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Properties

Label 35.21.c.a.34.1
Level $35$
Weight $21$
Character 35.34
Self dual yes
Analytic conductor $88.730$
Analytic rank $0$
Dimension $1$
CM discriminant -35
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] = [35,21,Mod(34,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 21, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.34");
 
S:= CuspForms(chi, 21);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 21 \)
Character orbit: \([\chi]\) \(=\) 35.c (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: \(88.7298177859\)
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 34.1
Character \(\chi\) \(=\) 35.34

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-12223.0 q^{3} +1.04858e6 q^{4} -9.76562e6 q^{5} -2.82475e8 q^{7} -3.33738e9 q^{9} +O(q^{10})\) \(q-12223.0 q^{3} +1.04858e6 q^{4} -9.76562e6 q^{5} -2.82475e8 q^{7} -3.33738e9 q^{9} -5.18361e10 q^{11} -1.28167e10 q^{12} -9.20780e10 q^{13} +1.19365e11 q^{15} +1.09951e12 q^{16} -2.83457e12 q^{17} -1.02400e13 q^{20} +3.45269e12 q^{21} +9.53674e13 q^{25} +8.34118e13 q^{27} -2.96197e14 q^{28} +4.98982e14 q^{29} +6.33592e14 q^{33} +2.75855e15 q^{35} -3.49950e15 q^{36} +1.12547e15 q^{39} -5.43541e16 q^{44} +3.25916e16 q^{45} -1.02681e17 q^{47} -1.34393e16 q^{48} +7.97923e16 q^{49} +3.46469e16 q^{51} -9.65507e16 q^{52} +5.06212e17 q^{55} +1.25164e17 q^{60} +9.42728e17 q^{63} +1.15292e18 q^{64} +8.99199e17 q^{65} -2.97226e18 q^{68} -6.44601e18 q^{71} -6.04212e18 q^{73} -1.16568e18 q^{75} +1.46424e19 q^{77} +8.15334e18 q^{79} -1.07374e19 q^{80} +1.06172e19 q^{81} +2.07767e19 q^{83} +3.62041e18 q^{84} +2.76813e19 q^{85} -6.09905e18 q^{87} +2.60097e19 q^{91} -1.15875e20 q^{97} +1.72997e20 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −12223.0 −0.206998 −0.103499 − 0.994630i \(-0.533004\pi\)
−0.103499 + 0.994630i \(0.533004\pi\)
\(4\) 1.04858e6 1.00000
\(5\) −9.76562e6 −1.00000
\(6\) 0 0
\(7\) −2.82475e8 −1.00000
\(8\) 0 0
\(9\) −3.33738e9 −0.957152
\(10\) 0 0
\(11\) −5.18361e10 −1.99851 −0.999253 − 0.0386575i \(-0.987692\pi\)
−0.999253 + 0.0386575i \(0.987692\pi\)
\(12\) −1.28167e10 −0.206998
\(13\) −9.20780e10 −0.667916 −0.333958 − 0.942588i \(-0.608384\pi\)
−0.333958 + 0.942588i \(0.608384\pi\)
\(14\) 0 0
\(15\) 1.19365e11 0.206998
\(16\) 1.09951e12 1.00000
\(17\) −2.83457e12 −1.40604 −0.703019 − 0.711171i \(-0.748165\pi\)
−0.703019 + 0.711171i \(0.748165\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −1.02400e13 −1.00000
\(21\) 3.45269e12 0.206998
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 9.53674e13 1.00000
\(26\) 0 0
\(27\) 8.34118e13 0.405126
\(28\) −2.96197e14 −1.00000
\(29\) 4.98982e14 1.18605 0.593027 − 0.805182i \(-0.297933\pi\)
0.593027 + 0.805182i \(0.297933\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 6.33592e14 0.413686
\(34\) 0 0
\(35\) 2.75855e15 1.00000
\(36\) −3.49950e15 −0.957152
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 1.12547e15 0.138257
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −5.43541e16 −1.99851
\(45\) 3.25916e16 0.957152
\(46\) 0 0
\(47\) −1.02681e17 −1.95215 −0.976074 − 0.217438i \(-0.930230\pi\)
−0.976074 + 0.217438i \(0.930230\pi\)
\(48\) −1.34393e16 −0.206998
