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Properties

Label 35.17.c.a.34.1
Level $35$
Weight $17$
Character 35.34
Self dual yes
Analytic conductor $56.814$
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] = [35,17,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, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.34");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 17 \)
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: \(56.8135903498\)
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)
 
\(f(q)\) \(=\) \(q-3007.00 q^{3} +65536.0 q^{4} -390625. q^{5} -5.76480e6 q^{7} -3.40047e7 q^{9} +O(q^{10})\) \(q-3007.00 q^{3} +65536.0 q^{4} -390625. q^{5} -5.76480e6 q^{7} -3.40047e7 q^{9} +1.45028e8 q^{11} -1.97067e8 q^{12} -1.57131e9 q^{13} +1.17461e9 q^{15} +4.29497e9 q^{16} +4.37239e9 q^{17} -2.56000e10 q^{20} +1.73348e10 q^{21} +1.52588e11 q^{25} +2.31694e11 q^{27} -3.77802e11 q^{28} -9.93242e11 q^{29} -4.36101e11 q^{33} +2.25188e12 q^{35} -2.22853e12 q^{36} +4.72492e12 q^{39} +9.50458e12 q^{44} +1.32831e13 q^{45} +4.28200e13 q^{47} -1.29150e13 q^{48} +3.32329e13 q^{49} -1.31478e13 q^{51} -1.02977e14 q^{52} -5.66517e13 q^{55} +7.69792e13 q^{60} +1.96030e14 q^{63} +2.81475e14 q^{64} +6.13791e14 q^{65} +2.86549e14 q^{68} +1.28332e15 q^{71} +4.91238e14 q^{73} -4.58832e14 q^{75} -8.36060e14 q^{77} +1.88480e15 q^{79} -1.67772e15 q^{80} +7.67087e14 q^{81} +1.56838e15 q^{83} +1.13605e15 q^{84} -1.70797e15 q^{85} +2.98668e15 q^{87} +9.05827e15 q^{91} -1.17531e16 q^{97} -4.93164e15 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) −3007.00 −0.458314 −0.229157 − 0.973389i \(-0.573597\pi\)
−0.229157 + 0.973389i \(0.573597\pi\)
\(4\) 65536.0 1.00000
\(5\) −390625. −1.00000
\(6\) 0 0
\(7\) −5.76480e6 −1.00000
\(8\) 0 0
\(9\) −3.40047e7 −0.789948
\(10\) 0 0
\(11\) 1.45028e8 0.676568 0.338284 − 0.941044i \(-0.390153\pi\)
0.338284 + 0.941044i \(0.390153\pi\)
\(12\) −1.97067e8 −0.458314
\(13\) −1.57131e9 −1.92626 −0.963128 − 0.269044i \(-0.913292\pi\)
−0.963128 + 0.269044i \(0.913292\pi\)
\(14\) 0 0
\(15\) 1.17461e9 0.458314
\(16\) 4.29497e9 1.00000
\(17\) 4.37239e9 0.626798 0.313399 − 0.949622i \(-0.398532\pi\)
0.313399 + 0.949622i \(0.398532\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −2.56000e10 −1.00000
\(21\) 1.73348e10 0.458314
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 1.52588e11 1.00000
\(26\) 0 0
\(27\) 2.31694e11 0.820359
\(28\) −3.77802e11 −1.00000
\(29\) −9.93242e11 −1.98551 −0.992753 − 0.120177i \(-0.961654\pi\)
−0.992753 + 0.120177i \(0.961654\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) −4.36101e11 −0.310081
\(34\) 0 0
\(35\) 2.25188e12 1.00000
\(36\) −2.22853e12 −0.789948
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 4.72492e12 0.882831
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 9.50458e12 0.676568
\(45\) 1.32831e13 0.789948
\(46\) 0 0
\(47\) 4.28200e13 1.79831 0.899153 − 0.437635i \(-0.144184\pi\)
0.899153 + 0.437635i \(0.144184\pi\)
\(48\) −1.29150e13 −0.458314
\(49\) 3.32329e13 1.00000
\(50\) 0 0
\(51\) −1.31478e13 −0.287270
\(52\) −1.02977e14 −1.92626
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −5.66517e13 −0.676568
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 7.69792e13 0.458314
