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Properties

Label 31.25.b.a.30.1
Level $31$
Weight $25$
Character 31.30
Self dual yes
Analytic conductor $113.140$
Analytic rank $0$
Dimension $1$
CM discriminant -31
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [31,25,Mod(30,31)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(31, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 25, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.30");
 
S:= CuspForms(chi, 25);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 25 \)
Character orbit: \([\chi]\) \(=\) 31.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: \(113.139817200\)
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 30.1
Character \(\chi\) \(=\) 31.30

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1217.00 q^{2} -1.52961e7 q^{4} +3.68218e8 q^{5} -2.51426e10 q^{7} -3.90333e10 q^{8} +2.82430e11 q^{9} +O(q^{10})\) \(q+1217.00 q^{2} -1.52961e7 q^{4} +3.68218e8 q^{5} -2.51426e10 q^{7} -3.90333e10 q^{8} +2.82430e11 q^{9} +4.48121e11 q^{10} -3.05986e13 q^{14} +2.09123e14 q^{16} +3.43717e14 q^{18} -4.07880e15 q^{19} -5.63230e15 q^{20} +7.59795e16 q^{25} +3.84585e17 q^{28} +7.87663e17 q^{31} +9.09372e17 q^{32} -9.25795e18 q^{35} -4.32008e18 q^{36} -4.96390e18 q^{38} -1.43727e19 q^{40} -1.11038e19 q^{41} +1.03995e20 q^{45} -1.69017e20 q^{47} +4.40570e20 q^{49} +9.24670e19 q^{50} +9.81398e20 q^{56} +7.49602e20 q^{59} +9.58586e20 q^{62} -7.10102e21 q^{63} -2.40180e21 q^{64} +1.02196e22 q^{67} -1.12669e22 q^{70} -2.03897e21 q^{71} -1.10241e22 q^{72} +6.23899e22 q^{76} +7.70027e22 q^{80} +7.97664e22 q^{81} -1.35134e22 q^{82} +1.26563e23 q^{90} -2.05694e23 q^{94} -1.50189e24 q^{95} +1.17576e24 q^{97} +5.36174e23 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1217.00 0.297119 0.148560 − 0.988903i \(-0.452536\pi\)
0.148560 + 0.988903i \(0.452536\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −1.52961e7 −0.911720
\(5\) 3.68218e8 1.50822 0.754109 − 0.656749i \(-0.228069\pi\)
0.754109 + 0.656749i \(0.228069\pi\)
\(6\) 0 0
\(7\) −2.51426e10 −1.81649 −0.908247 − 0.418434i \(-0.862579\pi\)
−0.908247 + 0.418434i \(0.862579\pi\)
\(8\) −3.90333e10 −0.568009
\(9\) 2.82430e11 1.00000
\(10\) 4.48121e11 0.448121
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −3.05986e13 −0.539715
\(15\) 0 0
\(16\) 2.09123e14 0.742954
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 3.43717e14 0.297119
\(19\) −4.07880e15 −1.84285 −0.921424 − 0.388560i \(-0.872973\pi\)
−0.921424 + 0.388560i \(0.872973\pi\)
\(20\) −5.63230e15 −1.37507
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 7.59795e16 1.27472
\(26\) 0 0
\(27\) 0 0
\(28\) 3.84585e17 1.65613
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 7.87663e17 1.00000
\(32\) 9.09372e17 0.788755
\(33\) 0 0
\(34\) 0 0
\(35\) −9.25795e18 −2.73967
\(36\) −4.32008e18 −0.911720
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −4.96390e18 −0.547545
\(39\) 0 0
\(40\) −1.43727e19 −0.856681
\(41\) −1.11038e19 −0.492115 −0.246057 − 0.969255i \(-0.579135\pi\)
−0.246057 + 0.969255i \(0.579135\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 1.03995e20 1.50822
\(46\) 0 0
\(47\) −1.69017e20 −1.45465 −0.727323 − 0.686295i \(-0.759236\pi\)
−0.727323 + 0.686295i \(0.759236\pi\)
\(48\) 0 0
\(49\) 4.40570e20 2.29965
\(50\) 9.24670e19 0.378745
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 9.81398e20 1.03178
\(57\) 0 0
\(58\) 0 0
\(59\) 7.49602e20 0.421315 0.210657 − 0.977560i \(-0.432440\pi\)
0.210657 + 0.977560i \(0.432440\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 9.58586e20 0.297119
\(63\) −7.10102e21 −1.81649
