0.580450\pi\)
\(150\) 0 0
\(151\) −2.96967e71 −0.368526 −0.184263 − 0.982877i \(-0.558990\pi\)
−0.184263 + 0.982877i \(0.558990\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −5.33907e71 −0.346164
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 3.35834e72 0.934202
\(159\) 0 0
\(160\) 0 0
\(161\) −8.62880e72 −1.29032
\(162\) −4.66461e72 −0.568617
\(163\) 1.92192e73 1.91225 0.956126 − 0.292955i \(-0.0946387\pi\)
0.956126 + 0.292955i \(0.0946387\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 3.31330e73 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 5.20492e73 0.878973
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −1.04775e74 −1.00000
\(176\) −1.03587e73 −0.0819192
\(177\) 0 0
\(178\) 0 0
\(179\) 3.92142e74 1.77539 0.887693 − 0.460435i \(-0.152307\pi\)
0.887693 + 0.460435i \(0.152307\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 6.74456e74 1.23017
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.24354e75 −1.72551 −0.862757 − 0.505619i \(-0.831264\pi\)
−0.862757 + 0.505619i \(0.831264\pi\)
\(192\) 0 0
\(193\) 4.34197e75 1.63793 0.818966 − 0.573842i \(-0.194548\pi\)
0.818966 + 0.573842i \(0.194548\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.98421e75 −0.676674
\(197\) 9.68820e75 1.85720 0.928598 − 0.371088i \(-0.121015\pi\)
0.928598 + 0.371088i \(0.121015\pi\)
\(198\) 2.13419e75 0.346164
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 8.18952e75 0.953386
\(201\) 0 0
\(202\) 0 0
\(203\) 2.72825e76 1.94319
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3.44920e76 1.29032
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 9.90766e76 1.97083 0.985417 − 0.170154i \(-0.0544267\pi\)
0.985417 + 0.170154i \(0.0544267\pi\)
\(212\) 7.60152e76 1.29365
\(213\) 0 0
\(214\) 1.88070e76 0.234782
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 1.45492e77 0.985770
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 3.72374e77 1.02990
\(225\) 4.18816e77 1.00000
\(226\) 5.50179e77 1.13482
\(227\) 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
\(232\) −2.13249e78 −1.85261
\(233\) 2.53800e78 1.91314 0.956569 − 0.291506i \(-0.0941564\pi\)
0.956569 + 0.291506i \(0.0941564\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.40530e78 0.457761 0.228880 − 0.973455i \(-0.426493\pi\)
0.228880 + 0.973455i \(0.426493\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 1.65822e78 0.357879
\(243\) 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 \(\pi\)
\(252\) 1.19288e79 0.676674
\(253\) −1.57810e79 −0.785522
\(254\) −6.68620e78 −0.292191
\(255\) 0 0
\(256\) −2.64069e79 −0.890838
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −7.56871e79 −1.73832
\(260\) 0 0
\(261\) −1.09056e80 −1.94319
\(262\) 0 0
\(263\) −2.58127e79 −0.357515 −0.178757 − 0.983893i \(-0.557208\pi\)
−0.178757 + 0.983893i \(0.557208\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.51828e80 −1.12955
\(269\) 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
\(274\) −2.08025e80 −0.745326
\(275\) −1.91620e80 −0.608782
\(276\) 0 0
\(277\) −2.22133e80 −0.555624 −0.277812 − 0.960636i \(-0.589609\pi\)
−0.277812 + 0.960636i \(0.589609\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.66904e80 0.883489 0.441744 − 0.897141i \(-0.354360\pi\)
0.441744 + 0.897141i \(0.354360\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 7.55661e80 0.829511
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.48850e81 −1.02990
\(289\) 1.62042e81 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
\(296\) 5.91595e81 1.65729
\(297\) 0 0
\(298\) 1.26774e81 0.284377
\(299\) 0 0
\(300\) 0 0
\(301\) 8.05917e81 1.29896
\(302\) 1.45050e81 0.209550
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) −5.45777e81 −0.411947
\(309\) 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
\(316\) 3.43300e82 1.11173
\(317\) 3.31180e82 0.966291 0.483145 − 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(318\) 0 0