\(49\) 7.97923e16 1.00000
\(50\) 0 0
\(51\) 3.46469e16 0.291047
\(52\) −9.65507e16 −0.667916
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 5.06212e17 1.99851
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 1.25164e17 0.206998
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 9.42728e17 0.957152
\(64\) 1.15292e18 1.00000
\(65\) 8.99199e17 0.667916
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −2.97226e18 −1.40604
\(69\) 0 0
\(70\) 0 0
\(71\) −6.44601e18 −1.98019 −0.990097 − 0.140384i \(-0.955166\pi\)
−0.990097 + 0.140384i \(0.955166\pi\)
\(72\) 0 0
\(73\) −6.04212e18 −1.40592 −0.702960 − 0.711230i \(-0.748139\pi\)
−0.702960 + 0.711230i \(0.748139\pi\)
\(74\) 0 0
\(75\) −1.16568e18 −0.206998
\(76\) 0 0
\(77\) 1.46424e19 1.99851
\(78\) 0 0
\(79\) 8.15334e18 0.861122 0.430561 − 0.902562i \(-0.358316\pi\)
0.430561 + 0.902562i \(0.358316\pi\)
\(80\) −1.07374e19 −1.00000
\(81\) 1.06172e19 0.873292
\(82\) 0 0
\(83\) 2.07767e19 1.33905 0.669524 − 0.742791i \(-0.266498\pi\)
0.669524 + 0.742791i \(0.266498\pi\)
\(84\) 3.62041e18 0.206998
\(85\) 2.76813e19 1.40604
\(86\) 0 0
\(87\) −6.09905e18 −0.245510
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 2.60097e19 0.667916
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.15875e20 −1.57135 −0.785675 − 0.618639i \(-0.787684\pi\)
−0.785675 + 0.618639i \(0.787684\pi\)
\(98\) 0 0
\(99\) 1.72997e20 1.91287
\(100\) 1.00000e20 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 2.32285e20 1.72842 0.864210 − 0.503130i \(-0.167818\pi\)
0.864210 + 0.503130i \(0.167818\pi\)
\(104\) 0 0
\(105\) −3.37177e19 −0.206998
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 8.74636e19 0.405126
\(109\) 4.78142e19 0.201972 0.100986 − 0.994888i \(-0.467800\pi\)
0.100986 + 0.994888i \(0.467800\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.10585e20 −1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.23220e20 1.18605
\(117\) 3.07299e20 0.639298
\(118\) 0 0
\(119\) 8.00695e20 1.40604
\(120\) 0 0
\(121\) 2.01423e21 2.99402
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.31323e20 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 6.64370e20 0.413686
\(133\) 0 0
\(134\) 0 0
\(135\) −8.14568e20 −0.405126
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 2.89255e21 1.00000
\(141\) 1.25507e21 0.404090
\(142\) 0 0
\(143\) 4.77296e21 1.33483
\(144\) −3.66949e21 −0.957152
\(145\) −4.87287e21 −1.18605
\(146\) 0 0
\(147\) −9.75301e20 −0.206998
\(148\) 0 0
\(149\) 2.70925e21 0.502326 0.251163 − 0.967945i \(-0.419187\pi\)
0.251163 + 0.967945i \(0.419187\pi\)
\(150\) 0 0
\(151\) −1.12003e22 −1.81743 −0.908717 − 0.417413i \(-0.862937\pi\)
−0.908717 + 0.417413i \(0.862937\pi\)
\(152\) 0 0
\(153\) 9.46003e21 1.34579
\(154\) 0 0
\(155\) 0 0
\(156\) 1.18014e21 0.138257
\(157\) −5.43718e21 −0.597554 −0.298777 − 0.954323i \(-0.596579\pi\)
−0.298777 + 0.954323i \(0.596579\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) −6.18743e21 −0.413686
\(166\) 0 0
\(167\) −3.37165e22 −1.99838 −0.999189 − 0.0402558i \(-0.987183\pi\)
−0.999189 + 0.0402558i \(0.987183\pi\)
\(168\) 0 0
\(169\) −1.05266e22 −0.553888
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.83714e22 1.59789 0.798944 − 0.601405i \(-0.205392\pi\)