\(64\) −2.40180e21 −0.508600
\(65\) 0 0
\(66\) 0 0
\(67\) 1.02196e22 1.24892 0.624461 − 0.781056i \(-0.285319\pi\)
0.624461 + 0.781056i \(0.285319\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −1.12669e22 −0.814009
\(71\) −2.03897e21 −0.124254 −0.0621270 − 0.998068i \(-0.519788\pi\)
−0.0621270 + 0.998068i \(0.519788\pi\)
\(72\) −1.10241e22 −0.568009
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 6.23899e22 1.68016
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 7.70027e22 1.12054
\(81\) 7.97664e22 1.00000
\(82\) −1.35134e22 −0.146217
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 1.26563e23 0.448121
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −2.05694e23 −0.432203
\(95\) −1.50189e24 −2.77942
\(96\) 0 0
\(97\) 1.17576e24 1.69456 0.847282 − 0.531144i \(-0.178238\pi\)
0.847282 + 0.531144i \(0.178238\pi\)
\(98\) 5.36174e23 0.683270
\(99\) 0 0
\(100\) −1.16219e24 −1.16219
\(101\) −1.58869e24 −1.40988 −0.704940 − 0.709267i \(-0.749026\pi\)
−0.704940 + 0.709267i \(0.749026\pi\)
\(102\) 0 0
\(103\) 1.41717e24 0.993978 0.496989 − 0.867757i \(-0.334439\pi\)
0.496989 + 0.867757i \(0.334439\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.41419e24 1.95995 0.979977 − 0.199113i \(-0.0638060\pi\)
0.979977 + 0.199113i \(0.0638060\pi\)
\(108\) 0 0
\(109\) 5.35875e24 1.90522 0.952610 − 0.304193i \(-0.0983868\pi\)
0.952610 + 0.304193i \(0.0983868\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −5.25790e24 −1.34957
\(113\) 8.57385e24 1.97804 0.989019 − 0.147790i \(-0.0472159\pi\)
0.989019 + 0.147790i \(0.0472159\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 9.12266e23 0.125181
\(119\) 0 0
\(120\) 0 0
\(121\) 9.84973e24 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −1.20482e25 −0.911720
\(125\) 6.02950e24 0.414344
\(126\) −8.64194e24 −0.539715
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −1.81797e25 −0.939869
\(129\) 0 0
\(130\) 0 0
\(131\) 4.75598e25 1.86202 0.931009 − 0.364995i \(-0.118929\pi\)
0.931009 + 0.364995i \(0.118929\pi\)
\(132\) 0 0
\(133\) 1.02552e26 3.34752
\(134\) 1.24372e25 0.371078
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 1.41611e26 2.49781
\(141\) 0 0
\(142\) −2.48142e24 −0.0369182
\(143\) 0 0
\(144\) 5.90625e25 0.742954
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.82542e26 −1.52450 −0.762249 − 0.647283i \(-0.775905\pi\)
−0.762249 + 0.647283i \(0.775905\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 1.59209e26 1.04675
\(153\) 0 0
\(154\) 0 0
\(155\) 2.90031e26 1.50822
\(156\) 0 0
\(157\) −3.19803e26 −1.42589 −0.712945 − 0.701220i \(-0.752639\pi\)
−0.712945 + 0.701220i \(0.752639\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 3.34847e26 1.18961
\(161\) 0 0
\(162\) 9.70758e25 0.297119
\(163\) 4.61261e26 1.31128 0.655640 − 0.755074i \(-0.272399\pi\)
0.655640 + 0.755074i \(0.272399\pi\)
\(164\) 1.69846e26 0.448671
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 5.42801e26 1.00000
\(170\) 0 0
\(171\) −1.15197e27 −1.84285
\(172\) 0 0
\(173\) 8.90852e26 1.23952 0.619758 − 0.784793i \(-0.287231\pi\)
0.619758 + 0.784793i \(0.287231\pi\)
\(174\) 0 0
\(175\) −1.91032e27 −2.31553
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −1.59073e27 −1.37507
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 2.58531e27 1.32623
\(189\) 0 0
\(190\) −1.82780e27 −0.825818
\(191\) 4.30076e27 1.82450 0.912252 − 0.409630i \(-0.134342\pi\)
0.912252 + 0.409630i \(0.134342\pi\)
\(192\) 0 0
\(193\) −1.21000e27 −0.453000 −0.226500 − 0.974011i \(-0.572728\pi\)
−0.226500 + 0.974011i \(0.572728\pi\)
\(194\) 1.43090e27 0.503487
\(195\) 0 0
\(196\) −6.73901e27 −2.09664
