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Properties

Label 35.19.c.b.34.1
Level $35$
Weight $19$
Character 35.34
Self dual yes
Analytic conductor $71.885$
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] = [35,19,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, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.34");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 19 \)
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: \(71.8851481984\)
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)
 
\(f(q)\) \(=\) \(q+39286.0 q^{3} +262144. q^{4} -1.95312e6 q^{5} +4.03536e7 q^{7} +1.15597e9 q^{9} +O(q^{10})\) \(q+39286.0 q^{3} +262144. q^{4} -1.95312e6 q^{5} +4.03536e7 q^{7} +1.15597e9 q^{9} +2.63751e9 q^{11} +1.02986e10 q^{12} -1.88226e10 q^{13} -7.67305e10 q^{15} +6.87195e10 q^{16} +5.41705e10 q^{17} -5.12000e11 q^{20} +1.58533e12 q^{21} +3.81470e12 q^{25} +3.01932e13 q^{27} +1.05785e13 q^{28} -1.46232e13 q^{29} +1.03617e14 q^{33} -7.88156e13 q^{35} +3.03030e14 q^{36} -7.39463e14 q^{39} +6.91407e14 q^{44} -2.25775e15 q^{45} -2.61033e14 q^{47} +2.69971e15 q^{48} +1.62841e15 q^{49} +2.12814e15 q^{51} -4.93422e15 q^{52} -5.15139e15 q^{55} -2.01144e16 q^{60} +4.66475e16 q^{63} +1.80144e16 q^{64} +3.67628e16 q^{65} +1.42005e16 q^{68} +1.15928e16 q^{71} -1.00715e17 q^{73} +1.49864e17 q^{75} +1.06433e17 q^{77} -1.26856e17 q^{79} -1.34218e17 q^{80} +7.38324e17 q^{81} -3.67225e17 q^{83} +4.15585e17 q^{84} -1.05802e17 q^{85} -5.74489e17 q^{87} -7.59558e17 q^{91} +1.51987e18 q^{97} +3.04888e18 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) 39286.0 1.99594 0.997968 − 0.0637204i \(-0.0202966\pi\)
0.997968 + 0.0637204i \(0.0202966\pi\)
\(4\) 262144. 1.00000
\(5\) −1.95312e6 −1.00000
\(6\) 0 0
\(7\) 4.03536e7 1.00000
\(8\) 0 0
\(9\) 1.15597e9 2.98376
\(10\) 0 0
\(11\) 2.63751e9 1.11856 0.559281 − 0.828978i \(-0.311077\pi\)
0.559281 + 0.828978i \(0.311077\pi\)
\(12\) 1.02986e10 1.99594
\(13\) −1.88226e10 −1.77496 −0.887480 − 0.460846i \(-0.847546\pi\)
−0.887480 + 0.460846i \(0.847546\pi\)
\(14\) 0 0
\(15\) −7.67305e10 −1.99594
\(16\) 6.87195e10 1.00000
\(17\) 5.41705e10 0.456796 0.228398 − 0.973568i \(-0.426651\pi\)
0.228398 + 0.973568i \(0.426651\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −5.12000e11 −1.00000
\(21\) 1.58533e12 1.99594
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 3.81470e12 1.00000
\(26\) 0 0
\(27\) 3.01932e13 3.95945
\(28\) 1.05785e13 1.00000
\(29\) −1.46232e13 −1.00800 −0.504001 − 0.863703i \(-0.668139\pi\)
−0.504001 + 0.863703i \(0.668139\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 1.03617e14 2.23258
\(34\) 0 0
\(35\) −7.88156e13 −1.00000
\(36\) 3.03030e14 2.98376
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −7.39463e14 −3.54271
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 6.91407e14 1.11856
\(45\) −2.25775e15 −2.98376
\(46\) 0 0
\(47\) −2.61033e14 −0.233246 −0.116623 − 0.993176i \(-0.537207\pi\)
−0.116623 + 0.993176i \(0.537207\pi\)
\(48\) 2.69971e15 1.99594
\(49\) 1.62841e15 1.00000
\(50\) 0 0
\(51\) 2.12814e15 0.911736
\(52\) −4.93422e15 −1.77496
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −5.15139e15 −1.11856
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) −2.01144e16 −1.99594