0.798944 + 0.601405i \(0.205392\pi\)
\(174\) 0 0
\(175\) −2.69389e22 −1.00000
\(176\) −5.69944e22 −1.99851
\(177\) 0 0
\(178\) 0 0
\(179\) −6.51993e22 −1.93069 −0.965346 − 0.260975i \(-0.915956\pi\)
−0.965346 + 0.260975i \(0.915956\pi\)
\(180\) 3.41748e22 0.957152
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.46933e23 2.80998
\(188\) −1.07669e23 −1.95215
\(189\) −2.35618e22 −0.405126
\(190\) 0 0
\(191\) −5.01206e22 −0.775679 −0.387840 − 0.921727i \(-0.626779\pi\)
−0.387840 + 0.921727i \(0.626779\pi\)
\(192\) −1.40922e22 −0.206998
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) −1.09909e22 −0.138257
\(196\) 8.36683e22 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.40950e23 −1.18605
\(204\) 3.63299e22 0.291047
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.01241e23 −0.667916
\(209\) 0 0
\(210\) 0 0
\(211\) 2.92989e23 1.67505 0.837523 − 0.546402i \(-0.184003\pi\)
0.837523 + 0.546402i \(0.184003\pi\)
\(212\) 0 0
\(213\) 7.87896e22 0.409895
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 7.38528e22 0.291022
\(220\) 5.30801e23 1.99851
\(221\) 2.61001e23 0.939117
\(222\) 0 0
\(223\) 1.09845e23 0.361185 0.180593 − 0.983558i \(-0.442198\pi\)
0.180593 + 0.983558i \(0.442198\pi\)
\(224\) 0 0
\(225\) −3.18278e23 −0.957152
\(226\) 0 0
\(227\) −4.21378e23 −1.15988 −0.579941 − 0.814659i \(-0.696924\pi\)
−0.579941 + 0.814659i \(0.696924\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) −1.78974e23 −0.413686
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 1.00275e24 1.95215
\(236\) 0 0
\(237\) −9.96583e22 −0.178250
\(238\) 0 0
\(239\) 1.13122e24 1.86024 0.930120 − 0.367257i \(-0.119703\pi\)
0.930120 + 0.367257i \(0.119703\pi\)
\(240\) 1.31243e23 0.206998
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −4.20613e23 −0.585895
\(244\) 0 0
\(245\) −7.79221e23 −1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −2.53954e23 −0.277180
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 9.88522e23 0.957152
\(253\) 0 0
\(254\) 0 0
\(255\) −3.38349e23 −0.291047
\(256\) 1.20893e24 1.00000
\(257\) 2.36769e24 1.88362 0.941809 − 0.336149i \(-0.109125\pi\)
0.941809 + 0.336149i \(0.109125\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 9.42878e23 0.667916
\(261\) −1.66529e24 −1.13523
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −3.11664e24 −1.40604
\(273\) −3.17917e23 −0.138257
\(274\) 0 0
\(275\) −4.94347e24 −1.99851
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.63935e24 −1.83724 −0.918620 − 0.395142i \(-0.870695\pi\)
−0.918620 + 0.395142i \(0.870695\pi\)
\(282\) 0 0
\(283\) 2.34938e22 0.00713000 0.00356500 − 0.999994i \(-0.498865\pi\)
0.00356500 + 0.999994i \(0.498865\pi\)
\(284\) −6.75914e24 −1.98019
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3.97053e24 0.976946
\(290\) 0 0
\(291\) 1.41634e24 0.325266
\(292\) −6.33562e24 −1.40592
\(293\) 8.84013e24 1.89576 0.947878 − 0.318635i \(-0.103224\pi\)
0.947878 + 0.318635i \(0.103224\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −4.32374e24 −0.809646
\(298\) 0 0
\(299\) 0 0
\(300\) −1.22230e24 −0.206998
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.12722e25 1.51573 0.757867 − 0.652409i \(-0.226242\pi\)
0.757867 + 0.652409i \(0.226242\pi\)
\(308\) 1.53537e25 1.99851