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −2.96573e27 −0.724054
\(201\) 0 0
\(202\) −1.93343e27 −0.418902
\(203\) 0 0
\(204\) 0 0
\(205\) −4.08862e27 −0.742217
\(206\) 1.72470e27 0.295330
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.49838e28 −1.92412 −0.962060 − 0.272838i \(-0.912038\pi\)
−0.962060 + 0.272838i \(0.912038\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 5.37207e27 0.582340
\(215\) 0 0
\(216\) 0 0
\(217\) −1.98039e28 −1.81649
\(218\) 6.52160e27 0.566078
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) −2.28640e28 −1.43277
\(225\) 2.14589e28 1.27472
\(226\) 1.04344e28 0.587713
\(227\) −3.52933e28 −1.88531 −0.942654 − 0.333771i \(-0.891679\pi\)
−0.942654 + 0.333771i \(0.891679\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.19848e28 1.63992 0.819958 − 0.572424i \(-0.193997\pi\)
0.819958 + 0.572424i \(0.193997\pi\)
\(234\) 0 0
\(235\) −6.22352e28 −2.19392
\(236\) −1.14660e28 −0.384121
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 1.19871e28 0.297119
\(243\) 0 0
\(244\) 0 0
\(245\) 1.62226e29 3.46838
\(246\) 0 0
\(247\) 0 0
\(248\) −3.07450e28 −0.568009
\(249\) 0 0
\(250\) 7.33791e27 0.123110
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.08618e29 1.65613
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.81707e28 0.229347
\(257\) −5.47859e28 −0.659889 −0.329945 − 0.944000i \(-0.607030\pi\)
−0.329945 + 0.944000i \(0.607030\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 5.78802e28 0.553241
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.24805e29 0.994613
\(267\) 0 0
\(268\) −1.56320e29 −1.13867
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 2.22459e29 1.00000
\(280\) 3.61368e29 1.55616
\(281\) 4.12639e29 1.70253 0.851264 − 0.524738i \(-0.175837\pi\)
0.851264 + 0.524738i \(0.175837\pi\)
\(282\) 0 0
\(283\) −5.07020e29 −1.92127 −0.960633 − 0.277820i \(-0.910388\pi\)
−0.960633 + 0.277820i \(0.910388\pi\)
\(284\) 3.11883e28 0.113285
\(285\) 0 0
\(286\) 0 0
\(287\) 2.79179e29 0.893923
\(288\) 2.56834e29 0.788755
\(289\) 3.39449e29 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.20902e29 −0.551809 −0.275904 − 0.961185i \(-0.588977\pi\)
−0.275904 + 0.961185i \(0.588977\pi\)
\(294\) 0 0
\(295\) 2.76017e29 0.635435
\(296\) 0 0
\(297\) 0 0
\(298\) −2.22153e29 −0.452958
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −8.52971e29 −1.36915
\(305\) 0 0
\(306\) 0 0
\(307\) −1.40121e30 −1.99913 −0.999566 − 0.0294667i \(-0.990619\pi\)
−0.999566 + 0.0294667i \(0.990619\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 3.52968e29 0.448121
\(311\) −1.60727e30 −1.96320 −0.981599 − 0.190954i \(-0.938842\pi\)
−0.981599 + 0.190954i \(0.938842\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) −3.89200e29 −0.423659
\(315\) −2.61472e30 −2.73967
\(316\) 0 0
\(317\) 1.17240e30 1.13858 0.569292 − 0.822136i \(-0.307217\pi\)
0.569292 + 0.822136i \(0.307217\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −8.84383e29 −0.767080
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.22012e30 −0.911720
\(325\) 0 0
\(326\) 5.61355e29 0.389606
\(327\) 0 0
\(328\) 4.33418e29 0.279525
\(329\) 4.24954e30 2.64236
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.76302e30 1.88365
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 6.60589e29 0.297119
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) −1.40195e30 −0.547545
\(343\) −6.26023e30 −2.36081
\(344\) 0 0
\(345\) 0 0
\(346\) 1.08417e30 0.368284
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −6.06064e30 −1.85615 −0.928075 − 0.372392i \(-0.878538\pi\)
−0.928075 + 0.372392i \(0.878538\pi\)
\(350\) −2.32486e30 −0.687988
