Properties

Label 16.22.a.d.1.2
Level $16$
Weight $22$
Character 16.1
Self dual yes
Analytic conductor $44.716$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [16,22,Mod(1,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

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: \(44.7163750859\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2161}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 540 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 7 \)
Twist minimal: no (minimal twist has level 4)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-22.7433\) of defining polynomial
Character \(\chi\) \(=\) 16.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+92135.9 q^{3} -1.86463e7 q^{5} -6.05236e8 q^{7} -1.97134e9 q^{9} +O(q^{10})\) \(q+92135.9 q^{3} -1.86463e7 q^{5} -6.05236e8 q^{7} -1.97134e9 q^{9} -1.12010e11 q^{11} +7.55030e11 q^{13} -1.71800e12 q^{15} +1.15238e13 q^{17} +4.15598e13 q^{19} -5.57640e13 q^{21} +8.14133e13 q^{23} -1.29151e14 q^{25} -1.14540e15 q^{27} +2.46131e15 q^{29} -6.30301e14 q^{31} -1.03201e16 q^{33} +1.12854e16 q^{35} +1.40093e16 q^{37} +6.95654e16 q^{39} +1.20777e17 q^{41} -1.67373e17 q^{43} +3.67582e16 q^{45} +2.73232e17 q^{47} -1.92235e17 q^{49} +1.06176e18 q^{51} +1.24102e18 q^{53} +2.08857e18 q^{55} +3.82915e18 q^{57} +3.96154e17 q^{59} +8.53713e18 q^{61} +1.19312e18 q^{63} -1.40785e19 q^{65} +1.76971e19 q^{67} +7.50108e18 q^{69} -1.03284e19 q^{71} -1.55389e19 q^{73} -1.18995e19 q^{75} +6.77923e19 q^{77} -7.30491e19 q^{79} -8.49120e19 q^{81} -8.17274e19 q^{83} -2.14877e20 q^{85} +2.26775e20 q^{87} -2.23635e20 q^{89} -4.56972e20 q^{91} -5.80733e19 q^{93} -7.74938e20 q^{95} -2.78850e20 q^{97} +2.20809e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 65640 q^{3} + 13689324 q^{5} + 260508080 q^{7} + 12461535162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 65640 q^{3} + 13689324 q^{5} + 260508080 q^{7} + 12461535162 q^{9} - 145435963320 q^{11} + 1428900417340 q^{13} - 6819782714352 q^{15} + 1840620576420 q^{17} + 16780743928568 q^{19} - 192357511002048 q^{21} + 319691925426960 q^{23} + 439606295919326 q^{25} - 17\!\cdots\!40 q^{27}+ \cdots - 26\!\cdots\!20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 92135.9 0.900856 0.450428 0.892813i \(-0.351271\pi\)
0.450428 + 0.892813i \(0.351271\pi\)
\(4\) 0 0
\(5\) −1.86463e7 −0.853903 −0.426951 0.904275i \(-0.640412\pi\)
−0.426951 + 0.904275i \(0.640412\pi\)
\(6\) 0 0
\(7\) −6.05236e8 −0.809833 −0.404916 0.914354i \(-0.632699\pi\)
−0.404916 + 0.914354i \(0.632699\pi\)
\(8\) 0 0
\(9\) −1.97134e9 −0.188458
\(10\) 0 0
\(11\) −1.12010e11 −1.30206 −0.651032 0.759050i \(-0.725664\pi\)
−0.651032 + 0.759050i \(0.725664\pi\)
\(12\) 0 0
\(13\) 7.55030e11 1.51901 0.759503 0.650504i \(-0.225442\pi\)
0.759503 + 0.650504i \(0.225442\pi\)
\(14\) 0 0
\(15\) −1.71800e12 −0.769244
\(16\) 0 0
\(17\) 1.15238e13 1.38638 0.693191 0.720754i \(-0.256204\pi\)
0.693191 + 0.720754i \(0.256204\pi\)
\(18\) 0 0
\(19\) 4.15598e13 1.55511 0.777554 0.628816i \(-0.216460\pi\)
0.777554 + 0.628816i \(0.216460\pi\)
\(20\) 0 0
\(21\) −5.57640e13 −0.729543
\(22\) 0 0
\(23\) 8.14133e13 0.409782 0.204891 0.978785i \(-0.434316\pi\)
0.204891 + 0.978785i \(0.434316\pi\)
\(24\) 0 0
\(25\) −1.29151e14 −0.270850
\(26\) 0 0
\(27\) −1.14540e15 −1.07063
\(28\) 0 0
\(29\) 2.46131e15 1.08639 0.543197 0.839605i \(-0.317214\pi\)
0.543197 + 0.839605i \(0.317214\pi\)
\(30\) 0 0
\(31\) −6.30301e14 −0.138118 −0.0690590 0.997613i \(-0.522000\pi\)
−0.0690590 + 0.997613i \(0.522000\pi\)
\(32\) 0 0
\(33\) −1.03201e16 −1.17297
\(34\) 0 0
\(35\) 1.12854e16 0.691519
\(36\) 0 0
\(37\) 1.40093e16 0.478959 0.239480 0.970901i \(-0.423023\pi\)
0.239480 + 0.970901i \(0.423023\pi\)
\(38\) 0 0
\(39\) 6.95654e16 1.36841
\(40\) 0 0
\(41\) 1.20777e17 1.40526 0.702628 0.711557i \(-0.252010\pi\)
0.702628 + 0.711557i \(0.252010\pi\)
\(42\) 0 0
\(43\) −1.67373e17 −1.18104 −0.590522 0.807022i \(-0.701078\pi\)
−0.590522 + 0.807022i \(0.701078\pi\)
\(44\) 0 0
\(45\) 3.67582e16 0.160925
\(46\) 0 0
\(47\) 2.73232e17 0.757712 0.378856 0.925456i \(-0.376318\pi\)
0.378856 + 0.925456i \(0.376318\pi\)
\(48\) 0 0
\(49\) −1.92235e17 −0.344171
\(50\) 0 0
\(51\) 1.06176e18 1.24893
