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Properties

Label 35.13.c.a.34.1
Level $35$
Weight $13$
Character 35.34
Self dual yes
Analytic conductor $31.990$
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] = [35,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.34");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 13 \)
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: \(31.9897836047\)
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)
 
\(f(q)\) \(=\) \(q-782.000 q^{3} +4096.00 q^{4} +15625.0 q^{5} +117649. q^{7} +80083.0 q^{9} +O(q^{10})\) \(q-782.000 q^{3} +4096.00 q^{4} +15625.0 q^{5} +117649. q^{7} +80083.0 q^{9} +2.81736e6 q^{11} -3.20307e6 q^{12} -1.95854e6 q^{13} -1.22188e7 q^{15} +1.67772e7 q^{16} -4.77066e7 q^{17} +6.40000e7 q^{20} -9.20015e7 q^{21} +2.44141e8 q^{25} +3.52962e8 q^{27} +4.81890e8 q^{28} +9.13676e8 q^{29} -2.20318e9 q^{33} +1.83827e9 q^{35} +3.28020e8 q^{36} +1.53158e9 q^{39} +1.15399e10 q^{44} +1.25130e9 q^{45} +9.29309e9 q^{47} -1.31198e10 q^{48} +1.38413e10 q^{49} +3.73066e10 q^{51} -8.02219e9 q^{52} +4.40213e10 q^{55} -5.00480e10 q^{60} +9.42168e9 q^{63} +6.87195e10 q^{64} -3.06022e10 q^{65} -1.95406e11 q^{68} -2.55286e11 q^{71} -4.83964e10 q^{73} -1.90918e11 q^{75} +3.31460e11 q^{77} +3.79436e11 q^{79} +2.62144e11 q^{80} -3.18576e11 q^{81} +6.48699e11 q^{83} -3.76838e11 q^{84} -7.45416e11 q^{85} -7.14495e11 q^{87} -2.30421e11 q^{91} -8.59766e11 q^{97} +2.25623e11 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) −782.000 −1.07270 −0.536351 − 0.843995i \(-0.680198\pi\)
−0.536351 + 0.843995i \(0.680198\pi\)
\(4\) 4096.00 1.00000
\(5\) 15625.0 1.00000
\(6\) 0 0
\(7\) 117649. 1.00000
\(8\) 0 0
\(9\) 80083.0 0.150690
\(10\) 0 0
\(11\) 2.81736e6 1.59033 0.795164 − 0.606395i \(-0.207385\pi\)
0.795164 + 0.606395i \(0.207385\pi\)
\(12\) −3.20307e6 −1.07270
\(13\) −1.95854e6 −0.405763 −0.202882 − 0.979203i \(-0.565031\pi\)
−0.202882 + 0.979203i \(0.565031\pi\)
\(14\) 0 0
\(15\) −1.22188e7 −1.07270
\(16\) 1.67772e7 1.00000
\(17\) −4.77066e7 −1.97645 −0.988223 − 0.153018i \(-0.951101\pi\)
−0.988223 + 0.153018i \(0.951101\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 6.40000e7 1.00000
\(21\) −9.20015e7 −1.07270
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 2.44141e8 1.00000
\(26\) 0 0
\(27\) 3.52962e8 0.911057
\(28\) 4.81890e8 1.00000
\(29\) 9.13676e8 1.53605 0.768023 − 0.640422i \(-0.221240\pi\)
0.768023 + 0.640422i \(0.221240\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) −2.20318e9 −1.70595
\(34\) 0 0
\(35\) 1.83827e9 1.00000
\(36\) 3.28020e8 0.150690
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 1.53158e9 0.435263
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 1.15399e10 1.59033
\(45\) 1.25130e9 0.150690
\(46\) 0 0
\(47\) 9.29309e9 0.862130 0.431065 − 0.902321i \(-0.358138\pi\)
0.431065 + 0.902321i \(0.358138\pi\)
\(48\) −1.31198e10 −1.07270
\(49\) 1.38413e10 1.00000
\(50\) 0 0
\(51\) 3.73066e10 2.12014
\(52\) −8.02219e9 −0.405763
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 4.40213e10 1.59033
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) −5.00480e10 −1.07270
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 9.42168e9 0.150690
\(64\) 6.87195e10 1.00000