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 1.96030e14 0.789948
\(64\) 2.81475e14 1.00000
\(65\) 6.13791e14 1.92626
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 2.86549e14 0.626798
\(69\) 0 0
\(70\) 0 0
\(71\) 1.28332e15 1.98732 0.993658 − 0.112441i \(-0.0358671\pi\)
0.993658 + 0.112441i \(0.0358671\pi\)
\(72\) 0 0
\(73\) 4.91238e14 0.609129 0.304564 − 0.952492i \(-0.401489\pi\)
0.304564 + 0.952492i \(0.401489\pi\)
\(74\) 0 0
\(75\) −4.58832e14 −0.458314
\(76\) 0 0
\(77\) −8.36060e14 −0.676568
\(78\) 0 0
\(79\) 1.88480e15 1.24236 0.621182 − 0.783666i \(-0.286653\pi\)
0.621182 + 0.783666i \(0.286653\pi\)
\(80\) −1.67772e15 −1.00000
\(81\) 7.67087e14 0.413966
\(82\) 0 0
\(83\) 1.56838e15 0.696350 0.348175 − 0.937430i \(-0.386801\pi\)
0.348175 + 0.937430i \(0.386801\pi\)
\(84\) 1.13605e15 0.458314
\(85\) −1.70797e15 −0.626798
\(86\) 0 0
\(87\) 2.98668e15 0.909985
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 9.05827e15 1.92626
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.17531e16 −1.49961 −0.749805 − 0.661659i \(-0.769853\pi\)
−0.749805 + 0.661659i \(0.769853\pi\)
\(98\) 0 0
\(99\) −4.93164e15 −0.534454
\(100\) 1.00000e16 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 1.26041e16 0.994982 0.497491 − 0.867469i \(-0.334255\pi\)
0.497491 + 0.867469i \(0.334255\pi\)
\(104\) 0 0
\(105\) −6.77139e15 −0.458314
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1.51843e16 0.820359
\(109\) −3.40284e16 −1.70777 −0.853885 − 0.520462i \(-0.825760\pi\)
−0.853885 + 0.520462i \(0.825760\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.47596e16 −1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.50931e16 −1.98551
\(117\) 5.34318e16 1.52164
\(118\) 0 0
\(119\) −2.52060e16 −0.626798
\(120\) 0 0
\(121\) −2.49165e16 −0.542255
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.96046e16 −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 \(\pi\)
\(132\) −2.85803e16 −0.310081
\(133\) 0 0
\(134\) 0 0
\(135\) −9.05053e16 −0.820359
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 1.47579e17 1.00000
\(141\) −1.28760e17 −0.824189
\(142\) 0 0
\(143\) −2.27884e17 −1.30324
\(144\) −1.46049e17 −0.789948
\(145\) 3.87985e17 1.98551
\(146\) 0 0
\(147\) −9.99314e16 −0.458314
\(148\) 0 0
\(149\) −3.27391e17 −1.34765 −0.673824 − 0.738892i \(-0.735350\pi\)
−0.673824 + 0.738892i \(0.735350\pi\)
\(150\) 0 0
\(151\) 5.08805e17 1.88250 0.941252 − 0.337705i \(-0.109651\pi\)
0.941252 + 0.337705i \(0.109651\pi\)
\(152\) 0 0
\(153\) −1.48682e17 −0.495138
\(154\) 0 0
\(155\) 0 0
\(156\) 3.09652e17 0.882831
\(157\) 6.84083e17 1.85315 0.926577 − 0.376106i \(-0.122737\pi\)
0.926577 + 0.376106i \(0.122737\pi\)
\(158\) 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
\(165\) 1.70352e17 0.310081
\(166\) 0 0
\(167\) −3.36634e17 −0.556451 −0.278225 − 0.960516i \(-0.589746\pi\)
−0.278225 + 0.960516i \(0.589746\pi\)
\(168\) 0 0
\(169\) 1.80359e18 2.71046
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.18457e18 −1.47636 −0.738178 − 0.674606i \(-0.764314\pi\)
−0.738178 + 0.674606i \(0.764314\pi\)
\(174\) 0 0
\(175\) −8.79639e17 −1.00000
\(176\) 6.22892e17 0.676568
\(177\) 0 0
\(178\) 0 0
\(179\) 1.05722e18 1.00309 0.501546 − 0.865131i \(-0.332765\pi\)