\(319\) 4.98963e82 1.18298
\(320\) 0 0
\(321\) 0 0
\(322\) 4.21464e82 0.733697
\(323\) 0 0
\(324\) −4.76832e82 −0.676674
\(325\) 0 0
\(326\) −9.38742e82 −1.08734
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.79223e83 −1.95716 −0.978582 − 0.205856i \(-0.934002\pi\)
−0.978582 + 0.205856i \(0.934002\pi\)
\(332\) 0 0
\(333\) 3.02544e83 1.73832
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.94893e83 −1.91745 −0.958725 − 0.284334i \(-0.908228\pi\)
−0.958725 + 0.284334i \(0.908228\pi\)
\(338\) −1.61835e83 −0.568617
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −4.62068e83 −1.00000
\(344\) −6.29931e83 −1.23841
\(345\) 0 0
\(346\) 0 0
\(347\) −9.27664e83 −1.36936 −0.684680 − 0.728843i \(-0.740058\pi\)
−0.684680 + 0.728843i \(0.740058\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 5.11759e83 0.568617
\(351\) 0 0
\(352\) 6.81028e83 0.626985
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −1.91537e84 −1.00952
\(359\) −4.07697e83 −0.195984 −0.0979921 − 0.995187i \(-0.531242\pi\)
−0.0979921 + 0.995187i \(0.531242\pi\)
\(360\) 0 0
\(361\) 2.49884e84 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 \(\pi\)
\(368\) 8.17708e83 0.173628
\(369\) 0 0
\(370\) 0 0
\(371\) 1.17700e85 1.91178
\(372\) 0 0
\(373\) 9.73957e84 1.32479 0.662397 − 0.749153i \(-0.269539\pi\)
0.662397 + 0.749153i \(0.269539\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 6.33815e84 0.509177 0.254588 − 0.967050i \(-0.418060\pi\)
0.254588 + 0.967050i \(0.418060\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.58427e85 0.981157
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.12079e85 −0.931357
\(387\) −3.22150e85 −1.29896
\(388\) 0 0
\(389\) 3.54897e85 1.20716 0.603581 − 0.797302i \(-0.293740\pi\)
0.603581 + 0.797302i \(0.293740\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 3.61168e85 0.953386
\(393\) 0 0
\(394\) −4.73209e85 −1.05603
\(395\) 0 0
\(396\) 2.18164e85 0.411947
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 9.92895e84 0.134562
\(401\) 9.33535e85 1.16511 0.582554 − 0.812792i \(-0.302053\pi\)
0.582554 + 0.812792i \(0.302053\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −1.33258e86 −1.10493
\(407\) −1.38423e86 −1.05826
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −1.68472e86 −0.733697
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −2.09669e86 −0.525079 −0.262539 − 0.964921i \(-0.584560\pi\)
−0.262539 + 0.964921i \(0.584560\pi\)
\(422\) −4.83929e86 −1.12065
\(423\) 0 0
\(424\) −9.19982e86 −1.82266
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 1.92251e86 0.279399
\(429\) 0 0
\(430\) 0 0
\(431\) −7.66199e85 −0.0884283 −0.0442142 − 0.999022i \(-0.514078\pi\)
−0.0442142 + 0.999022i \(0.514078\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.48727e87 1.17310
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.84703e87 1.00000
\(442\) 0 0
\(443\) −1.31051e87 −0.611109 −0.305555 − 0.952175i \(-0.598842\pi\)
−0.305555 + 0.952175i \(0.598842\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.40090e87 −0.451057
\(449\) 6.68368e87 1.99934 0.999670 − 0.0256999i \(-0.00818144\pi\)
0.999670 + 0.0256999i \(0.00818144\pi\)
\(450\) −2.04566e87 −0.568617
\(451\) 0 0
\(452\) 5.62410e87 1.35047
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.18704e88 1.98259 0.991293 − 0.131672i \(-0.0420345\pi\)
0.991293 + 0.131672i \(0.0420345\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 1.80564e88 1.96092 0.980459 − 0.196723i \(-0.0630301\pi\)
0.980459 + 0.196723i \(0.0630301\pi\)
\(464\) −2.58542e87 −0.261480
\(465\) 0 0
\(466\) −1.23966e88 −1.08784
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −2.35087e88 −1.66926
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.47393e88 0.790784
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −4.70483e88 −1.91178
\(478\) −6.86403e87 −0.260291
