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Properties

Label 56.9.h.b.13.1
Level $56$
Weight $9$
Character 56.13
Self dual yes
Analytic conductor $22.813$
Analytic rank $0$
Dimension $1$
CM discriminant -56
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [56,9,Mod(13,56)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(56, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.13");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 56.h (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: \(22.8132021634\)
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 13.1
Character \(\chi\) \(=\) 56.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{2} +62.0000 q^{3} +256.000 q^{4} +766.000 q^{5} +992.000 q^{6} +2401.00 q^{7} +4096.00 q^{8} -2717.00 q^{9} +O(q^{10})\) \(q+16.0000 q^{2} +62.0000 q^{3} +256.000 q^{4} +766.000 q^{5} +992.000 q^{6} +2401.00 q^{7} +4096.00 q^{8} -2717.00 q^{9} +12256.0 q^{10} +15872.0 q^{12} -38978.0 q^{13} +38416.0 q^{14} +47492.0 q^{15} +65536.0 q^{16} -43472.0 q^{18} -161858. q^{19} +196096. q^{20} +148862. q^{21} +358082. q^{23} +253952. q^{24} +196131. q^{25} -623648. q^{26} -575236. q^{27} +614656. q^{28} +759872. q^{30} +1.04858e6 q^{32} +1.83917e6 q^{35} -695552. q^{36} -2.58973e6 q^{38} -2.41664e6 q^{39} +3.13754e6 q^{40} +2.38179e6 q^{42} -2.08122e6 q^{45} +5.72931e6 q^{46} +4.06323e6 q^{48} +5.76480e6 q^{49} +3.13810e6 q^{50} -9.97837e6 q^{52} -9.20378e6 q^{54} +9.83450e6 q^{56} -1.00352e7 q^{57} +2.29578e7 q^{59} +1.21580e7 q^{60} -2.66252e7 q^{61} -6.52352e6 q^{63} +1.67772e7 q^{64} -2.98571e7 q^{65} +2.22011e7 q^{69} +2.94267e7 q^{70} -4.67510e7 q^{71} -1.11288e7 q^{72} +1.21601e7 q^{75} -4.14356e7 q^{76} -3.86662e7 q^{78} -5.83814e7 q^{79} +5.02006e7 q^{80} -1.78384e7 q^{81} -7.74803e7 q^{83} +3.81087e7 q^{84} -3.32996e7 q^{90} -9.35862e7 q^{91} +9.16690e7 q^{92} -1.23983e8 q^{95} +6.50117e7 q^{96} +9.22368e7 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 1.00000
\(3\) 62.0000 0.765432 0.382716 − 0.923866i \(-0.374989\pi\)
0.382716 + 0.923866i \(0.374989\pi\)
\(4\) 256.000 1.00000
\(5\) 766.000 1.22560 0.612800 − 0.790238i \(-0.290043\pi\)
0.612800 + 0.790238i \(0.290043\pi\)
\(6\) 992.000 0.765432
\(7\) 2401.00 1.00000
\(8\) 4096.00 1.00000
\(9\) −2717.00 −0.414114
\(10\) 12256.0 1.22560
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 15872.0 0.765432
\(13\) −38978.0 −1.36473 −0.682364 − 0.731013i \(-0.739048\pi\)
−0.682364 + 0.731013i \(0.739048\pi\)
\(14\) 38416.0 1.00000
\(15\) 47492.0 0.938114
\(16\) 65536.0 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −43472.0 −0.414114
\(19\) −161858. −1.24199 −0.620997 − 0.783813i \(-0.713272\pi\)
−0.620997 + 0.783813i \(0.713272\pi\)
\(20\) 196096. 1.22560
\(21\) 148862. 0.765432
\(22\) 0 0
\(23\) 358082. 1.27959 0.639795 − 0.768545i \(-0.279019\pi\)
0.639795 + 0.768545i \(0.279019\pi\)
\(24\) 253952. 0.765432
\(25\) 196131. 0.502095
\(26\) −623648. −1.36473
\(27\) −575236. −1.08241
\(28\) 614656. 1.00000
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 759872. 0.938114
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1.04858e6 1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 1.83917e6 1.22560
\(36\) −695552. −0.414114
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −2.58973e6 −1.24199
\(39\) −2.41664e6 −1.04461
\(40\) 3.13754e6 1.22560
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 2.38179e6 0.765432
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −2.08122e6 −0.507538
\(46\) 5.72931e6 1.27959
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 4.06323e6 0.765432
\(49\) 5.76480e6 1.00000
\(50\) 3.13810e6 0.502095
\(51\) 0 0
\(52\) −9.97837e6 −1.36473
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −9.20378e6 −1.08241
\(55\) 0 0