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −7.50784e29 −0.187402
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7.03997e30 −1.53616 −0.768082 − 0.640351i \(-0.778789\pi\)
−0.768082 + 0.640351i \(0.778789\pi\)
\(360\) −4.05928e30 −0.856681
\(361\) 1.17379e31 2.39609
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −3.13605e30 −0.492115
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 5.87157e30 0.809559 0.404780 − 0.914414i \(-0.367348\pi\)
0.404780 + 0.914414i \(0.367348\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.59730e30 0.826251
\(377\) 0 0
\(378\) 0 0
\(379\) 1.63109e31 1.85698 0.928492 − 0.371354i \(-0.121106\pi\)
0.928492 + 0.371354i \(0.121106\pi\)
\(380\) 2.29730e31 2.53405
\(381\) 0 0
\(382\) 5.23402e30 0.542095
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.47257e30 −0.134595
\(387\) 0 0
\(388\) −1.79846e31 −1.54497
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.71969e31 −1.30622
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 9.51441e30 0.620720 0.310360 − 0.950619i \(-0.399550\pi\)
0.310360 + 0.950619i \(0.399550\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.58891e31 0.947061
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 2.43008e31 1.28542
\(405\) 2.93714e31 1.50822
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) −4.97585e30 −0.220527
\(411\) 0 0
\(412\) −2.16773e31 −0.906230
\(413\) −1.88470e31 −0.765316
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.01322e31 0.346048 0.173024 − 0.984918i \(-0.444646\pi\)
0.173024 + 0.984918i \(0.444646\pi\)
\(420\) 0 0
\(421\) 6.01664e31 1.94075 0.970374 − 0.241609i \(-0.0776751\pi\)
0.970374 + 0.241609i \(0.0776751\pi\)
\(422\) −1.82353e31 −0.571693
\(423\) −4.77355e31 −1.45465
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −6.75200e31 −1.78693
\(429\) 0 0
\(430\) 0 0
\(431\) 7.99551e31 1.94589 0.972946 − 0.231032i \(-0.0742101\pi\)
0.972946 + 0.231032i \(0.0742101\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) −2.41013e31 −0.539715
\(435\) 0 0
\(436\) −8.19681e31 −1.73703
\(437\) 0 0
\(438\) 0 0
\(439\) 6.86642e31 1.34016 0.670079 − 0.742290i \(-0.266260\pi\)
0.670079 + 0.742290i \(0.266260\pi\)
\(440\) 0 0
\(441\) 1.24430e32 2.29965
\(442\) 0 0
\(443\) −1.14255e32 −2.00000 −1.00000 0.000714469i \(-0.999773\pi\)
−1.00000 0.000714469i \(0.999773\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 6.03874e31 0.923869
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 2.61154e31 0.378745
\(451\) 0 0
\(452\) −1.31147e32 −1.80342
\(453\) 0 0
\(454\) −4.29520e31 −0.560161
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 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
\(466\) 5.10956e31 0.487250
\(467\) −2.02413e32 −1.88120 −0.940600 − 0.339517i \(-0.889736\pi\)
−0.940600 + 0.339517i \(0.889736\pi\)
\(468\) 0 0
\(469\) −2.56947e32 −2.26866
\(470\) −7.57402e31 −0.651857
\(471\) 0 0
\(472\) −2.92594e31 −0.239310
\(473\) 0 0
\(474\) 0 0
\(475\) −3.09905e32 −2.34912
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6.11180e31 0.418932 0.209466 − 0.977816i \(-0.432828\pi\)
0.209466 + 0.977816i \(0.432828\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −1.50663e32 −0.911720
\(485\) 4.32935e32 2.55577
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 1.97428e32 1.03052
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.64718e32 0.742954
\(497\) 5.12650e31 0.225707
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −9.22281e31 −0.377766
\(501\) 0 0
\(502\) 0 0
\(503\) 5.09474e32 1.94225 0.971127 − 0.238565i \(-0.0766769\pi\)
0.971127 + 0.238565i \(0.0766769\pi\)
