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Properties

Label 4.45.b.a.3.1
Level $4$
Weight $45$
Character 4.3
Self dual yes
Analytic conductor $49.048$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4,45,Mod(3,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 45, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 45);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 45 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(49.0478939917\)
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 3.1
Character \(\chi\) \(=\) 4.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.19430e6 q^{2} +1.75922e13 q^{4} +9.46815e13 q^{5} -7.37870e19 q^{8} +9.84771e20 q^{9} +O(q^{10})\) \(q-4.19430e6 q^{2} +1.75922e13 q^{4} +9.46815e13 q^{5} -7.37870e19 q^{8} +9.84771e20 q^{9} -3.97123e20 q^{10} +4.74640e24 q^{13} +3.09485e26 q^{16} -5.04652e26 q^{17} -4.13043e27 q^{18} +1.66565e27 q^{20} -5.67538e30 q^{25} -1.99078e31 q^{26} -1.52161e32 q^{29} -1.29807e33 q^{32} +2.11666e33 q^{34} +1.73243e34 q^{36} +3.50881e34 q^{37} -6.98626e33 q^{40} -9.44265e34 q^{41} +9.32396e34 q^{45} +1.52867e37 q^{49} +2.38043e37 q^{50} +8.34995e37 q^{52} +1.63043e38 q^{53} +6.38209e38 q^{58} +2.50657e39 q^{61} +5.44452e39 q^{64} +4.49396e38 q^{65} -8.87793e39 q^{68} -7.26633e40 q^{72} -1.96251e41 q^{73} -1.47170e41 q^{74} +2.93025e40 q^{80} +9.69774e41 q^{81} +3.96054e41 q^{82} -4.77812e40 q^{85} +1.30963e43 q^{89} -3.91075e41 q^{90} +9.22466e43 q^{97} -6.41171e43 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.19430e6 −1.00000
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1.75922e13 1.00000
\(5\) 9.46815e13 0.0397123 0.0198561 − 0.999803i \(-0.493679\pi\)
0.0198561 + 0.999803i \(0.493679\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −7.37870e19 −1.00000
\(9\) 9.84771e20 1.00000
\(10\) −3.97123e20 −0.0397123
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 4.74640e24 1.47778 0.738891 − 0.673825i \(-0.235350\pi\)
0.738891 + 0.673825i \(0.235350\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.09485e26 1.00000
\(17\) −5.04652e26 −0.429651 −0.214825 − 0.976652i \(-0.568918\pi\)
−0.214825 + 0.976652i \(0.568918\pi\)
\(18\) −4.13043e27 −1.00000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.66565e27 0.0397123
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −5.67538e30 −0.998423
\(26\) −1.99078e31 −1.47778
\(27\) 0 0
\(28\) 0 0
\(29\) −1.52161e32 −1.02223 −0.511113 − 0.859514i \(-0.670767\pi\)
−0.511113 + 0.859514i \(0.670767\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.29807e33 −1.00000
\(33\) 0 0
\(34\) 2.11666e33 0.429651
\(35\) 0 0
\(36\) 1.73243e34 1.00000
\(37\) 3.50881e34 1.10846 0.554232 − 0.832362i \(-0.313012\pi\)
0.554232 + 0.832362i \(0.313012\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −6.98626e33 −0.0397123
\(41\) −9.44265e34 −0.311781 −0.155890 − 0.987774i \(-0.549825\pi\)
−0.155890 + 0.987774i \(0.549825\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 9.32396e34 0.0397123
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.52867e37 1.00000
\(50\) 2.38043e37 0.998423
\(51\) 0 0
\(52\) 8.34995e37 1.47778
\(53\) 1.63043e38 1.89772 0.948862 − 0.315692i \(-0.102237\pi\)
0.948862 + 0.315692i \(0.102237\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 6.38209e38 1.02223
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 2.50657e39 1.32380 0.661902 − 0.749590i \(-0.269749\pi\)
0.661902 + 0.749590i \(0.269749\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 5.44452e39 1.00000
\(65\) 4.49396e38 0.0586861
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −8.87793e39 −0.429651
