Properties

Label 8.35.d.a.3.1
Level $8$
Weight $35$
Character 8.3
Self dual yes
Analytic conductor $58.581$
Analytic rank $0$
Dimension $1$
CM discriminant -8
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] = [8,35,Mod(3,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 35, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 35);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 35 \)
Character orbit: \([\chi]\) \(=\) 8.d (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: \(58.5805232341\)
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 \(\chi\) \(=\) 8.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-131072. q^{2} +1.25174e8 q^{3} +1.71799e10 q^{4} -1.64069e13 q^{6} -2.25180e15 q^{8} -1.00855e15 q^{9} +O(q^{10})\) \(q-131072. q^{2} +1.25174e8 q^{3} +1.71799e10 q^{4} -1.64069e13 q^{6} -2.25180e15 q^{8} -1.00855e15 q^{9} -7.52887e17 q^{11} +2.15048e18 q^{12} +2.95148e20 q^{16} +1.39271e21 q^{17} +1.32193e20 q^{18} +1.78091e20 q^{19} +9.86824e22 q^{22} -2.81868e23 q^{24} +5.82077e23 q^{25} -2.21380e24 q^{27} -3.86856e25 q^{32} -9.42422e25 q^{33} -1.82546e26 q^{34} -1.73268e25 q^{36} -2.33427e25 q^{38} +3.36735e27 q^{41} +4.15836e27 q^{43} -1.29345e28 q^{44} +3.69450e28 q^{48} +5.41170e28 q^{49} -7.62939e28 q^{50} +1.74332e29 q^{51} +2.90167e29 q^{54} +2.22924e28 q^{57} -1.20794e30 q^{59} +5.07060e30 q^{64} +1.23525e31 q^{66} +2.09499e31 q^{67} +2.39266e31 q^{68} +2.27106e30 q^{72} +6.28862e31 q^{73} +7.28611e31 q^{75} +3.05958e30 q^{76} -2.60291e32 q^{81} -4.41366e32 q^{82} +4.66308e32 q^{83} -5.45045e32 q^{86} +1.69535e33 q^{88} -1.65328e33 q^{89} -4.84245e33 q^{96} -8.75827e33 q^{97} -7.09322e33 q^{98} +7.59326e32 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −131072. −1.00000
\(3\) 1.25174e8 0.969291 0.484646 0.874711i \(-0.338949\pi\)
0.484646 + 0.874711i \(0.338949\pi\)
\(4\) 1.71799e10 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −1.64069e13 −0.969291
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −2.25180e15 −1.00000
\(9\) −1.00855e15 −0.0604750
\(10\) 0 0
\(11\) −7.52887e17 −1.48955 −0.744774 0.667317i \(-0.767443\pi\)
−0.744774 + 0.667317i \(0.767443\pi\)
\(12\) 2.15048e18 0.969291
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.95148e20 1.00000
\(17\) 1.39271e21 1.68357 0.841783 0.539816i \(-0.181506\pi\)
0.841783 + 0.539816i \(0.181506\pi\)
\(18\) 1.32193e20 0.0604750
\(19\) 1.78091e20 0.0324961 0.0162480 0.999868i \(-0.494828\pi\)
0.0162480 + 0.999868i \(0.494828\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 9.86824e22 1.48955
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −2.81868e23 −0.969291
\(25\) 5.82077e23 1.00000
\(26\) 0 0
\(27\) −2.21380e24 −1.02791
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −3.86856e25 −1.00000
\(33\) −9.42422e25 −1.44380
\(34\) −1.82546e26 −1.68357
\(35\) 0 0
\(36\) −1.73268e25 −0.0604750
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −2.33427e25 −0.0324961
\(39\) 0 0
\(40\) 0 0
\(41\) 3.36735e27 1.28814 0.644070 0.764966i \(-0.277244\pi\)
0.644070 + 0.764966i \(0.277244\pi\)
\(42\) 0 0
\(43\) 4.15836e27 0.707878 0.353939 0.935268i \(-0.384842\pi\)
0.353939 + 0.935268i \(0.384842\pi\)
\(44\) −1.29345e28 −1.48955
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 3.69450e28 0.969291
\(49\) 5.41170e28 1.00000