\(52\) 0 0
\(53\) 1.24102e18 0.974726 0.487363 0.873199i \(-0.337959\pi\)
0.487363 + 0.873199i \(0.337959\pi\)
\(54\) 0 0
\(55\) 2.08857e18 1.11184
\(56\) 0 0
\(57\) 3.82915e18 1.40093
\(58\) 0 0
\(59\) 3.96154e17 0.100906 0.0504531 0.998726i \(-0.483933\pi\)
0.0504531 + 0.998726i \(0.483933\pi\)
\(60\) 0 0
\(61\) 8.53713e18 1.53232 0.766158 0.642652i \(-0.222166\pi\)
0.766158 + 0.642652i \(0.222166\pi\)
\(62\) 0 0
\(63\) 1.19312e18 0.152619
\(64\) 0 0
\(65\) −1.40785e19 −1.29708
\(66\) 0 0
\(67\) 1.76971e19 1.18609 0.593043 0.805171i \(-0.297926\pi\)
0.593043 + 0.805171i \(0.297926\pi\)
\(68\) 0 0
\(69\) 7.50108e18 0.369155
\(70\) 0 0
\(71\) −1.03284e19 −0.376548 −0.188274 0.982117i \(-0.560289\pi\)
−0.188274 + 0.982117i \(0.560289\pi\)
\(72\) 0 0
\(73\) −1.55389e19 −0.423185 −0.211593 0.977358i \(-0.567865\pi\)
−0.211593 + 0.977358i \(0.567865\pi\)
\(74\) 0 0
\(75\) −1.18995e19 −0.243997
\(76\) 0 0
\(77\) 6.77923e19 1.05445
\(78\) 0 0
\(79\) −7.30491e19 −0.868022 −0.434011 0.900908i \(-0.642902\pi\)
−0.434011 + 0.900908i \(0.642902\pi\)
\(80\) 0 0
\(81\) −8.49120e19 −0.776026
\(82\) 0 0
\(83\) −8.17274e19 −0.578160 −0.289080 0.957305i \(-0.593349\pi\)
−0.289080 + 0.957305i \(0.593349\pi\)
\(84\) 0 0
\(85\) −2.14877e20 −1.18384
\(86\) 0 0
\(87\) 2.26775e20 0.978685
\(88\) 0 0
\(89\) −2.23635e20 −0.760229 −0.380114 0.924939i \(-0.624115\pi\)
−0.380114 + 0.924939i \(0.624115\pi\)
\(90\) 0 0
\(91\) −4.56972e20 −1.23014
\(92\) 0 0
\(93\) −5.80733e19 −0.124424
\(94\) 0 0
\(95\) −7.74938e20 −1.32791
\(96\) 0 0
\(97\) −2.78850e20 −0.383943 −0.191972 0.981400i \(-0.561488\pi\)
−0.191972 + 0.981400i \(0.561488\pi\)
\(98\) 0 0
\(99\) 2.20809e20 0.245384
\(100\) 0 0
\(101\) 1.66280e21 1.49784 0.748920 0.662660i \(-0.230573\pi\)
0.748920 + 0.662660i \(0.230573\pi\)
\(102\) 0 0
\(103\) 1.51135e21 1.10809 0.554046 0.832486i \(-0.313083\pi\)
0.554046 + 0.832486i \(0.313083\pi\)
\(104\) 0 0
\(105\) 1.03979e21 0.622959
\(106\) 0 0
\(107\) 2.26434e21 1.11279 0.556393 0.830920i \(-0.312185\pi\)
0.556393 + 0.830920i \(0.312185\pi\)
\(108\) 0 0
\(109\) −3.92748e21 −1.58904 −0.794522 0.607235i \(-0.792279\pi\)
−0.794522 + 0.607235i \(0.792279\pi\)
\(110\) 0 0
\(111\) 1.29076e21 0.431474
\(112\) 0 0
\(113\) 6.00234e21 1.66340 0.831701 0.555224i \(-0.187367\pi\)
0.831701 + 0.555224i \(0.187367\pi\)
\(114\) 0 0
\(115\) −1.51806e21 −0.349914
\(116\) 0 0
\(117\) −1.48842e21 −0.286268
\(118\) 0 0
\(119\) −6.97464e21 −1.12274
\(120\) 0 0
\(121\) 5.14593e21 0.695372
\(122\) 0 0
\(123\) 1.11279e22 1.26593
\(124\) 0 0
\(125\) 1.12995e22 1.08518
\(126\) 0 0
\(127\) 1.32453e22 1.07677 0.538385 0.842699i \(-0.319035\pi\)
0.538385 + 0.842699i \(0.319035\pi\)
\(128\) 0 0
\(129\) −1.54210e22 −1.06395
\(130\) 0 0
\(131\) −2.38495e22 −1.40001 −0.700004 0.714139i \(-0.746818\pi\)
−0.700004 + 0.714139i \(0.746818\pi\)
\(132\) 0 0
\(133\) −2.51535e22 −1.25938
\(134\) 0 0
\(135\) 2.13576e22 0.914214
\(136\) 0 0
\(137\) −2.16852e21 −0.0795423 −0.0397712 0.999209i \(-0.512663\pi\)
−0.0397712 + 0.999209i \(0.512663\pi\)
\(138\) 0 0
\(139\) −2.92783e22 −0.922337 −0.461169 0.887313i \(-0.652570\pi\)
−0.461169 + 0.887313i \(0.652570\pi\)
\(140\) 0 0
\(141\) 2.51745e22 0.682590
\(142\) 0 0
\(143\) −8.45707e22 −1.97784
\(144\) 0 0
\(145\) −4.58944e22 −0.927675
\(146\) 0 0
\(147\) −1.77117e22 −0.310048
\(148\) 0 0
\(149\) −2.50216e22 −0.380066 −0.190033 0.981778i \(-0.560860\pi\)
−0.190033 + 0.981778i \(0.560860\pi\)
\(150\) 0 0
\(151\) 7.09599e22 0.937033 0.468517 0.883455i \(-0.344789\pi\)
0.468517 + 0.883455i \(0.344789\pi\)
\(152\) 0 0
\(153\) −2.27173e22 −0.261275
\(154\) 0 0
\(155\) 1.17528e22 0.117939
\(156\) 0 0
\(157\) 4.23299e22 0.371280 0.185640 0.982618i \(-0.440564\pi\)
0.185640 + 0.982618i \(0.440564\pi\)
\(158\) 0 0
\(159\) 1.14343e23 0.878088
\(160\) 0 0
\(161\) −4.92743e22 −0.331855
\(162\) 0 0
\(163\) 2.03046e22 0.120123 0.0600614 0.998195i \(-0.480870\pi\)
0.0600614 + 0.998195i \(0.480870\pi\)
\(164\) 0 0
\(165\) 1.92432e23 1.00161
\(166\) 0 0
\(167\) 1.47752e19 6.77660e−5 0 3.38830e−5 1.00000i \(-0.499989\pi\)
3.38830e−5 1.00000i \(0.499989\pi\)
\(168\) 0 0
\(169\) 3.23006e23 1.30738
\(170\) 0 0
\(171\) −8.19283e22 −0.293072
\(172\) 0 0