0.501546 + 0.865131i \(0.332765\pi\)
\(180\) 8.70520e17 0.789948
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.34121e17 0.424072
\(188\) 2.80625e18 1.79831
\(189\) −1.33567e18 −0.820359
\(190\) 0 0
\(191\) −3.37388e18 −1.90486 −0.952428 − 0.304763i \(-0.901423\pi\)
−0.952428 + 0.304763i \(0.901423\pi\)
\(192\) −8.46395e17 −0.458314
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) −1.84567e18 −0.882831
\(196\) 2.17795e18 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
\(203\) 5.72584e18 1.98551
\(204\) −8.61653e17 −0.287270
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −6.74871e18 −1.92626
\(209\) 0 0
\(210\) 0 0
\(211\) −3.62867e18 −0.923609 −0.461804 − 0.886982i \(-0.652798\pi\)
−0.461804 + 0.886982i \(0.652798\pi\)
\(212\) 0 0
\(213\) −3.85893e18 −0.910816
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.47715e18 −0.279172
\(220\) −3.71273e18 −0.676568
\(221\) −6.87036e18 −1.20737
\(222\) 0 0
\(223\) 1.08317e19 1.77117 0.885584 − 0.464480i \(-0.153759\pi\)
0.885584 + 0.464480i \(0.153759\pi\)
\(224\) 0 0
\(225\) −5.18870e18 −0.789948
\(226\) 0 0
\(227\) 6.10218e18 0.865521 0.432761 − 0.901509i \(-0.357539\pi\)
0.432761 + 0.901509i \(0.357539\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 2.51403e18 0.310081
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −1.67265e19 −1.79831
\(236\) 0 0
\(237\) −5.66760e18 −0.569394
\(238\) 0 0
\(239\) 2.83447e17 0.0266249 0.0133125 − 0.999911i \(-0.495762\pi\)
0.0133125 + 0.999911i \(0.495762\pi\)
\(240\) 5.04491e18 0.458314
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −1.22803e19 −1.01009
\(244\) 0 0
\(245\) −1.29816e19 −1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −4.71613e18 −0.319147
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.28470e19 0.789948
\(253\) 0 0
\(254\) 0 0
\(255\) 5.13585e18 0.287270
\(256\) 1.84467e19 1.00000
\(257\) 2.35829e19 1.23918 0.619588 − 0.784927i \(-0.287300\pi\)
0.619588 + 0.784927i \(0.287300\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 4.02254e19 1.92626
\(261\) 3.37749e19 1.56845
\(262\) 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 \(\pi\)
\(272\) 1.87793e19 0.626798
\(273\) −2.72382e19 −0.882831
\(274\) 0 0
\(275\) 2.21296e19 0.676568
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.41968e19 −1.90869 −0.954344 − 0.298710i \(-0.903444\pi\)
−0.954344 + 0.298710i \(0.903444\pi\)
\(282\) 0 0
\(283\) −2.52042e19 −0.612607 −0.306303 − 0.951934i \(-0.599092\pi\)
−0.306303 + 0.951934i \(0.599092\pi\)
\(284\) 8.41034e19 1.98732
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.95434e19 −0.607124
\(290\) 0 0
\(291\) 3.53415e19 0.687292
\(292\) 3.21938e19 0.609129
\(293\) −1.05000e20 −1.93307 −0.966536 − 0.256531i \(-0.917421\pi\)
−0.966536 + 0.256531i \(0.917421\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.36022e19 0.555029
\(298\) 0 0
\(299\) 0 0
\(300\) −3.00700e19 −0.458314
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.57530e20 1.99644 0.998218 − 0.0596760i \(-0.0190068\pi\)
0.998218 + 0.0596760i \(0.0190068\pi\)
\(308\) −5.47920e19 −0.676568
\(309\) −3.79006e19 −0.456014
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 1.78279e20 1.93528 0.967641 − 0.252330i \(-0.0811969\pi\)
0.967641 + 0.252330i \(0.0811969\pi\)