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 4.66475e16 2.98376
\(64\) 1.80144e16 1.00000
\(65\) 3.67628e16 1.77496
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 1.42005e16 0.456796
\(69\) 0 0
\(70\) 0 0
\(71\) 1.15928e16 0.252851 0.126426 − 0.991976i \(-0.459650\pi\)
0.126426 + 0.991976i \(0.459650\pi\)
\(72\) 0 0
\(73\) −1.00715e17 −1.71076 −0.855378 − 0.518004i \(-0.826675\pi\)
−0.855378 + 0.518004i \(0.826675\pi\)
\(74\) 0 0
\(75\) 1.49864e17 1.99594
\(76\) 0 0
\(77\) 1.06433e17 1.11856
\(78\) 0 0
\(79\) −1.26856e17 −1.05844 −0.529221 − 0.848484i \(-0.677516\pi\)
−0.529221 + 0.848484i \(0.677516\pi\)
\(80\) −1.34218e17 −1.00000
\(81\) 7.38324e17 4.91906
\(82\) 0 0
\(83\) −3.67225e17 −1.96440 −0.982198 − 0.187849i \(-0.939848\pi\)
−0.982198 + 0.187849i \(0.939848\pi\)
\(84\) 4.15585e17 1.99594
\(85\) −1.05802e17 −0.456796
\(86\) 0 0
\(87\) −5.74489e17 −2.01191
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −7.59558e17 −1.77496
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.51987e18 1.99922 0.999608 − 0.0280065i \(-0.00891590\pi\)
0.999608 + 0.0280065i \(0.00891590\pi\)
\(98\) 0 0
\(99\) 3.04888e18 3.33752
\(100\) 1.00000e18 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 1.83921e18 1.40960 0.704800 − 0.709406i \(-0.251037\pi\)
0.704800 + 0.709406i \(0.251037\pi\)
\(104\) 0 0
\(105\) −3.09635e18 −1.99594
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 7.91497e18 3.95945
\(109\) 3.67502e18 1.69208 0.846041 − 0.533118i \(-0.178980\pi\)
0.846041 + 0.533118i \(0.178980\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.77308e18 1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.83339e18 −1.00800
\(117\) −2.17583e19 −5.29605
\(118\) 0 0
\(119\) 2.18598e18 0.456796
\(120\) 0 0
\(121\) 1.39654e18 0.251180
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −7.45058e18 −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 \(\pi\)
\(132\) 2.71626e19 2.23258
\(133\) 0 0
\(134\) 0 0
\(135\) −5.89711e19 −3.95945
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −2.06610e19 −1.00000
\(141\) −1.02549e19 −0.465544
\(142\) 0 0
\(143\) −4.96447e19 −1.98540
\(144\) 7.94376e19 2.98376
\(145\) 2.85610e19 1.00800
\(146\) 0 0
\(147\) 6.39739e19 1.99594
\(148\) 0 0
\(149\) 1.74014e19 0.480737 0.240368 − 0.970682i \(-0.422732\pi\)
0.240368 + 0.970682i \(0.422732\pi\)
\(150\) 0 0
\(151\) −3.08468e19 −0.755818 −0.377909 − 0.925843i \(-0.623357\pi\)
−0.377909 + 0.925843i \(0.623357\pi\)
\(152\) 0 0
\(153\) 6.26195e19 1.36297
\(154\) 0 0
\(155\) 0 0
\(156\) −1.93846e20 −3.54271
\(157\) −8.52595e19 −1.47111 −0.735556 − 0.677463i \(-0.763079\pi\)
−0.735556 + 0.677463i \(0.763079\pi\)
\(158\) 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
\(165\) −2.02377e20 −2.23258
\(166\) 0 0
\(167\) −1.59059e20 −1.57438 −0.787189 − 0.616712i \(-0.788464\pi\)
−0.787189 + 0.616712i \(0.788464\pi\)
\(168\) 0 0
\(169\) 2.41833e20 2.15048
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.32056e19 −0.0951358 −0.0475679 − 0.998868i \(-0.515147\pi\)
−0.0475679 + 0.998868i \(0.515147\pi\)
\(174\) 0 0
\(175\) 1.53937e20 1.00000
\(176\) 1.81248e20 1.11856
\(177\) 0 0
\(178\) 0 0
\(179\) −1.43692e20 −0.761652 −0.380826 − 0.924647i \(-0.624360\pi\)