\(309\) −2.83922e24 −0.357779
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −1.61209e25 −1.78626 −0.893130 − 0.449798i \(-0.851496\pi\)
−0.893130 + 0.449798i \(0.851496\pi\)
\(314\) 0 0
\(315\) −9.20633e24 −0.957152
\(316\) 8.54940e24 0.861122
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) −2.58653e25 −2.37034
\(320\) −1.12590e25 −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.11329e25 0.873292
\(325\) −8.78124e24 −0.667916
\(326\) 0 0
\(327\) −5.84433e23 −0.0418078
\(328\) 0 0
\(329\) 2.90049e25 1.95215
\(330\) 0 0
\(331\) −8.69407e24 −0.550736 −0.275368 − 0.961339i \(-0.588800\pi\)
−0.275368 + 0.961339i \(0.588800\pi\)
\(332\) 2.17860e25 1.33905
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 3.79628e24 0.206998
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 2.90260e25 1.40604
\(341\) 0 0
\(342\) 0 0
\(343\) −2.25393e25 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) −6.39532e24 −0.245510
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) −7.68039e24 −0.270590
\(352\) 0 0
\(353\) −2.40055e25 −0.799033 −0.399516 − 0.916726i \(-0.630822\pi\)
−0.399516 + 0.916726i \(0.630822\pi\)
\(354\) 0 0
\(355\) 6.29494e25 1.98019
\(356\) 0 0
\(357\) −9.78689e24 −0.291047
\(358\) 0 0
\(359\) 7.09538e25 1.99541 0.997703 − 0.0677366i \(-0.0215778\pi\)
0.997703 + 0.0677366i \(0.0215778\pi\)
\(360\) 0 0
\(361\) 3.75900e25 1.00000
\(362\) 0 0
\(363\) −2.46199e25 −0.619755
\(364\) 2.72732e25 0.667916
\(365\) 5.90050e25 1.40592
\(366\) 0 0
\(367\) 6.97436e25 1.57342 0.786709 − 0.617324i \(-0.211783\pi\)
0.786709 + 0.617324i \(0.211783\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 1.13836e25 0.206998
\(376\) 0 0
\(377\) −4.59452e25 −0.792186
\(378\) 0 0
\(379\) −7.06162e25 −1.15481 −0.577407 − 0.816456i \(-0.695936\pi\)
−0.577407 + 0.816456i \(0.695936\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6.49052e25 0.955635 0.477817 − 0.878459i \(-0.341428\pi\)
0.477817 + 0.878459i \(0.341428\pi\)
\(384\) 0 0
\(385\) −1.42992e26 −1.99851
\(386\) 0 0
\(387\) 0 0
\(388\) −1.21504e26 −1.57135
\(389\) −1.57329e26 −1.98295 −0.991476 − 0.130289i \(-0.958409\pi\)
−0.991476 + 0.130289i \(0.958409\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −7.96225e25 −0.861122
\(396\) 1.81400e26 1.91287
\(397\) 7.71114e25 0.792891 0.396445 − 0.918058i \(-0.370244\pi\)
0.396445 + 0.918058i \(0.370244\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.04858e26 1.00000
\(401\) −1.31714e26 −1.22514 −0.612572 − 0.790415i \(-0.709865\pi\)
−0.612572 + 0.790415i \(0.709865\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.03684e26 −0.873292
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.43569e26 1.72842
\(413\) 0 0
\(414\) 0 0
\(415\) −2.02898e26 −1.33905
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) −3.53556e25 −0.206998
\(421\) −3.10366e26 −1.77441 −0.887205 − 0.461376i \(-0.847356\pi\)
−0.887205 + 0.461376i \(0.847356\pi\)
\(422\) 0 0
\(423\) 3.42687e26 1.86850
\(424\) 0 0
\(425\) −2.70325e26 −1.40604
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −5.83399e25 −0.276308
\(430\) 0 0
\(431\) 5.49293e25 0.248331 0.124166 − 0.992262i \(-0.460375\pi\)
0.124166 + 0.992262i \(0.460375\pi\)
\(432\) 9.17122e25 0.405126