\(314\) 0 0
\(315\) −7.65743e19 −0.789948
\(316\) 1.23522e20 1.24236
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) −1.44048e20 −1.34333
\(320\) −1.09951e20 −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 5.02718e19 0.413966
\(325\) −2.39762e20 −1.92626
\(326\) 0 0
\(327\) 1.02323e20 0.782695
\(328\) 0 0
\(329\) −2.46849e20 −1.79831
\(330\) 0 0
\(331\) −2.64845e20 −1.83809 −0.919047 − 0.394148i \(-0.871040\pi\)
−0.919047 + 0.394148i \(0.871040\pi\)
\(332\) 1.02786e20 0.696350
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 7.44522e19 0.458314
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −1.11933e20 −0.626798
\(341\) 0 0
\(342\) 0 0
\(343\) −1.91581e20 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 1.95735e20 0.909985
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) −3.64061e20 −1.58022
\(352\) 0 0
\(353\) 7.05531e18 0.0292630 0.0146315 − 0.999893i \(-0.495342\pi\)
0.0146315 + 0.999893i \(0.495342\pi\)
\(354\) 0 0
\(355\) −5.01296e20 −1.98732
\(356\) 0 0
\(357\) 7.57943e19 0.287270
\(358\) 0 0
\(359\) 1.98713e20 0.720228 0.360114 − 0.932908i \(-0.382738\pi\)
0.360114 + 0.932908i \(0.382738\pi\)
\(360\) 0 0
\(361\) 2.88441e20 1.00000
\(362\) 0 0
\(363\) 7.49239e19 0.248523
\(364\) 5.93643e20 1.92626
\(365\) −1.91890e20 −0.609129
\(366\) 0 0
\(367\) 1.42425e20 0.432770 0.216385 − 0.976308i \(-0.430573\pi\)
0.216385 + 0.976308i \(0.430573\pi\)
\(368\) 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
\(375\) 1.79231e20 0.458314
\(376\) 0 0
\(377\) 1.56069e21 3.82459
\(378\) 0 0
\(379\) 7.50256e20 1.76236 0.881182 − 0.472777i \(-0.156748\pi\)
0.881182 + 0.472777i \(0.156748\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.53448e20 −1.84326 −0.921631 − 0.388068i \(-0.873142\pi\)
−0.921631 + 0.388068i \(0.873142\pi\)
\(384\) 0 0
\(385\) 3.26586e20 0.676568
\(386\) 0 0
\(387\) 0 0
\(388\) −7.70250e20 −1.49961
\(389\) −7.79426e20 −1.48655 −0.743273 − 0.668988i \(-0.766728\pi\)
−0.743273 + 0.668988i \(0.766728\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −7.36251e20 −1.24236
\(396\) −3.23200e20 −0.534454
\(397\) 7.13405e20 1.15614 0.578072 − 0.815985i \(-0.303805\pi\)
0.578072 + 0.815985i \(0.303805\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 6.55360e20 1.00000
\(401\) 9.97144e20 1.49143 0.745715 − 0.666265i \(-0.232108\pi\)
0.745715 + 0.666265i \(0.232108\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −2.99643e20 −0.413966
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.26024e20 0.994982
\(413\) 0 0
\(414\) 0 0
\(415\) −6.12650e20 −0.696350
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) −4.43770e20 −0.458314
\(421\) 1.26818e21 1.28506 0.642530 − 0.766260i \(-0.277885\pi\)
0.642530 + 0.766260i \(0.277885\pi\)
\(422\) 0 0
\(423\) −1.45608e21 −1.42057
\(424\) 0 0
\(425\) 6.67174e20 0.626798
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 6.85247e20 0.597295
\(430\) 0 0
\(431\) 9.57464e20 0.804089 0.402045 − 0.915620i \(-0.368300\pi\)
0.402045 + 0.915620i \(0.368300\pi\)
\(432\) 9.95116e20 0.820359
\(433\) 1.39240e21 1.12684 0.563418 − 0.826172i \(-0.309486\pi\)
0.563418 + 0.826172i \(0.309486\pi\)
\(434\) 0 0
\(435\) −1.16667e21 −0.909985
\(436\) −2.23008e21 −1.70777