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −7.26633e40 −1.00000
\(73\) −1.96251e41 −1.99393 −0.996964 − 0.0778673i \(-0.975189\pi\)
−0.996964 + 0.0778673i \(0.975189\pi\)
\(74\) −1.47170e41 −1.10846
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 2.93025e40 0.0397123
\(81\) 9.69774e41 1.00000
\(82\) 3.96054e41 0.311781
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −4.77812e40 −0.0170624
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.30963e43 1.70046 0.850232 − 0.526409i \(-0.176462\pi\)
0.850232 + 0.526409i \(0.176462\pi\)
\(90\) −3.91075e41 −0.0397123
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.22466e43 1.80290 0.901451 − 0.432880i \(-0.142503\pi\)
0.901451 + 0.432880i \(0.142503\pi\)
\(98\) −6.41171e43 −1.00000
\(99\) 0 0
\(100\) −9.98423e43 −0.998423
\(101\) −7.99539e43 −0.642347 −0.321173 − 0.947020i \(-0.604077\pi\)
−0.321173 + 0.947020i \(0.604077\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −3.50222e44 −1.47778
\(105\) 0 0
\(106\) −6.83852e44 −1.89772
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 1.28774e45 1.93394 0.966971 − 0.254885i \(-0.0820375\pi\)
0.966971 + 0.254885i \(0.0820375\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.87652e45 1.95497 0.977487 − 0.210996i \(-0.0676706\pi\)
0.977487 + 0.210996i \(0.0676706\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.67684e45 −1.02223
\(117\) 4.67411e45 1.47778
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 6.62641e45 1.00000
\(122\) −1.05133e46 −1.32380
\(123\) 0 0
\(124\) 0 0
\(125\) −1.07556e45 −0.0793620
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −2.28360e46 −1.00000
\(129\) 0 0
\(130\) −1.88490e45 −0.0586861
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 3.72367e46 0.429651
\(137\) −1.90448e47 −1.87035 −0.935177 − 0.354182i \(-0.884759\pi\)
−0.935177 + 0.354182i \(0.884759\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 3.04772e47 1.00000
\(145\) −1.44068e46 −0.0405949
\(146\) 8.23137e47 1.99393
\(147\) 0 0
\(148\) 6.17276e47 1.10846
\(149\) −2.16485e47 −0.335220 −0.167610 − 0.985853i \(-0.553605\pi\)
−0.167610 + 0.985853i \(0.553605\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −4.96967e47 −0.429651
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −4.08072e48 −1.99960 −0.999802 − 0.0198915i \(-0.993668\pi\)
−0.999802 + 0.0198915i \(0.993668\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.22904e47 −0.0397123
\(161\) 0 0
\(162\) −4.06753e48 −1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −1.66117e48 −0.311781
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.22124e49 1.18384
\(170\) 2.00409e47 0.0170624
\(171\) 0 0
\(172\) 0 0
\(173\) 3.13267e49 1.81510 0.907549 − 0.419945i \(-0.137951\pi\)
0.907549 + 0.419945i \(0.137951\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −5.49297e49 −1.70046
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 1.64029e48 0.0397123
\(181\) 6.31189e49 1.35279 0.676396 − 0.736538i \(-0.263541\pi\)
0.676396 + 0.736538i \(0.263541\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.32219e48 0.0440196
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −1.22927e50 −0.641780 −0.320890 − 0.947116i \(-0.603982\pi\)
−0.320890 + 0.947116i \(0.603982\pi\)
\(194\) −3.86910e50 −1.80290
\(195\) 0 0
\(196\) 2.68926e50 1.00000
\(197\) −6.01566e50 −1.99998 −0.999992 − 0.00406419i \(-0.998706\pi\)
−0.999992 + 0.00406419i \(0.998706\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 4.18769e50 0.998423
\(201\) 0 0
\(202\) 3.35351e50 0.642347
\(203\) 0 0
\(204\) 0 0
\(205\) −8.94044e48 −0.0123815
\(206\) 0 0
\(207\) 0 0
\(208\) 1.46894e51 1.47778