\(65\) −3.06022e10 −0.405763
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −1.95406e11 −1.97645
\(69\) 0 0
\(70\) 0 0
\(71\) −2.55286e11 −1.99286 −0.996431 − 0.0844093i \(-0.973100\pi\)
−0.996431 + 0.0844093i \(0.973100\pi\)
\(72\) 0 0
\(73\) −4.83964e10 −0.319798 −0.159899 − 0.987133i \(-0.551117\pi\)
−0.159899 + 0.987133i \(0.551117\pi\)
\(74\) 0 0
\(75\) −1.90918e11 −1.07270
\(76\) 0 0
\(77\) 3.31460e11 1.59033
\(78\) 0 0
\(79\) 3.79436e11 1.56090 0.780451 − 0.625217i \(-0.214989\pi\)
0.780451 + 0.625217i \(0.214989\pi\)
\(80\) 2.62144e11 1.00000
\(81\) −3.18576e11 −1.12798
\(82\) 0 0
\(83\) 6.48699e11 1.98415 0.992075 − 0.125648i \(-0.0401011\pi\)
0.992075 + 0.125648i \(0.0401011\pi\)
\(84\) −3.76838e11 −1.07270
\(85\) −7.45416e11 −1.97645
\(86\) 0 0
\(87\) −7.14495e11 −1.64772
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −2.30421e11 −0.405763
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.59766e11 −1.03217 −0.516084 − 0.856538i \(-0.672611\pi\)
−0.516084 + 0.856538i \(0.672611\pi\)
\(98\) 0 0
\(99\) 2.25623e11 0.239647
\(100\) 1.00000e12 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 5.18485e9 0.00434223 0.00217112 − 0.999998i \(-0.499309\pi\)
0.00217112 + 0.999998i \(0.499309\pi\)
\(104\) 0 0
\(105\) −1.43752e12 −1.07270
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1.44573e12 0.911057
\(109\) 3.12129e12 1.86112 0.930561 − 0.366137i \(-0.119320\pi\)
0.930561 + 0.366137i \(0.119320\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.97382e12 1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.74242e12 1.53605
\(117\) −1.56846e11 −0.0611446
\(118\) 0 0
\(119\) −5.61264e12 −1.97645
\(120\) 0 0
\(121\) 4.79910e12 1.52914
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.81470e12 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 \(\pi\)
\(132\) −9.02421e12 −1.70595
\(133\) 0 0
\(134\) 0 0
\(135\) 5.51503e12 0.911057
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 7.52954e12 1.00000
\(141\) −7.26719e12 −0.924809
\(142\) 0 0
\(143\) −5.51792e12 −0.645297
\(144\) 1.34357e12 0.150690
\(145\) 1.42762e13 1.53605
\(146\) 0 0
\(147\) −1.08239e13 −1.07270
\(148\) 0 0
\(149\) −2.15991e13 −1.97387 −0.986934 − 0.161125i \(-0.948488\pi\)
−0.986934 + 0.161125i \(0.948488\pi\)
\(150\) 0 0
\(151\) −2.29210e13 −1.93362 −0.966811 − 0.255493i \(-0.917762\pi\)
−0.966811 + 0.255493i \(0.917762\pi\)
\(152\) 0 0
\(153\) −3.82049e12 −0.297831
\(154\) 0 0
\(155\) 0 0
\(156\) 6.27335e12 0.435263
\(157\) 2.63398e13 1.75879 0.879395 − 0.476092i \(-0.157947\pi\)
0.879395 + 0.476092i \(0.157947\pi\)
\(158\) 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
\(165\) −3.44246e13 −1.70595
\(166\) 0 0
\(167\) −3.57041e13 −1.64596 −0.822980 − 0.568070i \(-0.807690\pi\)
−0.822980 + 0.568070i \(0.807690\pi\)
\(168\) 0 0
\(169\) −1.94622e13 −0.835356
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.82681e13 1.05444 0.527218 − 0.849730i \(-0.323235\pi\)
0.527218 + 0.849730i \(0.323235\pi\)
\(174\) 0 0
\(175\) 2.87229e13 1.00000
\(176\) 4.72675e13 1.59033
\(177\) 0 0
\(178\) 0 0
\(179\) 4.64569e13 1.41232 0.706159 − 0.708053i \(-0.250426\pi\)
0.706159 + 0.708053i \(0.250426\pi\)
\(180\) 5.12531e12 0.150690
