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Properties

Label 7.55.b.a.6.1
Level $7$
Weight $55$
Character 7.6
Self dual yes
Analytic conductor $129.275$
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] = [7,55,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, 55); 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, 55, names="a")
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 55 \)
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: \(129.275273337\)
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)
 
\(f(q)\) \(=\) \(q-2.11357e8 q^{2} +2.66576e16 q^{4} -6.57124e22 q^{7} -1.82680e24 q^{8} +5.81497e25 q^{9} +2.41922e28 q^{11} +1.38888e31 q^{14} -9.41128e31 q^{16} -1.22904e34 q^{18} -5.11321e36 q^{22} +1.15717e37 q^{23} +5.55112e37 q^{25} -1.75173e39 q^{28} +4.81899e39 q^{29} +5.28001e40 q^{32} +1.55013e42 q^{36} +4.00275e42 q^{37} -2.30201e44 q^{43} +6.44906e44 q^{44} -2.44575e45 q^{46} +4.31811e45 q^{49} -1.17327e46 q^{50} +7.17962e46 q^{53} +1.20043e47 q^{56} -1.01853e48 q^{58} -3.82116e48 q^{63} -9.46431e48 q^{64} -2.91139e49 q^{67} -6.04957e48 q^{71} -1.06228e50 q^{72} -8.46010e50 q^{74} -1.58973e51 q^{77} -3.43359e51 q^{79} +3.38139e51 q^{81} +4.86547e52 q^{86} -4.41943e52 q^{88} +3.08472e53 q^{92} -9.12666e53 q^{98} +1.40677e54 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.11357e8 −1.57474 −0.787368 − 0.616484i \(-0.788557\pi\)
−0.787368 + 0.616484i \(0.788557\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 2.66576e16 1.47979
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −6.57124e22 −1.00000
\(8\) −1.82680e24 −0.755546
\(9\) 5.81497e25 1.00000
\(10\) 0 0
\(11\) 2.41922e28 1.84533 0.922664 − 0.385604i \(-0.126007\pi\)
0.922664 + 0.385604i \(0.126007\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 1.38888e31 1.57474
\(15\) 0 0
\(16\) −9.41128e31 −0.290007
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −1.22904e34 −1.57474
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.11321e36 −2.90590
\(23\) 1.15717e37 1.98036 0.990179 − 0.139804i \(-0.0446474\pi\)
0.990179 + 0.139804i \(0.0446474\pi\)
\(24\) 0 0
\(25\) 5.55112e37 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −1.75173e39 −1.47979
\(29\) 4.81899e39 1.57837 0.789187 − 0.614154i \(-0.210502\pi\)
0.789187 + 0.614154i \(0.210502\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 5.28001e40 1.21223
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.55013e42 1.47979
\(37\) 4.00275e42 1.82352 0.911762 − 0.410719i \(-0.134722\pi\)
0.911762 + 0.410719i \(0.134722\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −2.30201e44 −1.81326 −0.906628 − 0.421930i \(-0.861353\pi\)
−0.906628 + 0.421930i \(0.861353\pi\)
\(44\) 6.44906e44 2.73070
\(45\) 0 0
\(46\) −2.44575e45 −3.11854
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 4.31811e45 1.00000
\(50\) −1.17327e46 −1.57474
\(51\) 0 0
\(52\) 0 0
\(53\) 7.17962e46 1.99826 0.999132 − 0.0416651i \(-0.0132662\pi\)
0.999132 + 0.0416651i \(0.0132662\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.20043e47 0.755546
\(57\) 0 0
\(58\) −1.01853e48 −2.48552
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −3.82116e48 −1.00000
\(64\) −9.46431e48 −1.61894
\(65\) 0 0
\(66\) 0 0
\(67\) −2.91139e49 −1.44571 −0.722853 − 0.691002i \(-0.757170\pi\)
−0.722853 + 0.691002i \(0.757170\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.04957e48 −0.0627696 −0.0313848 − 0.999507i \(-0.509992\pi\)
−0.0313848 + 0.999507i \(0.509992\pi\)
\(72\) −1.06228e50 −0.755546