\(50\) −7.62939e28 −1.00000
\(51\) 1.74332e29 1.63187
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 2.90167e29 1.02791
\(55\) 0 0
\(56\) 0 0
\(57\) 2.22924e28 0.0314981
\(58\) 0 0
\(59\) −1.20794e30 −0.949648 −0.474824 0.880081i \(-0.657488\pi\)
−0.474824 + 0.880081i \(0.657488\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 5.07060e30 1.00000
\(65\) 0 0
\(66\) 1.23525e31 1.44380
\(67\) 2.09499e31 1.89631 0.948156 0.317805i \(-0.102946\pi\)
0.948156 + 0.317805i \(0.102946\pi\)
\(68\) 2.39266e31 1.68357
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 2.27106e30 0.0604750
\(73\) 6.28862e31 1.32455 0.662273 0.749263i \(-0.269592\pi\)
0.662273 + 0.749263i \(0.269592\pi\)
\(74\) 0 0
\(75\) 7.28611e31 0.969291
\(76\) 3.05958e30 0.0324961
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −2.60291e32 −0.935868
\(82\) −4.41366e32 −1.28814
\(83\) 4.66308e32 1.10750 0.553752 0.832681i \(-0.313195\pi\)
0.553752 + 0.832681i \(0.313195\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −5.45045e32 −0.707878
\(87\) 0 0
\(88\) 1.69535e33 1.48955
\(89\) −1.65328e33 −1.19872 −0.599359 0.800480i \(-0.704578\pi\)
−0.599359 + 0.800480i \(0.704578\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −4.84245e33 −0.969291
\(97\) −8.75827e33 −1.46994 −0.734969 0.678101i \(-0.762803\pi\)
−0.734969 + 0.678101i \(0.762803\pi\)
\(98\) −7.09322e33 −1.00000
\(99\) 7.59326e32 0.0900803
\(100\) 1.00000e34 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) −2.28501e34 −1.63187
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.28557e34 1.67328 0.836639 0.547755i \(-0.184517\pi\)
0.836639 + 0.547755i \(0.184517\pi\)
\(108\) −3.80328e34 −1.02791
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.55544e35 1.94769 0.973844 0.227217i \(-0.0729625\pi\)
0.973844 + 0.227217i \(0.0729625\pi\)
\(114\) −2.92191e33 −0.0314981
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 1.58328e35 0.949648
\(119\) 0 0
\(120\) 0 0
\(121\) 3.11363e35 1.21875
\(122\) 0 0
\(123\) 4.21507e35 1.24858
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −6.64614e35 −1.00000
\(129\) 5.20520e35 0.686140
\(130\) 0 0
\(131\) −1.56777e36 −1.59100 −0.795501 0.605953i \(-0.792792\pi\)
−0.795501 + 0.605953i \(0.792792\pi\)
\(132\) −1.61907e36 −1.44380
\(133\) 0 0
\(134\) −2.74594e36 −1.89631
\(135\) 0 0
\(136\) −3.13611e36 −1.68357
\(137\) 3.45386e36 1.63702 0.818512 0.574489i \(-0.194799\pi\)
0.818512 + 0.574489i \(0.194799\pi\)
\(138\) 0 0
\(139\) 5.39172e36 1.99745 0.998725 0.0504853i \(-0.0160768\pi\)
0.998725 + 0.0504853i \(0.0160768\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −2.97672e35 −0.0604750
\(145\) 0 0
\(146\) −8.24262e36 −1.32455
\(147\) 6.77406e36 0.969291
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) −9.55005e36 −0.969291
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) −4.01025e35 −0.0324961
\(153\) −1.40462e36 −0.101814
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 3.41169e37 0.935868
\(163\) 7.19599e37 1.77787 0.888937 0.458029i \(-0.151444\pi\)
0.888937 + 0.458029i \(0.151444\pi\)
\(164\) 5.78507e37 1.28814
\(165\) 0 0
\(166\) −6.11200e37 −1.10750
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 7.48297e37 1.00000
\(170\) 0 0
\(171\) −1.79614e35 −0.00196520
\(172\) 7.14401e37 0.707878
