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Properties

Label 7.11.b.a.6.1
Level $7$
Weight $11$
Character 7.6
Self dual yes
Analytic conductor $4.448$
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] = [7,11,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, 11); 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, 11, names="a")
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 11 \)
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: \(4.44750076872\)
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)
 
\(f(q)\) \(=\) \(q+57.0000 q^{2} +2225.00 q^{4} -16807.0 q^{7} +68457.0 q^{8} +59049.0 q^{9} -316326. q^{11} -957999. q^{14} +1.62365e6 q^{16} +3.36579e6 q^{18} -1.80306e7 q^{22} +1.16496e7 q^{23} +9.76562e6 q^{25} -3.73956e7 q^{28} +1.20052e7 q^{29} +2.24480e7 q^{32} +1.31384e8 q^{36} -5.97269e7 q^{37} -1.56001e8 q^{43} -7.03825e8 q^{44} +6.64028e8 q^{46} +2.82475e8 q^{49} +5.56641e8 q^{50} -2.33688e8 q^{53} -1.15056e9 q^{56} +6.84298e8 q^{58} -9.92437e8 q^{63} -3.83079e8 q^{64} +2.11111e9 q^{67} -3.60351e9 q^{71} +4.04232e9 q^{72} -3.40443e9 q^{74} +5.31649e9 q^{77} +2.72795e8 q^{79} +3.48678e9 q^{81} -8.89208e9 q^{86} -2.16547e10 q^{88} +2.59204e10 q^{92} +1.61011e10 q^{98} -1.86787e10 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 57.0000 1.78125 0.890625 − 0.454739i \(-0.150267\pi\)
0.890625 + 0.454739i \(0.150267\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 2225.00 2.17285
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −16807.0 −1.00000
\(8\) 68457.0 2.08914
\(9\) 59049.0 1.00000
\(10\) 0 0
\(11\) −316326. −1.96414 −0.982068 − 0.188528i \(-0.939628\pi\)
−0.982068 + 0.188528i \(0.939628\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −957999. −1.78125
\(15\) 0 0
\(16\) 1.62365e6 1.54843
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 3.36579e6 1.78125
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.80306e7 −3.49862
\(23\) 1.16496e7 1.80997 0.904987 − 0.425438i \(-0.139880\pi\)
0.904987 + 0.425438i \(0.139880\pi\)
\(24\) 0 0
\(25\) 9.76562e6 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −3.73956e7 −2.17285
\(29\) 1.20052e7 0.585302 0.292651 − 0.956219i \(-0.405463\pi\)
0.292651 + 0.956219i \(0.405463\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 2.24480e7 0.669003
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.31384e8 2.17285
\(37\) −5.97269e7 −0.861314 −0.430657 − 0.902516i \(-0.641718\pi\)
−0.430657 + 0.902516i \(0.641718\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −1.56001e8 −1.06117 −0.530586 − 0.847631i \(-0.678028\pi\)
−0.530586 + 0.847631i \(0.678028\pi\)
\(44\) −7.03825e8 −4.26778
\(45\) 0 0
\(46\) 6.64028e8 3.22402
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 2.82475e8 1.00000
\(50\) 5.56641e8 1.78125
\(51\) 0 0
\(52\) 0 0
\(53\) −2.33688e8 −0.558802 −0.279401 − 0.960175i \(-0.590136\pi\)
−0.279401 + 0.960175i \(0.590136\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.15056e9 −2.08914
\(57\) 0 0
\(58\) 6.84298e8 1.04257
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −9.92437e8 −1.00000
\(64\) −3.83079e8 −0.356770
\(65\) 0 0
\(66\) 0 0
\(67\) 2.11111e9 1.56364 0.781822 − 0.623502i \(-0.214291\pi\)
0.781822 + 0.623502i \(0.214291\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.60351e9 −1.99726 −0.998629 − 0.0523408i \(-0.983332\pi\)
−0.998629 + 0.0523408i \(0.983332\pi\)
\(72\) 4.04232e9 2.08914
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −3.40443e9 −1.53422
\(75\) 0 0
\(76\) 0 0
\(77\) 5.31649e9 1.96414
\(78\) 0 0