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 2.86828e51 1.89772
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −5.40115e51 −1.93394
\(219\) 0 0
\(220\) 0 0
\(221\) −2.39528e51 −0.634930
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −5.58895e51 −0.998423
\(226\) −1.20650e52 −1.95497
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 1.48483e52 1.80009 0.900047 − 0.435792i \(-0.143532\pi\)
0.900047 + 0.435792i \(0.143532\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.12275e52 1.02223
\(233\) 1.57557e52 1.30499 0.652497 − 0.757792i \(-0.273722\pi\)
0.652497 + 0.757792i \(0.273722\pi\)
\(234\) −1.96047e52 −1.47778
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 2.30244e52 0.907432 0.453716 − 0.891146i \(-0.350098\pi\)
0.453716 + 0.891146i \(0.350098\pi\)
\(242\) −2.77932e52 −1.00000
\(243\) 0 0
\(244\) 4.40961e52 1.32380
\(245\) 1.44737e51 0.0397123
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 4.51121e51 0.0793620
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 9.57810e52 1.00000
\(257\) −1.92632e53 −1.84587 −0.922933 − 0.384960i \(-0.874215\pi\)
−0.922933 + 0.384960i \(0.874215\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 7.90586e51 0.0586861
\(261\) −1.49844e53 −1.02223
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 1.54372e52 0.0753630
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.77620e53 −1.67700 −0.838499 − 0.544903i \(-0.816566\pi\)
−0.838499 + 0.544903i \(0.816566\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −1.56182e53 −0.429651
\(273\) 0 0
\(274\) 7.98796e53 1.87035
\(275\) 0 0
\(276\) 0 0
\(277\) 1.08537e54 1.99996 0.999981 − 0.00610785i \(-0.00194420\pi\)
0.999981 + 0.00610785i \(0.00194420\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.07789e54 1.44890 0.724448 − 0.689330i \(-0.242095\pi\)
0.724448 + 0.689330i \(0.242095\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.27831e54 −1.00000
\(289\) −1.12492e54 −0.815400
\(290\) 6.04265e52 0.0405949
\(291\) 0 0
\(292\) −3.45249e54 −1.99393
\(293\) 1.59159e54 0.852595 0.426298 − 0.904583i \(-0.359818\pi\)
0.426298 + 0.904583i \(0.359818\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.58904e54 −1.10846
\(297\) 0 0
\(298\) 9.08004e53 0.335220
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.37326e53 0.0525713
\(306\) 2.08443e54 0.429651
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 2.98682e54 0.374311 0.187156 − 0.982330i \(-0.440073\pi\)
0.187156 + 0.982330i \(0.440073\pi\)
\(314\) 1.71158e55 1.99960
\(315\) 0 0
\(316\) 0 0
\(317\) −1.08945e55 −1.03253 −0.516265 − 0.856429i \(-0.672678\pi\)
−0.516265 + 0.856429i \(0.672678\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 5.15495e53 0.0397123
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.70604e55 1.00000
\(325\) −2.69376e55 −1.47545
\(326\) 0 0
\(327\) 0 0
\(328\) 6.96745e54 0.311781
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 3.45537e55 1.10846
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −6.89490e55 −1.70087 −0.850433 − 0.526083i \(-0.823660\pi\)
−0.850433 + 0.526083i \(0.823660\pi\)
\(338\) −5.12224e55 −1.18384
\(339\) 0 0
\(340\) −8.40576e53 −0.0170624
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −1.31394e56 −1.81510
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 1.39922e56 1.59855 0.799275 − 0.600966i \(-0.205217\pi\)
0.799275 + 0.600966i \(0.205217\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.97026e56 1.75178 0.875891 − 0.482510i \(-0.160275\pi\)
0.875891 + 0.482510i \(0.160275\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.30392e56 1.70046
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −6.87987e54 −0.0397123