\(504\) 2.77176e32 1.03178
\(505\) −5.84982e32 −2.12641
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 3.27119e32 1.00801
\(513\) 0 0
\(514\) −6.66744e31 −0.196066
\(515\) 5.21829e32 1.49914
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.43675e32 1.10921 0.554603 − 0.832115i \(-0.312870\pi\)
0.554603 + 0.832115i \(0.312870\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −7.27480e32 −1.69764
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 4.80251e32 1.00000
\(530\) 0 0
\(531\) 2.11710e32 0.421315
\(532\) −1.56864e33 −3.05200
\(533\) 0 0
\(534\) 0 0
\(535\) 1.62538e33 2.95604
\(536\) −3.98903e32 −0.709398
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 4.20676e32 0.669238 0.334619 − 0.942354i \(-0.391392\pi\)
0.334619 + 0.942354i \(0.391392\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.97318e33 2.87349
\(546\) 0 0
\(547\) −3.86500e32 −0.538643 −0.269322 − 0.963050i \(-0.586799\pi\)
−0.269322 + 0.963050i \(0.586799\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 2.70733e32 0.297119
\(559\) 0 0
\(560\) −1.93605e33 −2.03545
\(561\) 0 0
\(562\) 5.02181e32 0.505854
\(563\) −1.97085e32 −0.194336 −0.0971681 − 0.995268i \(-0.530978\pi\)
−0.0971681 + 0.995268i \(0.530978\pi\)
\(564\) 0 0
\(565\) 3.15704e33 2.98331
\(566\) −6.17043e32 −0.570845
\(567\) −2.00554e33 −1.81649
\(568\) 7.95875e31 0.0705773
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 3.39761e32 0.265602
\(575\) 0 0
\(576\) −6.78338e32 −0.508600
\(577\) −2.66506e33 −1.95703 −0.978514 − 0.206182i \(-0.933896\pi\)
−0.978514 + 0.206182i \(0.933896\pi\)
\(578\) 4.13109e32 0.297119
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −2.68838e32 −0.163953
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) −3.21272e33 −1.84285
\(590\) 3.35912e32 0.188800
\(591\) 0 0
\(592\) 0 0
\(593\) −2.46617e33 −1.30427 −0.652134 − 0.758104i \(-0.726126\pi\)
−0.652134 + 0.758104i \(0.726126\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.79218e33 1.38992
\(597\) 0 0
\(598\) 0 0
\(599\) −4.17128e33 −1.95501 −0.977503 − 0.210923i \(-0.932353\pi\)
−0.977503 + 0.210923i \(0.932353\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 2.88631e33 1.24892
\(604\) 0 0
\(605\) 3.62684e33 1.50822
\(606\) 0 0
\(607\) 3.44209e33 1.37581 0.687904 − 0.725802i \(-0.258531\pi\)
0.687904 + 0.725802i \(0.258531\pi\)
\(608\) −3.70915e33 −1.45355
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −1.70527e33 −0.593980
\(615\) 0 0
\(616\) 0 0
\(617\) 9.03629e31 0.0296871 0.0148436 − 0.999890i \(-0.495275\pi\)
0.0148436 + 0.999890i \(0.495275\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) −4.43635e33 −1.37507
\(621\) 0 0
\(622\) −1.95605e33 −0.583304
\(623\) 0 0
\(624\) 0 0
\(625\) −2.30856e33 −0.649802
\(626\) 0 0
\(627\) 0 0
\(628\) 4.89174e33 1.30001
\(629\) 0 0
\(630\) −3.18211e33 −0.814009
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 1.42681e33 0.338295
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −5.75865e32 −0.124254
\(640\) −6.69409e33 −1.41753
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 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 \(\pi\)
\(648\) −3.11354e33 −0.568009
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −7.05551e33 −1.19552
\(653\) −2.08383e33 −0.346661 −0.173330 − 0.984864i \(-0.555453\pi\)
−0.173330 + 0.984864i \(0.555453\pi\)
\(654\) 0 0
\(655\) 1.75123e34 2.80833
\(656\) −2.32206e33 −0.365619
\(657\) 0 0
\(658\) 5.17169e33 0.785094
\(659\) 1.24629e34 1.85777 0.928886 − 0.370365i \(-0.120767\pi\)
0.928886 + 0.370365i \(0.120767\pi\)
\(660\) 0 0
\(661\) −1.15309e34 −1.65747 −0.828737 − 0.559638i \(-0.810940\pi\)