\(433\) −1.93136e26 −0.833653 −0.416826 − 0.908986i \(-0.636858\pi\)
−0.416826 + 0.908986i \(0.636858\pi\)
\(434\) 0 0
\(435\) 5.95611e25 0.245510
\(436\) 5.01368e25 0.201972
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −2.66297e26 −0.957152
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −3.31151e25 −0.103980
\(448\) −3.25672e26 −1.00000
\(449\) −2.79121e26 −0.838165 −0.419082 − 0.907948i \(-0.637648\pi\)
−0.419082 + 0.907948i \(0.637648\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.36901e26 0.376204
\(454\) 0 0
\(455\) −2.54001e26 −0.667916
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) −2.36436e26 −0.569623
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 5.48636e26 1.18605
\(465\) 0 0
\(466\) 0 0
\(467\) 8.40026e26 1.70264 0.851322 − 0.524643i \(-0.175801\pi\)
0.851322 + 0.524643i \(0.175801\pi\)
\(468\) 3.22227e26 0.639298
\(469\) 0 0
\(470\) 0 0
\(471\) 6.64586e25 0.123692
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 8.39589e26 1.40604
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.11207e27 2.99402
\(485\) 1.13159e27 1.57135
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7.50684e26 −0.921812 −0.460906 − 0.887449i \(-0.652475\pi\)
−0.460906 + 0.887449i \(0.652475\pi\)
\(492\) 0 0
\(493\) −1.41440e27 −1.66764
\(494\) 0 0
\(495\) −1.68942e27 −1.91287
\(496\) 0 0
\(497\) 1.82084e27 1.98019
\(498\) 0 0
\(499\) −5.96039e26 −0.622686 −0.311343 − 0.950298i \(-0.600779\pi\)
−0.311343 + 0.950298i \(0.600779\pi\)
\(500\) −9.76563e26 −1.00000
\(501\) 4.12117e26 0.413660
\(502\) 0 0
\(503\) −4.05147e26 −0.390781 −0.195390 − 0.980726i \(-0.562597\pi\)
−0.195390 + 0.980726i \(0.562597\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.28667e26 0.114653
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 1.70675e27 1.40592
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.26841e27 −1.72842
\(516\) 0 0
\(517\) 5.32260e27 3.90138
\(518\) 0 0
\(519\) −4.69014e26 −0.330759
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −3.06111e27 −1.99923 −0.999613 − 0.0278358i \(-0.991138\pi\)
−0.999613 + 0.0278358i \(0.991138\pi\)
\(524\) 0 0
\(525\) 3.29275e26 0.206998
\(526\) 0 0
\(527\) 0 0
\(528\) 6.96642e26 0.413686
\(529\) 1.71616e27 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 7.96931e26 0.399648
\(538\) 0 0
\(539\) −4.13612e27 −1.99851
\(540\) −8.54137e26 −0.405126
\(541\) 1.07381e26 0.0499982 0.0249991 − 0.999687i \(-0.492042\pi\)
0.0249991 + 0.999687i \(0.492042\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.66935e26 −0.201972
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −2.30312e27 −0.861122
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 3.03305e27 1.00000
\(561\) −1.79596e27 −0.581658
\(562\) 0 0
\(563\) −5.49299e27 −1.71682 −0.858411 − 0.512963i \(-0.828548\pi\)
−0.858411 + 0.512963i \(0.828548\pi\)
\(564\) 1.31604e27 0.404090
\(565\) 0 0
\(566\) 0 0
\(567\) −2.99909e27 −0.873292
\(568\) 0 0
\(569\) −5.79217e27 −1.62824 −0.814121 − 0.580696i \(-0.802781\pi\)
−0.814121 + 0.580696i \(0.802781\pi\)
\(570\) 0 0
\(571\) −3.76114e27 −1.02084 −0.510421 − 0.859925i \(-0.670510\pi\)
−0.510421 + 0.859925i \(0.670510\pi\)
\(572\) 5.00481e27 1.33483
\(573\) 6.12624e26 0.160564
\(574\) 0 0
\(575\) 0 0
\(576\) −3.84774e27 −0.957152