\(361\) 1.84144e56 1.00000
\(362\) −2.64740e56 −1.35279
\(363\) 0 0
\(364\) 0 0
\(365\) −1.85814e55 −0.0791834
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −9.29885e55 −0.311781
\(370\) −1.39343e55 −0.0440196
\(371\) 0 0
\(372\) 0 0
\(373\) −6.19257e56 −1.63786 −0.818929 − 0.573894i \(-0.805432\pi\)
−0.818929 + 0.573894i \(0.805432\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.22216e56 −1.51063
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5.15593e56 0.641780
\(387\) 0 0
\(388\) 1.62282e57 1.80290
\(389\) −3.14960e56 −0.330647 −0.165323 − 0.986239i \(-0.552867\pi\)
−0.165323 + 0.986239i \(0.552867\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.12796e57 −1.00000
\(393\) 0 0
\(394\) 2.52315e57 1.99998
\(395\) 0 0
\(396\) 0 0
\(397\) 1.87120e57 1.25525 0.627623 − 0.778517i \(-0.284028\pi\)
0.627623 + 0.778517i \(0.284028\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.75644e57 −0.998423
\(401\) −2.18202e57 −1.17404 −0.587020 − 0.809572i \(-0.699699\pi\)
−0.587020 + 0.809572i \(0.699699\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −1.40656e57 −0.642347
\(405\) 9.18196e55 0.0397123
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 5.53553e57 1.92862 0.964310 − 0.264777i \(-0.0852983\pi\)
0.964310 + 0.264777i \(0.0852983\pi\)
\(410\) 3.74989e55 0.0123815
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −6.16118e57 −1.47778
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −5.87042e56 −0.108259 −0.0541294 − 0.998534i \(-0.517238\pi\)
−0.0541294 + 0.998534i \(0.517238\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −1.20304e58 −1.89772
\(425\) 2.86409e57 0.428973
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −6.74282e57 −0.670051 −0.335026 − 0.942209i \(-0.608745\pi\)
−0.335026 + 0.942209i \(0.608745\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.26541e58 1.93394
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.50539e58 1.00000
\(442\) 1.00465e58 0.634930
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 1.23997e57 0.0675293
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.79953e58 −1.25217 −0.626087 − 0.779753i \(-0.715345\pi\)
−0.626087 + 0.779753i \(0.715345\pi\)
\(450\) 2.34417e58 0.998423
\(451\) 0 0
\(452\) 5.06042e58 1.95497
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2.73568e58 −0.829676 −0.414838 − 0.909895i \(-0.636162\pi\)
−0.414838 + 0.909895i \(0.636162\pi\)
\(458\) −6.22782e58 −1.80009
\(459\) 0 0
\(460\) 0 0
\(461\) −6.24086e58 −1.56251 −0.781257 − 0.624210i \(-0.785421\pi\)
−0.781257 + 0.624210i \(0.785421\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −4.70915e58 −1.02223
\(465\) 0 0
\(466\) −6.60843e58 −1.30499
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 8.22279e58 1.47778
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.60560e59 1.89772
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 1.66542e59 1.63807
\(482\) −9.65715e58 −0.907432
\(483\) 0 0
\(484\) 1.16573e59 1.00000
\(485\) 8.73405e57 0.0715974
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) −1.84952e59 −1.32380
\(489\) 0 0
\(490\) −6.07070e57 −0.0397123
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 7.67882e58 0.439200
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −1.89214e58 −0.0793620
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −7.57016e57 −0.0255091
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.48020e59 −1.83568 −0.917840 − 0.396951i \(-0.870068\pi\)
−0.917840 + 0.396951i \(0.870068\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −4.01735e59 −1.00000
\(513\) 0 0
\(514\) 8.07958e59 1.84587