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −8.46010e50 −2.87157
\(75\) 0 0
\(76\) 0 0
\(77\) −1.58973e51 −1.84533
\(78\) 0 0
\(79\) −3.43359e51 −1.99442 −0.997210 − 0.0746437i \(-0.976218\pi\)
−0.997210 + 0.0746437i \(0.976218\pi\)
\(80\) 0 0
\(81\) 3.38139e51 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.86547e52 2.85540
\(87\) 0 0
\(88\) −4.41943e52 −1.39423
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3.08472e53 2.93052
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −9.12666e53 −1.57474
\(99\) 1.40677e54 1.84533
\(100\) 1.47979e54 1.47979
\(101\) 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
\(106\) −1.51747e55 −3.14674
\(107\) 2.10423e54 0.338635 0.169318 − 0.985562i \(-0.445844\pi\)
0.169318 + 0.985562i \(0.445844\pi\)
\(108\) 0 0
\(109\) −1.86495e55 −1.82034 −0.910168 − 0.414240i \(-0.864047\pi\)
−0.910168 + 0.414240i \(0.864047\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 6.18437e54 0.290007
\(113\) −1.98601e55 −0.732594 −0.366297 − 0.930498i \(-0.619375\pi\)
−0.366297 + 0.930498i \(0.619375\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.28462e56 2.33566
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 4.13393e56 2.40524
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 8.07630e56 1.57474
\(127\) −2.67928e56 −0.422005 −0.211003 − 0.977485i \(-0.567673\pi\)
−0.211003 + 0.977485i \(0.567673\pi\)
\(128\) 1.04919e57 1.33716
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 6.15343e57 2.27661
\(135\) 0 0
\(136\) 0 0
\(137\) −2.64097e57 −0.537414 −0.268707 − 0.963222i \(-0.586596\pi\)
−0.268707 + 0.963222i \(0.586596\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.27862e57 0.0988455
\(143\) 0 0
\(144\) −5.47263e57 −0.290007
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 1.06703e59 2.69844
\(149\) 5.74055e58 1.21039 0.605193 − 0.796079i \(-0.293096\pi\)
0.605193 + 0.796079i \(0.293096\pi\)
\(150\) 0 0
\(151\) 8.99377e58 1.32301 0.661506 − 0.749939i \(-0.269917\pi\)
0.661506 + 0.749939i \(0.269917\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 3.36001e59 2.90590
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 7.25714e59 3.14068
\(159\) 0 0
\(160\) 0 0
\(161\) −7.60401e59 −1.98036
\(162\) −7.14682e59 −1.57474
\(163\) 1.07487e59 0.200581 0.100290 − 0.994958i \(-0.468023\pi\)
0.100290 + 0.994958i \(0.468023\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.42214e60 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −6.13660e60 −2.68324
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −3.64777e60 −1.00000
\(176\) −2.27680e60 −0.535159
\(177\) 0 0
\(178\) 0 0
\(179\) 3.30545e60 0.492265 0.246133 − 0.969236i \(-0.420840\pi\)
0.246133 + 0.969236i \(0.420840\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2.11391e61 −1.49625
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.65860e61 1.97810 0.989051 − 0.147577i \(-0.0471474\pi\)
0.989051 + 0.147577i \(0.0471474\pi\)
\(192\) 0 0
\(193\) −9.61251e61 −1.87408 −0.937042 − 0.349216i \(-0.886448\pi\)
−0.937042 + 0.349216i \(0.886448\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.15110e62 1.47979
\(197\) −7.82570e61 −0.876870 −0.438435 − 0.898763i \(-0.644467\pi\)
−0.438435 + 0.898763i \(0.644467\pi\)
\(198\) −2.97332e62 −2.90590
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −1.01408e62 −0.755546
\(201\) 0 0
\(202\) 0 0
\(203\) −3.16667e62 −1.57837
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.72888e62 1.98036
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 6.13194e62 1.07640 0.538198 − 0.842819i \(-0.319105\pi\)