−0.380826 + 0.924647i \(0.624360\pi\)
\(180\) −5.91856e20 −2.98376
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.42875e20 0.510955
\(188\) −6.84281e19 −0.233246
\(189\) 1.21840e21 3.95945
\(190\) 0 0
\(191\) −4.99144e20 −1.47545 −0.737727 − 0.675099i \(-0.764101\pi\)
−0.737727 + 0.675099i \(0.764101\pi\)
\(192\) 7.07714e20 1.99594
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 1.44426e21 3.54271
\(196\) 4.26879e20 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
\(203\) −5.90100e20 −1.00800
\(204\) 5.57880e20 0.911736
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.29348e21 −1.77496
\(209\) 0 0
\(210\) 0 0
\(211\) 1.22838e21 1.48181 0.740904 − 0.671611i \(-0.234397\pi\)
0.740904 + 0.671611i \(0.234397\pi\)
\(212\) 0 0
\(213\) 4.55436e20 0.504674
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3.95669e21 −3.41456
\(220\) −1.35041e21 −1.11856
\(221\) −1.01963e21 −0.810795
\(222\) 0 0
\(223\) −1.61720e21 −1.18582 −0.592911 − 0.805268i \(-0.702021\pi\)
−0.592911 + 0.805268i \(0.702021\pi\)
\(224\) 0 0
\(225\) 4.40967e21 2.98376
\(226\) 0 0
\(227\) −2.73502e20 −0.170895 −0.0854473 − 0.996343i \(-0.527232\pi\)
−0.0854473 + 0.996343i \(0.527232\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 4.18133e21 2.23258
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 5.09829e20 0.233246
\(236\) 0 0
\(237\) −4.98366e21 −2.11258
\(238\) 0 0
\(239\) −2.89021e21 −1.13592 −0.567960 − 0.823056i \(-0.692267\pi\)
−0.567960 + 0.823056i \(0.692267\pi\)
\(240\) −5.27288e21 −1.99594
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 1.73083e22 5.85867
\(244\) 0 0
\(245\) −3.18050e21 −1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −1.44268e22 −3.92081
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.22284e22 2.98376
\(253\) 0 0
\(254\) 0 0
\(255\) −4.15653e21 −0.911736
\(256\) 4.72237e21 1.00000
\(257\) −7.94598e21 −1.62461 −0.812306 − 0.583232i \(-0.801788\pi\)
−0.812306 + 0.583232i \(0.801788\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 9.63715e21 1.77496
\(261\) −1.69040e22 −3.00764
\(262\) 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 \(\pi\)
\(272\) 3.72257e21 0.456796
\(273\) −2.98400e22 −3.54271
\(274\) 0 0
\(275\) 1.00613e22 1.11856
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.18179e22 −1.99735 −0.998677 − 0.0514180i \(-0.983626\pi\)
−0.998677 + 0.0514180i \(0.983626\pi\)
\(282\) 0 0
\(283\) −3.56903e21 −0.306529 −0.153265 − 0.988185i \(-0.548979\pi\)
−0.153265 + 0.988185i \(0.548979\pi\)
\(284\) 3.03899e21 0.252851
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.11286e22 −0.791337
\(290\) 0 0
\(291\) 5.97094e22 3.99031
\(292\) −2.64018e22 −1.71076
\(293\) −9.15862e21 −0.575468 −0.287734 − 0.957710i \(-0.592902\pi\)
−0.287734 + 0.957710i \(0.592902\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 7.96349e22 4.42889
\(298\) 0 0
\(299\) 0 0
\(300\) 3.92860e22 1.99594
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4.59036e22 −1.89497 −0.947483 − 0.319805i \(-0.896383\pi\)
−0.947483 + 0.319805i \(0.896383\pi\)
\(308\) 2.79008e22 1.11856
\(309\) 7.22552e22 2.81347
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −6.08556e21 −0.211058 −0.105529 − 0.994416i \(-0.533654\pi\)