\(173\) 1.87075e22 0.0592287 0.0296143 0.999561i \(-0.490572\pi\)
0.0296143 + 0.999561i \(0.490572\pi\)
\(174\) 0 0
\(175\) 7.81671e22 0.219343
\(176\) 0 0
\(177\) 3.65000e22 0.0909020
\(178\) 0 0
\(179\) 2.14720e23 0.475244 0.237622 0.971358i \(-0.423632\pi\)
0.237622 + 0.971358i \(0.423632\pi\)
\(180\) 0 0
\(181\) −5.51937e23 −1.08709 −0.543544 0.839381i \(-0.682918\pi\)
−0.543544 + 0.839381i \(0.682918\pi\)
\(182\) 0 0
\(183\) 7.86576e23 1.38040
\(184\) 0 0
\(185\) −2.61222e23 −0.408985
\(186\) 0 0
\(187\) −1.29078e24 −1.80516
\(188\) 0 0
\(189\) 6.93240e23 0.867031
\(190\) 0 0
\(191\) 1.45180e24 1.62577 0.812883 0.582428i \(-0.197897\pi\)
0.812883 + 0.582428i \(0.197897\pi\)
\(192\) 0 0
\(193\) 5.88032e23 0.590268 0.295134 0.955456i \(-0.404636\pi\)
0.295134 + 0.955456i \(0.404636\pi\)
\(194\) 0 0
\(195\) −1.29714e24 −1.16849
\(196\) 0 0
\(197\) −1.28721e24 −1.04172 −0.520862 0.853641i \(-0.674390\pi\)
−0.520862 + 0.853641i \(0.674390\pi\)
\(198\) 0 0
\(199\) −1.02349e24 −0.744948 −0.372474 0.928043i \(-0.621490\pi\)
−0.372474 + 0.928043i \(0.621490\pi\)
\(200\) 0 0
\(201\) 1.63054e24 1.06849
\(202\) 0 0
\(203\) −1.48967e24 −0.879798
\(204\) 0 0
\(205\) −2.25206e24 −1.19995
\(206\) 0 0
\(207\) −1.60493e23 −0.0772267
\(208\) 0 0
\(209\) −4.65510e24 −2.02485
\(210\) 0 0
\(211\) −9.95941e23 −0.391984 −0.195992 0.980606i \(-0.562793\pi\)
−0.195992 + 0.980606i \(0.562793\pi\)
\(212\) 0 0
\(213\) −9.51616e23 −0.339216
\(214\) 0 0
\(215\) 3.12089e24 1.00850
\(216\) 0 0
\(217\) 3.81481e23 0.111852
\(218\) 0 0
\(219\) −1.43169e24 −0.381229
\(220\) 0 0
\(221\) 8.70084e24 2.10592
\(222\) 0 0
\(223\) 4.42854e24 0.975122 0.487561 0.873089i \(-0.337887\pi\)
0.487561 + 0.873089i \(0.337887\pi\)
\(224\) 0 0
\(225\) 2.54601e23 0.0510438
\(226\) 0 0
\(227\) −4.23549e24 −0.773806 −0.386903 0.922120i \(-0.626455\pi\)
−0.386903 + 0.922120i \(0.626455\pi\)
\(228\) 0 0
\(229\) 6.57364e24 1.09530 0.547650 0.836708i \(-0.315523\pi\)
0.547650 + 0.836708i \(0.315523\pi\)
\(230\) 0 0
\(231\) 6.24611e24 0.949912
\(232\) 0 0
\(233\) −1.03875e24 −0.144302 −0.0721509 0.997394i \(-0.522986\pi\)
−0.0721509 + 0.997394i \(0.522986\pi\)
\(234\) 0 0
\(235\) −5.09478e24 −0.647012
\(236\) 0 0
\(237\) −6.73044e24 −0.781963
\(238\) 0 0
\(239\) 5.83947e24 0.621149 0.310574 0.950549i \(-0.399479\pi\)
0.310574 + 0.950549i \(0.399479\pi\)
\(240\) 0 0
\(241\) 3.84920e24 0.375138 0.187569 0.982251i \(-0.439939\pi\)
0.187569 + 0.982251i \(0.439939\pi\)
\(242\) 0 0
\(243\) 4.15790e24 0.371542
\(244\) 0 0
\(245\) 3.58448e24 0.293888
\(246\) 0 0
\(247\) 3.13789e25 2.36222
\(248\) 0 0
\(249\) −7.53002e24 −0.520839
\(250\) 0 0
\(251\) −2.86677e25 −1.82313 −0.911567 0.411151i \(-0.865127\pi\)
−0.911567 + 0.411151i \(0.865127\pi\)
\(252\) 0 0
\(253\) −9.11908e24 −0.533563
\(254\) 0 0
\(255\) −1.97979e25 −1.06647
\(256\) 0 0
\(257\) 3.67250e25 1.82249 0.911243 0.411868i \(-0.135124\pi\)
0.911243 + 0.411868i \(0.135124\pi\)
\(258\) 0 0
\(259\) −8.47894e24 −0.387877
\(260\) 0 0
\(261\) −4.85207e24 −0.204740
\(262\) 0 0
\(263\) 1.79474e25 0.698983 0.349492 0.936939i \(-0.386354\pi\)
0.349492 + 0.936939i \(0.386354\pi\)
\(264\) 0 0
\(265\) −2.31405e25 −0.832321
\(266\) 0 0
\(267\) −2.06048e25 −0.684857
\(268\) 0 0
\(269\) −6.85890e24 −0.210793 −0.105396 0.994430i \(-0.533611\pi\)
−0.105396 + 0.994430i \(0.533611\pi\)
\(270\) 0 0
\(271\) −9.95982e24 −0.283187 −0.141594 0.989925i \(-0.545223\pi\)
−0.141594 + 0.989925i \(0.545223\pi\)
\(272\) 0 0
\(273\) −4.21035e25 −1.10818
\(274\) 0 0
\(275\) 1.44662e25 0.352664
\(276\) 0 0
\(277\) −1.13250e25 −0.255858 −0.127929 0.991783i \(-0.540833\pi\)
−0.127929 + 0.991783i \(0.540833\pi\)
\(278\) 0 0
\(279\) 1.24253e24 0.0260294
\(280\) 0 0
\(281\) −1.74755e23 −0.00339636 −0.00169818 0.999999i \(-0.500541\pi\)
−0.00169818 + 0.999999i \(0.500541\pi\)
\(282\) 0 0
\(283\) 2.85617e23 0.00515260 0.00257630 0.999997i \(-0.499180\pi\)
0.00257630 + 0.999997i \(0.499180\pi\)
\(284\) 0 0
\(285\) −7.13996e25 −1.19626
\(286\) 0 0
\(287\) −7.30989e25 −1.13802
\(288\) 0 0
\(289\) 6.37066e25 0.922055
\(290\) 0 0
\(291\) −2.56921e25 −0.345878
\(292\) 0 0
\(293\) 2.21385e25 0.277357 0.138678 0.990337i \(-0.455715\pi\)