\(56\) 9.83450e6 1.00000
\(57\) −1.00352e7 −0.950663
\(58\) 0 0
\(59\) 2.29578e7 1.89462 0.947311 − 0.320315i \(-0.103789\pi\)
0.947311 + 0.320315i \(0.103789\pi\)
\(60\) 1.21580e7 0.938114
\(61\) −2.66252e7 −1.92298 −0.961488 − 0.274847i \(-0.911373\pi\)
−0.961488 + 0.274847i \(0.911373\pi\)
\(62\) 0 0
\(63\) −6.52352e6 −0.414114
\(64\) 1.67772e7 1.00000
\(65\) −2.98571e7 −1.67261
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 2.22011e7 0.979440
\(70\) 2.94267e7 1.22560
\(71\) −4.67510e7 −1.83975 −0.919873 − 0.392216i \(-0.871708\pi\)
−0.919873 + 0.392216i \(0.871708\pi\)
\(72\) −1.11288e7 −0.414114
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.21601e7 0.384320
\(76\) −4.14356e7 −1.24199
\(77\) 0 0
\(78\) −3.86662e7 −1.04461
\(79\) −5.83814e7 −1.49888 −0.749439 − 0.662073i \(-0.769677\pi\)
−0.749439 + 0.662073i \(0.769677\pi\)
\(80\) 5.02006e7 1.22560
\(81\) −1.78384e7 −0.414396
\(82\) 0 0
\(83\) −7.74803e7 −1.63260 −0.816298 − 0.577631i \(-0.803977\pi\)
−0.816298 + 0.577631i \(0.803977\pi\)
\(84\) 3.81087e7 0.765432
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) −3.32996e7 −0.507538
\(91\) −9.35862e7 −1.36473
\(92\) 9.16690e7 1.27959
\(93\) 0 0
\(94\) 0 0
\(95\) −1.23983e8 −1.52219
\(96\) 6.50117e7 0.765432
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 9.22368e7 1.00000
\(99\) 0 0
\(100\) 5.02095e7 0.502095
\(101\) 1.76873e8 1.69971 0.849856 − 0.527015i \(-0.176689\pi\)
0.849856 + 0.527015i \(0.176689\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −1.59654e8 −1.36473
\(105\) 1.14028e8 0.938114
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −1.47260e8 −1.08241
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.57352e8 1.00000
\(113\) 2.92024e8 1.79104 0.895520 − 0.445021i \(-0.146804\pi\)
0.895520 + 0.445021i \(0.146804\pi\)
\(114\) −1.60563e8 −0.950663
\(115\) 2.74291e8 1.56827
\(116\) 0 0
\(117\) 1.05903e8 0.565153
\(118\) 3.67325e8 1.89462
\(119\) 0 0
\(120\) 1.94527e8 0.938114
\(121\) 2.14359e8 1.00000
\(122\) −4.26003e8 −1.92298
\(123\) 0 0
\(124\) 0 0
\(125\) −1.48982e8 −0.610232
\(126\) −1.04376e8 −0.414114
\(127\) 3.94289e8 1.51565 0.757827 − 0.652456i \(-0.226261\pi\)
0.757827 + 0.652456i \(0.226261\pi\)
\(128\) 2.68435e8 1.00000
\(129\) 0 0
\(130\) −4.77714e8 −1.67261
\(131\) −9.90131e7 −0.336207 −0.168104 − 0.985769i \(-0.553764\pi\)
−0.168104 + 0.985769i \(0.553764\pi\)
\(132\) 0 0
\(133\) −3.88621e8 −1.24199
\(134\) 0 0
\(135\) −4.40631e8 −1.32660
\(136\) 0 0
\(137\) 5.23111e8 1.48495 0.742474 − 0.669875i \(-0.233652\pi\)
0.742474 + 0.669875i \(0.233652\pi\)
\(138\) 3.55217e8 0.979440
\(139\) −1.86700e8 −0.500134 −0.250067 − 0.968229i \(-0.580453\pi\)
−0.250067 + 0.968229i \(0.580453\pi\)
\(140\) 4.70826e8 1.22560
\(141\) 0 0
\(142\) −7.48017e8 −1.83975
\(143\) 0 0
\(144\) −1.78061e8 −0.414114
\(145\) 0 0
\(146\) 0 0
\(147\) 3.57418e8 0.765432
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 1.94562e8 0.384320
\(151\) −1.01675e9 −1.95572 −0.977860 − 0.209261i \(-0.932894\pi\)
−0.977860 + 0.209261i \(0.932894\pi\)
\(152\) −6.62970e8 −1.24199
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −6.18659e8 −1.04461
\(157\) −1.21239e9 −1.99546 −0.997729 − 0.0673598i \(-0.978542\pi\)
−0.997729 + 0.0673598i \(0.978542\pi\)
\(158\) −9.34103e8 −1.49888
\(159\) 0 0
\(160\) 8.03209e8 1.22560
\(161\) 8.59755e8 1.27959
\(162\) −2.85414e8 −0.414396
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −1.23968e9 −1.63260
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 6.09739e8 0.765432
\(169\) 7.03554e8 0.862483
\(170\) 0 0
\(171\) 4.39768e8 0.514327
\(172\) 0 0
\(173\) 1.73717e9 1.93936 0.969681 − 0.244376i \(-0.0785830\pi\)
0.969681 + 0.244376i \(0.0785830\pi\)
\(174\) 0 0
\(175\) 4.70911e8 0.502095
\(176\) 0 0
\(177\) 1.42338e9 1.45020