−0.105529 + 0.994416i \(0.533654\pi\)
\(314\) 0 0
\(315\) −9.11085e22 −2.98376
\(316\) −3.32545e22 −1.05844
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) −3.85689e22 −1.12751
\(320\) −3.51844e22 −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.93547e23 4.91906
\(325\) −7.18024e22 −1.77496
\(326\) 0 0
\(327\) 1.44377e23 3.37729
\(328\) 0 0
\(329\) −1.05336e22 −0.233246
\(330\) 0 0
\(331\) 6.30614e22 1.32225 0.661123 − 0.750278i \(-0.270080\pi\)
0.661123 + 0.750278i \(0.270080\pi\)
\(332\) −9.62657e22 −1.96440
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 1.08943e23 1.99594
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −2.77353e22 −0.456796
\(341\) 0 0
\(342\) 0 0
\(343\) 6.57124e22 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) −1.50599e23 −2.01191
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) −5.68314e23 −7.02787
\(352\) 0 0
\(353\) −1.66377e23 −1.95488 −0.977441 − 0.211208i \(-0.932260\pi\)
−0.977441 + 0.211208i \(0.932260\pi\)
\(354\) 0 0
\(355\) −2.26423e22 −0.252851
\(356\) 0 0
\(357\) 8.58782e22 0.911736
\(358\) 0 0
\(359\) 1.67066e23 1.68670 0.843350 − 0.537364i \(-0.180580\pi\)
0.843350 + 0.537364i \(0.180580\pi\)
\(360\) 0 0
\(361\) 1.04127e23 1.00000
\(362\) 0 0
\(363\) 5.48646e22 0.501340
\(364\) −1.99114e23 −1.77496
\(365\) 1.96709e23 1.71076
\(366\) 0 0
\(367\) 2.27434e23 1.88305 0.941523 − 0.336949i \(-0.109395\pi\)
0.941523 + 0.336949i \(0.109395\pi\)
\(368\) 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
\(375\) −2.92704e23 −1.99594
\(376\) 0 0
\(377\) 2.75247e23 1.78916
\(378\) 0 0
\(379\) −7.40065e22 −0.458688 −0.229344 − 0.973345i \(-0.573658\pi\)
−0.229344 + 0.973345i \(0.573658\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.17288e23 −0.661402 −0.330701 − 0.943736i \(-0.607285\pi\)
−0.330701 + 0.943736i \(0.607285\pi\)
\(384\) 0 0
\(385\) −2.07877e23 −1.11856
\(386\) 0 0
\(387\) 0 0
\(388\) 3.98424e23 1.99922
\(389\) −3.70488e23 −1.81647 −0.908233 − 0.418464i \(-0.862569\pi\)
−0.908233 + 0.418464i \(0.862569\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.47766e23 1.05844
\(396\) 7.99246e23 3.33752
\(397\) −4.87276e23 −1.98912 −0.994559 − 0.104172i \(-0.966781\pi\)
−0.994559 + 0.104172i \(0.966781\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 2.62144e23 1.00000
\(401\) −3.92166e23 −1.46275 −0.731376 − 0.681974i \(-0.761122\pi\)
−0.731376 + 0.681974i \(0.761122\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.44204e24 −4.91906
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.82138e23 1.40960
\(413\) 0 0
\(414\) 0 0
\(415\) 7.17236e23 1.96440
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) −8.11690e23 −1.99594
\(421\) −1.62416e23 −0.390922 −0.195461 − 0.980712i \(-0.562620\pi\)
−0.195461 + 0.980712i \(0.562620\pi\)
\(422\) 0 0
\(423\) −3.01746e23 −0.695949
\(424\) 0 0
\(425\) 2.06644e23 0.456796
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.95034e24 −3.96274
\(430\) 0 0
\(431\) −2.72906e23 −0.531762 −0.265881 − 0.964006i \(-0.585663\pi\)
−0.265881 + 0.964006i \(0.585663\pi\)
\(432\) 2.07486e24 3.95945
\(433\) 9.00837e22 0.168366 0.0841832 − 0.996450i \(-0.473172\pi\)
0.0841832 + 0.996450i \(0.473172\pi\)
\(434\) 0 0
\(435\) 1.12205e24 2.01191