\(79\) 2.72795e8 0.0886547 0.0443273 − 0.999017i \(-0.485886\pi\)
0.0443273 + 0.999017i \(0.485886\pi\)
\(80\) 0 0
\(81\) 3.48678e9 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.89208e9 −1.89021
\(87\) 0 0
\(88\) −2.16547e10 −4.10336
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.59204e10 3.93281
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 1.61011e10 1.78125
\(99\) −1.86787e10 −1.96414
\(100\) 2.17285e10 2.17285
\(101\) 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
\(106\) −1.33202e10 −0.995366
\(107\) −2.38390e10 −1.69969 −0.849845 − 0.527033i \(-0.823304\pi\)
−0.849845 + 0.527033i \(0.823304\pi\)
\(108\) 0 0
\(109\) 1.74438e10 1.13373 0.566863 − 0.823812i \(-0.308157\pi\)
0.566863 + 0.823812i \(0.308157\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.72887e10 −1.54843
\(113\) −2.16700e10 −1.17616 −0.588080 − 0.808803i \(-0.700116\pi\)
−0.588080 + 0.808803i \(0.700116\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.67116e10 1.27178
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.41247e10 2.85783
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −5.65689e10 −1.78125
\(127\) 2.60080e9 0.0787206 0.0393603 − 0.999225i \(-0.487468\pi\)
0.0393603 + 0.999225i \(0.487468\pi\)
\(128\) −4.48223e10 −1.30450
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.20334e11 2.78524
\(135\) 0 0
\(136\) 0 0
\(137\) 2.84035e10 0.588530 0.294265 − 0.955724i \(-0.404925\pi\)
0.294265 + 0.955724i \(0.404925\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2.05400e11 −3.55762
\(143\) 0 0
\(144\) 9.58748e10 1.54843
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −1.32892e11 −1.87151
\(149\) −3.32984e10 −0.453411 −0.226706 − 0.973963i \(-0.572796\pi\)
−0.226706 + 0.973963i \(0.572796\pi\)
\(150\) 0 0
\(151\) −8.85583e10 −1.12809 −0.564046 − 0.825743i \(-0.690756\pi\)
−0.564046 + 0.825743i \(0.690756\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 3.03040e11 3.49862
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 1.55493e10 0.157916
\(159\) 0 0
\(160\) 0 0
\(161\) −1.95795e11 −1.80997
\(162\) 1.98747e11 1.78125
\(163\) 2.09574e11 1.82138 0.910689 − 0.413092i \(-0.135551\pi\)
0.910689 + 0.413092i \(0.135551\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.37858e11 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −3.47103e11 −2.30577
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −1.64131e11 −1.00000
\(176\) −5.13602e11 −3.04133
\(177\) 0 0
\(178\) 0 0
\(179\) −2.58169e10 −0.140488 −0.0702440 − 0.997530i \(-0.522378\pi\)
−0.0702440 + 0.997530i \(0.522378\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 7.97498e11 3.78129
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.97715e11 1.95801 0.979004 − 0.203843i \(-0.0653431\pi\)
0.979004 + 0.203843i \(0.0653431\pi\)
\(192\) 0 0
\(193\) −3.86650e11 −1.44388 −0.721941 − 0.691955i \(-0.756750\pi\)
−0.721941 + 0.691955i \(0.756750\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 6.28507e11 2.17285
\(197\) −4.44566e11 −1.49832 −0.749161 − 0.662388i \(-0.769543\pi\)
−0.749161 + 0.662388i \(0.769543\pi\)
\(198\) −1.06469e12 −3.49862
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 6.68525e11 2.08914
\(201\) 0 0
\(202\) 0 0
\(203\) −2.01772e11 −0.585302
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.87898e11 1.80997
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 3.67069e11 0.877678 0.438839 − 0.898566i \(-0.355390\pi\)
0.438839 + 0.898566i \(0.355390\pi\)
\(212\) −5.19957e11 −1.21419
\(213\) 0 0
\(214\) −1.35882e12 −3.02757
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 9.94295e11 2.01945
