Properties

Label 35.15.c.b.34.1
Level $35$
Weight $15$
Character 35.34
Self dual yes
Analytic conductor $43.515$
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] = [35,15,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, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.34");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 15 \)
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: \(43.5151388532\)
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)
 
\(f(q)\) \(=\) \(q+4031.00 q^{3} +16384.0 q^{4} +78125.0 q^{5} -823543. q^{7} +1.14660e7 q^{9} +O(q^{10})\) \(q+4031.00 q^{3} +16384.0 q^{4} +78125.0 q^{5} -823543. q^{7} +1.14660e7 q^{9} -3.73792e7 q^{11} +6.60439e7 q^{12} +6.52796e7 q^{13} +3.14922e8 q^{15} +2.68435e8 q^{16} +6.26193e8 q^{17} +1.28000e9 q^{20} -3.31970e9 q^{21} +6.10352e9 q^{25} +2.69393e10 q^{27} -1.34929e10 q^{28} -9.77565e9 q^{29} -1.50675e11 q^{33} -6.43393e10 q^{35} +1.87859e11 q^{36} +2.63142e11 q^{39} -6.12420e11 q^{44} +8.95781e11 q^{45} -7.19082e11 q^{47} +1.08206e12 q^{48} +6.78223e11 q^{49} +2.52419e12 q^{51} +1.06954e12 q^{52} -2.92025e12 q^{55} +5.15968e12 q^{60} -9.44274e12 q^{63} +4.39805e12 q^{64} +5.09997e12 q^{65} +1.02596e13 q^{68} -1.79056e12 q^{71} -2.20336e13 q^{73} +2.46033e13 q^{75} +3.07834e13 q^{77} -2.70883e13 q^{79} +2.09715e13 q^{80} +5.37507e13 q^{81} -3.37268e13 q^{83} -5.43900e13 q^{84} +4.89213e13 q^{85} -3.94056e13 q^{87} -5.37606e13 q^{91} -2.45876e13 q^{97} -4.28589e14 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) 4031.00 1.84316 0.921582 0.388184i \(-0.126897\pi\)
0.921582 + 0.388184i \(0.126897\pi\)
\(4\) 16384.0 1.00000
\(5\) 78125.0 1.00000
\(6\) 0 0
\(7\) −823543. −1.00000
\(8\) 0 0
\(9\) 1.14660e7 2.39725
\(10\) 0 0
\(11\) −3.73792e7 −1.91814 −0.959071 0.283165i \(-0.908616\pi\)
−0.959071 + 0.283165i \(0.908616\pi\)
\(12\) 6.60439e7 1.84316
\(13\) 6.52796e7 1.04034 0.520169 0.854064i \(-0.325869\pi\)
0.520169 + 0.854064i \(0.325869\pi\)
\(14\) 0 0
\(15\) 3.14922e8 1.84316
\(16\) 2.68435e8 1.00000
\(17\) 6.26193e8 1.52604 0.763020 0.646375i \(-0.223716\pi\)
0.763020 + 0.646375i \(0.223716\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.28000e9 1.00000
\(21\) −3.31970e9 −1.84316
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 6.10352e9 1.00000
\(26\) 0 0
\(27\) 2.69393e10 2.57537
\(28\) −1.34929e10 −1.00000
\(29\) −9.77565e9 −0.566708 −0.283354 0.959015i \(-0.591447\pi\)
−0.283354 + 0.959015i \(0.591447\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) −1.50675e11 −3.53545
\(34\) 0 0
\(35\) −6.43393e10 −1.00000
\(36\) 1.87859e11 2.39725
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 2.63142e11 1.91751
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −6.12420e11 −1.91814
\(45\) 8.95781e11 2.39725
\(46\) 0 0
\(47\) −7.19082e11 −1.41936 −0.709681 0.704523i \(-0.751161\pi\)
−0.709681 + 0.704523i \(0.751161\pi\)
\(48\) 1.08206e12 1.84316
\(49\) 6.78223e11 1.00000
\(50\) 0 0
\(51\) 2.52419e12 2.81274
\(52\) 1.06954e12 1.04034
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −2.92025e12 −1.91814
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 5.15968e12 1.84316
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −9.44274e12 −2.39725