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −3.31596e58 −0.0586861
\(521\) −1.16829e60 −1.98208 −0.991042 − 0.133552i \(-0.957362\pi\)
−0.991042 + 0.133552i \(0.957362\pi\)
\(522\) 6.28489e59 1.02223
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 8.24185e59 1.00000
\(530\) −6.47481e58 −0.0753630
\(531\) 0 0
\(532\) 0 0
\(533\) −4.48186e59 −0.460744
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 2.00328e60 1.67700
\(539\) 0 0
\(540\) 0 0
\(541\) 2.05657e60 1.52337 0.761683 − 0.647950i \(-0.224373\pi\)
0.761683 + 0.647950i \(0.224373\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 6.55076e59 0.429651
\(545\) 1.21925e59 0.0768013
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −3.35039e60 −1.87035
\(549\) 2.46840e60 1.32380
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −4.55236e60 −1.99996
\(555\) 0 0
\(556\) 0 0
\(557\) −4.84488e60 −1.89003 −0.945014 − 0.327029i \(-0.893953\pi\)
−0.945014 + 0.327029i \(0.893953\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −4.52101e60 −1.44890
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 2.72353e59 0.0776365
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7.98244e60 1.94835 0.974173 − 0.225804i \(-0.0725010\pi\)
0.974173 + 0.225804i \(0.0725010\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 5.36160e60 1.00000
\(577\) −2.87986e60 −0.517016 −0.258508 − 0.966009i \(-0.583231\pi\)
−0.258508 + 0.966009i \(0.583231\pi\)
\(578\) 4.71827e60 0.815400
\(579\) 0 0
\(580\) −2.53447e59 −0.0405949
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 1.44808e61 1.99393
\(585\) 4.42552e59 0.0586861
\(586\) −6.67560e60 −0.852595
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.08592e61 1.10846
\(593\) −1.13345e61 −1.11481 −0.557403 − 0.830242i \(-0.688202\pi\)
−0.557403 + 0.830242i \(0.688202\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.80844e60 −0.335220
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −2.52833e61 −1.85180 −0.925902 − 0.377765i \(-0.876693\pi\)
−0.925902 + 0.377765i \(0.876693\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 6.27398e59 0.0397123
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −9.95418e59 −0.0525713
\(611\) 0 0
\(612\) −8.74273e60 −0.429651
\(613\) −1.30294e61 −0.617722 −0.308861 − 0.951107i \(-0.599948\pi\)
−0.308861 + 0.951107i \(0.599948\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.95972e61 1.62701 0.813506 − 0.581556i \(-0.197556\pi\)
0.813506 + 0.581556i \(0.197556\pi\)
\(618\) 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
\(625\) 3.21589e61 0.995271
\(626\) −1.25276e61 −0.374311
\(627\) 0 0
\(628\) −7.17887e61 −1.99960
\(629\) −1.77073e61 −0.476252
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 4.56947e61 1.03253
\(635\) 0 0
\(636\) 0 0
\(637\) 7.25568e61 1.47778
\(638\) 0 0
\(639\) 0 0
\(640\) −2.16214e60 −0.0397123
\(641\) 8.63649e61 1.53271 0.766356 − 0.642416i \(-0.222068\pi\)
0.766356 + 0.642416i \(0.222068\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −7.15567e61 −1.00000
\(649\) 0 0
\(650\) 1.12984e62 1.47545
\(651\) 0 0
\(652\) 0 0
\(653\) −1.33386e61 −0.157406 −0.0787030 − 0.996898i \(-0.525078\pi\)
−0.0787030 + 0.996898i \(0.525078\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.92236e61 −0.311781
\(657\) −1.93262e62 −1.99393
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.30808e62 −1.18087 −0.590433 − 0.807087i \(-0.701043\pi\)
−0.590433 + 0.807087i \(0.701043\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −1.44929e62 −1.10846
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.21606e62 −1.34664 −0.673319 − 0.739353i \(-0.735132\pi\)
−0.673319 + 0.739353i \(0.735132\pi\)
\(674\) 2.89193e62 1.70087