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −1.13007e21 −0.789948
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 9.84465e20 0.617647
\(448\) −1.62265e21 −1.00000
\(449\) −3.17287e21 −1.92079 −0.960397 − 0.278634i \(-0.910118\pi\)
−0.960397 + 0.278634i \(0.910118\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −1.52998e21 −0.862779
\(454\) 0 0
\(455\) −3.53839e21 −1.92626
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 1.01305e21 0.514199
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −4.26594e21 −1.98551
\(465\) 0 0
\(466\) 0 0
\(467\) 4.44600e21 1.96533 0.982663 − 0.185403i \(-0.0593591\pi\)
0.982663 + 0.185403i \(0.0593591\pi\)
\(468\) 3.50170e21 1.52164
\(469\) 0 0
\(470\) 0 0
\(471\) −2.05704e21 −0.849327
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −1.65190e21 −0.626798
\(477\) 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
\(484\) −1.63293e21 −0.542255
\(485\) 4.59105e21 1.49961
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.33866e21 1.28441 0.642205 − 0.766533i \(-0.278020\pi\)
0.642205 + 0.766533i \(0.278020\pi\)
\(492\) 0 0
\(493\) −4.34284e21 −1.24451
\(494\) 0 0
\(495\) 1.92642e21 0.534454
\(496\) 0 0
\(497\) −7.39807e21 −1.98732
\(498\) 0 0
\(499\) 7.44307e21 1.93619 0.968095 − 0.250585i \(-0.0806230\pi\)
0.968095 + 0.250585i \(0.0806230\pi\)
\(500\) −3.90625e21 −1.00000
\(501\) 1.01226e21 0.255029
\(502\) 0 0
\(503\) 7.30313e21 1.78224 0.891118 − 0.453772i \(-0.149922\pi\)
0.891118 + 0.453772i \(0.149922\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −5.42338e21 −1.24224
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −2.83189e21 −0.609129
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.92349e21 −0.994982
\(516\) 0 0
\(517\) 6.21011e21 1.21668
\(518\) 0 0
\(519\) 3.56200e21 0.676635
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 9.20165e21 1.64381 0.821906 − 0.569623i \(-0.192911\pi\)
0.821906 + 0.569623i \(0.192911\pi\)
\(524\) 0 0
\(525\) 2.64507e21 0.458314
\(526\) 0 0
\(527\) 0 0
\(528\) −1.87304e21 −0.310081
\(529\) 6.13261e21 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.17906e21 −0.459731
\(538\) 0 0
\(539\) 4.81972e21 0.676568
\(540\) −5.93135e21 −0.820359
\(541\) 1.46731e22 1.99960 0.999800 − 0.0200000i \(-0.00636664\pi\)
0.999800 + 0.0200000i \(0.00636664\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.32923e22 1.70777
\(546\) 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
\(553\) −1.08655e22 −1.24236
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 9.67173e21 1.00000
\(561\) −1.90680e21 −0.194358
\(562\) 0 0
\(563\) −1.36880e22 −1.35604 −0.678020 − 0.735043i \(-0.737162\pi\)
−0.678020 + 0.735043i \(0.737162\pi\)
\(564\) −8.43839e21 −0.824189
\(565\) 0 0
\(566\) 0 0
\(567\) −4.42210e21 −0.413966
\(568\) 0 0
\(569\) −9.49520e21 −0.864184 −0.432092 − 0.901830i \(-0.642224\pi\)
−0.432092 + 0.901830i \(0.642224\pi\)
\(570\) 0 0
\(571\) 1.52843e22 1.35256 0.676278 − 0.736646i \(-0.263592\pi\)
0.676278 + 0.736646i \(0.263592\pi\)
\(572\) −1.49346e22 −1.30324
\(573\) 1.01452e22 0.873023
\(574\) 0 0
\(575\) 0 0
\(576\) −9.57146e21 −0.789948
\(577\) −2.40768e22 −1.95971 −0.979855 − 0.199709i \(-0.936000\pi\)
−0.979855 + 0.199709i \(0.936000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 2.54270e22 1.98551