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.69509e88 0.425888
\(485\) 0 0
\(486\) 0 0
\(487\) −9.46709e88 −1.93982 −0.969908 − 0.243471i \(-0.921714\pi\)
−0.969908 + 0.243471i \(0.921714\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −5.79446e87 −0.0906409 −0.0453204 − 0.998973i \(-0.514431\pi\)
−0.0453204 + 0.998973i \(0.514431\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.17005e89 1.22586
\(498\) 0 0
\(499\) −1.19724e89 −1.09866 −0.549331 − 0.835605i \(-0.685117\pi\)
−0.549331 + 0.835605i \(0.685117\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) −1.44370e89 −0.953386
\(505\) 0 0
\(506\) 7.70807e88 0.446662
\(507\) 0 0
\(508\) −6.83484e88 −0.347717
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −6.79545e88 −0.266876
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 3.69685e89 0.988438
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 5.32673e89 1.10493
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 1.26079e89 0.203289
\(527\) 0 0
\(528\) 0 0
\(529\) 4.97515e89 0.664919
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 1.83752e90 1.59145
\(537\) 0 0
\(538\) 0 0
\(539\) −8.45067e89 −0.608782
\(540\) 0 0
\(541\) 2.42373e90 1.54516 0.772582 − 0.634915i \(-0.218965\pi\)
0.772582 + 0.634915i \(0.218965\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.94089e90 −1.30286 −0.651428 − 0.758711i \(-0.725830\pi\)
−0.651428 + 0.758711i \(0.725830\pi\)
\(548\) −2.12649e90 −0.886964
\(549\) 0 0
\(550\) 9.35946e89 0.346164
\(551\) 0 0
\(552\) 0 0
\(553\) 5.31557e90 1.64294
\(554\) 1.08498e90 0.315937
\(555\) 0 0
\(556\) 0 0
\(557\) −5.81597e90 −1.41710 −0.708550 − 0.705660i \(-0.750651\pi\)
−0.708550 + 0.705660i \(0.750651\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −2.76898e90 −0.502367
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −7.38314e90 −1.00000
\(568\) −9.14548e90 −1.16872
\(569\) −1.61974e91 −1.95317 −0.976587 − 0.215124i \(-0.930984\pi\)
−0.976587 + 0.215124i \(0.930984\pi\)
\(570\) 0 0
\(571\) −1.35912e91 −1.45971 −0.729854 − 0.683603i \(-0.760412\pi\)
−0.729854 + 0.683603i \(0.760412\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.51264e91 1.29032
\(576\) 5.59981e90 0.451057
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −7.91478e90 −0.568617
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2.15259e91 1.16386
\(584\) 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
\(592\) 7.17248e90 0.233912
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.29593e91 0.338419
\(597\) 0 0
\(598\) 0 0
\(599\) 8.99586e91 1.99048 0.995240 − 0.0974580i \(-0.0310711\pi\)
0.995240 + 0.0974580i \(0.0310711\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) −3.93641e91 −0.738611
\(603\) 9.39713e91 1.66926
\(604\) 1.48275e91 0.249372
\(605\) 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
\(613\) −7.25737e91 −0.749176 −0.374588 − 0.927191i \(-0.622216\pi\)
−0.374588 + 0.927191i \(0.622216\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 6.60533e91 0.580405
\(617\) −5.32938e91 −0.443880 −0.221940 − 0.975060i \(-0.571239\pi\)
−0.221940 + 0.975060i \(0.571239\pi\)
\(618\) 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
\(625\) 1.83671e92 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 4.15780e92 1.65156 0.825780 − 0.563992i \(-0.190735\pi\)
0.825780 + 0.563992i \(0.190735\pi\)
\(632\) −4.15483e92 −1.56635
\(633\) 0 0
\(634\) −1.61761e92 −0.549450
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −2.43713e92 −0.672661
\(639\) −4.67704e92 −1.22586
\(640\) 0 0
\(641\) −8.31054e92 −1.96478 −0.982391 − 0.186836i \(-0.940177\pi\)
−0.982391 + 0.186836i \(0.940177\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 4.30834e92 0.873125
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 5.77091e92 0.953386
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −9.59612e92 −1.29397