\(675\) 0 0
\(676\) 2.14842e62 1.18384
\(677\) −4.51391e61 −0.240770 −0.120385 − 0.992727i \(-0.538413\pi\)
−0.120385 + 0.992727i \(0.538413\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 3.52563e60 0.0170624
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −1.80319e61 −0.0742760
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7.73867e62 2.80442
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 5.51105e62 1.81510
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 4.76525e61 0.133957
\(698\) −5.86875e62 −1.59855
\(699\) 0 0
\(700\) 0 0
\(701\) −3.90745e62 −0.968495 −0.484248 − 0.874931i \(-0.660907\pi\)
−0.484248 + 0.874931i \(0.660907\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −8.26387e62 −1.75178
\(707\) 0 0
\(708\) 0 0
\(709\) 3.77534e62 0.729019 0.364510 − 0.931200i \(-0.381237\pi\)
0.364510 + 0.931200i \(0.381237\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −9.66334e62 −1.70046
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 2.88563e61 0.0397123
\(721\) 0 0
\(722\) −7.72357e62 −1.00000
\(723\) 0 0
\(724\) 1.11040e63 1.35279
\(725\) 8.63570e62 1.02061
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 9.55005e62 1.00000
\(730\) 7.79358e61 0.0791834
\(731\) 0 0
\(732\) 0 0
\(733\) −2.14092e63 −1.98753 −0.993765 − 0.111493i \(-0.964437\pi\)
−0.993765 + 0.111493i \(0.964437\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 3.90022e62 0.311781
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 5.84446e61 0.0440196
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −2.04971e61 −0.0133124
\(746\) 2.59735e63 1.63786
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 3.02919e63 1.51063
\(755\) 0 0
\(756\) 0 0
\(757\) −2.19583e63 −1.00343 −0.501717 − 0.865032i \(-0.667298\pi\)
−0.501717 + 0.865032i \(0.667298\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.10596e63 −0.857011 −0.428506 − 0.903539i \(-0.640960\pi\)
−0.428506 + 0.903539i \(0.640960\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −4.70535e61 −0.0170624
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 4.12282e63 1.33295 0.666477 − 0.745526i \(-0.267801\pi\)
0.666477 + 0.745526i \(0.267801\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.16255e63 −0.641780
\(773\) −5.42303e63 −1.56420 −0.782102 − 0.623151i \(-0.785852\pi\)
−0.782102 + 0.623151i \(0.785852\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −6.80660e63 −1.80290
\(777\) 0 0
\(778\) 1.32104e63 0.330647
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 4.73100e63 1.00000
\(785\) −3.86368e62 −0.0794089
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −1.05829e64 −1.99998
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.18972e64 1.95630
\(794\) −7.84838e63 −1.25525
\(795\) 0 0
\(796\) 0 0
\(797\) 4.42617e63 0.651546 0.325773 − 0.945448i \(-0.394375\pi\)
0.325773 + 0.945448i \(0.394375\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 7.36706e63 0.998423
\(801\) 1.28968e64 1.70046
\(802\) 9.15206e63 1.17404
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 5.89956e63 0.642347
\(809\) 1.48570e63 0.157421 0.0787105 − 0.996898i \(-0.474920\pi\)
0.0787105 + 0.996898i \(0.474920\pi\)
\(810\) −3.85119e62 −0.0397123
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −2.32177e64 −1.92862
\(819\) 0 0
\(820\) −1.57282e62 −0.0123815
\(821\) −2.32627e64 −1.78283 −0.891415 − 0.453187i \(-0.850287\pi\)
−0.891415 + 0.453187i \(0.850287\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −3.21377e64 −1.98982 −0.994912 − 0.100743i \(-0.967878\pi\)
−0.994912 + 0.100743i \(0.967878\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2.58418e64 1.47778