\(581\) −9.04142e21 −0.696350
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −2.08718e22 −1.52164
\(586\) 0 0
\(587\) 2.77804e22 1.97076 0.985381 − 0.170367i \(-0.0544954\pi\)
0.985381 + 0.170367i \(0.0544954\pi\)
\(588\) −6.54911e21 −0.458314
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.60745e21 0.628305 0.314152 − 0.949373i \(-0.398280\pi\)
0.314152 + 0.949373i \(0.398280\pi\)
\(594\) 0 0
\(595\) 9.84608e21 0.626798
\(596\) −2.14559e22 −1.34765
\(597\) 0 0
\(598\) 0 0
\(599\) −1.90746e22 −1.15091 −0.575453 − 0.817835i \(-0.695174\pi\)
−0.575453 + 0.817835i \(0.695174\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 3.33451e22 1.88250
\(605\) 9.73300e21 0.542255
\(606\) 0 0
\(607\) −3.32163e22 −1.80236 −0.901181 − 0.433444i \(-0.857298\pi\)
−0.901181 + 0.433444i \(0.857298\pi\)
\(608\) 0 0
\(609\) −1.72176e22 −0.909985
\(610\) 0 0
\(611\) −6.72833e22 −3.46400
\(612\) −9.74400e21 −0.495138
\(613\) 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
\(624\) 2.02934e22 0.882831
\(625\) 2.32831e22 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 4.48320e22 1.85315
\(629\) 0 0
\(630\) 0 0
\(631\) 4.39637e22 1.74928 0.874639 − 0.484774i \(-0.161098\pi\)
0.874639 + 0.484774i \(0.161098\pi\)
\(632\) 0 0
\(633\) 1.09114e22 0.423303
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −5.22191e22 −1.92626
\(638\) 0 0
\(639\) −4.36388e22 −1.56988
\(640\) 0 0
\(641\) 5.08637e22 1.78461 0.892306 − 0.451431i \(-0.149086\pi\)
0.892306 + 0.451431i \(0.149086\pi\)
\(642\) 0 0
\(643\) 3.06355e22 1.04842 0.524212 − 0.851588i \(-0.324360\pi\)
0.524212 + 0.851588i \(0.324360\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5.28603e22 −1.72145 −0.860726 − 0.509068i \(-0.829990\pi\)
−0.860726 + 0.509068i \(0.829990\pi\)
\(648\) 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
\(657\) −1.67044e22 −0.481180
\(658\) 0 0
\(659\) −1.94527e22 −0.546885 −0.273443 − 0.961888i \(-0.588162\pi\)
−0.273443 + 0.961888i \(0.588162\pi\)
\(660\) 1.11642e22 0.310081
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 2.06592e22 0.553356
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −2.20617e22 −0.556451
\(669\) −3.25711e22 −0.811751
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 3.53536e22 0.820359
\(676\) 1.18200e23 2.71046
\(677\) 6.79762e22 1.54045 0.770225 − 0.637772i \(-0.220144\pi\)
0.770225 + 0.637772i \(0.220144\pi\)
\(678\) 0 0
\(679\) 6.77542e22 1.49961
\(680\) 0 0
\(681\) −1.83492e22 −0.396681
\(682\) 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 \(\pi\)
\(692\) −7.76319e22 −1.47636
\(693\) 2.84300e22 0.534454
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −5.76480e22 −1.00000
\(701\) −2.66326e22 −0.456740 −0.228370 − 0.973574i \(-0.573340\pi\)
−0.228370 + 0.973574i \(0.573340\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.08219e22 0.676568
\(705\) 5.02967e22 0.824189
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.21981e23 1.91040 0.955199 − 0.295963i \(-0.0956407\pi\)
0.955199 + 0.295963i \(0.0956407\pi\)
\(710\) 0 0
\(711\) −6.40921e22 −0.981404
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 8.90172e22 1.30324
\(716\) 6.92859e22 1.00309
\(717\) −8.52325e20 −0.0122026