\(653\) 5.65304e92 0.724682 0.362341 − 0.932046i \(-0.381978\pi\)
0.362341 + 0.932046i \(0.381978\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −8.16226e92 −0.773734 −0.386867 − 0.922136i \(-0.626443\pi\)
−0.386867 + 0.922136i \(0.626443\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 1.36383e93 1.11288
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −1.47774e93 −0.988438
\(667\) −3.93879e93 −2.50733
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 4.17485e93 1.97762 0.988812 − 0.149166i \(-0.0476588\pi\)
0.988812 + 0.149166i \(0.0476588\pi\)
\(674\) 2.41725e93 1.09030
\(675\) 0 0
\(676\) −1.65432e93 −0.676674
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.30023e93 1.83431 0.917155 − 0.398530i \(-0.130480\pi\)
0.917155 + 0.398530i \(0.130480\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 2.25692e93 0.568617
\(687\) 0 0
\(688\) −7.63727e92 −0.174791
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 3.37799e93 0.608782
\(694\) 4.53107e93 0.778642
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 5.23136e93 0.676674
\(701\) 1.29203e94 1.59433 0.797163 − 0.603764i \(-0.206333\pi\)
0.797163 + 0.603764i \(0.206333\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −2.56207e93 −0.274595
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −9.22436e93 −0.782727 −0.391364 − 0.920236i \(-0.627997\pi\)
−0.391364 + 0.920236i \(0.627997\pi\)
\(710\) 0 0
\(711\) −2.12480e94 −1.64294
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −1.95795e94 −1.20136
\(717\) 0 0
\(718\) 1.99135e93 0.111440
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.22053e94 −0.568617
\(723\) 0 0
\(724\) 0 0
\(725\) −4.78265e94 −1.94319
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2.95127e94 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
\(736\) −5.37601e94 −1.32890
\(737\) −4.29945e94 −1.01622
\(738\) 0 0
\(739\) 7.04279e94 1.52222 0.761111 − 0.648622i \(-0.224654\pi\)
0.761111 + 0.648622i \(0.224654\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −5.74892e94 −1.08707
\(743\) 6.31163e94 1.14159 0.570795 − 0.821092i \(-0.306635\pi\)
0.570795 + 0.821092i \(0.306635\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −4.75718e94 −0.753301
\(747\) 0 0
\(748\) 0 0
\(749\) 2.97677e94 0.412900
\(750\) 0 0
\(751\) −4.07232e94 −0.517278 −0.258639 − 0.965974i \(-0.583274\pi\)
−0.258639 + 0.965974i \(0.583274\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00385e95 1.95750 0.978749 − 0.205060i \(-0.0657390\pi\)
0.978749 + 0.205060i \(0.0657390\pi\)
\(758\) −3.09580e94 −0.289527
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 2.30285e95 1.73363
\(764\) 1.61949e95 1.16761
\(765\) 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
\(772\) −2.16794e95 −1.10835
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 1.57350e95 0.738611
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −1.73345e95 −0.686413
\(779\) 0 0
\(780\) 0 0
\(781\) 2.13988e95 0.746284
\(782\) 0 0
\(783\) 0 0
\(784\) 4.37879e94 0.134562
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −4.83729e95 −1.25672
\(789\) 0 0
\(790\) 0 0
\(791\) 8.70822e95 1.99575
\(792\) −2.64035e95 −0.580405
\(793\) 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
\(800\) −6.52777e95 −1.02990
\(801\) 0 0
\(802\) −4.55975e95 −0.662501
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 7.37839e95 0.804749 0.402375 − 0.915475i \(-0.368185\pi\)
0.402375 + 0.915475i \(0.368185\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −1.36221e96 −1.31490
\(813\) 0 0
\(814\) 6.76109e95 0.601744
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.91645e96 −1.28580 −0.642898 − 0.765952i \(-0.722268\pi\)
−0.642898 + 0.765952i \(0.722268\pi\)
\(822\) 0 0
\(823\) −7.09693e95 −0.439416 −0.219708 − 0.975566i \(-0.570511\pi\)
−0.219708 + 0.975566i \(0.570511\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.23079e96 −1.17700 −0.588501 − 0.808497i \(-0.700282\pi\)