0.538198 + 0.842819i \(0.319105\pi\)
\(212\) 1.91391e63 2.95701
\(213\) 0 0
\(214\) −4.44746e62 −0.533261
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 3.94171e63 2.86655
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) −3.46962e63 −1.21223
\(225\) 3.22796e63 1.00000
\(226\) 4.19758e63 1.15364
\(227\) 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
\(232\) −8.80331e63 −1.19253
\(233\) −1.43271e64 −1.72802 −0.864009 − 0.503477i \(-0.832054\pi\)
−0.864009 + 0.503477i \(0.832054\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.50530e64 0.913859 0.456929 − 0.889503i \(-0.348949\pi\)
0.456929 + 0.889503i \(0.348949\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −8.73737e64 −3.78761
\(243\) 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 \(\pi\)
\(252\) −1.01863e65 −1.47979
\(253\) 2.79944e65 3.65441
\(254\) 5.66286e64 0.664547
\(255\) 0 0
\(256\) −5.12602e64 −0.486745
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −2.63030e65 −1.82352
\(260\) 0 0
\(261\) 2.80223e65 1.57837
\(262\) 0 0
\(263\) −3.66076e65 −1.67790 −0.838950 − 0.544209i \(-0.816830\pi\)
−0.838950 + 0.544209i \(0.816830\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −7.76105e65 −2.13935
\(269\) 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
\(274\) 5.58189e65 0.846285
\(275\) 1.34294e66 1.84533
\(276\) 0 0
\(277\) 1.27164e66 1.43684 0.718421 − 0.695609i \(-0.244865\pi\)
0.718421 + 0.695609i \(0.244865\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.81099e66 1.38946 0.694730 − 0.719270i \(-0.255524\pi\)
0.694730 + 0.719270i \(0.255524\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −1.61267e65 −0.0928859
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 3.07031e66 1.21223
\(289\) 2.78126e66 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
\(296\) −7.31221e66 −1.37776
\(297\) 0 0
\(298\) −1.21331e67 −1.90604
\(299\) 0 0
\(300\) 0 0
\(301\) 1.51271e67 1.81326
\(302\) −1.90090e67 −2.08340
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) −4.23783e67 −2.73070
\(309\) 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
\(316\) −9.15311e67 −2.95133
\(317\) −5.62419e67 −1.66517 −0.832587 − 0.553895i \(-0.813141\pi\)
−0.832587 + 0.553895i \(0.813141\pi\)
\(318\) 0 0
\(319\) 1.16582e68 2.91262
\(320\) 0 0
\(321\) 0 0
\(322\) 1.60716e68 3.11854
\(323\) 0 0
\(324\) 9.01397e67 1.47979
\(325\) 0 0
\(326\) −2.27181e67 −0.315861
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −5.82275e66 −0.0536750 −0.0268375 − 0.999640i \(-0.508544\pi\)
−0.0268375 + 0.999640i \(0.508544\pi\)
\(332\) 0 0
\(333\) 2.32759e68 1.82352
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −6.74218e67 −0.382642 −0.191321 − 0.981528i \(-0.561277\pi\)
−0.191321 + 0.981528i \(0.561277\pi\)
\(338\) −3.00579e68 −1.57474
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −2.83754e68 −1.00000
\(344\) 4.20531e68 1.37000
\(345\) 0 0
\(346\) 0 0
\(347\) −6.90461e68 −1.77928 −0.889638 − 0.456667i \(-0.849043\pi\)
−0.889638 + 0.456667i \(0.849043\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 7.70983e68 1.57474
\(351\) 0 0
\(352\) 1.27735e69 2.23696
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −6.98632e68 −0.775188
\(359\) −9.16186e68 −0.942830 −0.471415 − 0.881912i \(-0.656257\pi\)
−0.471415 + 0.881912i \(0.656257\pi\)
\(360\) 0 0
\(361\) 1.12900e69 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 \(\pi\)
\(368\) −1.08904e69 −0.574319
\(369\) 0 0
\(370\) 0 0
\(371\) −4.71789e69 −1.99826
\(372\) 0 0