\(64\) 4.39805e12 1.00000
\(65\) 5.09997e12 1.04034
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 1.02596e13 1.52604
\(69\) 0 0
\(70\) 0 0
\(71\) −1.79056e12 −0.196870 −0.0984351 0.995143i \(-0.531384\pi\)
−0.0984351 + 0.995143i \(0.531384\pi\)
\(72\) 0 0
\(73\) −2.20336e13 −1.99446 −0.997230 0.0743777i \(-0.976303\pi\)
−0.997230 + 0.0743777i \(0.976303\pi\)
\(74\) 0 0
\(75\) 2.46033e13 1.84316
\(76\) 0 0
\(77\) 3.07834e13 1.91814
\(78\) 0 0
\(79\) −2.70883e13 −1.41056 −0.705281 0.708928i \(-0.749179\pi\)
−0.705281 + 0.708928i \(0.749179\pi\)
\(80\) 2.09715e13 1.00000
\(81\) 5.37507e13 2.34957
\(82\) 0 0
\(83\) −3.37268e13 −1.24288 −0.621438 0.783463i \(-0.713451\pi\)
−0.621438 + 0.783463i \(0.713451\pi\)
\(84\) −5.43900e13 −1.84316
\(85\) 4.89213e13 1.52604
\(86\) 0 0
\(87\) −3.94056e13 −1.04454
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −5.37606e13 −1.04034
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.45876e13 −0.304308 −0.152154 0.988357i \(-0.548621\pi\)
−0.152154 + 0.988357i \(0.548621\pi\)
\(98\) 0 0
\(99\) −4.28589e14 −4.59828
\(100\) 1.00000e14 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −6.30609e13 −0.512743 −0.256372 0.966578i \(-0.582527\pi\)
−0.256372 + 0.966578i \(0.582527\pi\)
\(104\) 0 0
\(105\) −2.59352e14 −1.84316
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 4.41373e14 2.57537
\(109\) −3.15014e13 −0.172323 −0.0861617 0.996281i \(-0.527460\pi\)
−0.0861617 + 0.996281i \(0.527460\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.21068e14 −1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.60164e14 −0.566708
\(117\) 7.48495e14 2.49395
\(118\) 0 0
\(119\) −5.15697e14 −1.52604
\(120\) 0 0
\(121\) 1.01745e15 2.67927
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.76837e14 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 \(\pi\)
\(132\) −2.46867e15 −3.53545
\(133\) 0 0
\(134\) 0 0
\(135\) 2.10463e15 2.57537
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −1.05414e15 −1.00000
\(141\) −2.89862e15 −2.61612
\(142\) 0 0
\(143\) −2.44010e15 −1.99551
\(144\) 3.07788e15 2.39725
\(145\) −7.63723e14 −0.566708
\(146\) 0 0
\(147\) 2.73392e15 1.84316
\(148\) 0 0
\(149\) 3.07982e15 1.88895 0.944477 0.328577i \(-0.106569\pi\)
0.944477 + 0.328577i \(0.106569\pi\)
\(150\) 0 0
\(151\) 1.06277e15 0.593748 0.296874 0.954917i \(-0.404056\pi\)
0.296874 + 0.954917i \(0.404056\pi\)
\(152\) 0 0
\(153\) 7.17993e15 3.65831
\(154\) 0 0
\(155\) 0 0
\(156\) 4.31132e15 1.91751
\(157\) 2.59952e14 0.110560 0.0552798 0.998471i \(-0.482395\pi\)
0.0552798 + 0.998471i \(0.482395\pi\)
\(158\) 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
\(165\) −1.17715e16 −3.53545
\(166\) 0 0
\(167\) −2.43217e15 −0.671396 −0.335698 0.941970i \(-0.608972\pi\)
−0.335698 + 0.941970i \(0.608972\pi\)
\(168\) 0 0
\(169\) 3.24051e14 0.0823013
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.18825e15 −1.98112 −0.990559 0.137089i \(-0.956225\pi\)
−0.990559 + 0.137089i \(0.956225\pi\)
\(174\) 0 0
\(175\) −5.02651e15 −1.00000
\(176\) −1.00339e16 −1.91814
\(177\) 0 0
\(178\) 0 0
\(179\) 1.16777e16 1.98330 0.991648 0.128976i \(-0.0411692\pi\)
0.991648 + 0.128976i \(0.0411692\pi\)