\(436\) 9.63385e23 1.69208
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.88240e24 2.98376
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.83630e23 0.959519
\(448\) 7.26946e23 1.00000
\(449\) 1.12457e24 1.51625 0.758123 − 0.652112i \(-0.226117\pi\)
0.758123 + 0.652112i \(0.226117\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −1.21185e24 −1.50856
\(454\) 0 0
\(455\) 1.48351e24 1.77496
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 1.63558e24 1.80866
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −1.00490e24 −1.00800
\(465\) 0 0
\(466\) 0 0
\(467\) 2.07767e24 1.96664 0.983319 − 0.181889i \(-0.0582210\pi\)
0.983319 + 0.181889i \(0.0582210\pi\)
\(468\) −5.70381e24 −5.29605
\(469\) 0 0
\(470\) 0 0
\(471\) −3.34951e24 −2.93625
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 5.73040e23 0.456796
\(477\) 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
\(484\) 3.66095e23 0.251180
\(485\) −2.96849e24 −1.99922
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.75967e24 1.66389 0.831944 − 0.554859i \(-0.187228\pi\)
0.831944 + 0.554859i \(0.187228\pi\)
\(492\) 0 0
\(493\) −7.92149e23 −0.460452
\(494\) 0 0
\(495\) −5.95485e24 −3.33752
\(496\) 0 0
\(497\) 4.67813e23 0.252851
\(498\) 0 0
\(499\) 1.84084e24 0.959646 0.479823 − 0.877365i \(-0.340701\pi\)
0.479823 + 0.877365i \(0.340701\pi\)
\(500\) −1.95313e24 −1.00000
\(501\) −6.24878e24 −3.14236
\(502\) 0 0
\(503\) 3.28613e24 1.59431 0.797154 − 0.603775i \(-0.206338\pi\)
0.797154 + 0.603775i \(0.206338\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.50067e24 4.29223
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −4.06421e24 −1.71076
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.59220e24 −1.40960
\(516\) 0 0
\(517\) −6.88476e23 −0.260900
\(518\) 0 0
\(519\) −5.18796e23 −0.189885
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 1.66929e24 0.570187 0.285093 − 0.958500i \(-0.407975\pi\)
0.285093 + 0.958500i \(0.407975\pi\)
\(524\) 0 0
\(525\) 6.04756e24 1.99594
\(526\) 0 0
\(527\) 0 0
\(528\) 7.12052e24 2.23258
\(529\) 3.24415e24 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −5.64510e24 −1.52021
\(538\) 0 0
\(539\) 4.29496e24 1.11856
\(540\) −1.54589e25 −3.95945
\(541\) 5.73913e24 1.44567 0.722837 − 0.691018i \(-0.242838\pi\)
0.722837 + 0.691018i \(0.242838\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −7.17778e24 −1.69208
\(546\) 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
\(553\) −5.11910e24 −1.05844
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −5.41617e24 −1.00000
\(561\) 5.61300e24 1.01983
\(562\) 0 0
\(563\) −1.12490e25 −1.97943 −0.989715 − 0.143056i \(-0.954307\pi\)
−0.989715 + 0.143056i \(0.954307\pi\)
\(564\) −2.68827e24 −0.465544
\(565\) 0 0
\(566\) 0 0
\(567\) 2.97940e25 4.91906
\(568\) 0 0
\(569\) 8.04569e24 1.28692 0.643462 − 0.765478i \(-0.277498\pi\)
0.643462 + 0.765478i \(0.277498\pi\)
\(570\) 0 0
\(571\) −1.03572e25 −1.60516 −0.802582 − 0.596542i \(-0.796541\pi\)
−0.802582 + 0.596542i \(0.796541\pi\)
\(572\) −1.30141e25 −1.98540
\(573\) −1.96094e25 −2.94491
\(574\) 0 0
\(575\) 0 0
\(576\) 2.08241e25 2.98376
\(577\) 3.17968e24 0.448540 0.224270 − 0.974527i \(-0.428000\pi\)
0.224270 + 0.974527i \(0.428000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 7.48710e24 1.00800