0.138678 + 0.990337i \(0.455715\pi\)
\(294\) 0 0
\(295\) −7.38682e24 −0.0861641
\(296\) 0 0
\(297\) 1.28296e26 1.39403
\(298\) 0 0
\(299\) 6.14695e25 0.622461
\(300\) 0 0
\(301\) 1.01300e26 0.956448
\(302\) 0 0
\(303\) 1.53204e26 1.34934
\(304\) 0 0
\(305\) −1.59186e26 −1.30845
\(306\) 0 0
\(307\) 1.00481e26 0.771137 0.385569 0.922679i \(-0.374005\pi\)
0.385569 + 0.922679i \(0.374005\pi\)
\(308\) 0 0
\(309\) 1.39250e26 0.998231
\(310\) 0 0
\(311\) 1.29128e26 0.865043 0.432521 0.901624i \(-0.357624\pi\)
0.432521 + 0.901624i \(0.357624\pi\)
\(312\) 0 0
\(313\) −1.59492e26 −0.998906 −0.499453 0.866341i \(-0.666466\pi\)
−0.499453 + 0.866341i \(0.666466\pi\)
\(314\) 0 0
\(315\) −2.22474e25 −0.130322
\(316\) 0 0
\(317\) 8.73329e25 0.478692 0.239346 0.970934i \(-0.423067\pi\)
0.239346 + 0.970934i \(0.423067\pi\)
\(318\) 0 0
\(319\) −2.75691e26 −1.41456
\(320\) 0 0
\(321\) 2.08627e26 1.00246
\(322\) 0 0
\(323\) 4.78928e26 2.15598
\(324\) 0 0
\(325\) −9.75132e25 −0.411423
\(326\) 0 0
\(327\) −3.61862e26 −1.43150
\(328\) 0 0
\(329\) −1.65370e26 −0.613620
\(330\) 0 0
\(331\) −5.50377e26 −1.91631 −0.958157 0.286245i \(-0.907593\pi\)
−0.958157 + 0.286245i \(0.907593\pi\)
\(332\) 0 0
\(333\) −2.76170e25 −0.0902636
\(334\) 0 0
\(335\) −3.29986e26 −1.01280
\(336\) 0 0
\(337\) 5.51815e25 0.159103 0.0795517 0.996831i \(-0.474651\pi\)
0.0795517 + 0.996831i \(0.474651\pi\)
\(338\) 0 0
\(339\) 5.53031e26 1.49849
\(340\) 0 0
\(341\) 7.05998e25 0.179838
\(342\) 0 0
\(343\) 4.54400e26 1.08855
\(344\) 0 0
\(345\) −1.39868e26 −0.315222
\(346\) 0 0
\(347\) 4.14487e26 0.879126 0.439563 0.898212i \(-0.355133\pi\)
0.439563 + 0.898212i \(0.355133\pi\)
\(348\) 0 0
\(349\) −4.30244e26 −0.859108 −0.429554 0.903041i \(-0.641329\pi\)
−0.429554 + 0.903041i \(0.641329\pi\)
\(350\) 0 0
\(351\) −8.64815e26 −1.62629
\(352\) 0 0
\(353\) −4.04943e26 −0.717398 −0.358699 0.933453i \(-0.616780\pi\)
−0.358699 + 0.933453i \(0.616780\pi\)
\(354\) 0 0
\(355\) 1.92587e26 0.321535
\(356\) 0 0
\(357\) −6.42614e26 −1.01143
\(358\) 0 0
\(359\) −1.27535e27 −1.89294 −0.946469 0.322794i \(-0.895378\pi\)
−0.946469 + 0.322794i \(0.895378\pi\)
\(360\) 0 0
\(361\) 1.01301e27 1.41836
\(362\) 0 0
\(363\) 4.74125e26 0.626431
\(364\) 0 0
\(365\) 2.89744e26 0.361359
\(366\) 0 0
\(367\) −4.71084e26 −0.554759 −0.277380 0.960760i \(-0.589466\pi\)
−0.277380 + 0.960760i \(0.589466\pi\)
\(368\) 0 0
\(369\) −2.38093e26 −0.264832
\(370\) 0 0
\(371\) −7.51111e26 −0.789365
\(372\) 0 0
\(373\) −9.52311e26 −0.945880 −0.472940 0.881095i \(-0.656807\pi\)
−0.472940 + 0.881095i \(0.656807\pi\)
\(374\) 0 0
\(375\) 1.04109e27 0.977593
\(376\) 0 0
\(377\) 1.85836e27 1.65024
\(378\) 0 0
\(379\) 6.39167e26 0.536912 0.268456 0.963292i \(-0.413487\pi\)
0.268456 + 0.963292i \(0.413487\pi\)
\(380\) 0 0
\(381\) 1.22037e27 0.970016
\(382\) 0 0
\(383\) 2.07948e27 1.56447 0.782237 0.622981i \(-0.214078\pi\)
0.782237 + 0.622981i \(0.214078\pi\)
\(384\) 0 0
\(385\) −1.26408e27 −0.900402
\(386\) 0 0
\(387\) 3.29948e26 0.222577
\(388\) 0 0
\(389\) 9.59448e26 0.613127 0.306564 0.951850i \(-0.400821\pi\)
0.306564 + 0.951850i \(0.400821\pi\)
\(390\) 0 0
\(391\) 9.38193e26 0.568115
\(392\) 0 0
\(393\) −2.19739e27 −1.26121
\(394\) 0 0
\(395\) 1.36210e27 0.741206
\(396\) 0 0
\(397\) 1.07265e27 0.553551 0.276776 0.960935i \(-0.410734\pi\)
0.276776 + 0.960935i \(0.410734\pi\)
\(398\) 0 0
\(399\) −2.31754e27 −1.13452
\(400\) 0 0
\(401\) 3.20976e27 1.49093 0.745464 0.666546i \(-0.232228\pi\)
0.745464 + 0.666546i \(0.232228\pi\)
\(402\) 0 0
\(403\) −4.75896e26 −0.209802
\(404\) 0 0
\(405\) 1.58330e27 0.662651
\(406\) 0 0
\(407\) −1.56918e27 −0.623636
\(408\) 0 0
\(409\) −1.21280e27 −0.457820 −0.228910 0.973448i \(-0.573516\pi\)
−0.228910 + 0.973448i \(0.573516\pi\)
\(410\) 0 0
\(411\) −1.99799e26 −0.0716562
\(412\) 0 0
\(413\) −2.39767e26 −0.0817172
\(414\) 0 0
\(415\) 1.52392e27 0.493692
\(416\) 0 0
\(417\) −2.69758e27 −0.830893
\(418\) 0 0
\(419\) −1.87689e27 −0.549784 −0.274892 0.961475i \(-0.588642\pi\)
−0.274892 + 0.961475i \(0.588642\pi\)
\(420\) 0 0
\(421\) −1.61930e27 −0.451195 −0.225598 0.974221i \(-0.572433\pi\)
−0.225598 + 0.974221i \(0.572433\pi\)
\(422\) 0 0
\(423\) −5.38632e26 −0.142797
\(424\) 0 0