\(180\) 1.46765e16 2.39725
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.34066e16 −2.92716
\(188\) −1.17814e16 −1.41936
\(189\) −2.21856e16 −2.57537
\(190\) 0 0
\(191\) −1.84093e16 −1.98520 −0.992601 0.121421i \(-0.961255\pi\)
−0.992601 + 0.121421i \(0.961255\pi\)
\(192\) 1.77285e16 1.84316
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 2.05580e16 1.91751
\(196\) 1.11120e16 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
\(203\) 8.05067e15 0.566708
\(204\) 4.13562e16 2.81274
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1.75234e16 1.04034
\(209\) 0 0
\(210\) 0 0
\(211\) −3.63121e16 −1.95018 −0.975089 0.221815i \(-0.928802\pi\)
−0.975089 + 0.221815i \(0.928802\pi\)
\(212\) 0 0
\(213\) −7.21774e15 −0.362864
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −8.88174e16 −3.67612
\(220\) −4.78453e16 −1.91814
\(221\) 4.08777e16 1.58760
\(222\) 0 0
\(223\) 5.48238e16 1.99910 0.999551 0.0299629i \(-0.00953891\pi\)
0.999551 + 0.0299629i \(0.00953891\pi\)
\(224\) 0 0
\(225\) 6.99829e16 2.39725
\(226\) 0 0
\(227\) −5.16108e16 −1.66173 −0.830863 0.556476i \(-0.812153\pi\)
−0.830863 + 0.556476i \(0.812153\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 1.24088e17 3.53545
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −5.61783e16 −1.41936
\(236\) 0 0
\(237\) −1.09193e17 −2.59990
\(238\) 0 0
\(239\) 4.86280e16 1.09169 0.545847 0.837885i \(-0.316208\pi\)
0.545847 + 0.837885i \(0.316208\pi\)
\(240\) 8.45362e16 1.84316
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 8.78194e16 1.75528
\(244\) 0 0
\(245\) 5.29862e16 1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −1.35953e17 −2.29083
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −1.54710e17 −2.39725
\(253\) 0 0
\(254\) 0 0
\(255\) 1.97202e17 2.81274
\(256\) 7.20576e16 1.00000
\(257\) 5.60794e16 0.757307 0.378653 0.925539i \(-0.376387\pi\)
0.378653 + 0.925539i \(0.376387\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 8.35579e16 1.04034
\(261\) −1.12088e17 −1.35854
\(262\) 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 \(\pi\)
\(272\) 1.68092e17 1.52604
\(273\) −2.16709e17 −1.91751
\(274\) 0 0
\(275\) −2.28144e17 −1.91814
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.18891e17 −1.58228 −0.791142 0.611632i \(-0.790513\pi\)
−0.791142 + 0.611632i \(0.790513\pi\)
\(282\) 0 0
\(283\) 1.31355e17 0.903531 0.451766 0.892137i \(-0.350794\pi\)
0.451766 + 0.892137i \(0.350794\pi\)
\(284\) −2.93365e16 −0.196870
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.23740e17 1.32880
\(290\) 0 0
\(291\) −9.91125e16 −0.560890
\(292\) −3.60998e17 −1.99446
\(293\) −8.34444e16 −0.450115 −0.225058 0.974345i \(-0.572257\pi\)
−0.225058 + 0.974345i \(0.572257\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.00697e18 −4.93992
\(298\) 0 0
\(299\) 0 0
\(300\) 4.03100e17 1.84316
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.63992e17 1.80527 0.902634 0.430410i \(-0.141631\pi\)
0.902634 + 0.430410i \(0.141631\pi\)
\(308\) 5.04355e17 1.91814
\(309\) −2.54199e17 −0.945070
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 3.40046e17 1.15539 0.577694 0.816254i \(-0.303953\pi\)
0.577694 + 0.816254i \(0.303953\pi\)
\(314\) 0 0