−0.588501 + 0.808497i \(0.700282\pi\)
\(828\) −1.72218e96 −0.873125
\(829\) 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
\(841\) 9.15551e96 2.77597
\(842\) 1.02410e96 0.298569
\(843\) 0 0
\(844\) −4.94687e96 −1.33361
\(845\) 0 0
\(846\) 0 0
\(847\) 2.62463e96 0.629384
\(848\) −1.11538e96 −0.257253
\(849\) 0 0
\(850\) 0 0
\(851\) 1.09270e97 2.24298
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2.32674e96 −0.393653
\(857\) 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
\(862\) 3.74241e95 0.0502819
\(863\) 1.41838e97 1.83415 0.917077 − 0.398711i \(-0.130542\pi\)
0.917077 + 0.398711i \(0.130542\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 9.72155e96 1.00019
\(870\) 0 0
\(871\) 0 0
\(872\) −1.79998e97 −1.65282
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.46175e97 1.87179 0.935894 − 0.352282i \(-0.114594\pi\)
0.935894 + 0.352282i \(0.114594\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −9.02160e96 −0.568617
\(883\) −2.16822e97 −1.31644 −0.658218 − 0.752827i \(-0.728690\pi\)
−0.658218 + 0.752827i \(0.728690\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 6.40106e96 0.347487
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −1.05829e97 −0.513862
\(890\) 0 0
\(891\) −1.35029e97 −0.608782
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −2.06339e97 −0.773422
\(897\) 0 0
\(898\) −3.26457e97 −1.13686
\(899\) 0 0
\(900\) −2.09114e97 −0.676674
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −6.80663e97 −1.90272
\(905\) 0 0
\(906\) 0 0
\(907\) 7.08832e97 1.77624 0.888122 − 0.459607i \(-0.152010\pi\)
0.888122 + 0.459607i \(0.152010\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3.81563e97 −0.826912 −0.413456 − 0.910524i \(-0.635678\pi\)
−0.413456 + 0.910524i \(0.635678\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −5.79798e97 −1.12733
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.01289e98 1.64494 0.822471 − 0.568807i \(-0.192595\pi\)
0.822471 + 0.568807i \(0.192595\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.32680e98 1.73832
\(926\) −8.81946e97 −1.11501
\(927\) 0 0
\(928\) 1.69978e98 2.00129
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.26722e98 −1.29457
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 1.14825e98 0.949170
\(939\) 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
\(946\) −7.19922e97 −0.449653
\(947\) 3.27977e98 1.97831 0.989153 − 0.146888i \(-0.0469258\pi\)
0.989153 + 0.146888i \(0.0469258\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.84092e98 −0.901509 −0.450755 − 0.892648i \(-0.648845\pi\)
−0.450755 + 0.892648i \(0.648845\pi\)
\(954\) 2.29802e98 1.08707
\(955\) 0 0
\(956\) −7.01663e97 −0.309755
\(957\) 0 0
\(958\) 0 0
\(959\) −3.29261e98 −1.31077
\(960\) 0 0
\(961\) 2.69074e98 1.00000
\(962\) 0 0
\(963\) −1.18991e98 −0.412900
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 3.48456e98 1.05457 0.527283 − 0.849690i \(-0.323211\pi\)
0.527283 + 0.849690i \(0.323211\pi\)
\(968\) −2.05150e98 −0.600046
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 4.62409e98 1.10301
\(975\) 0 0
\(976\) 0 0
\(977\) 3.63537e98 0.783481 0.391740 − 0.920076i \(-0.371873\pi\)
0.391740 + 0.920076i \(0.371873\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −9.20519e98 −1.73363
\(982\) 2.83024e97 0.0515400
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.16351e99 −1.67607
\(990\) 0 0
\(991\) −1.15499e99 −1.55649 −0.778246 − 0.627960i \(-0.783890\pi\)
−0.778246 + 0.627960i \(0.783890\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −5.71496e98 −0.697048
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 5.84779e98 0.624718
\(999\) 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 7.67.b.a.6.1 ✓ 1
7.6 odd 2 CM 7.67.b.a.6.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.67.b.a.6.1 ✓ 1 1.1 even 1 trivial
7.67.b.a.6.1 ✓ 1 7.6 odd 2 CM