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.34407e14 −3.14320
\(188\) 3.80645e13 0.862130
\(189\) 4.15256e13 0.911057
\(190\) 0 0
\(191\) −8.26229e13 −1.70177 −0.850884 − 0.525354i \(-0.823933\pi\)
−0.850884 + 0.525354i \(0.823933\pi\)
\(192\) −5.37386e13 −1.07270
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 2.39309e13 0.435263
\(196\) 5.66939e13 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
\(203\) 1.07493e14 1.53605
\(204\) 1.52808e14 2.12014
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −3.28589e13 −0.405763
\(209\) 0 0
\(210\) 0 0
\(211\) −5.76635e12 −0.0653441 −0.0326720 − 0.999466i \(-0.510402\pi\)
−0.0326720 + 0.999466i \(0.510402\pi\)
\(212\) 0 0
\(213\) 1.99634e14 2.13775
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3.78460e13 0.343048
\(220\) 1.80311e14 1.59033
\(221\) 9.34354e13 0.801970
\(222\) 0 0
\(223\) −2.24267e14 −1.82363 −0.911815 − 0.410601i \(-0.865319\pi\)
−0.911815 + 0.410601i \(0.865319\pi\)
\(224\) 0 0
\(225\) 1.95515e13 0.150690
\(226\) 0 0
\(227\) −2.73199e14 −1.99675 −0.998374 − 0.0570035i \(-0.981845\pi\)
−0.998374 + 0.0570035i \(0.981845\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) −2.59202e14 −1.70595
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 1.45204e14 0.862130
\(236\) 0 0
\(237\) −2.96719e14 −1.67438
\(238\) 0 0
\(239\) −3.42934e14 −1.84003 −0.920013 − 0.391889i \(-0.871822\pi\)
−0.920013 + 0.391889i \(0.871822\pi\)
\(240\) −2.04997e14 −1.07270
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 6.15477e13 0.298933
\(244\) 0 0
\(245\) 2.16270e14 1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −5.07282e14 −2.12840
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 3.85912e13 0.150690
\(253\) 0 0
\(254\) 0 0
\(255\) 5.82915e14 2.12014
\(256\) 2.81475e14 1.00000
\(257\) −6.23825e13 −0.216503 −0.108252 − 0.994124i \(-0.534525\pi\)
−0.108252 + 0.994124i \(0.534525\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.25347e14 −0.405763
\(261\) 7.31699e13 0.231467
\(262\) 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 \(\pi\)
\(272\) −8.00384e14 −1.97645
\(273\) 1.80189e14 0.435263
\(274\) 0 0
\(275\) 6.87833e14 1.59033
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.21256e14 −1.05880 −0.529400 − 0.848373i \(-0.677583\pi\)
−0.529400 + 0.848373i \(0.677583\pi\)
\(282\) 0 0
\(283\) 6.02124e14 1.17211 0.586053 − 0.810272i \(-0.300681\pi\)
0.586053 + 0.810272i \(0.300681\pi\)
\(284\) −1.04565e15 −1.99286
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.69330e15 2.90634
\(290\) 0 0
\(291\) 6.72337e14 1.10721
\(292\) −1.98231e14 −0.319798
\(293\) −1.24154e15 −1.96226 −0.981130 − 0.193348i \(-0.938065\pi\)
−0.981130 + 0.193348i \(0.938065\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 9.94422e14 1.44888
\(298\) 0 0
\(299\) 0 0
\(300\) −7.82000e14 −1.07270
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1.12979e15 −1.34948 −0.674742 − 0.738054i \(-0.735745\pi\)
−0.674742 + 0.738054i \(0.735745\pi\)
\(308\) 1.35766e15 1.59033
\(309\) −4.05455e12 −0.00465792
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 1.05266e15 1.11950 0.559749 − 0.828662i \(-0.310898\pi\)
0.559749 + 0.828662i \(0.310898\pi\)
\(314\) 0 0
\(315\) 1.47214e14 0.150690
\(316\) 1.55417e15 1.56090