\(581\) −1.48188e25 −1.96440
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 4.24967e25 5.29605
\(586\) 0 0
\(587\) −1.62188e25 −1.96009 −0.980047 − 0.198764i \(-0.936307\pi\)
−0.980047 + 0.198764i \(0.936307\pi\)
\(588\) 1.67704e25 1.99594
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.76595e25 1.94754 0.973771 − 0.227529i \(-0.0730646\pi\)
0.973771 + 0.227529i \(0.0730646\pi\)
\(594\) 0 0
\(595\) −4.26948e24 −0.456796
\(596\) 4.56167e24 0.480737
\(597\) 0 0
\(598\) 0 0
\(599\) 1.25563e25 1.26479 0.632397 − 0.774644i \(-0.282071\pi\)
0.632397 + 0.774644i \(0.282071\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −8.08630e24 −0.755818
\(605\) −2.72762e24 −0.251180
\(606\) 0 0
\(607\) 6.20551e24 0.554727 0.277363 − 0.960765i \(-0.410539\pi\)
0.277363 + 0.960765i \(0.410539\pi\)
\(608\) 0 0
\(609\) −2.31827e25 −2.01191
\(610\) 0 0
\(611\) 4.91330e24 0.414002
\(612\) 1.64153e25 1.36297
\(613\) 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
\(624\) −5.08155e25 −3.54271
\(625\) 1.45519e25 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −2.23503e25 −1.47111
\(629\) 0 0
\(630\) 0 0
\(631\) 3.09799e25 1.95351 0.976756 − 0.214354i \(-0.0687645\pi\)
0.976756 + 0.214354i \(0.0687645\pi\)
\(632\) 0 0
\(633\) 4.82583e25 2.95759
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.06509e25 −1.77496
\(638\) 0 0
\(639\) 1.34010e25 0.754447
\(640\) 0 0
\(641\) −3.53248e25 −1.93356 −0.966779 − 0.255616i \(-0.917722\pi\)
−0.966779 + 0.255616i \(0.917722\pi\)
\(642\) 0 0
\(643\) −2.74006e25 −1.45835 −0.729173 − 0.684329i \(-0.760095\pi\)
−0.729173 + 0.684329i \(0.760095\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.90600e25 1.96604 0.983022 − 0.183488i \(-0.0587389\pi\)
0.983022 + 0.183488i \(0.0587389\pi\)
\(648\) 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
\(657\) −1.16423e26 −5.10448
\(658\) 0 0
\(659\) −2.28020e25 −0.972758 −0.486379 − 0.873748i \(-0.661683\pi\)
−0.486379 + 0.873748i \(0.661683\pi\)
\(660\) −5.30520e25 −2.23258
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) −4.00571e25 −1.61830
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −4.16963e25 −1.57438
\(669\) −6.35332e25 −2.36682
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 1.15178e26 3.95945
\(676\) 6.33952e25 2.15048
\(677\) 5.53670e25 1.85333 0.926666 − 0.375887i \(-0.122662\pi\)
0.926666 + 0.375887i \(0.122662\pi\)
\(678\) 0 0
\(679\) 6.13321e25 1.99922
\(680\) 0 0
\(681\) −1.07448e25 −0.341095
\(682\) 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 \(\pi\)
\(692\) −3.46177e24 −0.0951358
\(693\) 1.23033e26 3.33752
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 4.03536e25 1.00000
\(701\) −7.34137e25 −1.79604 −0.898018 − 0.439959i \(-0.854993\pi\)
−0.898018 + 0.439959i \(0.854993\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.75132e25 1.11856
\(705\) 2.00292e25 0.465544
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −8.54177e25 −1.88682 −0.943412 − 0.331622i \(-0.892404\pi\)
−0.943412 + 0.331622i \(0.892404\pi\)
\(710\) 0 0
\(711\) −1.46642e26 −3.15814
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 9.69623e25 1.98540
\(716\) −3.76681e25 −0.761652
\(717\) −1.13545e26 −2.26722