\(833\) −7.71446e63 −0.429651
\(834\) 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
\(841\) 9.95854e62 0.0449453
\(842\) 2.46223e63 0.108259
\(843\) 0 0
\(844\) 0 0
\(845\) 1.15629e63 0.0470130
\(846\) 0 0
\(847\) 0 0
\(848\) 5.04594e64 1.89772
\(849\) 0 0
\(850\) −1.20129e64 −0.428973
\(851\) 0 0
\(852\) 0 0
\(853\) −3.55138e64 −1.17360 −0.586798 − 0.809733i \(-0.699612\pi\)
−0.586798 + 0.809733i \(0.699612\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.49590e64 1.93669 0.968347 − 0.249606i \(-0.0803012\pi\)
0.968347 + 0.249606i \(0.0803012\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 2.96606e63 0.0720817
\(866\) 2.82814e64 0.670051
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −9.50181e64 −1.93394
\(873\) 9.08418e64 1.80290
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.00439e65 −1.80264 −0.901321 − 0.433152i \(-0.857401\pi\)
−0.901321 + 0.433152i \(0.857401\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.22745e65 1.99312 0.996561 − 0.0828657i \(-0.0264072\pi\)
0.996561 + 0.0828657i \(0.0264072\pi\)
\(882\) −6.31406e64 −1.00000
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −4.21382e64 −0.634930
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −5.20083e63 −0.0675293
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 1.17421e65 1.25217
\(899\) 0 0
\(900\) −9.83218e64 −0.998423
\(901\) −8.22800e64 −0.815359
\(902\) 0 0
\(903\) 0 0
\(904\) −2.12249e65 −1.95497
\(905\) 5.97619e63 0.0537225
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −7.87363e64 −0.642347
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1.14743e65 0.829676
\(915\) 0 0
\(916\) 2.61214e65 1.80009
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 2.61760e65 1.56251
\(923\) 0 0
\(924\) 0 0
\(925\) −1.99138e65 −1.10672
\(926\) 0 0
\(927\) 0 0
\(928\) 1.97516e65 1.02223
\(929\) 3.94543e65 1.99411 0.997053 − 0.0767160i \(-0.0244435\pi\)
0.997053 + 0.0767160i \(0.0244435\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.77178e65 1.30499
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −3.44889e65 −1.47778
\(937\) −1.77699e65 −0.743729 −0.371864 − 0.928287i \(-0.621281\pi\)
−0.371864 + 0.928287i \(0.621281\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −2.39292e65 −0.911918 −0.455959 − 0.890001i \(-0.650704\pi\)
−0.455959 + 0.890001i \(0.650704\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −9.31486e65 −2.94659
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.67308e65 1.92434 0.962171 − 0.272445i \(-0.0878324\pi\)
0.962171 + 0.272445i \(0.0878324\pi\)
\(954\) −6.73437e65 −1.89772
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 4.16787e65 1.00000
\(962\) −6.98527e65 −1.63807
\(963\) 0 0
\(964\) 4.05050e65 0.907432
\(965\) −1.16389e64 −0.0254866
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −4.88943e65 −1.00000
\(969\) 0 0
\(970\) −3.66333e64 −0.0715974
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 7.75747e65 1.32380
\(977\) 9.63641e65 1.60781 0.803906 − 0.594757i \(-0.202752\pi\)
0.803906 + 0.594757i \(0.202752\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.54624e64 0.0397123
\(981\) 1.26812e66 1.93394
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −5.69572e64 −0.0794239
\(986\) −3.22073e65 −0.439200
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −9.91460e65 −1.05921 −0.529605 − 0.848245i \(-0.677660\pi\)
−0.529605 + 0.848245i \(0.677660\pi\)
\(998\) 0 0
\(999\) 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 4.45.b.a.3.1 ✓ 1
4.3 odd 2 CM 4.45.b.a.3.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.45.b.a.3.1 ✓ 1 1.1 even 1 trivial
4.45.b.a.3.1 ✓ 1 4.3 odd 2 CM