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.22213e38 −1.48955
\(177\) −1.51204e38 −0.920485
\(178\) 2.16699e38 1.19872
\(179\) −3.97254e38 −1.99787 −0.998935 0.0461494i \(-0.985305\pi\)
−0.998935 + 0.0461494i \(0.985305\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.04856e39 −2.50775
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 6.34710e38 0.969291
\(193\) −1.34453e39 −1.87973 −0.939865 0.341545i \(-0.889050\pi\)
−0.939865 + 0.341545i \(0.889050\pi\)
\(194\) 1.14796e39 1.46994
\(195\) 0 0
\(196\) 9.29722e38 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −9.95264e37 −0.0900803
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −1.31072e39 −1.00000
\(201\) 2.62239e39 1.83808
\(202\) 0 0
\(203\) 0 0
\(204\) 2.99500e39 1.63187
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.34082e38 −0.0484044
\(210\) 0 0
\(211\) 1.18424e39 0.363612 0.181806 0.983334i \(-0.441806\pi\)
0.181806 + 0.983334i \(0.441806\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −6.92791e39 −1.67328
\(215\) 0 0
\(216\) 4.98504e39 1.02791
\(217\) 0 0
\(218\) 0 0
\(219\) 7.87175e39 1.28387
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −5.87054e38 −0.0604750
\(226\) −2.03875e40 −1.94769
\(227\) −2.25333e40 −1.99703 −0.998515 0.0544835i \(-0.982649\pi\)
−0.998515 + 0.0544835i \(0.982649\pi\)
\(228\) 3.82981e38 0.0314981
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.51122e40 1.99714 0.998568 0.0534881i \(-0.0170339\pi\)
0.998568 + 0.0534881i \(0.0170339\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2.07523e40 −0.949648
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 4.29481e40 1.37611 0.688057 0.725657i \(-0.258464\pi\)
0.688057 + 0.725657i \(0.258464\pi\)
\(242\) −4.08109e40 −1.21875
\(243\) 4.33814e39 0.120781
\(244\) 0 0
\(245\) 0 0
\(246\) −5.52477e40 −1.24858
\(247\) 0 0
\(248\) 0 0
\(249\) 5.83699e40 1.07349
\(250\) 0 0
\(251\) −1.22250e41 −1.96244 −0.981220 0.192891i \(-0.938214\pi\)
−0.981220 + 0.192891i \(0.938214\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 8.71123e40 1.00000
\(257\) 1.77576e41 1.90775 0.953874 0.300208i \(-0.0970561\pi\)
0.953874 + 0.300208i \(0.0970561\pi\)
\(258\) −6.82256e40 −0.686140
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 2.05491e41 1.59100
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 2.12215e41 1.44380
\(265\) 0 0
\(266\) 0 0
\(267\) −2.06949e41 −1.16191
\(268\) 3.59916e41 1.89631
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 4.11056e41 1.68357
\(273\) 0 0
\(274\) −4.52704e41 −1.63702
\(275\) −4.38238e41 −1.48955
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −7.06704e41 −1.99745
\(279\) 0 0
\(280\) 0 0
\(281\) −7.75436e41 −1.82616 −0.913079 0.407782i \(-0.866302\pi\)
−0.913079 + 0.407782i \(0.866302\pi\)
\(282\) 0 0
\(283\) −5.94949e41 −1.24197 −0.620984 0.783824i \(-0.713267\pi\)
−0.620984 + 0.783824i \(0.713267\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 3.90165e40 0.0604750
\(289\) 1.25532e42 1.83439
\(290\) 0 0
\(291\) −1.09631e42 −1.42480
\(292\) 1.08038e42 1.32455
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) −8.87889e41 −0.969291
\(295\) 0 0
\(296\) 0 0
\(297\) 1.66674e42 1.53112
\(298\) 0 0
\(299\) 0 0
\(300\) 1.25174e42 0.969291
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 5.25632e40 0.0324961