\(315\) −7.37714e17 −2.39725
\(316\) −4.43815e17 −1.41056
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 3.65406e17 1.08703
\(320\) 3.43597e17 1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 8.80651e17 2.34957
\(325\) 3.98435e17 1.04034
\(326\) 0 0
\(327\) −1.26982e17 −0.317620
\(328\) 0 0
\(329\) 5.92195e17 1.41936
\(330\) 0 0
\(331\) 8.69979e17 1.99854 0.999270 0.0381916i \(-0.0121597\pi\)
0.999270 + 0.0381916i \(0.0121597\pi\)
\(332\) −5.52579e17 −1.24288
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −8.91126e17 −1.84316
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 8.01527e17 1.52604
\(341\) 0 0
\(342\) 0 0
\(343\) −5.58546e17 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) −6.45622e17 −1.04454
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 1.75858e18 2.67925
\(352\) 0 0
\(353\) −1.34306e18 −1.96641 −0.983203 0.182518i \(-0.941575\pi\)
−0.983203 + 0.182518i \(0.941575\pi\)
\(354\) 0 0
\(355\) −1.39887e17 −0.196870
\(356\) 0 0
\(357\) −2.07877e18 −2.81274
\(358\) 0 0
\(359\) −4.05103e17 −0.527113 −0.263557 0.964644i \(-0.584896\pi\)
−0.263557 + 0.964644i \(0.584896\pi\)
\(360\) 0 0
\(361\) 7.99007e17 1.00000
\(362\) 0 0
\(363\) 4.10135e18 4.93834
\(364\) −8.80813e17 −1.04034
\(365\) −1.72137e18 −1.99446
\(366\) 0 0
\(367\) 1.27513e18 1.42197 0.710987 0.703205i \(-0.248248\pi\)
0.710987 + 0.703205i \(0.248248\pi\)
\(368\) 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
\(375\) 1.92213e18 1.84316
\(376\) 0 0
\(377\) −6.38151e17 −0.589568
\(378\) 0 0
\(379\) −2.10679e18 −1.87563 −0.937816 0.347134i \(-0.887155\pi\)
−0.937816 + 0.347134i \(0.887155\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.19092e18 1.81232 0.906161 0.422934i \(-0.139000\pi\)
0.906161 + 0.422934i \(0.139000\pi\)
\(384\) 0 0
\(385\) 2.40495e18 1.91814
\(386\) 0 0
\(387\) 0 0
\(388\) −4.02843e17 −0.304308
\(389\) −1.37870e18 −1.02287 −0.511437 0.859321i \(-0.670887\pi\)
−0.511437 + 0.859321i \(0.670887\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.11627e18 −1.41056
\(396\) −7.02201e18 −4.59828
\(397\) −1.33090e18 −0.856271 −0.428136 0.903714i \(-0.640829\pi\)
−0.428136 + 0.903714i \(0.640829\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.63840e18 1.00000
\(401\) 1.98615e18 1.19124 0.595622 0.803265i \(-0.296905\pi\)
0.595622 + 0.803265i \(0.296905\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 4.19927e18 2.34957
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.03319e18 −0.512743
\(413\) 0 0
\(414\) 0 0
\(415\) −2.63490e18 −1.24288
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) −4.24922e18 −1.84316
\(421\) 4.68710e18 1.99954 0.999768 0.0215211i \(-0.00685092\pi\)
0.999768 + 0.0215211i \(0.00685092\pi\)
\(422\) 0 0
\(423\) −8.24498e18 −3.40257
\(424\) 0 0
\(425\) 3.82198e18 1.52604
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −9.83603e18 −3.67806
\(430\) 0 0
\(431\) −2.92748e18 −1.05963 −0.529813 0.848115i \(-0.677738\pi\)
−0.529813 + 0.848115i \(0.677738\pi\)
\(432\) 7.23145e18 2.57537
\(433\) −5.11915e18 −1.79384 −0.896919 0.442196i \(-0.854200\pi\)
−0.896919 + 0.442196i \(0.854200\pi\)
\(434\) 0 0
\(435\) −3.07857e18 −1.04454
\(436\) −5.16119e17 −0.172323