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 2.57416e15 2.44282
\(320\) 1.07374e15 1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.30489e15 −1.12798
\(325\) −4.78160e14 −0.405763
\(326\) 0 0
\(327\) −2.44085e15 −1.99643
\(328\) 0 0
\(329\) 1.09332e15 0.862130
\(330\) 0 0
\(331\) −2.33115e15 −1.77256 −0.886282 − 0.463147i \(-0.846720\pi\)
−0.886282 + 0.463147i \(0.846720\pi\)
\(332\) 2.65707e15 1.98415
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −1.54353e15 −1.07270
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −3.05322e15 −1.97645
\(341\) 0 0
\(342\) 0 0
\(343\) 1.62841e15 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) −2.92657e15 −1.64772
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) −6.91291e14 −0.369673
\(352\) 0 0
\(353\) −1.44155e15 −0.745044 −0.372522 − 0.928023i \(-0.621507\pi\)
−0.372522 + 0.928023i \(0.621507\pi\)
\(354\) 0 0
\(355\) −3.98885e15 −1.99286
\(356\) 0 0
\(357\) 4.38908e15 2.12014
\(358\) 0 0
\(359\) −3.35862e15 −1.56889 −0.784447 − 0.620195i \(-0.787053\pi\)
−0.784447 + 0.620195i \(0.787053\pi\)
\(360\) 0 0
\(361\) 2.21331e15 1.00000
\(362\) 0 0
\(363\) −3.75290e15 −1.64031
\(364\) −9.43802e14 −0.405763
\(365\) −7.56193e14 −0.319798
\(366\) 0 0
\(367\) 4.75912e15 1.94773 0.973867 − 0.227117i \(-0.0729300\pi\)
0.973867 + 0.227117i \(0.0729300\pi\)
\(368\) 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
\(375\) −2.98309e15 −1.07270
\(376\) 0 0
\(377\) −1.78947e15 −0.623271
\(378\) 0 0
\(379\) 2.13974e15 0.721981 0.360991 − 0.932569i \(-0.382439\pi\)
0.360991 + 0.932569i \(0.382439\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.85893e15 0.588940 0.294470 − 0.955661i \(-0.404857\pi\)
0.294470 + 0.955661i \(0.404857\pi\)
\(384\) 0 0
\(385\) 5.17906e15 1.59033
\(386\) 0 0
\(387\) 0 0
\(388\) −3.52160e15 −1.03217
\(389\) −1.61872e15 −0.467170 −0.233585 − 0.972336i \(-0.575046\pi\)
−0.233585 + 0.972336i \(0.575046\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.92868e15 1.56090
\(396\) 9.24151e14 0.239647
\(397\) 7.81126e15 1.99516 0.997581 − 0.0695181i \(-0.0221462\pi\)
0.997581 + 0.0695181i \(0.0221462\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.09600e15 1.00000
\(401\) −7.10277e15 −1.70829 −0.854145 − 0.520035i \(-0.825919\pi\)
−0.854145 + 0.520035i \(0.825919\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −4.97774e15 −1.12798
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.12372e13 0.00434223
\(413\) 0 0
\(414\) 0 0
\(415\) 1.01359e16 1.98415
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) −5.88810e15 −1.07270
\(421\) 6.78129e15 1.21792 0.608961 − 0.793200i \(-0.291587\pi\)
0.608961 + 0.793200i \(0.291587\pi\)
\(422\) 0 0
\(423\) 7.44218e14 0.129915
\(424\) 0 0
\(425\) −1.16471e16 −1.97645
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 4.31501e15 0.692211
\(430\) 0 0
\(431\) 8.28846e15 1.29303 0.646517 − 0.762899i \(-0.276225\pi\)
0.646517 + 0.762899i \(0.276225\pi\)
\(432\) 5.92172e15 0.911057
\(433\) −1.31605e16 −1.99684 −0.998422 − 0.0561591i \(-0.982115\pi\)
−0.998422 + 0.0561591i \(0.982115\pi\)
\(434\) 0 0
\(435\) −1.11640e16 −1.64772
\(436\) 1.27848e16 1.86112
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.10845e15 0.150690