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −5.32793e8 −0.507538
\(181\) 2.13310e9 1.98745 0.993725 − 0.111848i \(-0.0356768\pi\)
0.993725 + 0.111848i \(0.0356768\pi\)
\(182\) −1.49738e9 −1.36473
\(183\) −1.65076e9 −1.47191
\(184\) 1.46670e9 1.27959
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.38114e9 −1.08241
\(190\) −1.98373e9 −1.52219
\(191\) 4.81221e8 0.361586 0.180793 − 0.983521i \(-0.442134\pi\)
0.180793 + 0.983521i \(0.442134\pi\)
\(192\) 1.04019e9 0.765432
\(193\) −2.19426e9 −1.58146 −0.790732 − 0.612162i \(-0.790300\pi\)
−0.790732 + 0.612162i \(0.790300\pi\)
\(194\) 0 0
\(195\) −1.85114e9 −1.28027
\(196\) 1.47579e9 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 8.03353e8 0.502095
\(201\) 0 0
\(202\) 2.82996e9 1.69971
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −9.72909e8 −0.529896
\(208\) −2.55446e9 −1.36473
\(209\) 0 0
\(210\) 1.82445e9 0.938114
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) −2.89856e9 −1.40820
\(214\) 0 0
\(215\) 0 0
\(216\) −2.35617e9 −1.08241
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 2.51763e9 1.00000
\(225\) −5.32888e8 −0.207925
\(226\) 4.67239e9 1.79104
\(227\) 2.17560e9 0.819360 0.409680 − 0.912229i \(-0.365640\pi\)
0.409680 + 0.912229i \(0.365640\pi\)
\(228\) −2.56901e9 −0.950663
\(229\) 3.53497e9 1.28542 0.642708 − 0.766111i \(-0.277811\pi\)
0.642708 + 0.766111i \(0.277811\pi\)
\(230\) 4.38865e9 1.56827
\(231\) 0 0
\(232\) 0 0
\(233\) −6.95434e7 −0.0235957 −0.0117978 − 0.999930i \(-0.503755\pi\)
−0.0117978 + 0.999930i \(0.503755\pi\)
\(234\) 1.69445e9 0.565153
\(235\) 0 0
\(236\) 5.87720e9 1.89462
\(237\) −3.61965e9 −1.14729
\(238\) 0 0
\(239\) −2.44982e9 −0.750831 −0.375415 − 0.926857i \(-0.622500\pi\)
−0.375415 + 0.926857i \(0.622500\pi\)
\(240\) 3.11244e9 0.938114
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 3.42974e9 1.00000
\(243\) 2.66814e9 0.765216
\(244\) −6.81606e9 −1.92298
\(245\) 4.41584e9 1.22560
\(246\) 0 0
\(247\) 6.30890e9 1.69499
\(248\) 0 0
\(249\) −4.80378e9 −1.24964
\(250\) −2.38372e9 −0.610232
\(251\) −7.59438e9 −1.91336 −0.956682 − 0.291134i \(-0.905967\pi\)
−0.956682 + 0.291134i \(0.905967\pi\)
\(252\) −1.67002e9 −0.414114
\(253\) 0 0
\(254\) 6.30863e9 1.51565
\(255\) 0 0
\(256\) 4.29497e9 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −7.64343e9 −1.67261
\(261\) 0 0
\(262\) −1.58421e9 −0.336207
\(263\) −5.56662e9 −1.16351 −0.581753 − 0.813366i \(-0.697633\pi\)
−0.581753 + 0.813366i \(0.697633\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.21794e9 −1.24199
\(267\) 0 0
\(268\) 0 0
\(269\) −3.62458e9 −0.692228 −0.346114 − 0.938192i \(-0.612499\pi\)
−0.346114 + 0.938192i \(0.612499\pi\)
\(270\) −7.05009e9 −1.32660
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) −5.80234e9 −1.04461
\(274\) 8.36977e9 1.48495
\(275\) 0 0
\(276\) 5.68348e9 0.979440
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −2.98721e9 −0.500134
\(279\) 0 0
\(280\) 7.53322e9 1.22560
\(281\) −1.05619e10 −1.69401 −0.847007 − 0.531581i \(-0.821598\pi\)
−0.847007 + 0.531581i \(0.821598\pi\)
\(282\) 0 0
\(283\) 9.03270e9 1.40822 0.704112 − 0.710088i \(-0.251345\pi\)
0.704112 + 0.710088i \(0.251345\pi\)
\(284\) −1.19683e10 −1.83975
\(285\) −7.68696e9 −1.16513
\(286\) 0 0
\(287\) 0 0
\(288\) −2.84898e9 −0.414114
\(289\) 6.97576e9 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.38068e10 1.87337 0.936683 − 0.350179i \(-0.113879\pi\)
0.936683 + 0.350179i \(0.113879\pi\)
\(294\) 5.71868e9 0.765432
\(295\) 1.75857e10 2.32205
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.39573e10 −1.74629
\(300\) 3.11299e9 0.384320
\(301\) 0 0
\(302\) −1.62680e10 −1.95572
\(303\) 1.09661e10 1.30101
\(304\) −1.06075e10 −1.24199
\(305\) −2.03949e10 −2.35680
\(306\) 0 0
\(307\) 1.75767e10 1.97872 0.989358 − 0.145501i \(-0.0464796\pi\)