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 7.77650e18 2.39725
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1.24148e19 3.48165
\(448\) −3.62198e18 −1.00000
\(449\) 7.28003e18 1.97883 0.989415 0.145110i \(-0.0463537\pi\)
0.989415 + 0.145110i \(0.0463537\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 4.28404e18 1.09438
\(454\) 0 0
\(455\) −4.20004e18 −1.04034
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 1.68692e19 3.93012
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −2.62413e18 −0.566708
\(465\) 0 0
\(466\) 0 0
\(467\) −8.22965e18 −1.69888 −0.849441 0.527683i \(-0.823061\pi\)
−0.849441 + 0.527683i \(0.823061\pi\)
\(468\) 1.22634e19 2.49395
\(469\) 0 0
\(470\) 0 0
\(471\) 1.04787e18 0.203779
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −8.44918e18 −1.52604
\(477\) 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
\(484\) 1.66699e19 2.67927
\(485\) −1.92090e18 −0.304308
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −9.52143e18 −1.38399 −0.691993 0.721904i \(-0.743267\pi\)
−0.691993 + 0.721904i \(0.743267\pi\)
\(492\) 0 0
\(493\) −6.12145e18 −0.864820
\(494\) 0 0
\(495\) −3.34835e19 −4.59828
\(496\) 0 0
\(497\) 1.47460e18 0.196870
\(498\) 0 0
\(499\) 1.30232e19 1.69049 0.845244 0.534380i \(-0.179455\pi\)
0.845244 + 0.534380i \(0.179455\pi\)
\(500\) 7.81250e18 1.00000
\(501\) −9.80407e18 −1.23749
\(502\) 0 0
\(503\) 3.16406e17 0.0388391 0.0194195 0.999811i \(-0.493818\pi\)
0.0194195 + 0.999811i \(0.493818\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.30625e18 0.151695
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 1.81456e19 1.99446
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.92664e18 −0.512743
\(516\) 0 0
\(517\) 2.68787e19 2.72254
\(518\) 0 0
\(519\) −3.70379e19 −3.65152
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 1.70696e19 1.59482 0.797410 0.603438i \(-0.206203\pi\)
0.797410 + 0.603438i \(0.206203\pi\)
\(524\) 0 0
\(525\) −2.02619e19 −1.84316
\(526\) 0 0
\(527\) 0 0
\(528\) −4.04466e19 −3.53545
\(529\) 1.15928e19 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4.70729e19 3.65554
\(538\) 0 0
\(539\) −2.53514e19 −1.91814
\(540\) 3.44823e19 2.57537
\(541\) 1.88435e19 1.38925 0.694624 0.719373i \(-0.255571\pi\)
0.694624 + 0.719373i \(0.255571\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.46104e18 −0.172323
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 2.23084e19 1.41056
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1.72709e19 −1.00000
\(561\) −9.43519e19 −5.39524
\(562\) 0 0
\(563\) 2.25342e18 0.125685 0.0628426 0.998023i \(-0.479983\pi\)
0.0628426 + 0.998023i \(0.479983\pi\)
\(564\) −4.74910e19 −2.61612
\(565\) 0 0
\(566\) 0 0
\(567\) −4.42660e19 −2.34957
\(568\) 0 0
\(569\) 7.46856e18 0.386768 0.193384 0.981123i \(-0.438054\pi\)
0.193384 + 0.981123i \(0.438054\pi\)
\(570\) 0 0
\(571\) 2.62350e19 1.32565 0.662825 0.748774i \(-0.269357\pi\)
0.662825 + 0.748774i \(0.269357\pi\)
\(572\) −3.99786e19 −1.99551
\(573\) −7.42081e19 −3.65905
\(574\) 0 0
\(575\) 0 0
\(576\) 5.04280e19 2.39725
\(577\) 7.45339e18 0.350045 0.175022 0.984564i \(-0.444000\pi\)
0.175022 + 0.984564i \(0.444000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −1.25128e19 −0.566708