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) −3.77284e11 −0.669003
\(225\) 5.76650e11 1.00000
\(226\) −1.23519e12 −2.09503
\(227\) 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
\(232\) 8.21842e11 1.22278
\(233\) 2.63675e11 0.383964 0.191982 − 0.981398i \(-0.438509\pi\)
0.191982 + 0.981398i \(0.438509\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.03612e12 −1.32868 −0.664339 − 0.747431i \(-0.731287\pi\)
−0.664339 + 0.747431i \(0.731287\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 4.22511e12 5.09051
\(243\) 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 \(\pi\)
\(252\) −2.20817e12 −2.17285
\(253\) −3.68508e12 −3.55504
\(254\) 1.48246e11 0.140221
\(255\) 0 0
\(256\) −2.16260e12 −1.96687
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 1.00383e12 0.861314
\(260\) 0 0
\(261\) 7.08897e11 0.585302
\(262\) 0 0
\(263\) −1.70917e11 −0.135833 −0.0679166 − 0.997691i \(-0.521635\pi\)
−0.0679166 + 0.997691i \(0.521635\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 4.69723e12 3.39757
\(269\) 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
\(274\) 1.61900e12 1.04832
\(275\) −3.08912e12 −1.96414
\(276\) 0 0
\(277\) 3.24831e12 1.99186 0.995930 − 0.0901313i \(-0.0287287\pi\)
0.995930 + 0.0901313i \(0.0287287\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.98819e12 1.70560 0.852799 − 0.522239i \(-0.174903\pi\)
0.852799 + 0.522239i \(0.174903\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −8.01782e12 −4.33975
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.32553e12 0.669003
\(289\) 2.01599e12 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
\(296\) −4.08873e12 −1.79941
\(297\) 0 0
\(298\) −1.89801e12 −0.807639
\(299\) 0 0
\(300\) 0 0
\(301\) 2.62192e12 1.06117
\(302\) −5.04783e12 −2.00941
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 1.18292e13 4.26778
\(309\) 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
\(316\) 6.06970e11 0.192633
\(317\) −6.35300e12 −1.98464 −0.992322 − 0.123683i \(-0.960529\pi\)
−0.992322 + 0.123683i \(0.960529\pi\)
\(318\) 0 0
\(319\) −3.79757e12 −1.14961
\(320\) 0 0
\(321\) 0 0
\(322\) −1.11603e13 −3.22402
\(323\) 0 0
\(324\) 7.75810e12 2.17285
\(325\) 0 0
\(326\) 1.19457e13 3.24433
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.83278e11 0.222309 0.111155 − 0.993803i \(-0.464545\pi\)
0.111155 + 0.993803i \(0.464545\pi\)
\(332\) 0 0
\(333\) −3.52681e12 −0.861314
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.60949e12 1.98074 0.990371 − 0.138437i \(-0.0442079\pi\)
0.990371 + 0.138437i \(0.0442079\pi\)
\(338\) 7.85793e12 1.78125
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −4.74756e12 −1.00000
\(344\) −1.06794e13 −2.21694
\(345\) 0 0
\(346\) 0 0
\(347\) −4.78028e12 −0.950180 −0.475090 − 0.879937i \(-0.657585\pi\)
−0.475090 + 0.879937i \(0.657585\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) −9.35546e12 −1.78125
\(351\) 0 0
\(352\) −7.10089e12 −1.31401
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −1.47156e12 −0.250244
\(359\) 6.29788e12 1.05614 0.528071 − 0.849200i \(-0.322915\pi\)
0.528071 + 0.849200i \(0.322915\pi\)
\(360\) 0 0
\(361\) 6.13107e12 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 1.89149e13 2.80262
\(369\) 0 0
\(370\) 0 0
\(371\) 3.92760e12 0.558802
\(372\) 0 0
\(373\) −1.40811e13 −1.95026 −0.975128 − 0.221644i \(-0.928858\pi\)
−0.975128 + 0.221644i \(0.928858\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.56390e13 1.99993 0.999963 − 0.00859646i \(-0.00273637\pi\)