0.989358 + 0.145501i \(0.0464796\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) −9.89854e9 −1.04461
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) −1.93982e10 −1.99546
\(315\) −4.99701e9 −0.507538
\(316\) −1.49456e10 −1.49888
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.28513e10 1.22560
\(321\) 0 0
\(322\) 1.37561e10 1.27959
\(323\) 0 0
\(324\) −4.56663e9 −0.414396
\(325\) −7.64479e9 −0.685224
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −1.98349e10 −1.63260
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 9.75582e9 0.765432
\(337\) 2.54892e10 1.97623 0.988113 − 0.153729i \(-0.0491283\pi\)
0.988113 + 0.153729i \(0.0491283\pi\)
\(338\) 1.12569e10 0.862483
\(339\) 1.81055e10 1.37092
\(340\) 0 0
\(341\) 0 0
\(342\) 7.03629e9 0.514327
\(343\) 1.38413e10 1.00000
\(344\) 0 0
\(345\) 1.70060e10 1.20040
\(346\) 2.77948e10 1.93936
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −2.26739e10 −1.52836 −0.764178 − 0.645006i \(-0.776855\pi\)
−0.764178 + 0.645006i \(0.776855\pi\)
\(350\) 7.53457e9 0.502095
\(351\) 2.24215e10 1.47719
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 2.27742e10 1.45020
\(355\) −3.58113e10 −2.25479
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9.77837e9 0.588693 0.294347 − 0.955699i \(-0.404898\pi\)
0.294347 + 0.955699i \(0.404898\pi\)
\(360\) −8.52469e9 −0.507538
\(361\) 9.21445e9 0.542551
\(362\) 3.41296e10 1.98745
\(363\) 1.32903e10 0.765432
\(364\) −2.39581e10 −1.36473
\(365\) 0 0
\(366\) −2.64122e10 −1.47191
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 2.34673e10 1.27959
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −9.23691e9 −0.467091
\(376\) 0 0
\(377\) 0 0
\(378\) −2.20983e10 −1.08241
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) −3.17397e10 −1.52219
\(381\) 2.44459e10 1.16013
\(382\) 7.69954e9 0.361586
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 1.66430e10 0.765432
\(385\) 0 0
\(386\) −3.51082e10 −1.58146
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) −2.96183e10 −1.28027
\(391\) 0 0
\(392\) 2.36126e10 1.00000
\(393\) −6.13881e9 −0.257344
\(394\) 0 0
\(395\) −4.47202e10 −1.83703
\(396\) 0 0
\(397\) 4.30731e10 1.73398 0.866991 − 0.498324i \(-0.166051\pi\)
0.866991 + 0.498324i \(0.166051\pi\)
\(398\) 0 0
\(399\) −2.40945e10 −0.950663
\(400\) 1.28536e10 0.502095
\(401\) −5.13490e10 −1.98589 −0.992944 − 0.118583i \(-0.962165\pi\)
−0.992944 + 0.118583i \(0.962165\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 4.52794e10 1.69971
\(405\) −1.36642e10 −0.507884
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 3.24329e10 1.13663
\(412\) 0 0
\(413\) 5.51217e10 1.89462
\(414\) −1.55665e10 −0.529896
\(415\) −5.93499e10 −2.00091
\(416\) −4.08714e10 −1.36473
\(417\) −1.15754e10 −0.382818
\(418\) 0 0
\(419\) 5.15451e10 1.67237 0.836183 − 0.548451i \(-0.184782\pi\)
0.836183 + 0.548451i \(0.184782\pi\)
\(420\) 2.91912e10 0.938114
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) −4.63770e10 −1.40820
\(427\) −6.39271e10 −1.92298
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.22775e10 −1.80477 −0.902385 − 0.430930i \(-0.858186\pi\)
−0.902385 + 0.430930i \(0.858186\pi\)
\(432\) −3.76987e10 −1.08241
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.79584e10 −1.58925
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −1.56630e10 −0.414114
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 4.02821e10 1.00000
\(449\) 8.12827e10 1.99992 0.999960 − 0.00890858i \(-0.00283573\pi\)
0.999960 + 0.00890858i \(0.00283573\pi\)
\(450\) −8.52621e9 −0.207925
\(451\) 0 0
\(452\) 7.47582e10 1.79104
\(453\) −6.30385e10 −1.49697
\(454\) 3.48095e10 0.819360
\(455\) −7.16870e10 −1.67261
\(456\) −4.11042e10 −0.950663
\(457\) 4.76720e10 1.09295 0.546473 − 0.837477i \(-0.315970\pi\)