\(581\) 2.77754e19 1.24288
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 5.84762e19 2.49395
\(586\) 0 0
\(587\) 4.11324e19 1.71284 0.856420 0.516280i \(-0.172683\pi\)
0.856420 + 0.516280i \(0.172683\pi\)
\(588\) 4.47925e19 1.84316
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.33074e19 1.29169 0.645845 0.763469i \(-0.276505\pi\)
0.645845 + 0.763469i \(0.276505\pi\)
\(594\) 0 0
\(595\) −4.02888e19 −1.52604
\(596\) 5.04598e19 1.88895
\(597\) 0 0
\(598\) 0 0
\(599\) −5.21665e19 −1.88540 −0.942700 0.333641i \(-0.891723\pi\)
−0.942700 + 0.333641i \(0.891723\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.74125e19 0.593748
\(605\) 7.94885e19 2.67927
\(606\) 0 0
\(607\) −5.60905e19 −1.84743 −0.923716 0.383078i \(-0.874864\pi\)
−0.923716 + 0.383078i \(0.874864\pi\)
\(608\) 0 0
\(609\) 3.24522e19 1.04454
\(610\) 0 0
\(611\) −4.69414e19 −1.47661
\(612\) 1.17636e20 3.65831
\(613\) 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
\(624\) 7.06367e19 1.91751
\(625\) 3.72529e19 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 4.25906e18 0.110560
\(629\) 0 0
\(630\) 0 0
\(631\) 2.67572e19 0.671792 0.335896 0.941899i \(-0.390961\pi\)
0.335896 + 0.941899i \(0.390961\pi\)
\(632\) 0 0
\(633\) −1.46374e20 −3.59450
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4.42741e19 1.04034
\(638\) 0 0
\(639\) −2.05305e19 −0.471948
\(640\) 0 0
\(641\) −3.26194e19 −0.733617 −0.366809 0.930296i \(-0.619550\pi\)
−0.366809 + 0.930296i \(0.619550\pi\)
\(642\) 0 0
\(643\) 2.56818e19 0.565129 0.282565 0.959248i \(-0.408815\pi\)
0.282565 + 0.959248i \(0.408815\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.96839e19 −0.625447 −0.312724 0.949844i \(-0.601241\pi\)
−0.312724 + 0.949844i \(0.601241\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −2.52637e20 −4.78123
\(658\) 0 0
\(659\) −4.96394e18 −0.0919663 −0.0459831 0.998942i \(-0.514642\pi\)
−0.0459831 + 0.998942i \(0.514642\pi\)
\(660\) −1.92865e20 −3.53545
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 1.64778e20 2.92620
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −3.98487e19 −0.671396
\(669\) 2.20995e20 3.68467
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 1.64424e20 2.57537
\(676\) 5.30926e18 0.0823013
\(677\) −7.06704e19 −1.08422 −0.542109 0.840308i \(-0.682374\pi\)
−0.542109 + 0.840308i \(0.682374\pi\)
\(678\) 0 0
\(679\) 2.02489e19 0.304308
\(680\) 0 0
\(681\) −2.08043e20 −3.06284
\(682\) 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 \(\pi\)
\(692\) −1.50540e20 −1.98112
\(693\) 3.52962e20 4.59828
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −8.23543e19 −1.00000
\(701\) 1.66360e20 1.99997 0.999986 0.00525306i \(-0.00167211\pi\)
0.999986 + 0.00525306i \(0.00167211\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.64395e20 −1.91814
\(705\) −2.26455e20 −2.61612
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.73928e20 −1.93128 −0.965639 0.259888i \(-0.916314\pi\)
−0.965639 + 0.259888i \(0.916314\pi\)
\(710\) 0 0
\(711\) −3.10594e20 −3.38147
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −1.90633e20 −1.99551
\(716\) 1.91328e20 1.98330
\(717\) 1.96019e20 2.01217