\(577\) 7.92365e27 1.93717 0.968583 − 0.248689i \(-0.0799996\pi\)
0.968583 + 0.248689i \(0.0799996\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −5.10957e27 −1.18605
\(581\) −5.86891e27 −1.33905
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −3.00097e27 −0.639298
\(586\) 0 0
\(587\) 8.17115e27 1.68229 0.841147 − 0.540806i \(-0.181881\pi\)
0.841147 + 0.540806i \(0.181881\pi\)
\(588\) −1.02268e27 −0.206998
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7.55280e27 1.40463 0.702314 − 0.711868i \(-0.252150\pi\)
0.702314 + 0.711868i \(0.252150\pi\)
\(594\) 0 0
\(595\) −7.81928e27 −1.40604
\(596\) 2.84085e27 0.502326
\(597\) 0 0
\(598\) 0 0
\(599\) 1.08997e28 1.83292 0.916462 − 0.400122i \(-0.131032\pi\)
0.916462 + 0.400122i \(0.131032\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.17443e28 −1.81743
\(605\) −1.96702e28 −2.99402
\(606\) 0 0
\(607\) 7.21826e27 1.06303 0.531515 − 0.847049i \(-0.321623\pi\)
0.531515 + 0.847049i \(0.321623\pi\)
\(608\) 0 0
\(609\) 1.72283e27 0.245510
\(610\) 0 0
\(611\) 9.45468e27 1.30387
\(612\) 9.91956e27 1.34579
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 1.23747e27 0.138257
\(625\) 9.09495e27 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −5.70130e27 −0.597554
\(629\) 0 0
\(630\) 0 0
\(631\) 1.18334e28 1.18254 0.591269 − 0.806475i \(-0.298627\pi\)
0.591269 + 0.806475i \(0.298627\pi\)
\(632\) 0 0
\(633\) −3.58121e27 −0.346731
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −7.34711e27 −0.667916
\(638\) 0 0
\(639\) 2.15128e28 1.89535
\(640\) 0 0
\(641\) 1.29422e28 1.10517 0.552583 − 0.833458i \(-0.313642\pi\)
0.552583 + 0.833458i \(0.313642\pi\)
\(642\) 0 0
\(643\) 2.13379e28 1.76621 0.883104 − 0.469178i \(-0.155450\pi\)
0.883104 + 0.469178i \(0.155450\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.59166e28 −1.23825 −0.619124 − 0.785293i \(-0.712512\pi\)
−0.619124 + 0.785293i \(0.712512\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.01649e28 1.34568
\(658\) 0 0
\(659\) −2.08071e28 −1.34697 −0.673484 − 0.739202i \(-0.735203\pi\)
−0.673484 + 0.739202i \(0.735203\pi\)
\(660\) −6.48799e27 −0.413686
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) −3.19022e27 −0.194395
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −3.53543e28 −1.99838
\(669\) −1.34263e27 −0.0747645
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 7.95477e27 0.405126
\(676\) −1.10380e28 −0.553888
\(677\) 4.03233e28 1.99375 0.996873 − 0.0790220i \(-0.0251797\pi\)
0.996873 + 0.0790220i \(0.0251797\pi\)
\(678\) 0 0
\(679\) 3.27319e28 1.57135
\(680\) 0 0
\(681\) 5.15051e27 0.240093
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 4.02353e28 1.59789
\(693\) −4.88673e28 −1.91287
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −2.82475e28 −1.00000
\(701\) 3.60658e28 1.25868 0.629339 − 0.777131i \(-0.283326\pi\)
0.629339 + 0.777131i \(0.283326\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −5.97629e28 −1.99851
\(705\) −1.22566e28 −0.404090
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 5.97191e28 1.86059 0.930295 − 0.366812i \(-0.119551\pi\)
0.930295 + 0.366812i \(0.119551\pi\)
\(710\) 0 0
\(711\) −2.72108e28 −0.824225
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −4.66109e28 −1.33483