\(305\) 0 0
\(306\) 1.84107e41 0.101814
\(307\) −1.70205e42 −0.890474 −0.445237 0.895413i \(-0.646880\pi\)
−0.445237 + 0.895413i \(0.646880\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 5.04774e42 1.90039 0.950195 0.311655i \(-0.100883\pi\)
0.950195 + 0.311655i \(0.100883\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 6.61619e42 1.62189
\(322\) 0 0
\(323\) 2.48030e41 0.0547092
\(324\) −4.47177e42 −0.935868
\(325\) 0 0
\(326\) −9.43192e42 −1.77787
\(327\) 0 0
\(328\) −7.58261e42 −1.28814
\(329\) 0 0
\(330\) 0 0
\(331\) 4.83570e42 0.703694 0.351847 0.936058i \(-0.385554\pi\)
0.351847 + 0.936058i \(0.385554\pi\)
\(332\) 8.01112e42 1.10750
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.38367e43 −1.48362 −0.741812 0.670608i \(-0.766033\pi\)
−0.741812 + 0.670608i \(0.766033\pi\)
\(338\) −9.80808e42 −1.00000
\(339\) 1.94701e43 1.88788
\(340\) 0 0
\(341\) 0 0
\(342\) 2.35424e40 0.00196520
\(343\) 0 0
\(344\) −9.36380e42 −0.707878
\(345\) 0 0
\(346\) 0 0
\(347\) −2.17221e43 −1.41677 −0.708386 0.705826i \(-0.750576\pi\)
−0.708386 + 0.705826i \(0.750576\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.91259e43 1.48955
\(353\) −4.10312e43 −1.99960 −0.999802 0.0198780i \(-0.993672\pi\)
−0.999802 + 0.0198780i \(0.993672\pi\)
\(354\) 1.98186e43 0.920485
\(355\) 0 0
\(356\) −2.84032e43 −1.19872
\(357\) 0 0
\(358\) 5.20689e43 1.99787
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −3.00029e43 −0.998944
\(362\) 0 0
\(363\) 3.89746e43 1.18132
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −3.39615e42 −0.0779002
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 1.37436e44 2.50775
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 5.15181e43 0.750055 0.375028 0.927014i \(-0.377633\pi\)
0.375028 + 0.927014i \(0.377633\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −8.31927e43 −0.969291
\(385\) 0 0
\(386\) 1.76230e44 1.87973
\(387\) −4.19392e42 −0.0428089
\(388\) −1.50466e44 −1.46994
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.21861e44 −1.00000
\(393\) −1.96245e44 −1.54214
\(394\) 0 0
\(395\) 0 0
\(396\) 1.30451e43 0.0900803
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.71799e44 1.00000
\(401\) −1.39769e44 −0.779751 −0.389876 0.920868i \(-0.627482\pi\)
−0.389876 + 0.920868i \(0.627482\pi\)
\(402\) −3.43722e44 −1.83808
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −3.92561e44 −1.63187
\(409\) −3.82421e44 −1.52491 −0.762457 0.647038i \(-0.776007\pi\)
−0.762457 + 0.647038i \(0.776007\pi\)
\(410\) 0 0
\(411\) 4.32335e44 1.58675
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.74905e44 1.93611
\(418\) 1.75744e43 0.0484044
\(419\) 7.36380e44 1.94744 0.973719 0.227753i \(-0.0731380\pi\)
0.973719 + 0.227753i \(0.0731380\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −1.55221e44 −0.363612
\(423\) 0 0
\(424\) 0 0
\(425\) 8.10666e44 1.68357
\(426\) 0 0
\(427\) 0 0
\(428\) 9.08055e44 1.67328
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −6.53399e44 −1.02791
\(433\) −1.89011e44 −0.285886 −0.142943 0.989731i \(-0.545657\pi\)
−0.142943 + 0.989731i \(0.545657\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −1.03177e45 −1.28387
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −5.45798e43 −0.0604750
\(442\) 0 0
\(443\) −1.27867e45 −1.31189 −0.655943 0.754811i \(-0.727729\pi\)