0.546473 + 0.837477i \(0.315970\pi\)
\(458\) 5.65595e10 1.28542
\(459\) 0 0
\(460\) 7.02184e10 1.56827
\(461\) −3.62989e10 −0.803692 −0.401846 − 0.915707i \(-0.631631\pi\)
−0.401846 + 0.915707i \(0.631631\pi\)
\(462\) 0 0
\(463\) 5.07204e10 1.10372 0.551860 − 0.833937i \(-0.313918\pi\)
0.551860 + 0.833937i \(0.313918\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −1.11269e9 −0.0235957
\(467\) 5.42206e10 1.13998 0.569989 − 0.821652i \(-0.306947\pi\)
0.569989 + 0.821652i \(0.306947\pi\)
\(468\) 2.71112e10 0.565153
\(469\) 0 0
\(470\) 0 0
\(471\) −7.51680e10 −1.52739
\(472\) 9.40352e10 1.89462
\(473\) 0 0
\(474\) −5.79144e10 −1.14729
\(475\) −3.17454e10 −0.623600
\(476\) 0 0
\(477\) 0 0
\(478\) −3.91971e10 −0.750831
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 4.97990e10 0.938114
\(481\) 0 0
\(482\) 0 0
\(483\) 5.33048e10 0.979440
\(484\) 5.48759e10 1.00000
\(485\) 0 0
\(486\) 4.26903e10 0.765216
\(487\) −4.82674e10 −0.858101 −0.429051 − 0.903280i \(-0.641152\pi\)
−0.429051 + 0.903280i \(0.641152\pi\)
\(488\) −1.09057e11 −1.92298
\(489\) 0 0
\(490\) 7.06534e10 1.22560
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 1.00942e11 1.69499
\(495\) 0 0
\(496\) 0 0
\(497\) −1.12249e11 −1.83975
\(498\) −7.68604e10 −1.24964
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −3.81395e10 −0.610232
\(501\) 0 0
\(502\) −1.21510e11 −1.91336
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) −2.67203e10 −0.414114
\(505\) 1.35484e11 2.08317
\(506\) 0 0
\(507\) 4.36203e10 0.660172
\(508\) 1.00938e11 1.51565
\(509\) −1.87817e9 −0.0279811 −0.0139905 − 0.999902i \(-0.504453\pi\)
−0.0139905 + 0.999902i \(0.504453\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 6.87195e10 1.00000
\(513\) 9.31065e10 1.34435
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 1.07705e11 1.48445
\(520\) −1.22295e11 −1.67261
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 1.10215e10 0.147310 0.0736552 − 0.997284i \(-0.476534\pi\)
0.0736552 + 0.997284i \(0.476534\pi\)
\(524\) −2.53473e10 −0.336207
\(525\) 2.91965e10 0.384320
\(526\) −8.90659e10 −1.16351
\(527\) 0 0
\(528\) 0 0
\(529\) 4.99117e10 0.637353
\(530\) 0 0
\(531\) −6.23764e10 −0.784589
\(532\) −9.94870e10 −1.24199
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −5.79933e10 −0.692228
\(539\) 0 0
\(540\) −1.12801e11 −1.32660
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 1.32252e11 1.52126
\(544\) 0 0
\(545\) 0 0
\(546\) −9.28375e10 −1.04461
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 1.33916e11 1.48495
\(549\) 7.23407e10 0.796331
\(550\) 0 0
\(551\) 0 0
\(552\) 9.09356e10 0.979440
\(553\) −1.40174e11 −1.49888
\(554\) 0 0
\(555\) 0 0
\(556\) −4.77953e10 −0.500134
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1.20532e11 1.22560
\(561\) 0 0
\(562\) −1.68991e11 −1.69401
\(563\) −2.45818e10 −0.244670 −0.122335 − 0.992489i \(-0.539038\pi\)
−0.122335 + 0.992489i \(0.539038\pi\)
\(564\) 0 0
\(565\) 2.23691e11 2.19510
\(566\) 1.44523e11 1.40822
\(567\) −4.28300e10 −0.414396
\(568\) −1.91492e11 −1.83975
\(569\) 1.86474e11 1.77898 0.889488 − 0.456959i \(-0.151061\pi\)
0.889488 + 0.456959i \(0.151061\pi\)
\(570\) −1.22991e11 −1.16513
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 2.98357e10 0.276769
\(574\) 0 0
\(575\) 7.02310e10 0.642477
\(576\) −4.55837e10 −0.414114
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 1.11612e11 1.00000
\(579\) −1.36044e11 −1.21050
\(580\) 0 0
\(581\) −1.86030e11 −1.63260
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 8.11219e10 0.692651
\(586\) 2.20909e11 1.87337
\(587\) −2.01093e11 −1.69373 −0.846866 − 0.531807i \(-0.821513\pi\)
−0.846866 + 0.531807i \(0.821513\pi\)
\(588\) 9.14989e10 0.765432
\(589\) 0 0
\(590\) 2.81371e11 2.32205
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −2.23317e11 −1.74629