\(425\) −1.48832e27 −0.375502
\(426\) 0 0
\(427\) −5.16698e27 −1.24092
\(428\) 0 0
\(429\) −7.79200e27 −1.78175
\(430\) 0 0
\(431\) 7.51808e27 1.63718 0.818589 0.574380i \(-0.194757\pi\)
0.818589 + 0.574380i \(0.194757\pi\)
\(432\) 0 0
\(433\) −5.53633e25 −0.0114842 −0.00574208 0.999984i \(-0.501828\pi\)
−0.00574208 + 0.999984i \(0.501828\pi\)
\(434\) 0 0
\(435\) −4.22852e27 −0.835702
\(436\) 0 0
\(437\) 3.38352e27 0.637256
\(438\) 0 0
\(439\) −2.79922e27 −0.502527 −0.251264 0.967919i \(-0.580846\pi\)
−0.251264 + 0.967919i \(0.580846\pi\)
\(440\) 0 0
\(441\) 3.78960e26 0.0648616
\(442\) 0 0
\(443\) 6.75198e27 1.10203 0.551014 0.834496i \(-0.314241\pi\)
0.551014 + 0.834496i \(0.314241\pi\)
\(444\) 0 0
\(445\) 4.16997e27 0.649162
\(446\) 0 0
\(447\) −2.30539e27 −0.342385
\(448\) 0 0
\(449\) 3.45343e27 0.489400 0.244700 0.969599i \(-0.421311\pi\)
0.244700 + 0.969599i \(0.421311\pi\)
\(450\) 0 0
\(451\) −1.35283e28 −1.82973
\(452\) 0 0
\(453\) 6.53795e27 0.844132
\(454\) 0 0
\(455\) 8.52085e27 1.05042
\(456\) 0 0
\(457\) −5.82771e27 −0.686085 −0.343042 0.939320i \(-0.611457\pi\)
−0.343042 + 0.939320i \(0.611457\pi\)
\(458\) 0 0
\(459\) −1.31994e28 −1.48430
\(460\) 0 0
\(461\) −9.26600e27 −0.995480 −0.497740 0.867326i \(-0.665837\pi\)
−0.497740 + 0.867326i \(0.665837\pi\)
\(462\) 0 0
\(463\) 1.43306e28 1.47117 0.735586 0.677431i \(-0.236907\pi\)
0.735586 + 0.677431i \(0.236907\pi\)
\(464\) 0 0
\(465\) 1.08285e27 0.106246
\(466\) 0 0
\(467\) 1.24220e28 1.16510 0.582552 0.812794i \(-0.302054\pi\)
0.582552 + 0.812794i \(0.302054\pi\)
\(468\) 0 0
\(469\) −1.07109e28 −0.960532
\(470\) 0 0
\(471\) 3.90010e27 0.334470
\(472\) 0 0
\(473\) 1.87474e28 1.53780
\(474\) 0 0
\(475\) −5.36751e27 −0.421201
\(476\) 0 0
\(477\) −2.44647e27 −0.183695
\(478\) 0 0
\(479\) 1.55141e28 1.11482 0.557410 0.830237i \(-0.311795\pi\)
0.557410 + 0.830237i \(0.311795\pi\)
\(480\) 0 0
\(481\) 1.05775e28 0.727542
\(482\) 0 0
\(483\) −4.53993e27 −0.298954
\(484\) 0 0
\(485\) 5.19953e27 0.327850
\(486\) 0 0
\(487\) 1.10212e26 0.00665542 0.00332771 0.999994i \(-0.498941\pi\)
0.00332771 + 0.999994i \(0.498941\pi\)
\(488\) 0 0
\(489\) 1.87079e27 0.108213
\(490\) 0 0
\(491\) 1.45763e28 0.807779 0.403890 0.914808i \(-0.367658\pi\)
0.403890 + 0.914808i \(0.367658\pi\)
\(492\) 0 0
\(493\) 2.83637e28 1.50616
\(494\) 0 0
\(495\) −4.11727e27 −0.209534
\(496\) 0 0
\(497\) 6.25112e27 0.304941
\(498\) 0 0
\(499\) −1.13652e28 −0.531522 −0.265761 0.964039i \(-0.585623\pi\)
−0.265761 + 0.964039i \(0.585623\pi\)
\(500\) 0 0
\(501\) 1.36133e24 6.10474e−5 0
\(502\) 0 0
\(503\) −3.78149e28 −1.62630 −0.813148 0.582057i \(-0.802248\pi\)
−0.813148 + 0.582057i \(0.802248\pi\)
\(504\) 0 0
\(505\) −3.10051e28 −1.27901
\(506\) 0 0
\(507\) 2.97605e28 1.17776
\(508\) 0 0
\(509\) −9.31841e27 −0.353839 −0.176919 0.984225i \(-0.556613\pi\)
−0.176919 + 0.984225i \(0.556613\pi\)
\(510\) 0 0
\(511\) 9.40471e27 0.342710
\(512\) 0 0
\(513\) −4.76028e28 −1.66495
\(514\) 0 0
\(515\) −2.81812e28 −0.946202
\(516\) 0 0
\(517\) −3.06046e28 −0.986590
\(518\) 0 0
\(519\) 1.72363e27 0.0533565
\(520\) 0 0
\(521\) −2.08795e25 −0.000620762 0 −0.000310381 1.00000i \(-0.500099\pi\)
−0.000310381 1.00000i \(0.500099\pi\)
\(522\) 0 0
\(523\) −1.53000e28 −0.436941 −0.218470 0.975844i \(-0.570107\pi\)
−0.218470 + 0.975844i \(0.570107\pi\)
\(524\) 0 0
\(525\) 7.20199e27 0.197597
\(526\) 0 0
\(527\) −7.26347e27 −0.191484
\(528\) 0 0
\(529\) −3.28435e28 −0.832079
\(530\) 0 0
\(531\) −7.80952e26 −0.0190166
\(532\) 0 0
\(533\) 9.11907e28 2.13459
\(534\) 0 0
\(535\) −4.22216e28 −0.950210
\(536\) 0 0
\(537\) 1.97834e28 0.428126
\(538\) 0 0
\(539\) 2.15322e28 0.448132
\(540\) 0 0
\(541\) −7.80431e28 −1.56230 −0.781148 0.624346i \(-0.785365\pi\)
−0.781148 + 0.624346i \(0.785365\pi\)
\(542\) 0 0
\(543\) −5.08532e28 −0.979310
\(544\) 0 0
\(545\) 7.32332e28 1.35689
\(546\) 0 0
\(547\) 4.32938e28 0.771897 0.385948 0.922520i \(-0.373874\pi\)
0.385948 + 0.922520i \(0.373874\pi\)
\(548\) 0 0
\(549\) −1.68295e28 −0.288777
\(550\) 0 0
\(551\) 1.02292e29 1.68946
\(552\) 0 0
\(553\) 4.42120e28 0.702953
\(554\) 0 0
\(555\) −2.40679e28 −0.368436
\(556\) 0 0
\(557\) 1.94436e28 0.286614 0.143307 0.989678i \(-0.454226\pi\)