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −1.55152e26 −2.98376
\(721\) 7.42187e25 1.40960
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5.57832e25 −1.00800
\(726\) 0 0
\(727\) 2.23528e25 0.394024 0.197012 − 0.980401i \(-0.436876\pi\)
0.197012 + 0.980401i \(0.436876\pi\)
\(728\) 0 0
\(729\) 3.93933e26 6.77447
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.02780e26 1.68256 0.841282 − 0.540597i \(-0.181801\pi\)
0.841282 + 0.540597i \(0.181801\pi\)
\(734\) 0 0
\(735\) −1.24949e26 −1.99594
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.13732e26 1.73015 0.865075 − 0.501642i \(-0.167270\pi\)
0.865075 + 0.501642i \(0.167270\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −3.39871e25 −0.480737
\(746\) 0 0
\(747\) −4.24500e26 −5.86128
\(748\) 3.74539e25 0.510955
\(749\) 0 0
\(750\) 0 0
\(751\) 5.94526e25 0.782369 0.391184 − 0.920312i \(-0.372065\pi\)
0.391184 + 0.920312i \(0.372065\pi\)
\(752\) −1.79380e25 −0.233246
\(753\) 0 0
\(754\) 0 0
\(755\) 6.02476e25 0.755818
\(756\) 3.19398e26 3.95945
\(757\) 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
\(763\) 1.48300e26 1.69208
\(764\) −1.30848e26 −1.47545
\(765\) −1.22304e26 −1.36297
\(766\) 0 0
\(767\) 0 0
\(768\) 1.85523e26 1.99594
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −3.12166e26 −3.24262
\(772\) 0 0
\(773\) 1.47642e26 1.49829 0.749144 − 0.662407i \(-0.230465\pi\)
0.749144 + 0.662407i \(0.230465\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 3.78605e26 3.54271
\(781\) 3.05762e25 0.282830
\(782\) 0 0
\(783\) −4.41523e26 −3.99114
\(784\) 1.11904e26 1.00000
\(785\) 1.66522e26 1.47111
\(786\) 0 0
\(787\) 1.18712e26 1.02499 0.512496 − 0.858690i \(-0.328721\pi\)
0.512496 + 0.858690i \(0.328721\pi\)
\(788\) 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
\(797\) 2.13770e26 1.64748 0.823742 − 0.566964i \(-0.191882\pi\)
0.823742 + 0.566964i \(0.191882\pi\)
\(798\) 0 0
\(799\) −1.41403e25 −0.106546
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.65637e26 −1.91359
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.52601e26 −1.02806 −0.514032 − 0.857771i \(-0.671849\pi\)
−0.514032 + 0.857771i \(0.671849\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −1.54691e26 −1.00800
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 1.46245e26 0.911736
\(817\) 0 0
\(818\) 0 0
\(819\) −8.78026e26 −5.29605
\(820\) 0 0
\(821\) −9.71710e25 −0.573387 −0.286694 − 0.958022i \(-0.592556\pi\)
−0.286694 + 0.958022i \(0.592556\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 3.95268e26 2.23258
\(826\) 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
\(832\) −3.39077e26 −1.77496
\(833\) 8.82120e25 0.456796
\(834\) 0 0
\(835\) 3.10661e26 1.57438
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 3.38186e24 0.0160691
\(842\) 0 0
\(843\) −8.57136e26 −3.98659
\(844\) 3.22013e26 1.48181
\(845\) −4.72331e26 −2.15048
\(846\) 0 0
\(847\) 5.63555e25 0.251180
\(848\) 0 0
\(849\) −1.40213e26 −0.611813
\(850\) 0 0
\(851\) 0 0
\(852\) 1.19390e26 0.504674
\(853\) 2.09653e26 0.876921 0.438461 − 0.898750i \(-0.355524\pi\)
0.438461 + 0.898750i \(0.355524\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.13309e26 1.25645 0.628226 − 0.778031i \(-0.283781\pi\)