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1.68905e16 2.11737
\(448\) 8.08478e15 1.00000
\(449\) −1.37644e16 −1.67988 −0.839942 − 0.542676i \(-0.817411\pi\)
−0.839942 + 0.542676i \(0.817411\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.79242e16 2.07420
\(454\) 0 0
\(455\) −3.60032e15 −0.405763
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) −1.68386e16 −1.80065
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 1.53289e16 1.53605
\(465\) 0 0
\(466\) 0 0
\(467\) −1.24813e16 −1.20325 −0.601627 − 0.798777i \(-0.705481\pi\)
−0.601627 + 0.798777i \(0.705481\pi\)
\(468\) −6.42441e14 −0.0611446
\(469\) 0 0
\(470\) 0 0
\(471\) −2.05977e16 −1.88666
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −2.29894e16 −1.97645
\(477\) 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
\(484\) 1.96571e16 1.52914
\(485\) −1.34338e16 −1.03217
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −2.22226e16 −1.58601 −0.793006 − 0.609214i \(-0.791485\pi\)
−0.793006 + 0.609214i \(0.791485\pi\)
\(492\) 0 0
\(493\) −4.35884e16 −3.03591
\(494\) 0 0
\(495\) 3.52536e15 0.239647
\(496\) 0 0
\(497\) −3.00342e16 −1.99286
\(498\) 0 0
\(499\) 5.83027e15 0.377646 0.188823 − 0.982011i \(-0.439533\pi\)
0.188823 + 0.982011i \(0.439533\pi\)
\(500\) 1.56250e16 1.00000
\(501\) 2.79206e16 1.76563
\(502\) 0 0
\(503\) 2.94141e16 1.81613 0.908066 − 0.418826i \(-0.137558\pi\)
0.908066 + 0.418826i \(0.137558\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.52194e16 0.896088
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −5.69378e15 −0.319798
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8.10133e13 0.00434223
\(516\) 0 0
\(517\) 2.61820e16 1.37107
\(518\) 0 0
\(519\) −2.21056e16 −1.13109
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 1.32964e16 0.649719 0.324859 − 0.945762i \(-0.394683\pi\)
0.324859 + 0.945762i \(0.394683\pi\)
\(524\) 0 0
\(525\) −2.24613e16 −1.07270
\(526\) 0 0
\(527\) 0 0
\(528\) −3.69632e16 −1.70595
\(529\) 2.19146e16 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.63293e16 −1.51500
\(538\) 0 0
\(539\) 3.89959e16 1.59033
\(540\) 2.25896e16 0.911057
\(541\) −7.52174e14 −0.0300009 −0.0150005 − 0.999887i \(-0.504775\pi\)
−0.0150005 + 0.999887i \(0.504775\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.87701e16 1.86112
\(546\) 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
\(553\) 4.46402e16 1.56090
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 3.08410e16 1.00000
\(561\) 1.05106e17 3.37172
\(562\) 0 0
\(563\) −3.69705e16 −1.16093 −0.580464 − 0.814286i \(-0.697129\pi\)
−0.580464 + 0.814286i \(0.697129\pi\)
\(564\) −2.97664e16 −0.924809
\(565\) 0 0
\(566\) 0 0
\(567\) −3.74801e16 −1.12798
\(568\) 0 0
\(569\) 3.90675e15 0.115118 0.0575588 − 0.998342i \(-0.481668\pi\)
0.0575588 + 0.998342i \(0.481668\pi\)
\(570\) 0 0
\(571\) −5.63732e16 −1.62651 −0.813254 − 0.581909i \(-0.802306\pi\)
−0.813254 + 0.581909i \(0.802306\pi\)
\(572\) −2.26014e16 −0.645297
\(573\) 6.46111e16 1.82549
\(574\) 0 0
\(575\) 0 0
\(576\) 5.50326e15 0.150690
\(577\) −7.29673e16 −1.97730 −0.988652 − 0.150225i \(-0.952000\pi\)
−0.988652 + 0.150225i \(0.952000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 5.84753e16 1.53605
\(581\) 7.63187e16 1.98415