\(599\) −7.48927e10 −0.581745 −0.290872 − 0.956762i \(-0.593945\pi\)
−0.290872 + 0.956762i \(0.593945\pi\)
\(600\) 4.98079e10 0.384320
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −2.60288e11 −1.95572
\(605\) 1.64199e11 1.22560
\(606\) 1.75458e11 1.30101
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) −1.69720e11 −1.24199
\(609\) 0 0
\(610\) −3.26319e11 −2.35680
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 2.81227e11 1.97872
\(615\) 0 0
\(616\) 0 0
\(617\) −1.95121e11 −1.34637 −0.673183 − 0.739476i \(-0.735073\pi\)
−0.673183 + 0.739476i \(0.735073\pi\)
\(618\) 0 0
\(619\) 7.34104e10 0.500029 0.250014 − 0.968242i \(-0.419565\pi\)
0.250014 + 0.968242i \(0.419565\pi\)
\(620\) 0 0
\(621\) −2.05982e11 −1.38504
\(622\) 0 0
\(623\) 0 0
\(624\) −1.58377e11 −1.04461
\(625\) −1.90734e11 −1.25000
\(626\) 0 0
\(627\) 0 0
\(628\) −3.10371e11 −1.99546
\(629\) 0 0
\(630\) −7.99522e10 −0.507538
\(631\) 2.97940e11 1.87936 0.939682 − 0.342048i \(-0.111121\pi\)
0.939682 + 0.342048i \(0.111121\pi\)
\(632\) −2.39130e11 −1.49888
\(633\) 0 0
\(634\) 0 0
\(635\) 3.02026e11 1.85759
\(636\) 0 0
\(637\) −2.24700e11 −1.36473
\(638\) 0 0
\(639\) 1.27023e11 0.761864
\(640\) 2.05622e11 1.22560
\(641\) −2.80459e11 −1.66126 −0.830631 − 0.556824i \(-0.812020\pi\)
−0.830631 + 0.556824i \(0.812020\pi\)
\(642\) 0 0
\(643\) 2.62396e11 1.53502 0.767508 − 0.641040i \(-0.221497\pi\)
0.767508 + 0.641040i \(0.221497\pi\)
\(644\) 2.20097e11 1.27959
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −7.30661e10 −0.414396
\(649\) 0 0
\(650\) −1.22317e11 −0.685224
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −7.58440e10 −0.412056
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.43289e11 −0.750596 −0.375298 − 0.926904i \(-0.622460\pi\)
−0.375298 + 0.926904i \(0.622460\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −3.17359e11 −1.63260
\(665\) −2.97684e11 −1.52219
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 1.56093e11 0.765432
\(673\) −2.01479e11 −0.982130 −0.491065 − 0.871123i \(-0.663392\pi\)
−0.491065 + 0.871123i \(0.663392\pi\)
\(674\) 4.07827e11 1.97623
\(675\) −1.12822e11 −0.543472
\(676\) 1.80110e11 0.862483
\(677\) −4.59893e10 −0.218928 −0.109464 − 0.993991i \(-0.534914\pi\)
−0.109464 + 0.993991i \(0.534914\pi\)
\(678\) 2.89688e11 1.37092
\(679\) 0 0
\(680\) 0 0
\(681\) 1.34887e11 0.627164
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 1.12581e11 0.514327
\(685\) 4.00703e11 1.81995
\(686\) 2.21461e11 1.00000
\(687\) 2.19168e11 0.983898
\(688\) 0 0
\(689\) 0 0
\(690\) 2.72096e11 1.20040
\(691\) −3.43993e11 −1.50882 −0.754410 − 0.656403i \(-0.772077\pi\)
−0.754410 + 0.656403i \(0.772077\pi\)
\(692\) 4.44716e11 1.93936
\(693\) 0 0
\(694\) 0 0
\(695\) −1.43013e11 −0.612964
\(696\) 0 0
\(697\) 0 0
\(698\) −3.62782e11 −1.52836
\(699\) −4.31169e9 −0.0180609
\(700\) 1.20553e11 0.502095
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 3.58745e11 1.47719
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.24671e11 1.69971
\(708\) 3.64387e11 1.45020
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) −5.72981e11 −2.25479
\(711\) 1.58622e11 0.620706
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1.51889e11 −0.574710
\(718\) 1.56454e11 0.588693
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −1.36395e11 −0.507538
\(721\) 0 0
\(722\) 1.47431e11 0.542551
\(723\) 0 0
\(724\) 5.46073e11 1.98745
\(725\) 0 0
\(726\) 2.12644e11 0.765432
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) −3.83329e11 −1.36473
\(729\) 2.82463e11 1.00012
\(730\) 0 0
\(731\) 0 0
\(732\) −4.22595e11 −1.47191
\(733\) −2.93933e11 −1.01820 −0.509099 − 0.860708i \(-0.670021\pi\)
−0.509099 + 0.860708i \(0.670021\pi\)
\(734\) 0 0
\(735\) 2.73782e11 0.938114
\(736\) 3.75476e11 1.27959