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 5.70504e22 0.789948
\(721\) −7.26603e22 −0.994982
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.51557e23 −1.98551
\(726\) 0 0
\(727\) 2.71850e20 0.00348381 0.00174190 − 0.999998i \(-0.499446\pi\)
0.00174190 + 0.999998i \(0.499446\pi\)
\(728\) 0 0
\(729\) 3.90621e21 0.0489706
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1.63658e23 −1.96384 −0.981920 − 0.189297i \(-0.939379\pi\)
−0.981920 + 0.189297i \(0.939379\pi\)
\(734\) 0 0
\(735\) 3.90357e22 0.458314
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.76666e23 −1.98608 −0.993041 − 0.117767i \(-0.962426\pi\)
−0.993041 + 0.117767i \(0.962426\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 1.27887e23 1.34765
\(746\) 0 0
\(747\) −5.33324e22 −0.550080
\(748\) 4.15578e22 0.424072
\(749\) 0 0
\(750\) 0 0
\(751\) −1.56086e23 −1.54257 −0.771286 − 0.636489i \(-0.780386\pi\)
−0.771286 + 0.636489i \(0.780386\pi\)
\(752\) 1.83910e23 1.79831
\(753\) 0 0
\(754\) 0 0
\(755\) −1.98752e23 −1.88250
\(756\) −8.75343e22 −0.820359
\(757\) 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
\(763\) 1.96167e23 1.70777
\(764\) −2.21110e23 −1.90486
\(765\) 5.80788e22 0.495138
\(766\) 0 0
\(767\) 0 0
\(768\) −5.54694e22 −0.458314
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −7.09139e22 −0.567932
\(772\) 0 0
\(773\) −2.03953e23 −1.59991 −0.799953 − 0.600063i \(-0.795142\pi\)
−0.799953 + 0.600063i \(0.795142\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) −1.20958e23 −0.882831
\(781\) 1.86117e23 1.34456
\(782\) 0 0
\(783\) −2.30128e23 −1.62883
\(784\) 1.42734e23 1.00000
\(785\) −2.67220e23 −1.85315
\(786\) 0 0
\(787\) 1.99204e23 1.35363 0.676816 − 0.736152i \(-0.263359\pi\)
0.676816 + 0.736152i \(0.263359\pi\)
\(788\) 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
\(797\) 2.06229e23 1.26673 0.633365 − 0.773853i \(-0.281673\pi\)
0.633365 + 0.773853i \(0.281673\pi\)
\(798\) 0 0
\(799\) 1.87226e23 1.12717
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7.12435e22 0.412117
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.25510e23 1.77409 0.887045 − 0.461684i \(-0.152754\pi\)
0.887045 + 0.461684i \(0.152754\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 3.75249e23 1.98551
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −5.64693e22 −0.287270
\(817\) 0 0
\(818\) 0 0
\(819\) −3.08023e23 −1.52164
\(820\) 0 0
\(821\) −3.46096e22 −0.167668 −0.0838342 − 0.996480i \(-0.526717\pi\)
−0.0838342 + 0.996480i \(0.526717\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) −6.65437e22 −0.310081
\(826\) 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
\(832\) −4.42283e23 −1.92626
\(833\) 1.45307e23 0.626798
\(834\) 0 0
\(835\) 1.31498e23 0.556451
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 7.36283e23 2.94223
\(842\) 0 0
\(843\) 2.23110e23 0.874779
\(844\) −2.37809e23 −0.923609
\(845\) −7.04526e23 −2.71046
\(846\) 0 0
\(847\) 1.43639e23 0.542255
\(848\) 0 0
\(849\) 7.57891e22 0.280766
\(850\) 0 0
\(851\) 0 0
\(852\) −2.52899e23 −0.910816
\(853\) −5.15553e23 −1.83942 −0.919711 − 0.392597i \(-0.871577\pi\)
−0.919711 + 0.392597i \(0.871577\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 5.63159e23 1.93546 0.967731 − 0.251984i \(-0.0810831\pi\)
0.967731 + 0.251984i \(0.0810831\pi\)