\(373\) −1.42691e69 −0.522705 −0.261353 − 0.965243i \(-0.584169\pi\)
−0.261353 + 0.965243i \(0.584169\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.96373e69 0.705635 0.352818 − 0.935692i \(-0.385224\pi\)
0.352818 + 0.935692i \(0.385224\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.61870e70 −3.11499
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.03167e70 2.95119
\(387\) −1.33861e70 −1.81326
\(388\) 0 0
\(389\) −8.58113e69 −1.01136 −0.505680 − 0.862721i \(-0.668758\pi\)
−0.505680 + 0.862721i \(0.668758\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −7.88832e69 −0.755546
\(393\) 0 0
\(394\) 1.65402e70 1.38084
\(395\) 0 0
\(396\) 3.75011e70 2.73070
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −5.22431e69 −0.290007
\(401\) 8.53127e69 0.442706 0.221353 − 0.975194i \(-0.428953\pi\)
0.221353 + 0.975194i \(0.428953\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 6.69299e70 2.48552
\(407\) 9.68354e70 3.36500
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −1.42220e71 −3.11854
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −1.14552e71 −1.59730 −0.798648 − 0.601799i \(-0.794451\pi\)
−0.798648 + 0.601799i \(0.794451\pi\)
\(422\) −1.29603e71 −1.69504
\(423\) 0 0
\(424\) −1.31157e71 −1.50978
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 5.60938e70 0.501110
\(429\) 0 0
\(430\) 0 0
\(431\) 2.49815e71 1.84813 0.924066 − 0.382233i \(-0.124845\pi\)
0.924066 + 0.382233i \(0.124845\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.97150e71 −2.69372
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 2.51097e71 1.00000
\(442\) 0 0
\(443\) 1.78927e70 0.0630630 0.0315315 − 0.999503i \(-0.489962\pi\)
0.0315315 + 0.999503i \(0.489962\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 6.21922e71 1.61894
\(449\) 6.77020e71 1.65940 0.829699 − 0.558211i \(-0.188512\pi\)
0.829699 + 0.558211i \(0.188512\pi\)
\(450\) −6.82253e71 −1.57474
\(451\) 0 0
\(452\) −5.29422e71 −1.08409
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.26294e71 −0.496445 −0.248223 − 0.968703i \(-0.579846\pi\)
−0.248223 + 0.968703i \(0.579846\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 3.59150e70 0.0384228 0.0192114 − 0.999815i \(-0.493884\pi\)
0.0192114 + 0.999815i \(0.493884\pi\)
\(464\) −4.53528e71 −0.457740
\(465\) 0 0
\(466\) 3.02814e72 2.72117
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 1.91314e72 1.44571
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.56908e72 −3.34605
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.17493e72 1.99826
\(478\) −3.18156e72 −1.43909
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.10200e73 3.55925
\(485\) 0 0
\(486\) 0 0
\(487\) −6.82158e72 −1.86467 −0.932337 − 0.361590i \(-0.882234\pi\)
−0.932337 + 0.361590i \(0.882234\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8.97770e72 1.96773 0.983863 − 0.178924i \(-0.0572616\pi\)
0.983863 + 0.178924i \(0.0572616\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.97532e71 0.0627696
\(498\) 0 0
\(499\) −1.33415e73 −1.89012 −0.945058 − 0.326903i \(-0.893995\pi\)
−0.945058 + 0.326903i \(0.893995\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 6.98048e72 0.755546
\(505\) 0 0
\(506\) −5.91683e73 −5.75473
\(507\) 0 0
\(508\) −7.14231e72 −0.624480
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −8.06629e72 −0.570670
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 5.55933e73 2.87157
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) −5.92272e73 −2.48552
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 7.73729e73 2.64225