−0.655943 + 0.754811i \(0.727729\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.91402e44 −0.319474 −0.159737 0.987160i \(-0.551065\pi\)
−0.159737 + 0.987160i \(0.551065\pi\)
\(450\) 7.69464e43 0.0604750
\(451\) −2.53524e45 −1.91875
\(452\) 2.67222e45 1.94769
\(453\) 0 0
\(454\) 2.95348e45 1.99703
\(455\) 0 0
\(456\) −5.01981e43 −0.0314981
\(457\) 3.21808e45 1.94546 0.972729 0.231945i \(-0.0745090\pi\)
0.972729 + 0.231945i \(0.0745090\pi\)
\(458\) 0 0
\(459\) −3.08319e45 −1.73055
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −4.60223e45 −1.99714
\(467\) −2.77331e45 −1.16041 −0.580205 0.814470i \(-0.697028\pi\)
−0.580205 + 0.814470i \(0.697028\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 2.72005e45 0.949648
\(473\) −3.13078e45 −1.05442
\(474\) 0 0
\(475\) 1.03663e44 0.0324961
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −5.62929e45 −1.37611
\(483\) 0 0
\(484\) 5.34917e45 1.21875
\(485\) 0 0
\(486\) −5.68609e44 −0.120781
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 9.00753e45 1.72328
\(490\) 0 0
\(491\) 5.30868e44 0.0947549 0.0473774 0.998877i \(-0.484914\pi\)
0.0473774 + 0.998877i \(0.484914\pi\)
\(492\) 7.24143e45 1.24858
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −7.65065e45 −1.07349
\(499\) 2.33034e45 0.316017 0.158008 0.987438i \(-0.449493\pi\)
0.158008 + 0.987438i \(0.449493\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1.60236e46 1.96244
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.36676e45 0.969291
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.14180e46 −1.00000
\(513\) −3.94258e44 −0.0334030
\(514\) −2.32752e46 −1.90775
\(515\) 0 0
\(516\) 8.94247e45 0.686140
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.02261e45 0.457359 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(522\) 0 0
\(523\) 1.13155e46 0.690470 0.345235 0.938516i \(-0.387799\pi\)
0.345235 + 0.938516i \(0.387799\pi\)
\(524\) −2.69341e46 −1.59100
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −2.78154e46 −1.44380
\(529\) 1.98951e46 1.00000
\(530\) 0 0
\(531\) 1.21827e45 0.0574299
\(532\) 0 0
\(533\) 0 0
\(534\) 2.71252e46 1.16191
\(535\) 0 0
\(536\) −4.71749e46 −1.89631
\(537\) −4.97261e46 −1.93652
\(538\) 0 0
\(539\) −4.07440e46 −1.48955
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −5.38780e46 −1.68357
\(545\) 0 0
\(546\) 0 0
\(547\) 7.00458e46 1.99341 0.996707 0.0810835i \(-0.0258381\pi\)
0.996707 + 0.0810835i \(0.0258381\pi\)
\(548\) 5.93369e46 1.63702
\(549\) 0 0
\(550\) 5.74407e46 1.48955
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 9.26291e46 1.99745
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −1.31252e47 −2.43074
\(562\) 1.01638e47 1.82616
\(563\) 3.32267e46 0.579221 0.289611 0.957145i \(-0.406474\pi\)
0.289611 + 0.957145i \(0.406474\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 7.79811e46 1.24197
\(567\) 0 0
\(568\) 0 0
\(569\) −6.31003e46 −0.918593 −0.459296 0.888283i \(-0.651898\pi\)
−0.459296 + 0.888283i \(0.651898\pi\)
\(570\) 0 0
\(571\) 1.23807e47 1.69798 0.848988 0.528411i \(-0.177212\pi\)
0.848988 + 0.528411i \(0.177212\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −5.11397e45 −0.0604750
\(577\) 5.13132e45 0.0589170 0.0294585 0.999566i \(-0.490622\pi\)
0.0294585 + 0.999566i \(0.490622\pi\)
