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Properties

Label 35.15.c.a.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)
 
\(f(q)\) \(=\) \(q-4031.00 q^{3} +16384.0 q^{4} -78125.0 q^{5} +823543. q^{7} +1.14660e7 q^{9} +O(q^{10})\) \(q-4031.00 q^{3} +16384.0 q^{4} -78125.0 q^{5} +823543. q^{7} +1.14660e7 q^{9} -3.73792e7 q^{11} -6.60439e7 q^{12} -6.52796e7 q^{13} +3.14922e8 q^{15} +2.68435e8 q^{16} -6.26193e8 q^{17} -1.28000e9 q^{20} -3.31970e9 q^{21} +6.10352e9 q^{25} -2.69393e10 q^{27} +1.34929e10 q^{28} -9.77565e9 q^{29} +1.50675e11 q^{33} -6.43393e10 q^{35} +1.87859e11 q^{36} +2.63142e11 q^{39} -6.12420e11 q^{44} -8.95781e11 q^{45} +7.19082e11 q^{47} -1.08206e12 q^{48} +6.78223e11 q^{49} +2.52419e12 q^{51} -1.06954e12 q^{52} +2.92025e12 q^{55} +5.15968e12 q^{60} +9.44274e12 q^{63} +4.39805e12 q^{64} +5.09997e12 q^{65} -1.02596e13 q^{68} -1.79056e12 q^{71} +2.20336e13 q^{73} -2.46033e13 q^{75} -3.07834e13 q^{77} -2.70883e13 q^{79} -2.09715e13 q^{80} +5.37507e13 q^{81} +3.37268e13 q^{83} -5.43900e13 q^{84} +4.89213e13 q^{85} +3.94056e13 q^{87} -5.37606e13 q^{91} +2.45876e13 q^{97} -4.28589e14 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) −4031.00 −1.84316 −0.921582 − 0.388184i \(-0.873103\pi\)
−0.921582 + 0.388184i \(0.873103\pi\)
\(4\) 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 \(-0.674131\pi\)
−0.520169 + 0.854064i \(0.674131\pi\)
\(14\) 0 0
\(15\) 3.14922e8 1.84316
\(16\) 2.68435e8 1.00000
\(17\) −6.26193e8 −1.52604 −0.763020 − 0.646375i \(-0.776284\pi\)
−0.763020 + 0.646375i \(0.776284\pi\)
\(18\) 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 \(-0.248839\pi\)
0.709681 + 0.704523i \(0.248839\pi\)
\(48\) −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 \(-0.0236970\pi\)
0.997230 + 0.0743777i \(0.0236970\pi\)
\(74\) 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 \(-0.286549\pi\)
0.621438 + 0.783463i \(0.286549\pi\)
\(84\) −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 \(-0.451379\pi\)
0.152154 + 0.988357i \(0.451379\pi\)
\(98\) 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 \(-0.417473\pi\)
0.256372 + 0.966578i \(0.417473\pi\)
\(104\) 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 \(-0.517605\pi\)
−0.0552798 + 0.998471i \(0.517605\pi\)
\(158\) 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 \(-0.391028\pi\)
0.335698 + 0.941970i \(0.391028\pi\)
\(168\) 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 \(-0.0437745\pi\)
0.990559 + 0.137089i \(0.0437745\pi\)
\(174\) 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 \(-0.990461\pi\)
−0.999551 + 0.0299629i \(0.990461\pi\)
\(224\) 0 0
\(225\) 6.99829e16 2.39725
\(226\) 0 0
\(227\) 5.16108e16 1.66173 0.830863 − 0.556476i \(-0.187847\pi\)
0.830863 + 0.556476i \(0.187847\pi\)
\(228\) 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 \(-0.623613\pi\)
−0.378653 + 0.925539i \(0.623613\pi\)
\(258\) 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 \(-0.649206\pi\)
−0.451766 + 0.892137i \(0.649206\pi\)
\(284\) −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 \(-0.427743\pi\)
0.225058 + 0.974345i \(0.427743\pi\)
\(294\) 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 \(-0.858369\pi\)
−0.902634 + 0.430410i \(0.858369\pi\)
\(308\) −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 \(-0.696047\pi\)
−0.577694 + 0.816254i \(0.696047\pi\)
\(314\) 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 \(-0.0584247\pi\)
0.983203 + 0.182518i \(0.0584247\pi\)