\(527\) 0 0
\(528\) 0 0
\(529\) 9.97600e73 2.92182
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 5.31851e73 1.09230
\(537\) 0 0
\(538\) 0 0
\(539\) 1.04465e74 1.84533
\(540\) 0 0
\(541\) −1.20351e74 −1.92364 −0.961818 − 0.273689i \(-0.911756\pi\)
−0.961818 + 0.273689i \(0.911756\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.66589e74 1.97692 0.988459 − 0.151487i \(-0.0484061\pi\)
0.988459 + 0.151487i \(0.0484061\pi\)
\(548\) −7.04019e73 −0.795261
\(549\) 0 0
\(550\) −2.83840e74 −2.90590
\(551\) 0 0
\(552\) 0 0
\(553\) 2.25629e74 1.99442
\(554\) −2.68770e74 −2.26265
\(555\) 0 0
\(556\) 0 0
\(557\) 2.57695e74 1.87507 0.937533 − 0.347897i \(-0.113104\pi\)
0.937533 + 0.347897i \(0.113104\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −3.82767e74 −2.18803
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.22199e74 −1.00000
\(568\) 1.10513e73 0.0474253
\(569\) −3.58059e74 −1.46529 −0.732645 − 0.680611i \(-0.761714\pi\)
−0.732645 + 0.680611i \(0.761714\pi\)
\(570\) 0 0
\(571\) 1.00118e74 0.372681 0.186340 − 0.982485i \(-0.440337\pi\)
0.186340 + 0.982485i \(0.440337\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.42356e74 1.98036
\(576\) −5.50347e74 −1.61894
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −5.87840e74 −1.57474
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1.73691e75 3.68745
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −3.76710e74 −0.528835
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.53029e75 1.79112
\(597\) 0 0
\(598\) 0 0
\(599\) −1.39535e75 −1.42613 −0.713067 − 0.701096i \(-0.752695\pi\)
−0.713067 + 0.701096i \(0.752695\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) −3.19722e75 −2.85540
\(603\) −1.69296e75 −1.44571
\(604\) 2.39752e75 1.95778
\(605\) 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
\(613\) 3.62762e75 1.98696 0.993480 − 0.114008i \(-0.0363689\pi\)
0.993480 + 0.114008i \(0.0363689\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 2.90411e75 1.39423
\(617\) 4.45285e74 0.204616 0.102308 − 0.994753i \(-0.467377\pi\)
0.102308 + 0.994753i \(0.467377\pi\)
\(618\) 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
\(625\) 3.08149e75 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.76981e75 −0.443745 −0.221872 − 0.975076i \(-0.571217\pi\)
−0.221872 + 0.975076i \(0.571217\pi\)
\(632\) 6.27247e75 1.50688
\(633\) 0 0
\(634\) 1.18871e76 2.62221
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −2.46405e76 −4.58660
\(639\) −3.51781e74 −0.0627696
\(640\) 0 0
\(641\) −3.30728e75 −0.542381 −0.271190 − 0.962526i \(-0.587417\pi\)
−0.271190 + 0.962526i \(0.587417\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −2.02704e76 −2.93052
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −6.17712e75 −0.755546
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 2.86533e75 0.296818
\(653\) 5.32535e75 0.529288 0.264644 − 0.964346i \(-0.414746\pi\)
0.264644 + 0.964346i \(0.414746\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.47695e76 −1.14673 −0.573365 − 0.819300i \(-0.694362\pi\)
−0.573365 + 0.819300i \(0.694362\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 1.23068e75 0.0845240
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −4.91953e76 −2.87157
\(667\) 5.57636e76 3.12574
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.36841e76 0.602297 0.301149 − 0.953577i \(-0.402630\pi\)
0.301149 + 0.953577i \(0.402630\pi\)
\(674\) 1.42501e76 0.602560
\(675\) 0 0
\(676\) 3.79107e76 1.47979
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.68038e76 1.38332 0.691659 − 0.722225i \(-0.256880\pi\)