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 3.91152e11 1.29740
\(742\) 0 0
\(743\) 2.75592e11 0.904298 0.452149 − 0.891942i \(-0.350657\pi\)
0.452149 + 0.891942i \(0.350657\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2.10514e11 0.676080
\(748\) 0 0
\(749\) 0 0
\(750\) −1.47791e11 −0.467091
\(751\) −6.18771e11 −1.94523 −0.972613 − 0.232431i \(-0.925332\pi\)
−0.972613 + 0.232431i \(0.925332\pi\)
\(752\) 0 0
\(753\) −4.70852e11 −1.46455
\(754\) 0 0
\(755\) −7.78831e11 −2.39693
\(756\) −3.53572e11 −1.08241
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) −5.07835e11 −1.52219
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 3.91135e11 1.16013
\(763\) 0 0
\(764\) 1.23193e11 0.361586
\(765\) 0 0
\(766\) 0 0
\(767\) −8.94850e11 −2.58564
\(768\) 2.66288e11 0.765432
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −5.61731e11 −1.58146
\(773\) 5.28140e11 1.47921 0.739607 − 0.673039i \(-0.235012\pi\)
0.739607 + 0.673039i \(0.235012\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) −4.73893e11 −1.28027
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.77802e11 1.00000
\(785\) −9.28688e11 −2.44563
\(786\) −9.82210e10 −0.257344
\(787\) −2.66951e11 −0.695878 −0.347939 − 0.937517i \(-0.613118\pi\)
−0.347939 + 0.937517i \(0.613118\pi\)
\(788\) 0 0
\(789\) −3.45130e11 −0.890585
\(790\) −7.15523e11 −1.83703
\(791\) 7.01150e11 1.79104
\(792\) 0 0
\(793\) 1.03780e12 2.62434
\(794\) 6.89170e11 1.73398
\(795\) 0 0
\(796\) 0 0
\(797\) −3.34006e11 −0.827792 −0.413896 − 0.910324i \(-0.635832\pi\)
−0.413896 + 0.910324i \(0.635832\pi\)
\(798\) −3.85512e11 −0.950663
\(799\) 0 0
\(800\) 2.05658e11 0.502095
\(801\) 0 0
\(802\) −8.21585e11 −1.98589
\(803\) 0 0
\(804\) 0 0
\(805\) 6.58572e11 1.56827
\(806\) 0 0
\(807\) −2.24724e11 −0.529853
\(808\) 7.24471e11 1.69971
\(809\) −7.08759e10 −0.165464 −0.0827322 − 0.996572i \(-0.526365\pi\)
−0.0827322 + 0.996572i \(0.526365\pi\)
\(810\) −2.18627e11 −0.507884
\(811\) 2.41236e11 0.557646 0.278823 − 0.960342i \(-0.410056\pi\)
0.278823 + 0.960342i \(0.410056\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 2.54274e11 0.565153
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 5.18926e11 1.13663
\(823\) −7.06186e11 −1.53929 −0.769644 − 0.638473i \(-0.779566\pi\)
−0.769644 + 0.638473i \(0.779566\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 8.81948e11 1.89462
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −2.49065e11 −0.529896
\(829\) −9.28602e11 −1.96613 −0.983064 − 0.183264i \(-0.941334\pi\)
−0.983064 + 0.183264i \(0.941334\pi\)
\(830\) −9.49598e11 −2.00091
\(831\) 0 0
\(832\) −6.53942e11 −1.36473
\(833\) 0 0
\(834\) −1.85207e11 −0.382818
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 8.24721e11 1.67237
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 4.67060e11 0.938114
\(841\) 5.00246e11 1.00000
\(842\) 0 0
\(843\) −6.54839e11 −1.29665
\(844\) 0 0
\(845\) 5.38922e11 1.05706
\(846\) 0 0
\(847\) 5.14676e11 1.00000
\(848\) 0 0
\(849\) 5.60028e11 1.07790
\(850\) 0 0
\(851\) 0 0
\(852\) −7.42032e11 −1.40820
\(853\) −5.11104e11 −0.965413 −0.482707 − 0.875782i \(-0.660346\pi\)
−0.482707 + 0.875782i \(0.660346\pi\)
\(854\) −1.02283e12 −1.92298
\(855\) 3.36862e11 0.630359
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 1.08697e12 1.99639 0.998195 − 0.0600503i \(-0.0191261\pi\)
0.998195 + 0.0600503i \(0.0191261\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −9.96440e11 −1.80477
\(863\) −6.42684e11 −1.15866 −0.579328 − 0.815095i \(-0.696685\pi\)
−0.579328 + 0.815095i \(0.696685\pi\)
\(864\) −6.03179e11 −1.08241
\(865\) 1.33067e12 2.37688
\(866\) 0 0
\(867\) 4.32497e11 0.765432
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) −9.27335e11 −1.58925
\(875\) −3.57707e11 −0.610232
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 8.56022e11 1.43393