\(354\) 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 \(-0.751752\pi\)
−0.710987 + 0.703205i \(0.751752\pi\)
\(368\) 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 \(-0.861000\pi\)
−0.906161 + 0.422934i \(0.861000\pi\)
\(384\) 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 \(-0.359171\pi\)
0.428136 + 0.903714i \(0.359171\pi\)
\(398\) 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 \(-0.145800\pi\)
0.896919 + 0.442196i \(0.145800\pi\)
\(434\) 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 \(-0.176939\pi\)
0.849441 + 0.527683i \(0.176939\pi\)
\(468\) −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 \(-0.506182\pi\)
−0.0194195 + 0.999811i \(0.506182\pi\)
\(504\) 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 \(-0.793797\pi\)
−0.797410 + 0.603438i \(0.793797\pi\)
\(524\) 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 \(-0.520017\pi\)
−0.0628426 + 0.998023i \(0.520017\pi\)
\(564\) −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 \(-0.556000\pi\)
−0.175022 + 0.984564i \(0.556000\pi\)
\(578\) 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 \(-0.827317\pi\)
−0.856420 + 0.516280i \(0.827317\pi\)
\(588\) −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 \(-0.723495\pi\)
−0.645845 + 0.763469i \(0.723495\pi\)
\(594\) 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 \(-0.125136\pi\)
0.923716 + 0.383078i \(0.125136\pi\)
\(608\) 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 \(-0.591185\pi\)
−0.282565 + 0.959248i \(0.591185\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.96839e19 0.625447 0.312724 − 0.949844i \(-0.398759\pi\)
0.312724 + 0.949844i \(0.398759\pi\)
\(648\) 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 \(-0.317626\pi\)
0.542109 + 0.840308i \(0.317626\pi\)
\(678\) 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 \(-0.562015\pi\)
−0.193595 + 0.981081i \(0.562015\pi\)
\(728\) 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 \(-0.303043\pi\)
0.580024 + 0.814600i \(0.303043\pi\)
\(734\) 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 \(-0.179251\pi\)
0.845587 + 0.533837i \(0.179251\pi\)
\(774\) 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 \(-0.855439\pi\)
−0.898634 + 0.438700i \(0.855439\pi\)
\(788\) 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 \(-0.371464\pi\)
0.392923 + 0.919572i \(0.371464\pi\)
\(798\) 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 \(-0.612370\pi\)
−0.345735 + 0.938332i \(0.612370\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.14755e20 0.337991 0.168995 − 0.985617i \(-0.445948\pi\)
0.168995 + 0.985617i \(0.445948\pi\)
\(858\) 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 \(-0.871217\pi\)
−0.919266 + 0.393637i \(0.871217\pi\)
\(888\) 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 \(-0.489280\pi\)
0.0336713 + 0.999433i \(0.489280\pi\)
\(938\) 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 \(-0.148078\pi\)
0.893731 + 0.448603i \(0.148078\pi\)
\(984\) 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 \(-0.326415\pi\)
0.518704 + 0.854954i \(0.326415\pi\)
\(998\) 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.a.34.1 ✓ 1
5.4 even 2 35.15.c.b.34.1 yes 1
7.6 odd 2 35.15.c.b.34.1 yes 1
35.34 odd 2 CM 35.15.c.a.34.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.15.c.a.34.1 ✓ 1 1.1 even 1 trivial
35.15.c.a.34.1 ✓ 1 35.34 odd 2 CM
35.15.c.b.34.1 yes 1 5.4 even 2
35.15.c.b.34.1 yes 1 7.6 odd 2