0.999963 + 0.00859646i \(0.00273637\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 2.83698e13 3.48770
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.20391e13 −2.57191
\(387\) −9.21173e12 −1.06117
\(388\) 0 0
\(389\) −1.62268e13 −1.82173 −0.910866 − 0.412703i \(-0.864585\pi\)
−0.910866 + 0.412703i \(0.864585\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.93374e13 2.08914
\(393\) 0 0
\(394\) −2.53403e13 −2.66889
\(395\) 0 0
\(396\) −4.15602e13 −4.26778
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.58559e13 1.54843
\(401\) 1.56589e13 1.51022 0.755109 − 0.655599i \(-0.227584\pi\)
0.755109 + 0.655599i \(0.227584\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −1.15010e13 −1.04257
\(407\) 1.88932e13 1.69174
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 3.92102e13 3.22402
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −2.21462e13 −1.67451 −0.837257 − 0.546809i \(-0.815842\pi\)
−0.837257 + 0.546809i \(0.815842\pi\)
\(422\) 2.09229e13 1.56336
\(423\) 0 0
\(424\) −1.59976e13 −1.16742
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −5.30418e13 −3.69317
\(429\) 0 0
\(430\) 0 0
\(431\) 2.41135e13 1.62134 0.810671 − 0.585503i \(-0.199103\pi\)
0.810671 + 0.585503i \(0.199103\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.88124e13 2.46342
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.66799e13 1.00000
\(442\) 0 0
\(443\) −8.06311e12 −0.472589 −0.236295 − 0.971681i \(-0.575933\pi\)
−0.236295 + 0.971681i \(0.575933\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 6.43841e12 0.356770
\(449\) −3.25050e13 −1.78123 −0.890613 − 0.454762i \(-0.849724\pi\)
−0.890613 + 0.454762i \(0.849724\pi\)
\(450\) 3.28691e13 1.78125
\(451\) 0 0
\(452\) −4.82157e13 −2.55562
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6.46199e12 −0.324180 −0.162090 − 0.986776i \(-0.551823\pi\)
−0.162090 + 0.986776i \(0.551823\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −3.42233e13 −1.60849 −0.804243 − 0.594301i \(-0.797429\pi\)
−0.804243 + 0.594301i \(0.797429\pi\)
\(464\) 1.94923e13 0.906301
\(465\) 0 0
\(466\) 1.50295e13 0.683935
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −3.54815e13 −1.56364
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.93473e13 2.08429
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.37991e13 −0.558802
\(478\) −5.90587e13 −2.36671
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.64927e14 6.20964
\(485\) 0 0
\(486\) 0 0
\(487\) −4.71610e13 −1.72162 −0.860811 − 0.508924i \(-0.830043\pi\)
−0.860811 + 0.508924i \(0.830043\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.08478e13 0.730553 0.365277 − 0.930899i \(-0.380974\pi\)
0.365277 + 0.930899i \(0.380974\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.05642e13 1.99726
\(498\) 0 0
\(499\) 6.17850e13 1.99701 0.998505 − 0.0546568i \(-0.0174065\pi\)
0.998505 + 0.0546568i \(0.0174065\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) −6.79392e13 −2.08914
\(505\) 0 0
\(506\) −2.10049e14 −6.33241
\(507\) 0 0
\(508\) 5.78678e12 0.171048
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −7.73700e13 −2.19899
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 5.72183e13 1.53422
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 4.04071e13 1.04257
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −9.74226e12 −0.241953
\(527\) 0 0
\(528\) 0 0
\(529\) 9.42871e13 2.27601
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 1.44521e14 3.26667
\(537\) 0 0
\(538\) 0 0
\(539\) −8.93543e13 −1.96414
\(540\) 0 0
\(541\) 3.11042e13 0.671171 0.335585 − 0.942010i \(-0.391066\pi\)