−0.828737 + 0.559638i \(0.810940\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.77613e34 5.04879
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 4.57960e33 0.559667
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −8.30275e33 −0.911720
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −2.95617e34 −3.07816
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.05912e34 1.99817 0.999086 − 0.0427502i \(-0.0136120\pi\)
0.999086 + 0.0427502i \(0.0136120\pi\)
\(684\) 1.76207e34 1.68016
\(685\) 0 0
\(686\) −7.61870e33 −0.701441
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.33352e34 −1.12528 −0.562642 − 0.826701i \(-0.690215\pi\)
−0.562642 + 0.826701i \(0.690215\pi\)
\(692\) −1.36266e34 −1.13009
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −7.37580e33 −0.551498
\(699\) 0 0
\(700\) 2.92205e34 2.11111
\(701\) −1.88951e34 −1.34194 −0.670968 − 0.741486i \(-0.734121\pi\)
−0.670968 + 0.741486i \(0.734121\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.99437e34 2.56104
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) −9.13704e32 −0.0556808
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) −8.56765e33 −0.456424
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 2.17478e34 1.12054
\(721\) −3.56315e34 −1.80555
\(722\) 1.42850e34 0.711923
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.31949e34 0.605329 0.302665 − 0.953097i \(-0.402124\pi\)
0.302665 + 0.953097i \(0.402124\pi\)
\(728\) 0 0
\(729\) 2.25284e34 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 4.62842e34 1.92391 0.961956 − 0.273204i \(-0.0880833\pi\)
0.961956 + 0.273204i \(0.0880833\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −3.81657e33 −0.146217
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −6.72151e34 −2.29928
\(746\) 7.14571e33 0.240535
\(747\) 0 0
\(748\) 0 0
\(749\) −1.10984e35 −3.56024
\(750\) 0 0
\(751\) 4.55664e33 0.141568 0.0707840 − 0.997492i \(-0.477450\pi\)
0.0707840 + 0.997492i \(0.477450\pi\)
\(752\) −3.53454e34 −1.08073
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 1.98504e34 0.551745
\(759\) 0 0
\(760\) 5.86235e34 1.57873
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −1.34733e35 −3.46082
\(764\) −6.57850e34 −1.66344
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.83302e34 0.662420 0.331210 − 0.943557i \(-0.392543\pi\)
0.331210 + 0.943557i \(0.392543\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.85083e34 0.413009
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 5.98462e34 1.27472
\(776\) −4.58937e34 −0.962527
\(777\) 0 0
\(778\) 0 0
\(779\) 4.52903e34 0.906892
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 9.21333e34 1.70853
\(785\) −1.17757e35 −2.15055
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.15569e35 −3.59309
\(792\) 0 0
\(793\) 0 0
\(794\) 1.15790e34 0.184428
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 6.90936e34 1.00544
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 6.20116e34 0.800824
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 3.57450e34 0.448121
\(811\) 6.04041e34 0.746133 0.373067 − 0.927805i \(-0.378306\pi\)
0.373067 + 0.927805i \(0.378306\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.69844e35 1.97770
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 6.25401e34 0.676694
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) −5.53169e34 −0.564588
\(825\) 0 0
\(826\) −2.29368e34 −0.227390
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 1.23309e34 0.102818
\(839\) 2.20893e35 1.81568 0.907839 − 0.419319i \(-0.137731\pi\)
0.907839 + 0.419319i \(0.137731\pi\)
\(840\) 0 0
\(841\) 1.25185e35 1.00000