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −2.45072e15 −0.0611446
\(586\) 0 0
\(587\) −4.99707e16 −1.22148 −0.610741 − 0.791830i \(-0.709128\pi\)
−0.610741 + 0.791830i \(0.709128\pi\)
\(588\) −4.43346e16 −1.07270
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8.59515e16 1.97663 0.988314 − 0.152430i \(-0.0487098\pi\)
0.988314 + 0.152430i \(0.0487098\pi\)
\(594\) 0 0
\(595\) −8.76974e16 −1.97645
\(596\) −8.84699e16 −1.97387
\(597\) 0 0
\(598\) 0 0
\(599\) 6.20062e15 0.134238 0.0671188 − 0.997745i \(-0.478619\pi\)
0.0671188 + 0.997745i \(0.478619\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −9.38843e16 −1.93362
\(605\) 7.49859e16 1.52914
\(606\) 0 0
\(607\) 3.30057e16 0.659867 0.329933 − 0.944004i \(-0.392974\pi\)
0.329933 + 0.944004i \(0.392974\pi\)
\(608\) 0 0
\(609\) −8.40596e16 −1.64772
\(610\) 0 0
\(611\) −1.82009e16 −0.349821
\(612\) −1.56487e16 −0.297831
\(613\) 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
\(624\) 2.56956e16 0.435263
\(625\) 5.96046e16 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 1.07888e17 1.75879
\(629\) 0 0
\(630\) 0 0
\(631\) −4.67766e16 −0.741058 −0.370529 − 0.928821i \(-0.620824\pi\)
−0.370529 + 0.928821i \(0.620824\pi\)
\(632\) 0 0
\(633\) 4.50928e15 0.0700947
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.71087e16 −0.405763
\(638\) 0 0
\(639\) −2.04441e16 −0.300305
\(640\) 0 0
\(641\) −4.77373e16 −0.688192 −0.344096 − 0.938934i \(-0.611815\pi\)
−0.344096 + 0.938934i \(0.611815\pi\)
\(642\) 0 0
\(643\) −2.98840e15 −0.0422838 −0.0211419 − 0.999776i \(-0.506730\pi\)
−0.0211419 + 0.999776i \(0.506730\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5.72089e16 −0.779898 −0.389949 − 0.920836i \(-0.627507\pi\)
−0.389949 + 0.920836i \(0.627507\pi\)
\(648\) 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
\(657\) −3.87573e15 −0.0481904
\(658\) 0 0
\(659\) 3.01278e16 0.367837 0.183918 − 0.982942i \(-0.441122\pi\)
0.183918 + 0.982942i \(0.441122\pi\)
\(660\) −1.41003e17 −1.70595
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) −7.30665e16 −0.860275
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −1.46244e17 −1.64596
\(669\) 1.75377e17 1.95621
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 8.61724e16 0.911057
\(676\) −7.97172e16 −0.835356
\(677\) −5.07480e16 −0.527093 −0.263546 − 0.964647i \(-0.584892\pi\)
−0.263546 + 0.964647i \(0.584892\pi\)
\(678\) 0 0
\(679\) −1.01151e17 −1.03217
\(680\) 0 0
\(681\) 2.13641e17 2.14192
\(682\) 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 \(\pi\)
\(692\) 1.15786e17 1.05444
\(693\) 2.65443e16 0.239647
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.17649e17 1.00000
\(701\) −5.17668e16 −0.436258 −0.218129 − 0.975920i \(-0.569995\pi\)
−0.218129 + 0.975920i \(0.569995\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.93608e17 1.59033
\(705\) −1.13550e17 −0.924809
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.47620e17 1.94943 0.974716 − 0.223446i \(-0.0717305\pi\)
0.974716 + 0.223446i \(0.0717305\pi\)
\(710\) 0 0
\(711\) 3.03864e16 0.235213
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −8.62175e16 −0.645297
\(716\) 1.90288e17 1.41232
\(717\) 2.68175e17 1.97380