0.335585 + 0.942010i \(0.391066\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −9.29416e13 −1.89790 −0.948951 − 0.315425i \(-0.897853\pi\)
−0.948951 + 0.315425i \(0.897853\pi\)
\(548\) 6.31977e13 1.27879
\(549\) 0 0
\(550\) −1.76080e14 −3.49862
\(551\) 0 0
\(552\) 0 0
\(553\) −4.58487e12 −0.0886547
\(554\) 1.85154e14 3.54800
\(555\) 0 0
\(556\) 0 0
\(557\) −6.82963e13 −1.27386 −0.636929 − 0.770923i \(-0.719796\pi\)
−0.636929 + 0.770923i \(0.719796\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 1.70327e14 3.03810
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −5.86024e13 −1.00000
\(568\) −2.46686e14 −4.17256
\(569\) −9.58640e12 −0.160729 −0.0803645 − 0.996766i \(-0.525608\pi\)
−0.0803645 + 0.996766i \(0.525608\pi\)
\(570\) 0 0
\(571\) −8.12132e13 −1.33797 −0.668985 − 0.743276i \(-0.733271\pi\)
−0.668985 + 0.743276i \(0.733271\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.13766e14 1.80997
\(576\) −2.26204e13 −0.356770
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 1.14912e14 1.78125
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 7.39217e13 1.09756
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −9.69756e13 −1.33369
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −7.40890e13 −0.985195
\(597\) 0 0
\(598\) 0 0
\(599\) −1.02711e14 −1.33193 −0.665966 − 0.745982i \(-0.731980\pi\)
−0.665966 + 0.745982i \(0.731980\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 1.49449e14 1.89021
\(603\) 1.24659e14 1.56364
\(604\) −1.97042e14 −2.45118
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.34938e14 1.55894 0.779472 − 0.626437i \(-0.215487\pi\)
0.779472 + 0.626437i \(0.215487\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 3.63951e14 4.10336
\(617\) −1.27728e14 −1.42844 −0.714219 − 0.699922i \(-0.753218\pi\)
−0.714219 + 0.699922i \(0.753218\pi\)
\(618\) 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
\(625\) 9.53674e13 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.99076e14 1.99009 0.995043 − 0.0994453i \(-0.0317068\pi\)
0.995043 + 0.0994453i \(0.0317068\pi\)
\(632\) 1.86748e13 0.185212
\(633\) 0 0
\(634\) −3.62121e14 −3.53515
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −2.16461e14 −2.04775
\(639\) −2.12784e14 −1.99726
\(640\) 0 0
\(641\) −8.42670e13 −0.778695 −0.389348 − 0.921091i \(-0.627300\pi\)
−0.389348 + 0.921091i \(0.627300\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −4.35644e14 −3.93281
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 2.38695e14 2.08914
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 4.66303e14 3.95758
\(653\) −1.16963e14 −0.985105 −0.492552 − 0.870283i \(-0.663936\pi\)
−0.492552 + 0.870283i \(0.663936\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.60404e14 1.29059 0.645297 − 0.763932i \(-0.276734\pi\)
0.645297 + 0.763932i \(0.276734\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 5.03468e13 0.395988
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −2.01028e14 −1.53422
\(667\) 1.39856e14 1.05938
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.74208e14 −1.98612 −0.993060 − 0.117612i \(-0.962476\pi\)
−0.993060 + 0.117612i \(0.962476\pi\)
\(674\) 4.90741e14 3.52820
\(675\) 0 0
\(676\) 3.06735e14 2.17285
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.57065e14 1.05676 0.528379 − 0.849008i \(-0.322800\pi\)
0.528379 + 0.849008i \(0.322800\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.70611e14 −1.78125
\(687\) 0 0
\(688\) −2.53291e14 −1.64315
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 3.13933e14 1.96414
\(694\) −2.72476e14 −1.69251