0.143307 + 0.989678i \(0.454226\pi\)
\(558\) 0 0
\(559\) −1.26372e29 −1.79401
\(560\) 0 0
\(561\) −1.18927e29 −1.62619
\(562\) 0 0
\(563\) 1.12608e29 1.48331 0.741655 0.670781i \(-0.234041\pi\)
0.741655 + 0.670781i \(0.234041\pi\)
\(564\) 0 0
\(565\) −1.11922e29 −1.42038
\(566\) 0 0
\(567\) 5.13918e28 0.628451
\(568\) 0 0
\(569\) 6.23100e28 0.734309 0.367154 0.930160i \(-0.380332\pi\)
0.367154 + 0.930160i \(0.380332\pi\)
\(570\) 0 0
\(571\) 1.12398e29 1.27667 0.638334 0.769760i \(-0.279624\pi\)
0.638334 + 0.769760i \(0.279624\pi\)
\(572\) 0 0
\(573\) 1.33763e29 1.46458
\(574\) 0 0
\(575\) −1.05146e28 −0.110990
\(576\) 0 0
\(577\) 1.21433e29 1.23592 0.617960 0.786210i \(-0.287960\pi\)
0.617960 + 0.786210i \(0.287960\pi\)
\(578\) 0 0
\(579\) 5.41788e28 0.531747
\(580\) 0 0
\(581\) 4.94644e28 0.468213
\(582\) 0 0
\(583\) −1.39006e29 −1.26916
\(584\) 0 0
\(585\) 2.77535e28 0.244445
\(586\) 0 0
\(587\) −6.84042e28 −0.581276 −0.290638 0.956833i \(-0.593868\pi\)
−0.290638 + 0.956833i \(0.593868\pi\)
\(588\) 0 0
\(589\) −2.61952e28 −0.214788
\(590\) 0 0
\(591\) −1.18598e29 −0.938444
\(592\) 0 0
\(593\) −5.52013e28 −0.421575 −0.210787 0.977532i \(-0.567603\pi\)
−0.210787 + 0.977532i \(0.567603\pi\)
\(594\) 0 0
\(595\) 1.30051e29 0.958709
\(596\) 0 0
\(597\) −9.43001e28 −0.671091
\(598\) 0 0
\(599\) 1.03055e29 0.708091 0.354046 0.935228i \(-0.384806\pi\)
0.354046 + 0.935228i \(0.384806\pi\)
\(600\) 0 0
\(601\) 1.81265e29 1.20263 0.601314 0.799013i \(-0.294644\pi\)
0.601314 + 0.799013i \(0.294644\pi\)
\(602\) 0 0
\(603\) −3.48869e28 −0.223527
\(604\) 0 0
\(605\) −9.59527e28 −0.593780
\(606\) 0 0
\(607\) 3.69189e28 0.220682 0.110341 0.993894i \(-0.464806\pi\)
0.110341 + 0.993894i \(0.464806\pi\)
\(608\) 0 0
\(609\) −1.37252e29 −0.792572
\(610\) 0 0
\(611\) 2.06298e29 1.15097
\(612\) 0 0
\(613\) 1.42961e29 0.770696 0.385348 0.922771i \(-0.374081\pi\)
0.385348 + 0.922771i \(0.374081\pi\)
\(614\) 0 0
\(615\) −2.07495e29 −1.08098
\(616\) 0 0
\(617\) −3.41658e29 −1.72027 −0.860137 0.510063i \(-0.829622\pi\)
−0.860137 + 0.510063i \(0.829622\pi\)
\(618\) 0 0
\(619\) −1.26714e29 −0.616700 −0.308350 0.951273i \(-0.599777\pi\)
−0.308350 + 0.951273i \(0.599777\pi\)
\(620\) 0 0
\(621\) −9.32511e28 −0.438725
\(622\) 0 0
\(623\) 1.35352e29 0.615658
\(624\) 0 0
\(625\) −1.49109e29 −0.655790
\(626\) 0 0
\(627\) −4.28902e29 −1.82410
\(628\) 0 0
\(629\) 1.61441e29 0.664021
\(630\) 0 0
\(631\) −8.54527e28 −0.339952 −0.169976 0.985448i \(-0.554369\pi\)
−0.169976 + 0.985448i \(0.554369\pi\)
\(632\) 0 0
\(633\) −9.17619e28 −0.353121
\(634\) 0 0
\(635\) −2.46977e29 −0.919457
\(636\) 0 0
\(637\) −1.45143e29 −0.522797
\(638\) 0 0
\(639\) 2.03607e28 0.0709634
\(640\) 0 0
\(641\) 3.29678e29 1.11194 0.555969 0.831203i \(-0.312347\pi\)
0.555969 + 0.831203i \(0.312347\pi\)
\(642\) 0 0
\(643\) −2.26473e29 −0.739268 −0.369634 0.929177i \(-0.620517\pi\)
−0.369634 + 0.929177i \(0.620517\pi\)
\(644\) 0 0
\(645\) 2.87546e29 0.908511
\(646\) 0 0
\(647\) 1.40922e29 0.431005 0.215503 0.976503i \(-0.430861\pi\)
0.215503 + 0.976503i \(0.430861\pi\)
\(648\) 0 0
\(649\) −4.43731e28 −0.131386
\(650\) 0 0
\(651\) 3.51481e28 0.100763
\(652\) 0 0
\(653\) −6.00388e29 −1.66665 −0.833323 0.552786i \(-0.813565\pi\)
−0.833323 + 0.552786i \(0.813565\pi\)
\(654\) 0 0
\(655\) 4.44705e29 1.19547
\(656\) 0 0
\(657\) 3.06324e28 0.0797526
\(658\) 0 0
\(659\) 5.22410e29 1.31739 0.658695 0.752410i \(-0.271109\pi\)
0.658695 + 0.752410i \(0.271109\pi\)
\(660\) 0 0
\(661\) 4.46063e29 1.08963 0.544817 0.838555i \(-0.316599\pi\)
0.544817 + 0.838555i \(0.316599\pi\)
\(662\) 0 0
\(663\) 8.01659e29 1.89713
\(664\) 0 0
\(665\) 4.69021e29 1.07539
\(666\) 0 0
\(667\) 2.00383e29 0.445185
\(668\) 0 0
\(669\) 4.08027e29 0.878445
\(670\) 0 0
\(671\) −9.56241e29 −1.99518
\(672\) 0 0
\(673\) −7.24333e29 −1.46481 −0.732403 0.680871i \(-0.761601\pi\)
−0.732403 + 0.680871i \(0.761601\pi\)
\(674\) 0 0
\(675\) 1.47931e29 0.289980
\(676\) 0 0
\(677\) 2.20060e29 0.418176 0.209088 0.977897i \(-0.432950\pi\)
0.209088 + 0.977897i \(0.432950\pi\)
\(678\) 0 0
\(679\) 1.68770e29 0.310930
\(680\) 0 0
\(681\) −3.90240e29 −0.697088
\(682\) 0 0
\(683\) −7.92621e29 −1.37293 −0.686465 0.727163i \(-0.740838\pi\)