0.691659 + 0.722225i \(0.256880\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 5.99734e76 1.57474
\(687\) 0 0
\(688\) 2.16649e76 0.525858
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) −9.24424e76 −1.84533
\(694\) 1.45934e77 2.80189
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −9.72406e76 −1.47979
\(701\) 6.17669e76 0.904418 0.452209 − 0.891912i \(-0.350636\pi\)
0.452209 + 0.891912i \(0.350636\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −2.28963e77 −2.98747
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.82350e77 −1.96543 −0.982714 − 0.185128i \(-0.940730\pi\)
−0.982714 + 0.185128i \(0.940730\pi\)
\(710\) 0 0
\(711\) −1.99662e77 −1.99442
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 8.81154e76 0.728451
\(717\) 0 0
\(718\) 1.93643e77 1.48471
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −2.38623e77 −1.57474
\(723\) 0 0
\(724\) 0 0
\(725\) 2.67507e77 1.57837
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.96627e77 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
\(736\) 6.10984e77 2.40065
\(737\) −7.04330e77 −2.66780
\(738\) 0 0
\(739\) −5.44083e77 −1.91543 −0.957713 − 0.287725i \(-0.907101\pi\)
−0.957713 + 0.287725i \(0.907101\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 9.97162e77 3.14674
\(743\) −5.75975e77 −1.75269 −0.876346 − 0.481682i \(-0.840026\pi\)
−0.876346 + 0.481682i \(0.840026\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 3.01587e77 0.823123
\(747\) 0 0
\(748\) 0 0
\(749\) −1.38274e77 −0.338635
\(750\) 0 0
\(751\) −4.20483e77 −0.958230 −0.479115 − 0.877752i \(-0.659042\pi\)
−0.479115 + 0.877752i \(0.659042\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −5.64047e77 −1.03688 −0.518438 − 0.855115i \(-0.673486\pi\)
−0.518438 + 0.855115i \(0.673486\pi\)
\(758\) −6.26407e77 −1.11119
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.22550e78 1.82034
\(764\) 2.04160e78 2.92718
\(765\) 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
\(772\) −2.56246e78 −2.77326
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 2.82926e78 2.85540
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 1.81369e78 1.59262
\(779\) 0 0
\(780\) 0 0
\(781\) −1.46353e77 −0.115830
\(782\) 0 0
\(783\) 0 0
\(784\) −4.06390e77 −0.290007
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −2.08614e78 −1.29759
\(789\) 0 0
\(790\) 0 0
\(791\) 1.30505e78 0.732594
\(792\) −2.56989e78 −1.39423
\(793\) 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
\(800\) 2.93099e78 1.21223
\(801\) 0 0
\(802\) −1.80315e78 −0.697145
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.50635e78 0.460592 0.230296 − 0.973121i \(-0.426030\pi\)
0.230296 + 0.973121i \(0.426030\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −8.44157e78 −2.33566
\(813\) 0 0
\(814\) −2.04669e79 −5.29899
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 7.35578e78 1.51134 0.755672 − 0.654950i \(-0.227310\pi\)
0.755672 + 0.654950i \(0.227310\pi\)
\(822\) 0 0
\(823\) −8.95437e78 −1.72282 −0.861409 − 0.507912i \(-0.830418\pi\)
−0.861409 + 0.507912i \(0.830418\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.18047e79 1.99254 0.996270 − 0.0862964i \(-0.0275032\pi\)
0.996270 + 0.0862964i \(0.0275032\pi\)
\(828\) 1.79376e79 2.93052
\(829\) 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
\(841\) 1.39010e79 1.49126
\(842\) 2.42114e79 2.51532
\(843\) 0 0
\(844\) 1.63463e79 1.59284
\(845\) 0 0
\(846\) 0 0
\(847\) −2.71650e79 −2.40524
\(848\) −6.75694e78 −0.579511
\(849\) 0 0