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 2.40459e20 2.39725
\(721\) 5.19334e19 0.512743
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5.96658e19 −0.566708
\(726\) 0 0
\(727\) 4.15590e19 0.387190 0.193595 0.981081i \(-0.437985\pi\)
0.193595 + 0.981081i \(0.437985\pi\)
\(728\) 0 0
\(729\) 9.69120e19 0.885697
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1.31887e20 −1.16005 −0.580024 0.814600i \(-0.696957\pi\)
−0.580024 + 0.814600i \(0.696957\pi\)
\(734\) 0 0
\(735\) 2.13587e20 1.84316
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 2.30724e20 1.91682 0.958412 0.285390i \(-0.0921230\pi\)
0.958412 + 0.285390i \(0.0921230\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 2.40611e20 1.88895
\(746\) 0 0
\(747\) −3.86711e20 −2.97949
\(748\) −3.83494e20 −2.92716
\(749\) 0 0
\(750\) 0 0
\(751\) 2.63493e20 1.95565 0.977823 0.209434i \(-0.0671620\pi\)
0.977823 + 0.209434i \(0.0671620\pi\)
\(752\) −1.93027e20 −1.41936
\(753\) 0 0
\(754\) 0 0
\(755\) 8.30293e19 0.593748
\(756\) −3.63490e20 −2.57537
\(757\) 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
\(763\) 2.59427e19 0.172323
\(764\) −3.01619e20 −1.98520
\(765\) 5.60932e20 3.65831
\(766\) 0 0
\(767\) 0 0
\(768\) 2.90464e20 1.84316
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 2.26056e20 1.39584
\(772\) 0 0
\(773\) −2.78898e20 −1.69117 −0.845587 0.533837i \(-0.820749\pi\)
−0.845587 + 0.533837i \(0.820749\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 3.36822e20 1.91751
\(781\) 6.69296e19 0.377625
\(782\) 0 0
\(783\) −2.63349e20 −1.45948
\(784\) 1.82059e20 1.00000
\(785\) 2.03088e19 0.110560
\(786\) 0 0
\(787\) 3.36075e20 1.79727 0.898634 0.438700i \(-0.144561\pi\)
0.898634 + 0.438700i \(0.144561\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.60526e20 −0.785845 −0.392923 0.919572i \(-0.628536\pi\)
−0.392923 + 0.919572i \(0.628536\pi\)
\(798\) 0 0
\(799\) −4.50284e20 −2.16600
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 8.23598e20 3.82566
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.84920e20 0.815349 0.407674 0.913127i \(-0.366340\pi\)
0.407674 + 0.913127i \(0.366340\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 1.31902e20 0.566708
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 6.77581e20 2.81274
\(817\) 0 0
\(818\) 0 0
\(819\) −6.16418e20 −2.49395
\(820\) 0 0
\(821\) 6.16481e19 0.245198 0.122599 0.992456i \(-0.460877\pi\)
0.122599 + 0.992456i \(0.460877\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) −9.19650e20 −3.53545
\(826\) 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
\(832\) 2.87103e20 1.04034
\(833\) 4.24699e20 1.52604
\(834\) 0 0
\(835\) −1.90013e20 −0.671396
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −2.01995e20 −0.678842
\(842\) 0 0
\(843\) −8.82351e20 −2.91641
\(844\) −5.94937e20 −1.95018
\(845\) 2.53165e19 0.0823013
\(846\) 0 0
\(847\) −8.37916e20 −2.67927
\(848\) 0 0
\(849\) 5.29494e20 1.66536
\(850\) 0 0
\(851\) 0 0
\(852\) −1.18256e20 −0.362864
\(853\) 2.27204e20 0.691470 0.345735 0.938332i \(-0.387630\pi\)
0.345735 + 0.938332i \(0.387630\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.14755e20 −0.337991 −0.168995 0.985617i \(-0.554052\pi\)
−0.168995 + 0.985617i \(0.554052\pi\)