\(578\) −1.64538e47 −1.83439
\(579\) −1.68301e47 −1.82201
\(580\) 0 0
\(581\) 0 0
\(582\) 1.43696e47 1.42480
\(583\) 0 0
\(584\) −1.41607e47 −1.32455
\(585\) 0 0
\(586\) 0 0
\(587\) 1.10110e47 0.944013 0.472007 0.881595i \(-0.343530\pi\)
0.472007 + 0.881595i \(0.343530\pi\)
\(588\) 1.16377e47 0.969291
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.64527e47 −1.90783 −0.953914 0.300079i \(-0.902987\pi\)
−0.953914 + 0.300079i \(0.902987\pi\)
\(594\) −2.18463e47 −1.53112
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −1.64069e47 −0.969291
\(601\) 2.96967e47 1.70546 0.852731 0.522351i \(-0.174945\pi\)
0.852731 + 0.522351i \(0.174945\pi\)
\(602\) 0 0
\(603\) −2.11290e46 −0.114679
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) −6.88956e45 −0.0324961
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −2.41313e46 −0.101814
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 2.23092e47 0.890474
\(615\) 0 0
\(616\) 0 0
\(617\) 5.43586e47 1.99719 0.998597 0.0529492i \(-0.0168621\pi\)
0.998597 + 0.0529492i \(0.0168621\pi\)
\(618\) 0 0
\(619\) −5.68111e47 −1.97557 −0.987785 0.155822i \(-0.950197\pi\)
−0.987785 + 0.155822i \(0.950197\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 3.38813e47 1.00000
\(626\) −6.61617e47 −1.90039
\(627\) −1.67837e46 −0.0469180
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 1.48237e47 0.352446
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.77216e47 0.532391 0.266196 0.963919i \(-0.414233\pi\)
0.266196 + 0.963919i \(0.414233\pi\)
\(642\) −8.67197e47 −1.62189
\(643\) 9.25829e47 1.68634 0.843168 0.537650i \(-0.180688\pi\)
0.843168 + 0.537650i \(0.180688\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −3.25097e46 −0.0547092
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 5.86124e47 0.935868
\(649\) 9.09446e47 1.41455
\(650\) 0 0
\(651\) 0 0
\(652\) 1.23626e48 1.77787
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 9.93868e47 1.28814
\(657\) −6.34240e46 −0.0801018
\(658\) 0 0
\(659\) −1.54391e48 −1.85169 −0.925846 0.377901i \(-0.876646\pi\)
−0.925846 + 0.377901i \(0.876646\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −6.33825e47 −0.703694
\(663\) 0 0
\(664\) −1.05003e48 −1.10750
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.26946e48 −1.90400 −0.951998 0.306104i \(-0.900975\pi\)
−0.951998 + 0.306104i \(0.900975\pi\)
\(674\) 1.81360e48 1.48362
\(675\) −1.28860e48 −1.02791
\(676\) 1.28556e48 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) −2.55199e48 −1.88788
\(679\) 0 0
\(680\) 0 0
\(681\) −2.82059e48 −1.93570
\(682\) 0 0
\(683\) −2.91379e48 −1.90242 −0.951211 0.308542i \(-0.900159\pi\)
−0.951211 + 0.308542i \(0.900159\pi\)
\(684\) −3.08574e45 −0.00196520
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 1.22733e48 0.707878
\(689\) 0 0
\(690\) 0 0
\(691\) −3.49647e47 −0.187285 −0.0936426 0.995606i \(-0.529851\pi\)
−0.0936426 + 0.995606i \(0.529851\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 2.84715e48 1.41677
\(695\) 0 0
\(696\) 0 0
\(697\) 4.68976e48 2.16867
\(698\) 0 0
\(699\) 4.39515e48 1.93581
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −3.81759e48 −1.48955
\(705\) 0 0
\(706\) 5.37804e48 1.99960
\(707\) 0 0
\(708\) −2.59766e48 −0.920485
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 3.72286e48 1.19872