\(850\) 0 0
\(851\) 4.63184e79 3.61123
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −3.84401e78 −0.255854
\(857\) 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
\(862\) −5.28003e79 −2.91032
\(863\) −1.72334e79 −0.920620 −0.460310 − 0.887758i \(-0.652262\pi\)
−0.460310 + 0.887758i \(0.652262\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −8.30662e79 −3.68036
\(870\) 0 0
\(871\) 0 0
\(872\) 3.40688e79 1.37535
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.74662e79 1.29614 0.648069 − 0.761582i \(-0.275577\pi\)
0.648069 + 0.761582i \(0.275577\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −5.30713e79 −1.57474
\(883\) 1.15525e78 0.0332459 0.0166229 − 0.999862i \(-0.494709\pi\)
0.0166229 + 0.999862i \(0.494709\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −3.78175e78 −0.0993075
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.76062e79 0.422005
\(890\) 0 0
\(891\) 8.18035e79 1.84533
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −6.89447e79 −1.33716
\(897\) 0 0
\(898\) −1.43093e80 −2.61311
\(899\) 0 0
\(900\) 8.60495e79 1.47979
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 3.62804e79 0.553508
\(905\) 0 0
\(906\) 0 0
\(907\) 5.39963e78 0.0753299 0.0376649 − 0.999290i \(-0.488008\pi\)
0.0376649 + 0.999290i \(0.488008\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.51693e79 0.683441 0.341721 − 0.939802i \(-0.388990\pi\)
0.341721 + 0.939802i \(0.388990\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 6.89646e79 0.781770
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −1.21732e80 −1.19093 −0.595465 − 0.803381i \(-0.703032\pi\)
−0.595465 + 0.803381i \(0.703032\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.22197e80 1.82352
\(926\) −7.59090e78 −0.0605057
\(927\) 0 0
\(928\) 2.54443e80 1.91335
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −3.81925e80 −2.55711
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) −4.04356e80 −2.27661
\(939\) 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
\(946\) 1.17707e81 5.26915
\(947\) −2.57052e80 −1.11833 −0.559167 − 0.829055i \(-0.688879\pi\)
−0.559167 + 0.829055i \(0.688879\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 5.44606e80 1.99790 0.998951 − 0.0457925i \(-0.0145813\pi\)
0.998951 + 0.0457925i \(0.0145813\pi\)
\(954\) −8.82402e80 −3.14674
\(955\) 0 0
\(956\) 4.01276e80 1.35232
\(957\) 0 0
\(958\) 0 0
\(959\) 1.73545e80 0.537414
\(960\) 0 0
\(961\) 3.41611e80 1.00000
\(962\) 0 0
\(963\) 1.22361e80 0.338635
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −8.07975e80 −1.99932 −0.999662 − 0.0260161i \(-0.991718\pi\)
−0.999662 + 0.0260161i \(0.991718\pi\)
\(968\) −7.55185e80 −1.81727
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1.44179e81 2.93637
\(975\) 0 0
\(976\) 0 0
\(977\) −8.36201e80 −1.56732 −0.783659 − 0.621191i \(-0.786649\pi\)
−0.783659 + 0.621191i \(0.786649\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.08446e81 −1.82034
\(982\) −1.89750e81 −3.09865
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.66381e81 −3.59090
\(990\) 0 0
\(991\) 1.00732e81 1.28581 0.642907 − 0.765945i \(-0.277728\pi\)
0.642907 + 0.765945i \(0.277728\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −8.40213e79 −0.0988455
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 2.81982e81 2.97643
\(999\) 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 7.55.b.a.6.1 ✓ 1
7.6 odd 2 CM 7.55.b.a.6.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.55.b.a.6.1 ✓ 1 1.1 even 1 trivial
7.55.b.a.6.1 ✓ 1 7.6 odd 2 CM