\(842\) 7.32225e34 0.576633
\(843\) 0 0
\(844\) 2.29194e35 1.75426
\(845\) 1.99869e35 1.50822
\(846\) −5.80941e34 −0.432203
\(847\) −2.47648e35 −1.81649
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −2.96570e35 −1.99866 −0.999330 − 0.0366134i \(-0.988343\pi\)
−0.999330 + 0.0366134i \(0.988343\pi\)
\(854\) 0 0
\(855\) −4.24177e35 −2.77942
\(856\) −1.72300e35 −1.11327
\(857\) −1.00954e35 −0.643212 −0.321606 − 0.946874i \(-0.604223\pi\)
−0.321606 + 0.946874i \(0.604223\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 9.73053e34 0.578162
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 3.28027e35 1.86946
\(866\) 0 0
\(867\) 0 0
\(868\) 3.02923e35 1.65613
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −2.09169e35 −1.08218
\(873\) 3.32069e35 1.69456
\(874\) 0 0
\(875\) −1.51597e35 −0.752654
\(876\) 0 0
\(877\) 3.76954e35 1.82093 0.910464 − 0.413587i \(-0.135724\pi\)
0.910464 + 0.413587i \(0.135724\pi\)
\(878\) 8.35643e34 0.398187
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 1.51431e35 0.683270
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1.39048e35 −0.594238
\(887\) −7.11140e34 −0.299828 −0.149914 − 0.988699i \(-0.547900\pi\)
−0.149914 + 0.988699i \(0.547900\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.89389e35 2.68069
\(894\) 0 0
\(895\) 0 0
\(896\) 4.57086e35 1.70727
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −3.28237e35 −1.16219
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −3.34665e35 −1.12354
\(905\) 0 0
\(906\) 0 0
\(907\) −6.12926e35 −1.97752 −0.988760 − 0.149513i \(-0.952229\pi\)
−0.988760 + 0.149513i \(0.952229\pi\)
\(908\) 5.39851e35 1.71887
\(909\) −4.48692e35 −1.40988
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.19578e36 −3.38235
\(918\) 0 0
\(919\) −6.83548e35 −1.88357 −0.941787 − 0.336210i \(-0.890855\pi\)
−0.941787 + 0.336210i \(0.890855\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 4.00252e35 0.993978
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −1.79700e36 −4.23790
\(932\) −6.42205e35 −1.49514
\(933\) 0 0
\(934\) −2.46336e35 −0.558941
\(935\) 0 0
\(936\) 0 0
\(937\) 6.97782e35 1.52351 0.761754 − 0.647867i \(-0.224339\pi\)
0.761754 + 0.647867i \(0.224339\pi\)
\(938\) −3.12704e35 −0.674062
\(939\) 0 0
\(940\) 9.51957e35 2.00025
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.56759e35 0.313018
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −3.77155e35 −0.697969
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 1.58361e36 2.75175
\(956\) 0 0
\(957\) 0 0
\(958\) 7.43806e34 0.124473
\(959\) 0 0
\(960\) 0 0
\(961\) 6.20413e35 1.00000
\(962\) 0 0
\(963\) 1.24670e36 1.95995
\(964\) 0 0
\(965\) −4.45543e35 −0.683223
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −3.84467e35 −0.568009
\(969\) 0 0
\(970\) 5.26882e35 0.759369
\(971\) 2.61867e35 0.372777 0.186389 − 0.982476i \(-0.440322\pi\)
0.186389 + 0.982476i \(0.440322\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.51250e36 −1.99968 −0.999838 − 0.0179737i \(-0.994278\pi\)
−0.999838 + 0.0179737i \(0.994278\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −2.48142e36 −3.16219
\(981\) 1.51347e36 1.90522
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 7.16279e35 0.788755
\(993\) 0 0
\(994\) 6.23895e34 0.0670617
\(995\) 0 0
\(996\) 0 0
\(997\) −1.03201e36 −1.06990 −0.534950 − 0.844884i \(-0.679669\pi\)
−0.534950 + 0.844884i \(0.679669\pi\)
\(998\) 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 31.25.b.a.30.1 ✓ 1
31.30 odd 2 CM 31.25.b.a.30.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.25.b.a.30.1 ✓ 1 1.1 even 1 trivial
31.25.b.a.30.1 ✓ 1 31.30 odd 2 CM