\(716\) −6.83664e28 −1.93069
\(717\) −1.38269e28 −0.385065
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 3.58349e28 0.957152
\(721\) −6.56149e28 −1.72842
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.75866e28 1.18605
\(726\) 0 0
\(727\) 3.17314e28 0.769388 0.384694 − 0.923044i \(-0.374307\pi\)
0.384694 + 0.923044i \(0.374307\pi\)
\(728\) 0 0
\(729\) −3.18787e28 −0.752013
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −2.11175e28 −0.471631 −0.235816 − 0.971798i \(-0.575776\pi\)
−0.235816 + 0.971798i \(0.575776\pi\)
\(734\) 0 0
\(735\) 9.52442e27 0.206998
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −5.78538e28 −1.19093 −0.595467 − 0.803380i \(-0.703033\pi\)
−0.595467 + 0.803380i \(0.703033\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −2.64575e28 −0.502326
\(746\) 0 0
\(747\) −6.93399e28 −1.28167
\(748\) 1.54070e29 2.80998
\(749\) 0 0
\(750\) 0 0
\(751\) 8.78255e27 0.153894 0.0769469 − 0.997035i \(-0.475483\pi\)
0.0769469 + 0.997035i \(0.475483\pi\)
\(752\) −1.12899e29 −1.95215
\(753\) 0 0
\(754\) 0 0
\(755\) 1.09378e29 1.81743
\(756\) −2.47063e28 −0.405126
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −1.35063e28 −0.201972
\(764\) −5.25552e28 −0.775679
\(765\) −9.23831e28 −1.34579
\(766\) 0 0
\(767\) 0 0
\(768\) −1.47767e28 −0.206998
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −2.89402e28 −0.389904
\(772\) 0 0
\(773\) −1.05649e29 −1.38698 −0.693490 − 0.720466i \(-0.743928\pi\)
−0.693490 + 0.720466i \(0.743928\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) −1.15248e28 −0.138257
\(781\) 3.34136e29 3.95743
\(782\) 0 0
\(783\) 4.16210e28 0.480501
\(784\) 8.77325e28 1.00000
\(785\) 5.30975e28 0.597554
\(786\) 0 0
\(787\) −1.76682e29 −1.93841 −0.969206 − 0.246251i \(-0.920801\pi\)
−0.969206 + 0.246251i \(0.920801\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.96283e29 1.89801 0.949006 − 0.315259i \(-0.102091\pi\)
0.949006 + 0.315259i \(0.102091\pi\)
\(798\) 0 0
\(799\) 2.91057e29 2.74480
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.13200e29 2.80974
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.98238e29 −1.65082 −0.825412 − 0.564531i \(-0.809057\pi\)
−0.825412 + 0.564531i \(0.809057\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −1.47797e29 −1.18605
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 3.80947e28 0.291047
\(817\) 0 0
\(818\) 0 0
\(819\) −8.68045e28 −0.639298
\(820\) 0 0
\(821\) −1.32825e29 −0.954662 −0.477331 − 0.878723i \(-0.658396\pi\)
−0.477331 + 0.878723i \(0.658396\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 6.04241e28 0.413686
\(826\) 0 0
\(827\) 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
\(832\) −1.06159e29 −0.667916
\(833\) −2.26176e29 −1.40604
\(834\) 0 0
\(835\) 3.29263e29 1.99838
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 7.19883e28 0.406726
\(842\) 0 0
\(843\) 6.89298e28 0.380304
\(844\) 3.07221e29 1.67505
\(845\) 1.02799e29 0.553888
\(846\) 0 0
\(847\) −5.68970e29 −2.99402
\(848\) 0 0
\(849\) −2.87165e26 −0.00147589
\(850\) 0 0
\(851\) 0 0
\(852\) 8.26169e28 0.409895
\(853\) 3.57091e29 1.75101 0.875505 − 0.483210i \(-0.160529\pi\)
0.875505 + 0.483210i \(0.160529\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.92416e29 −0.900393 −0.450197 − 0.892929i \(-0.648646\pi\)