0.628226 + 0.778031i \(0.283781\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 2.57922e25 0.0951358
\(866\) 0 0
\(867\) −4.37200e26 −1.57946
\(868\) 0 0
\(869\) −3.34584e26 −1.18393
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.75692e27 5.96518
\(874\) 0 0
\(875\) −3.00658e26 −1.00000
\(876\) −1.03722e27 −3.41456
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) −3.59805e26 −1.14860
\(880\) −3.54001e26 −1.11856
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −2.67289e26 −0.810795
\(885\) 0 0
\(886\) 0 0
\(887\) 3.84828e26 1.13228 0.566141 − 0.824309i \(-0.308436\pi\)
0.566141 + 0.824309i \(0.308436\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.94734e27 5.50227
\(892\) −4.23939e26 −1.18582
\(893\) 0 0
\(894\) 0 0
\(895\) 2.80649e26 0.761652
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 1.15597e27 2.98376
\(901\) 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 \(\pi\)
\(908\) −7.16970e25 −0.170895
\(909\) 0 0
\(910\) 0 0
\(911\) 7.74282e26 1.79157 0.895785 − 0.444488i \(-0.146614\pi\)
0.895785 + 0.444488i \(0.146614\pi\)
\(912\) 0 0
\(913\) −9.68559e26 −2.19730
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 8.42991e26 1.80295 0.901474 − 0.432833i \(-0.142486\pi\)
0.901474 + 0.432833i \(0.142486\pi\)
\(920\) 0 0
\(921\) −1.80337e27 −3.78223
\(922\) 0 0
\(923\) −2.18207e26 −0.448801
\(924\) 1.09611e27 2.23258
\(925\) 0 0
\(926\) 0 0
\(927\) 2.12607e27 4.20591
\(928\) 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
\(935\) −2.79053e26 −0.510955
\(936\) 0 0
\(937\) −4.39247e26 −0.788955 −0.394477 − 0.918906i \(-0.629074\pi\)
−0.394477 + 0.918906i \(0.629074\pi\)
\(938\) 0 0
\(939\) −2.39077e26 −0.421258
\(940\) 1.33649e26 0.233246
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −2.37970e27 −3.95945
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) −1.30644e27 −2.11258
\(949\) 1.89571e27 3.03652
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 9.74891e26 1.47545
\(956\) −7.57651e26 −1.13592
\(957\) −1.51522e27 −2.25044
\(958\) 0 0
\(959\) 0 0
\(960\) −1.38225e27 −1.99594
\(961\) 6.99054e26 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 \(\pi\)
\(972\) 4.53728e27 5.85867
\(973\) 0 0
\(974\) 0 0
\(975\) −2.82083e27 −3.54271
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −8.33748e26 −1.00000
\(981\) 4.24821e27 5.04876
\(982\) 0 0
\(983\) −1.25612e27 −1.46572 −0.732859 − 0.680380i \(-0.761815\pi\)
−0.732859 + 0.680380i \(0.761815\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −4.13823e26 −0.465544
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −3.91782e26 −0.424993 −0.212496 − 0.977162i \(-0.568159\pi\)
−0.212496 + 0.977162i \(0.568159\pi\)
\(992\) 0 0
\(993\) 2.47743e27 2.63912
\(994\) 0 0
\(995\) 0 0
\(996\) −3.78190e27 −3.92081
\(997\) −1.17071e27 −1.20280 −0.601400 − 0.798948i \(-0.705390\pi\)
−0.601400 + 0.798948i \(0.705390\pi\)
\(998\) 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 35.19.c.b.34.1 yes 1
5.4 even 2 35.19.c.a.34.1 ✓ 1
7.6 odd 2 35.19.c.a.34.1 ✓ 1
35.34 odd 2 CM 35.19.c.b.34.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.19.c.a.34.1 ✓ 1 5.4 even 2
35.19.c.a.34.1 ✓ 1 7.6 odd 2
35.19.c.b.34.1 yes 1 1.1 even 1 trivial
35.19.c.b.34.1 yes 1 35.34 odd 2 CM