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −6.82478e48 −1.99787
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.93254e48 0.998944
\(723\) 5.37600e48 1.33385
\(724\) 0 0
\(725\) 0 0
\(726\) −5.10848e48 −1.18132
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 4.88395e48 1.05294
\(730\) 0 0
\(731\) 5.79141e48 1.19176
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.57729e49 −2.82465
\(738\) 4.45140e47 0.0779002
\(739\) 6.52587e48 1.11605 0.558024 0.829824i \(-0.311560\pi\)
0.558024 + 0.829824i \(0.311560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −4.70296e47 −0.0669763
\(748\) −1.80141e49 −2.50775
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −1.53026e49 −1.90218
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −6.75258e48 −0.750055
\(759\) 0 0
\(760\) 0 0
\(761\) 1.42370e49 1.47870 0.739349 0.673322i \(-0.235133\pi\)
0.739349 + 0.673322i \(0.235133\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.09042e49 0.969291
\(769\) −6.11390e48 −0.531582 −0.265791 0.964031i \(-0.585633\pi\)
−0.265791 + 0.964031i \(0.585633\pi\)
\(770\) 0 0
\(771\) 2.22280e49 1.84916
\(772\) −2.30988e49 −1.87973
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 5.49706e47 0.0428089
\(775\) 0 0
\(776\) 1.97219e49 1.46994
\(777\) 0 0
\(778\) 0 0
\(779\) 5.99695e47 0.0418595
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.59725e49 1.00000
\(785\) 0 0
\(786\) 2.57222e49 1.54214
\(787\) −3.25337e49 −1.90881 −0.954407 0.298508i \(-0.903511\pi\)
−0.954407 + 0.298508i \(0.903511\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −1.70985e48 −0.0900803
\(793\) 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
\(800\) −2.25180e49 −1.00000
\(801\) 1.66742e48 0.0724924
\(802\) 1.83198e49 0.779751
\(803\) −4.73462e49 −1.97297
\(804\) 4.50523e49 1.83808
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 4.90182e49 1.79983 0.899916 0.436063i \(-0.143627\pi\)
0.899916 + 0.436063i \(0.143627\pi\)
\(810\) 0 0
\(811\) −5.42916e49 −1.91152 −0.955759 0.294151i \(-0.904963\pi\)
−0.955759 + 0.294151i \(0.904963\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 5.14537e49 1.63187
\(817\) 7.40566e47 0.0230032
\(818\) 5.01247e49 1.52491
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) −5.66670e49 −1.58675
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) −5.48562e49 −1.44380
\(826\) 0 0
\(827\) 6.45448e49 1.63030 0.815150 0.579250i \(-0.196654\pi\)
0.815150 + 0.579250i \(0.196654\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7.53694e49 1.68357
\(834\) −8.84612e49 −1.93611
\(835\) 0 0
\(836\) −2.30352e48 −0.0484044
\(837\) 0 0
\(838\) −9.65188e49 −1.94744
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.26662e49 1.00000
\(842\) 0 0
\(843\) −9.70647e49 −1.77008
\(844\) 2.03451e49 0.363612
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −7.44724e49 −1.20383
\(850\) −1.06256e50 −1.68357
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.19021e50 −1.67328
\(857\) 8.73535e49 1.20394 0.601972 0.798517i \(-0.294382\pi\)
0.601972 + 0.798517i \(0.294382\pi\)
\(858\) 0 0
\(859\) 3.39348e49 0.449532 0.224766 0.974413i \(-0.427838\pi\)
0.224766 + 0.974413i \(0.427838\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 8.56423e49 1.02791
\(865\) 0 0
\(866\) 2.47740e49 0.285886