−0.450197 + 0.892929i \(0.648646\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −3.74721e29 −1.59789
\(866\) 0 0
\(867\) −4.85318e28 −0.202225
\(868\) 0 0
\(869\) −4.22637e29 −1.72096
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 3.86720e29 1.50402
\(874\) 0 0
\(875\) 2.63076e29 1.00000
\(876\) 7.74403e28 0.291022
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) −1.08053e29 −0.392417
\(880\) 5.56586e29 1.99851
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 2.73679e29 0.939117
\(885\) 0 0
\(886\) 0 0
\(887\) −5.72394e29 −1.89871 −0.949355 − 0.314205i \(-0.898262\pi\)
−0.949355 + 0.314205i \(0.898262\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −5.50354e29 −1.74528
\(892\) 1.15180e29 0.361185
\(893\) 0 0
\(894\) 0 0
\(895\) 6.36712e29 1.93069
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −3.33738e29 −0.957152
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) −4.41847e29 −1.15988
\(909\) 0 0
\(910\) 0 0
\(911\) −5.13245e29 −1.30359 −0.651796 − 0.758395i \(-0.725984\pi\)
−0.651796 + 0.758395i \(0.725984\pi\)
\(912\) 0 0
\(913\) −1.07698e30 −2.67609
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −7.32720e29 −1.70523 −0.852615 − 0.522540i \(-0.824984\pi\)
−0.852615 + 0.522540i \(0.824984\pi\)
\(920\) 0 0
\(921\) −1.37779e29 −0.313753
\(922\) 0 0
\(923\) 5.93536e29 1.32260
\(924\) −1.87668e29 −0.413686
\(925\) 0 0
\(926\) 0 0
\(927\) −7.75225e29 −1.65436
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.43489e30 −2.80998
\(936\) 0 0
\(937\) 6.10530e29 1.17034 0.585168 − 0.810912i \(-0.301028\pi\)
0.585168 + 0.810912i \(0.301028\pi\)
\(938\) 0 0
\(939\) 1.97045e29 0.369752
\(940\) 1.05146e30 1.95215
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 2.30095e29 0.405126
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) −1.04499e29 −0.178250
\(949\) 5.56346e29 0.939037
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 4.89459e29 0.775679
\(956\) 1.18617e30 1.86024
\(957\) 3.16151e29 0.490654
\(958\) 0 0
\(959\) 0 0
\(960\) 1.37619e29 0.206998
\(961\) 6.71791e29 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −4.41045e29 −0.585895
\(973\) 0 0
\(974\) 0 0
\(975\) 1.07333e29 0.138257
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −8.17073e29 −1.00000
\(981\) −1.59574e29 −0.193318
\(982\) 0 0
\(983\) 1.64579e30 1.95362 0.976811 − 0.214104i \(-0.0686830\pi\)
0.976811 + 0.214104i \(0.0686830\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −3.54527e29 −0.404090
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.52128e30 1.66523 0.832613 − 0.553855i \(-0.186844\pi\)
0.832613 + 0.553855i \(0.186844\pi\)
\(992\) 0 0
\(993\) 1.06268e29 0.114001
\(994\) 0 0
\(995\) 0 0
\(996\) −2.66290e29 −0.277180
\(997\) −1.83592e30 −1.89192 −0.945961 − 0.324280i \(-0.894878\pi\)
−0.945961 + 0.324280i \(0.894878\pi\)
\(998\) 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 35.21.c.a.34.1 ✓ 1
5.4 even 2 35.21.c.b.34.1 yes 1
7.6 odd 2 35.21.c.b.34.1 yes 1
35.34 odd 2 CM 35.21.c.a.34.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.21.c.a.34.1 ✓ 1 1.1 even 1 trivial
35.21.c.a.34.1 ✓ 1 35.34 odd 2 CM
35.21.c.b.34.1 yes 1 5.4 even 2
35.21.c.b.34.1 yes 1 7.6 odd 2