−0.686465 + 0.727163i \(0.740838\pi\)
\(684\) 0 0
\(685\) 4.04350e28 0.0679214
\(686\) 0 0
\(687\) 6.05668e29 0.986708
\(688\) 0 0
\(689\) 9.37009e29 1.48061
\(690\) 0 0
\(691\) −8.05813e29 −1.23514 −0.617568 0.786517i \(-0.711882\pi\)
−0.617568 + 0.786517i \(0.711882\pi\)
\(692\) 0 0
\(693\) −1.33641e29 −0.198720
\(694\) 0 0
\(695\) 5.45932e29 0.787586
\(696\) 0 0
\(697\) 1.39182e30 1.94822
\(698\) 0 0
\(699\) −9.57057e28 −0.129995
\(700\) 0 0
\(701\) −8.00523e29 −1.05520 −0.527600 0.849493i \(-0.676908\pi\)
−0.527600 + 0.849493i \(0.676908\pi\)
\(702\) 0 0
\(703\) 5.82224e29 0.744834
\(704\) 0 0
\(705\) −4.69412e29 −0.582865
\(706\) 0 0
\(707\) −1.00639e30 −1.21300
\(708\) 0 0
\(709\) −1.20024e30 −1.40437 −0.702185 0.711994i \(-0.747792\pi\)
−0.702185 + 0.711994i \(0.747792\pi\)
\(710\) 0 0
\(711\) 1.44004e29 0.163585
\(712\) 0 0
\(713\) −5.13149e28 −0.0565983
\(714\) 0 0
\(715\) 1.57693e30 1.68889
\(716\) 0 0
\(717\) 5.38025e29 0.559566
\(718\) 0 0
\(719\) 3.17808e29 0.321004 0.160502 0.987035i \(-0.448689\pi\)
0.160502 + 0.987035i \(0.448689\pi\)
\(720\) 0 0
\(721\) −9.14727e29 −0.897369
\(722\) 0 0
\(723\) 3.54649e29 0.337946
\(724\) 0 0
\(725\) −3.17882e29 −0.294250
\(726\) 0 0
\(727\) −1.42314e30 −1.27978 −0.639892 0.768465i \(-0.721021\pi\)
−0.639892 + 0.768465i \(0.721021\pi\)
\(728\) 0 0
\(729\) 1.27130e30 1.11073
\(730\) 0 0
\(731\) −1.92877e30 −1.63738
\(732\) 0 0
\(733\) −1.17834e30 −0.972031 −0.486016 0.873950i \(-0.661550\pi\)
−0.486016 + 0.873950i \(0.661550\pi\)
\(734\) 0 0
\(735\) 3.30259e29 0.264751
\(736\) 0 0
\(737\) −1.98225e30 −1.54436
\(738\) 0 0
\(739\) −1.73577e30 −1.31440 −0.657198 0.753718i \(-0.728258\pi\)
−0.657198 + 0.753718i \(0.728258\pi\)
\(740\) 0 0
\(741\) 2.89112e30 2.12802
\(742\) 0 0
\(743\) −4.69683e29 −0.336064 −0.168032 0.985782i \(-0.553741\pi\)
−0.168032 + 0.985782i \(0.553741\pi\)
\(744\) 0 0
\(745\) 4.66561e29 0.324540
\(746\) 0 0
\(747\) 1.61112e29 0.108959
\(748\) 0 0
\(749\) −1.37046e30 −0.901170
\(750\) 0 0
\(751\) 7.47353e29 0.477866 0.238933 0.971036i \(-0.423202\pi\)
0.238933 + 0.971036i \(0.423202\pi\)
\(752\) 0 0
\(753\) −2.64132e30 −1.64238
\(754\) 0 0
\(755\) −1.32314e30 −0.800135
\(756\) 0 0
\(757\) 2.10588e30 1.23859 0.619293 0.785160i \(-0.287419\pi\)
0.619293 + 0.785160i \(0.287419\pi\)
\(758\) 0 0
\(759\) −8.40194e29 −0.480664
\(760\) 0 0
\(761\) −9.10629e29 −0.506760 −0.253380 0.967367i \(-0.581542\pi\)
−0.253380 + 0.967367i \(0.581542\pi\)
\(762\) 0 0
\(763\) 2.37706e30 1.28686
\(764\) 0 0
\(765\) 4.23595e29 0.223103
\(766\) 0 0
\(767\) 2.99108e29 0.153277
\(768\) 0 0
\(769\) 2.59434e30 1.29360 0.646801 0.762659i \(-0.276106\pi\)
0.646801 + 0.762659i \(0.276106\pi\)
\(770\) 0 0
\(771\) 3.38369e30 1.64180
\(772\) 0 0
\(773\) −1.32692e30 −0.626559 −0.313279 0.949661i \(-0.601428\pi\)
−0.313279 + 0.949661i \(0.601428\pi\)
\(774\) 0 0
\(775\) 8.14042e28 0.0374092
\(776\) 0 0
\(777\) −7.81215e29 −0.349422
\(778\) 0 0
\(779\) 5.01949e30 2.18533
\(780\) 0 0
\(781\) 1.15688e30 0.490290
\(782\) 0 0
\(783\) −2.81920e30 −1.16313
\(784\) 0 0
\(785\) −7.89298e29 −0.317037
\(786\) 0 0
\(787\) 5.12807e29 0.200548 0.100274 0.994960i \(-0.468028\pi\)
0.100274 + 0.994960i \(0.468028\pi\)
\(788\) 0 0
\(789\) 1.65360e30 0.629684
\(790\) 0 0
\(791\) −3.63284e30 −1.34708
\(792\) 0 0
\(793\) 6.44579e30 2.32760
\(794\) 0 0
\(795\) −2.13207e30 −0.749802
\(796\) 0 0
\(797\) −1.27653e30 −0.437239 −0.218620 0.975810i \(-0.570155\pi\)
−0.218620 + 0.975810i \(0.570155\pi\)
\(798\) 0 0
\(799\) 3.14868e30 1.05048
\(800\) 0 0
\(801\) 4.40859e29 0.143271
\(802\) 0 0
\(803\) 1.74051e30 0.551015
\(804\) 0 0
\(805\) 9.18785e29 0.283372
\(806\) 0 0
\(807\) −6.31951e29 −0.189894
\(808\) 0 0
\(809\) 1.38086e30 0.404287 0.202144 0.979356i \(-0.435209\pi\)
0.202144 + 0.979356i \(0.435209\pi\)
\(810\) 0 0
\(811\) −4.03179e29 −0.115021 −0.0575107 0.998345i \(-0.518316\pi\)
−0.0575107 + 0.998345i \(0.518316\pi\)
\(812\) 0 0
\(813\) −9.17657e29 −0.255111
\(814\) 0 0
\(815\) −3.78607e29 −0.102573
\(816\) 0 0
\(817\) −6.95598e30 −1.83665
\(818\) 0 0
\(819\) 9.00845e29 0.231830
\(820\) 0 0
\(821\) −3.10056e29 −0.0777745 −0.0388873 0.999244i \(-0.512381\pi\)
−0.0388873 + 0.999244i \(0.512381\pi\)