\(858\) 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
\(865\) 4.62722e23 1.47636
\(866\) 0 0
\(867\) 8.88370e22 0.278254
\(868\) 0 0
\(869\) 2.73350e23 0.840545
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 3.99660e23 1.18461
\(874\) 0 0
\(875\) 3.43609e23 1.00000
\(876\) −9.68067e22 −0.279172
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 3.15735e23 0.885955
\(880\) −2.43317e23 −0.676568
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −4.50256e23 −1.20737
\(885\) 0 0
\(886\) 0 0
\(887\) 4.85897e23 1.26810 0.634052 − 0.773290i \(-0.281390\pi\)
0.634052 + 0.773290i \(0.281390\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.11249e23 0.280076
\(892\) 7.09869e23 1.77117
\(893\) 0 0
\(894\) 0 0
\(895\) −4.12976e23 −1.00309
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −3.40047e23 −0.789948
\(901\) 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 \(\pi\)
\(908\) 3.99912e23 0.865521
\(909\) 0 0
\(910\) 0 0
\(911\) −9.47077e23 −1.99636 −0.998179 − 0.0603222i \(-0.980787\pi\)
−0.998179 + 0.0603222i \(0.980787\pi\)
\(912\) 0 0
\(913\) 2.27460e23 0.471129
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −1.27572e23 −0.250744 −0.125372 − 0.992110i \(-0.540012\pi\)
−0.125372 + 0.992110i \(0.540012\pi\)
\(920\) 0 0
\(921\) −4.73692e23 −0.914995
\(922\) 0 0
\(923\) −2.01648e24 −3.82808
\(924\) 1.64760e23 0.310081
\(925\) 0 0
\(926\) 0 0
\(927\) −4.28599e23 −0.785984
\(928\) 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
\(935\) −2.47704e23 −0.424072
\(936\) 0 0
\(937\) 2.19658e23 0.369683 0.184842 − 0.982768i \(-0.440823\pi\)
0.184842 + 0.982768i \(0.440823\pi\)
\(938\) 0 0
\(939\) −5.36084e23 −0.886968
\(940\) −1.09619e24 −1.79831
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 5.21745e23 0.820359
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) −3.71432e23 −0.569394
\(949\) −7.71886e23 −1.17334
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 1.31792e24 1.90486
\(956\) 1.85760e22 0.0266249
\(957\) 4.33153e23 0.615667
\(958\) 0 0
\(959\) 0 0
\(960\) 3.30623e23 0.458314
\(961\) 7.27423e23 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 \(\pi\)
\(972\) −8.04800e23 −1.01009
\(973\) 0 0
\(974\) 0 0
\(975\) 7.20965e23 0.882831
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −8.50763e23 −1.00000
\(981\) 1.15712e24 1.34905
\(982\) 0 0
\(983\) −2.45972e23 −0.282136 −0.141068 − 0.990000i \(-0.545054\pi\)
−0.141068 + 0.990000i \(0.545054\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 7.42274e23 0.824189
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.83707e24 −1.97486 −0.987429 − 0.158063i \(-0.949475\pi\)
−0.987429 + 0.158063i \(0.949475\pi\)
\(992\) 0 0
\(993\) 7.96389e23 0.842425
\(994\) 0 0
\(995\) 0 0
\(996\) −3.09076e23 −0.319147
\(997\) −9.75045e22 −0.0998765 −0.0499383 − 0.998752i \(-0.515902\pi\)
−0.0499383 + 0.998752i \(0.515902\pi\)
\(998\) 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 35.17.c.a.34.1 ✓ 1
5.4 even 2 35.17.c.b.34.1 yes 1
7.6 odd 2 35.17.c.b.34.1 yes 1
35.34 odd 2 CM 35.17.c.a.34.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.17.c.a.34.1 ✓ 1 1.1 even 1 trivial
35.17.c.a.34.1 ✓ 1 35.34 odd 2 CM
35.17.c.b.34.1 yes 1 5.4 even 2
35.17.c.b.34.1 yes 1 7.6 odd 2