\(867\) 1.57134e50 1.77806
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 8.83317e48 0.0888944
\(874\) 0 0
\(875\) 0 0
\(876\) 1.35236e50 1.28387
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.15569e50 0.995979 0.497989 0.867183i \(-0.334072\pi\)
0.497989 + 0.867183i \(0.334072\pi\)
\(882\) 7.15388e48 0.0604750
\(883\) −9.97534e49 −0.827172 −0.413586 0.910465i \(-0.635724\pi\)
−0.413586 + 0.910465i \(0.635724\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.67598e50 1.31189
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.95970e50 1.39402
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 5.13018e49 0.319474
\(899\) 0 0
\(900\) −1.00855e49 −0.0604750
\(901\) 0 0
\(902\) 3.32299e50 1.91875
\(903\) 0 0
\(904\) −3.50254e50 −1.94769
\(905\) 0 0
\(906\) 0 0
\(907\) −3.11402e50 −1.63681 −0.818403 0.574644i \(-0.805140\pi\)
−0.818403 + 0.574644i \(0.805140\pi\)
\(908\) −3.87118e50 −1.99703
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 6.57956e48 0.0314981
\(913\) −3.51078e50 −1.64968
\(914\) −4.21801e50 −1.94546
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 4.04120e50 1.73055
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −2.13054e50 −0.863128
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −4.38576e50 −1.53383 −0.766915 0.641748i \(-0.778209\pi\)
−0.766915 + 0.641748i \(0.778209\pi\)
\(930\) 0 0
\(931\) 9.63774e48 0.0324961
\(932\) 6.03224e50 1.99714
\(933\) 0 0
\(934\) 3.63503e50 1.16041
\(935\) 0 0
\(936\) 0 0
\(937\) −2.58288e50 −0.780786 −0.390393 0.920648i \(-0.627661\pi\)
−0.390393 + 0.920648i \(0.627661\pi\)
\(938\) 0 0
\(939\) 6.31847e50 1.84203
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −3.56522e50 −0.949648
\(945\) 0 0
\(946\) 4.10357e50 1.05442
\(947\) −2.56016e49 −0.0646125 −0.0323063 0.999478i \(-0.510285\pi\)
−0.0323063 + 0.999478i \(0.510285\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −1.35873e49 −0.0324961
\(951\) 0 0
\(952\) 0 0
\(953\) −8.42069e50 −1.90883 −0.954417 0.298476i \(-0.903522\pi\)
−0.954417 + 0.298476i \(0.903522\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 5.08508e50 1.00000
\(962\) 0 0
\(963\) −5.33078e49 −0.101191
\(964\) 7.37843e50 1.37611
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −7.01126e50 −1.21875
\(969\) 3.10470e49 0.0530292
\(970\) 0 0
\(971\) 5.74217e50 0.946998 0.473499 0.880794i \(-0.342991\pi\)
0.473499 + 0.880794i \(0.342991\pi\)
\(972\) 7.45287e49 0.120781
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.33580e51 1.98397 0.991983 0.126369i \(-0.0403322\pi\)
0.991983 + 0.126369i \(0.0403322\pi\)
\(978\) −1.18064e51 −1.72328
\(979\) 1.24474e51 1.78555
\(980\) 0 0
\(981\) 0 0
\(982\) −6.95820e49 −0.0947549
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −9.49148e50 −1.24858
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 6.05306e50 0.682084
\(994\) 0 0
\(995\) 0 0
\(996\) 1.00279e51 1.07349
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −3.05443e50 −0.316017
\(999\) 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 8.35.d.a.3.1 1
4.3 odd 2 32.35.d.a.15.1 1
8.3 odd 2 CM 8.35.d.a.3.1 1
8.5 even 2 32.35.d.a.15.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.35.d.a.3.1 1 1.1 even 1 trivial
8.35.d.a.3.1 1 8.3 odd 2 CM
32.35.d.a.15.1 1 4.3 odd 2
32.35.d.a.15.1 1 8.5 even 2