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −3.65191e14 −2.17285
\(701\) −3.14675e14 −1.85897 −0.929484 − 0.368862i \(-0.879747\pi\)
−0.929484 + 0.368862i \(0.879747\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.21178e14 0.700745
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.53178e14 −0.855000 −0.427500 − 0.904015i \(-0.640606\pi\)
−0.427500 + 0.904015i \(0.640606\pi\)
\(710\) 0 0
\(711\) 1.61083e13 0.0886547
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −5.74427e13 −0.305260
\(717\) 0 0
\(718\) 3.58979e14 1.88125
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.49471e14 1.78125
\(723\) 0 0
\(724\) 0 0
\(725\) 1.17239e14 0.585302
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2.05891e14 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
\(736\) 2.61511e14 1.21088
\(737\) −6.67800e14 −3.07121
\(738\) 0 0
\(739\) −4.33794e14 −1.96816 −0.984082 − 0.177715i \(-0.943129\pi\)
−0.984082 + 0.177715i \(0.943129\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.23873e14 0.995366
\(743\) 1.69642e14 0.749185 0.374593 − 0.927190i \(-0.377783\pi\)
0.374593 + 0.927190i \(0.377783\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −8.02621e14 −3.47389
\(747\) 0 0
\(748\) 0 0
\(749\) 4.00662e14 1.69969
\(750\) 0 0
\(751\) 1.52668e14 0.639068 0.319534 − 0.947575i \(-0.396474\pi\)
0.319534 + 0.947575i \(0.396474\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.77230e14 −0.712947 −0.356474 − 0.934305i \(-0.616021\pi\)
−0.356474 + 0.934305i \(0.616021\pi\)
\(758\) 8.91425e14 3.56237
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −2.93178e14 −1.13373
\(764\) 1.10742e15 4.25446
\(765\) 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
\(772\) −8.60296e14 −3.13734
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −5.25068e14 −1.89021
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −9.24927e14 −3.24496
\(779\) 0 0
\(780\) 0 0
\(781\) 1.13988e15 3.92289
\(782\) 0 0
\(783\) 0 0
\(784\) 4.58641e14 1.54843
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −9.89160e14 −3.25563
\(789\) 0 0
\(790\) 0 0
\(791\) 3.64207e14 1.17616
\(792\) −1.27869e15 −4.10336
\(793\) 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
\(800\) 2.19219e14 0.669003
\(801\) 0 0
\(802\) 8.92558e14 2.69008
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −3.55581e14 −1.02611 −0.513057 − 0.858354i \(-0.671487\pi\)
−0.513057 + 0.858354i \(0.671487\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −4.48942e14 −1.27178
\(813\) 0 0
\(814\) 1.07691e15 3.01341
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 5.20501e14 1.39542 0.697711 − 0.716379i \(-0.254202\pi\)
0.697711 + 0.716379i \(0.254202\pi\)
\(822\) 0 0
\(823\) −7.04921e14 −1.86699 −0.933493 − 0.358595i \(-0.883256\pi\)
−0.933493 + 0.358595i \(0.883256\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5.84635e14 1.51132 0.755662 − 0.654962i \(-0.227315\pi\)
0.755662 + 0.654962i \(0.227315\pi\)
\(828\) 1.53057e15 3.93281
\(829\) 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
\(841\) −2.76582e14 −0.657421
\(842\) −1.26233e15 −2.98273
\(843\) 0 0
\(844\) 8.16728e14 1.90706
\(845\) 0 0
\(846\) 0 0
\(847\) −1.24581e15 −2.85783
\(848\) −3.79428e14 −0.865267
\(849\) 0 0
\(850\) 0 0
\(851\) −6.95796e14 −1.55896
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.63195e15 −3.55089
\(857\) 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
\(862\) 1.37447e15 2.88801
\(863\) −1.94832e14 −0.407012 −0.203506 − 0.979074i \(-0.565234\pi\)