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −2.50607e11 −0.414114
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 1.09031e12 1.77737
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 9.46689e11 1.51565
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 6.44514e11 1.00000
\(897\) −8.65354e11 −1.33667
\(898\) 1.30052e12 1.99992
\(899\) 0 0
\(900\) −1.36419e11 −0.207925
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 1.19613e12 1.79104
\(905\) 1.63395e12 2.43582
\(906\) −1.00862e12 −1.49697
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 5.56952e11 0.819360
\(909\) −4.80563e11 −0.703874
\(910\) −1.14699e12 −1.67261
\(911\) −7.21830e11 −1.04800 −0.524000 − 0.851718i \(-0.675561\pi\)
−0.524000 + 0.851718i \(0.675561\pi\)
\(912\) −6.57667e11 −0.950663
\(913\) 0 0
\(914\) 7.62752e11 1.09295
\(915\) −1.26448e12 −1.80397
\(916\) 9.04952e11 1.28542
\(917\) −2.37730e11 −0.336207
\(918\) 0 0
\(919\) −9.17658e11 −1.28653 −0.643264 − 0.765645i \(-0.722420\pi\)
−0.643264 + 0.765645i \(0.722420\pi\)
\(920\) 1.12350e12 1.56827
\(921\) 1.08975e12 1.51457
\(922\) −5.80782e11 −0.803692
\(923\) 1.82226e12 2.51075
\(924\) 0 0
\(925\) 0 0
\(926\) 8.11527e11 1.10372
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −9.33079e11 −1.24199
\(932\) −1.78031e10 −0.0235957
\(933\) 0 0
\(934\) 8.67529e11 1.13998
\(935\) 0 0
\(936\) 4.33780e11 0.565153
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.28153e12 1.63445 0.817224 − 0.576320i \(-0.195512\pi\)
0.817224 + 0.576320i \(0.195512\pi\)
\(942\) −1.20269e12 −1.52739
\(943\) 0 0
\(944\) 1.50456e12 1.89462
\(945\) −1.05795e12 −1.32660
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) −9.26630e11 −1.14729
\(949\) 0 0
\(950\) −5.07926e11 −0.623600
\(951\) 0 0
\(952\) 0 0
\(953\) −1.01556e12 −1.23122 −0.615611 − 0.788051i \(-0.711091\pi\)
−0.615611 + 0.788051i \(0.711091\pi\)
\(954\) 0 0
\(955\) 3.68615e11 0.443159
\(956\) −6.27153e11 −0.750831
\(957\) 0 0
\(958\) 0 0
\(959\) 1.25599e12 1.48495
\(960\) 7.96784e11 0.938114
\(961\) 8.52891e11 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.68080e12 −1.93824
\(966\) 8.52877e11 0.979440
\(967\) −1.71826e12 −1.96509 −0.982543 − 0.186033i \(-0.940437\pi\)
−0.982543 + 0.186033i \(0.940437\pi\)
\(968\) 8.78014e11 1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) −1.77763e12 −1.99969 −0.999847 − 0.0174766i \(-0.994437\pi\)
−0.999847 + 0.0174766i \(0.994437\pi\)
\(972\) 6.83045e11 0.765216
\(973\) −4.48268e11 −0.500134
\(974\) −7.72279e11 −0.858101
\(975\) −4.73977e11 −0.524492
\(976\) −1.74491e12 −1.92298
\(977\) −1.26298e12 −1.38618 −0.693090 − 0.720851i \(-0.743751\pi\)
−0.693090 + 0.720851i \(0.743751\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.13045e12 1.22560
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 1.61508e12 1.69499
\(989\) 0 0
\(990\) 0 0
\(991\) 6.05695e11 0.628000 0.314000 − 0.949423i \(-0.398331\pi\)
0.314000 + 0.949423i \(0.398331\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −1.79599e12 −1.83975
\(995\) 0 0
\(996\) −1.22977e12 −1.24964
\(997\) −1.71333e12 −1.73404 −0.867021 − 0.498272i \(-0.833968\pi\)
−0.867021 + 0.498272i \(0.833968\pi\)
\(998\) 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 56.9.h.b.13.1 yes 1
4.3 odd 2 224.9.h.a.209.1 1
7.6 odd 2 56.9.h.a.13.1 ✓ 1
8.3 odd 2 224.9.h.b.209.1 1
8.5 even 2 56.9.h.a.13.1 ✓ 1
28.27 even 2 224.9.h.b.209.1 1
56.13 odd 2 CM 56.9.h.b.13.1 yes 1
56.27 even 2 224.9.h.a.209.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.9.h.a.13.1 ✓ 1 7.6 odd 2
56.9.h.a.13.1 ✓ 1 8.5 even 2
56.9.h.b.13.1 yes 1 1.1 even 1 trivial
56.9.h.b.13.1 yes 1 56.13 odd 2 CM
224.9.h.a.209.1 1 4.3 odd 2
224.9.h.a.209.1 1 56.27 even 2
224.9.h.b.209.1 1 8.3 odd 2
224.9.h.b.209.1 1 28.27 even 2