\(858\) 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
\(865\) −7.17832e20 −1.98112
\(866\) 0 0
\(867\) 9.01897e20 2.44919
\(868\) 0 0
\(869\) 1.01254e21 2.70566
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −2.81921e20 −0.729504
\(874\) 0 0
\(875\) −3.92696e20 −1.00000
\(876\) −1.45518e21 −3.67612
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) −3.36364e20 −0.829637
\(880\) −7.83898e20 −1.91814
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 6.69739e20 1.58760
\(885\) 0 0
\(886\) 0 0
\(887\) 7.94212e20 1.83853 0.919266 0.393637i \(-0.128783\pi\)
0.919266 + 0.393637i \(0.128783\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −2.00916e21 −4.50682
\(892\) 8.98233e20 1.99910
\(893\) 0 0
\(894\) 0 0
\(895\) 9.12323e20 1.98330
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 1.14660e21 2.39725
\(901\) 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 \(\pi\)
\(908\) −8.45591e20 −1.66173
\(909\) 0 0
\(910\) 0 0
\(911\) 9.81917e20 1.88559 0.942793 0.333379i \(-0.108189\pi\)
0.942793 + 0.333379i \(0.108189\pi\)
\(912\) 0 0
\(913\) 1.26068e21 2.38401
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −8.47543e20 −1.53092 −0.765461 0.643482i \(-0.777489\pi\)
−0.765461 + 0.643482i \(0.777489\pi\)
\(920\) 0 0
\(921\) 1.87035e21 3.32740
\(922\) 0 0
\(923\) −1.16887e20 −0.204811
\(924\) 2.03305e21 3.53545
\(925\) 0 0
\(926\) 0 0
\(927\) −7.23056e20 −1.22918
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.82864e21 −2.92716
\(936\) 0 0
\(937\) −4.27038e19 −0.0673425 −0.0336713 0.999433i \(-0.510720\pi\)
−0.0336713 + 0.999433i \(0.510720\pi\)
\(938\) 0 0
\(939\) 1.37073e21 2.12957
\(940\) −9.20424e20 −1.41936
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −1.73325e21 −2.57537
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) −1.78902e21 −2.59990
\(949\) −1.43834e21 −2.07491
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −1.43823e21 −1.98520
\(956\) 7.96721e20 1.09169
\(957\) 1.47295e21 2.00357
\(958\) 0 0
\(959\) 0 0
\(960\) 1.38504e21 1.84316
\(961\) 7.56944e20 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 \(\pi\)
\(972\) 1.43883e21 1.75528
\(973\) 0 0
\(974\) 0 0
\(975\) 1.60609e21 1.91751
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 8.68126e20 1.00000
\(981\) −3.61195e20 −0.413103
\(982\) 0 0
\(983\) −1.58530e21 −1.78746 −0.893731 0.448603i \(-0.851922\pi\)
−0.893731 + 0.448603i \(0.851922\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 2.38714e21 2.61612
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −9.51633e20 −1.01380 −0.506902 0.862004i \(-0.669209\pi\)
−0.506902 + 0.862004i \(0.669209\pi\)
\(992\) 0 0
\(993\) 3.50689e21 3.68364
\(994\) 0 0
\(995\) 0 0
\(996\) −2.22745e21 −2.29083
\(997\) −1.01582e21 −1.03741 −0.518704 0.854954i \(-0.673585\pi\)
−0.518704 + 0.854954i \(0.673585\pi\)
\(998\) 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 35.15.c.b.34.1 yes 1
5.4 even 2 35.15.c.a.34.1 1
7.6 odd 2 35.15.c.a.34.1 1
35.34 odd 2 CM 35.15.c.b.34.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.15.c.a.34.1 1 5.4 even 2
35.15.c.a.34.1 1 7.6 odd 2
35.15.c.b.34.1 yes 1 1.1 even 1 trivial
35.15.c.b.34.1 yes 1 35.34 odd 2 CM