−0.203506 + 0.979074i \(0.565234\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −8.62923e13 −0.174130
\(870\) 0 0
\(871\) 0 0
\(872\) 1.19415e15 2.36851
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.40999e14 1.62105 0.810527 − 0.585702i \(-0.199181\pi\)
0.810527 + 0.585702i \(0.199181\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 9.50753e14 1.78125
\(883\) 4.78865e14 0.892092 0.446046 − 0.895010i \(-0.352832\pi\)
0.446046 + 0.895010i \(0.352832\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4.59597e14 −0.841800
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −4.37117e13 −0.0787206
\(890\) 0 0
\(891\) −1.10296e15 −1.96414
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 7.53328e14 1.30450
\(897\) 0 0
\(898\) −1.85279e15 −3.17281
\(899\) 0 0
\(900\) 1.28305e15 2.17285
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −1.48346e15 −2.45716
\(905\) 0 0
\(906\) 0 0
\(907\) 7.82523e14 1.27486 0.637428 − 0.770510i \(-0.279999\pi\)
0.637428 + 0.770510i \(0.279999\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.12504e15 1.79298 0.896489 − 0.443066i \(-0.146109\pi\)
0.896489 + 0.443066i \(0.146109\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −3.68334e14 −0.577445
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.25660e15 1.91699 0.958494 − 0.285113i \(-0.0920311\pi\)
0.958494 + 0.285113i \(0.0920311\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −5.83271e14 −0.861314
\(926\) −1.95073e15 −2.86511
\(927\) 0 0
\(928\) 2.69494e14 0.391569
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 5.86678e14 0.834296
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) −2.02245e15 −2.78524
\(939\) 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
\(946\) 2.81280e15 3.71264
\(947\) 1.50182e15 1.97182 0.985911 − 0.167273i \(-0.0534961\pi\)
0.985911 + 0.167273i \(0.0534961\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.47273e15 −1.87351 −0.936757 − 0.349979i \(-0.886189\pi\)
−0.936757 + 0.349979i \(0.886189\pi\)
\(954\) −7.86547e14 −0.995366
\(955\) 0 0
\(956\) −2.30536e15 −2.88702
\(957\) 0 0
\(958\) 0 0
\(959\) −4.77377e14 −0.588530
\(960\) 0 0
\(961\) 8.19628e14 1.00000
\(962\) 0 0
\(963\) −1.40767e15 −1.69969
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.29018e15 −1.52588 −0.762938 − 0.646471i \(-0.776244\pi\)
−0.762938 + 0.646471i \(0.776244\pi\)
\(968\) 5.07436e15 5.97041
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.68818e15 −3.06664
\(975\) 0 0
\(976\) 0 0
\(977\) 7.17832e14 0.806399 0.403200 − 0.915112i \(-0.367898\pi\)
0.403200 + 0.915112i \(0.367898\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.03004e15 1.13373
\(982\) 1.18832e15 1.30130
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.81736e15 −1.92070
\(990\) 0 0
\(991\) 1.37334e15 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 3.45216e15 3.55762
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 3.52175e15 3.55717
\(999\) 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 7.11.b.a.6.1 ✓ 1
3.2 odd 2 63.11.d.a.55.1 1
4.3 odd 2 112.11.c.a.97.1 1
7.2 even 3 49.11.d.a.31.1 2
7.3 odd 6 49.11.d.a.19.1 2
7.4 even 3 49.11.d.a.19.1 2
7.5 odd 6 49.11.d.a.31.1 2
7.6 odd 2 CM 7.11.b.a.6.1 ✓ 1
21.20 even 2 63.11.d.a.55.1 1
28.27 even 2 112.11.c.a.97.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.11.b.a.6.1 ✓ 1 1.1 even 1 trivial
7.11.b.a.6.1 ✓ 1 7.6 odd 2 CM
49.11.d.a.19.1 2 7.3 odd 6
49.11.d.a.19.1 2 7.4 even 3
49.11.d.a.31.1 2 7.2 even 3
49.11.d.a.31.1 2 7.5 odd 6
63.11.d.a.55.1 1 3.2 odd 2
63.11.d.a.55.1 1 21.20 even 2
112.11.c.a.97.1 1 4.3 odd 2
112.11.c.a.97.1 1 28.27 even 2