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 2.09933e16 0.150690
\(721\) 6.09993e14 0.00434223
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.23066e17 1.53605
\(726\) 0 0
\(727\) 1.12643e17 0.762952 0.381476 − 0.924379i \(-0.375416\pi\)
0.381476 + 0.924379i \(0.375416\pi\)
\(728\) 0 0
\(729\) 1.21174e17 0.807316
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −4.41580e16 −0.284699 −0.142349 − 0.989816i \(-0.545466\pi\)
−0.142349 + 0.989816i \(0.545466\pi\)
\(734\) 0 0
\(735\) −1.69123e17 −1.07270
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −2.49811e17 −1.53371 −0.766857 − 0.641818i \(-0.778180\pi\)
−0.766857 + 0.641818i \(0.778180\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −3.37486e17 −1.97387
\(746\) 0 0
\(747\) 5.19497e16 0.298992
\(748\) −5.50530e17 −3.14320
\(749\) 0 0
\(750\) 0 0
\(751\) −3.46011e17 −1.92863 −0.964317 − 0.264751i \(-0.914710\pi\)
−0.964317 + 0.264751i \(0.914710\pi\)
\(752\) 1.55912e17 0.862130
\(753\) 0 0
\(754\) 0 0
\(755\) −3.58140e17 −1.93362
\(756\) 1.70089e17 0.911057
\(757\) 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
\(763\) 3.67216e17 1.86112
\(764\) −3.38423e17 −1.70177
\(765\) −5.96951e16 −0.297831
\(766\) 0 0
\(767\) 0 0
\(768\) −2.20113e17 −1.07270
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 4.87831e16 0.232243
\(772\) 0 0
\(773\) 3.77937e17 1.77150 0.885752 − 0.464159i \(-0.153643\pi\)
0.885752 + 0.464159i \(0.153643\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 9.80211e16 0.435263
\(781\) −7.19234e17 −3.16930
\(782\) 0 0
\(783\) 3.22493e17 1.39943
\(784\) 2.32218e17 1.00000
\(785\) 4.11559e17 1.75879
\(786\) 0 0
\(787\) 7.79934e16 0.328254 0.164127 − 0.986439i \(-0.447519\pi\)
0.164127 + 0.986439i \(0.447519\pi\)
\(788\) 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
\(797\) −6.22406e16 −0.242842 −0.121421 − 0.992601i \(-0.538745\pi\)
−0.121421 + 0.992601i \(0.538745\pi\)
\(798\) 0 0
\(799\) −4.43342e17 −1.70395
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.36350e17 −0.508583
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.24762e17 −1.87185 −0.935925 − 0.352199i \(-0.885434\pi\)
−0.935925 + 0.352199i \(0.885434\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 4.40292e17 1.53605
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 6.25901e17 2.12014
\(817\) 0 0
\(818\) 0 0
\(819\) −1.84528e16 −0.0611446
\(820\) 0 0
\(821\) 1.98323e17 0.647612 0.323806 − 0.946123i \(-0.395037\pi\)
0.323806 + 0.946123i \(0.395037\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) −5.37885e17 −1.70595
\(826\) 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
\(832\) −1.34590e17 −0.405763
\(833\) −6.60321e17 −1.97645
\(834\) 0 0
\(835\) −5.57877e17 −1.64596
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 4.80990e17 1.35944
\(842\) 0 0
\(843\) 4.07623e17 1.13578
\(844\) −2.36190e16 −0.0653441
\(845\) −3.04097e17 −0.835356
\(846\) 0 0
\(847\) 5.64609e17 1.52914
\(848\) 0 0
\(849\) −4.70861e17 −1.25732
\(850\) 0 0
\(851\) 0 0
\(852\) 8.17700e17 2.13775
\(853\) −7.35412e17 −1.90914 −0.954568 − 0.297994i \(-0.903683\pi\)
−0.954568 + 0.297994i \(0.903683\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.56469e17 1.65702 0.828512 − 0.559971i \(-0.189188\pi\)
0.828512 + 0.559971i \(0.189188\pi\)
\(858\) 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
\(865\) 4.41689e17 1.05444
\(866\) 0 0
\(867\) −1.32416e18 −3.11764
\(868\) 0 0
\(869\) 1.06901e18 2.48235
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −6.88527e16 −0.155538
\(874\) 0 0
\(875\) 4.48795e17 1.00000
\(876\) 1.55017e17 0.343048
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 9.70888e17 2.10492
\(880\) 7.38555e17 1.59033
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 3.82711e17 0.801970
\(885\) 0 0
\(886\) 0 0
\(887\) 1.18912e17 0.244165 0.122082 − 0.992520i \(-0.461043\pi\)
0.122082 + 0.992520i \(0.461043\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −8.97543e17 −1.79386
\(892\) −9.18599e17 −1.82363
\(893\) 0 0
\(894\) 0 0
\(895\) 7.25890e17 1.41232
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 8.00830e16 0.150690
\(901\) 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 \(\pi\)
\(908\) −1.11902e18 −1.99675
\(909\) 0 0
\(910\) 0 0
\(911\) −8.44153e17 −1.47676 −0.738382 − 0.674383i \(-0.764410\pi\)
−0.738382 + 0.674383i \(0.764410\pi\)
\(912\) 0 0
\(913\) 1.82762e18 3.15545
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.15157e18 1.91160 0.955802 − 0.294012i \(-0.0949907\pi\)
0.955802 + 0.294012i \(0.0949907\pi\)
\(920\) 0 0
\(921\) 8.83496e17 1.44759
\(922\) 0 0
\(923\) 4.99989e17 0.808630
\(924\) −1.06169e18 −1.70595
\(925\) 0 0
\(926\) 0 0
\(927\) 4.15218e14 0.000654332 0
\(928\) 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
\(935\) −2.10011e18 −3.14320
\(936\) 0 0
\(937\) 3.39150e17 0.501134 0.250567 − 0.968099i \(-0.419383\pi\)
0.250567 + 0.968099i \(0.419383\pi\)
\(938\) 0 0
\(939\) −8.23182e17 −1.20089
\(940\) 5.94758e17 0.862130
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 6.48838e17 0.911057
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) −1.21536e18 −1.67438
\(949\) 9.47863e16 0.129762
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −1.29098e18 −1.70177
\(956\) −1.40466e18 −1.84003
\(957\) −2.01299e18 −2.62042
\(958\) 0 0
\(959\) 0 0
\(960\) −8.39666e17 −1.07270
\(961\) 7.87663e17 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 \(\pi\)
\(972\) 2.52099e17 0.298933
\(973\) 0 0
\(974\) 0 0
\(975\) 3.73921e17 0.435263
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 8.85842e17 1.00000
\(981\) 2.49962e17 0.280453
\(982\) 0 0
\(983\) 1.58457e18 1.75626 0.878131 − 0.478420i \(-0.158790\pi\)
0.878131 + 0.478420i \(0.158790\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −8.54978e17 −0.924809
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.17097e18 1.23624 0.618122 − 0.786082i \(-0.287894\pi\)
0.618122 + 0.786082i \(0.287894\pi\)
\(992\) 0 0
\(993\) 1.82296e18 1.90143
\(994\) 0 0
\(995\) 0 0
\(996\) −2.07783e18 −2.12840
\(997\) −1.78532e18 −1.81779 −0.908896 − 0.417024i \(-0.863073\pi\)
−0.908896 + 0.417024i \(0.863073\pi\)
\(998\) 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 35.13.c.a.34.1 ✓ 1
5.4 even 2 35.13.c.b.34.1 yes 1
7.6 odd 2 35.13.c.b.34.1 yes 1
35.34 odd 2 CM 35.13.c.a.34.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.13.c.a.34.1 ✓ 1 1.1 even 1 trivial
35.13.c.a.34.1 ✓ 1 35.34 odd 2 CM
35.13.c.b.34.1 yes 1 5.4 even 2
35.13.c.b.34.1 yes 1 7.6 odd 2