\(822\) 0 0
\(823\) −1.10242e30 −0.269555 −0.134777 0.990876i \(-0.543032\pi\)
−0.134777 + 0.990876i \(0.543032\pi\)
\(824\) 0 0
\(825\) 1.33286e30 0.317700
\(826\) 0 0
\(827\) 9.82951e29 0.228415 0.114207 0.993457i \(-0.463567\pi\)
0.114207 + 0.993457i \(0.463567\pi\)
\(828\) 0 0
\(829\) −5.60776e30 −1.27048 −0.635239 0.772316i \(-0.719098\pi\)
−0.635239 + 0.772316i \(0.719098\pi\)
\(830\) 0 0
\(831\) −1.04344e30 −0.230492
\(832\) 0 0
\(833\) −2.21528e30 −0.477152
\(834\) 0 0
\(835\) −2.75504e26 −5.78655e−5 0
\(836\) 0 0
\(837\) 7.21949e29 0.147873
\(838\) 0 0
\(839\) −3.92515e30 −0.784071 −0.392036 0.919950i \(-0.628229\pi\)
−0.392036 + 0.919950i \(0.628229\pi\)
\(840\) 0 0
\(841\) 9.25212e29 0.180253
\(842\) 0 0
\(843\) −1.61012e28 −0.00305963
\(844\) 0 0
\(845\) −6.02288e30 −1.11637
\(846\) 0 0
\(847\) −3.11450e30 −0.563135
\(848\) 0 0
\(849\) 2.63155e28 0.00464175
\(850\) 0 0
\(851\) 1.14054e30 0.196269
\(852\) 0 0
\(853\) 9.01715e30 1.51393 0.756963 0.653458i \(-0.226683\pi\)
0.756963 + 0.653458i \(0.226683\pi\)
\(854\) 0 0
\(855\) 1.52766e30 0.250255
\(856\) 0 0
\(857\) 7.89609e30 1.26216 0.631079 0.775719i \(-0.282612\pi\)
0.631079 + 0.775719i \(0.282612\pi\)
\(858\) 0 0
\(859\) −3.91414e30 −0.610531 −0.305266 0.952267i \(-0.598745\pi\)
−0.305266 + 0.952267i \(0.598745\pi\)
\(860\) 0 0
\(861\) −6.73503e30 −1.02520
\(862\) 0 0
\(863\) −1.32795e31 −1.97273 −0.986365 0.164572i \(-0.947376\pi\)
−0.986365 + 0.164572i \(0.947376\pi\)
\(864\) 0 0
\(865\) −3.48826e29 −0.0505755
\(866\) 0 0
\(867\) 5.86966e30 0.830639
\(868\) 0 0
\(869\) 8.18221e30 1.13022
\(870\) 0 0
\(871\) 1.33618e31 1.80167
\(872\) 0 0
\(873\) 5.49707e29 0.0723571
\(874\) 0 0
\(875\) −6.83884e30 −0.878816
\(876\) 0 0
\(877\) −1.33474e31 −1.67456 −0.837278 0.546778i \(-0.815854\pi\)
−0.837278 + 0.546778i \(0.815854\pi\)
\(878\) 0 0
\(879\) 2.03975e30 0.249859
\(880\) 0 0
\(881\) −1.47240e31 −1.76108 −0.880538 0.473976i \(-0.842818\pi\)
−0.880538 + 0.473976i \(0.842818\pi\)
\(882\) 0 0
\(883\) −4.20820e29 −0.0491484 −0.0245742 0.999698i \(-0.507823\pi\)
−0.0245742 + 0.999698i \(0.507823\pi\)
\(884\) 0 0
\(885\) −6.80591e29 −0.0776215
\(886\) 0 0
\(887\) −3.56495e30 −0.397060 −0.198530 0.980095i \(-0.563617\pi\)
−0.198530 + 0.980095i \(0.563617\pi\)
\(888\) 0 0
\(889\) −8.01654e30 −0.872004
\(890\) 0 0
\(891\) 9.51097e30 1.01044
\(892\) 0 0
\(893\) 1.13555e31 1.17832
\(894\) 0 0
\(895\) −4.00375e30 −0.405812
\(896\) 0 0
\(897\) 5.66355e30 0.560748
\(898\) 0 0
\(899\) −1.55137e30 −0.150051
\(900\) 0 0
\(901\) 1.43013e31 1.35134
\(902\) 0 0
\(903\) 9.33337e30 0.861623
\(904\) 0 0
\(905\) 1.02916e31 0.928267
\(906\) 0 0
\(907\) 8.63550e30 0.761046 0.380523 0.924771i \(-0.375744\pi\)
0.380523 + 0.924771i \(0.375744\pi\)
\(908\) 0 0
\(909\) −3.27794e30 −0.282280
\(910\) 0 0
\(911\) 2.20841e31 1.85839 0.929197 0.369586i \(-0.120500\pi\)
0.929197 + 0.369586i \(0.120500\pi\)
\(912\) 0 0
\(913\) 9.15426e30 0.752801
\(914\) 0 0
\(915\) −1.46668e31 −1.17872
\(916\) 0 0
\(917\) 1.44346e31 1.13377
\(918\) 0 0
\(919\) −2.56943e30 −0.197253 −0.0986265 0.995125i \(-0.531445\pi\)
−0.0986265 + 0.995125i \(0.531445\pi\)
\(920\) 0 0
\(921\) 9.25792e30 0.694684
\(922\) 0 0
\(923\) −7.79825e30 −0.571978
\(924\) 0 0
\(925\) −1.80932e30 −0.129726
\(926\) 0 0
\(927\) −2.97939e30 −0.208828
\(928\) 0 0
\(929\) −1.85518e31 −1.27122 −0.635611 0.772010i \(-0.719252\pi\)
−0.635611 + 0.772010i \(0.719252\pi\)
\(930\) 0 0
\(931\) −7.98925e30 −0.535223
\(932\) 0 0
\(933\) 1.18974e31 0.779279
\(934\) 0 0
\(935\) 2.40683e31 1.54143
\(936\) 0 0
\(937\) −5.89731e30 −0.369308 −0.184654 0.982804i \(-0.559116\pi\)
−0.184654 + 0.982804i \(0.559116\pi\)
\(938\) 0 0
\(939\) −1.46950e31 −0.899871
\(940\) 0 0
\(941\) −1.82883e31 −1.09517 −0.547587 0.836749i \(-0.684453\pi\)
−0.547587 + 0.836749i \(0.684453\pi\)
\(942\) 0 0
\(943\) 9.83289e30 0.575849
\(944\) 0 0
\(945\) −1.29264e31 −0.740360
\(946\) 0 0
\(947\) 5.73590e30 0.321312 0.160656 0.987010i \(-0.448639\pi\)
0.160656 + 0.987010i \(0.448639\pi\)
\(948\) 0 0
\(949\) −1.17324e31 −0.642821
\(950\) 0 0
\(951\) 8.04649e30 0.431232
\(952\) 0 0
\(953\) 1.27548e31 0.668647 0.334324 0.942458i \(-0.391492\pi\)
0.334324 + 0.942458i \(0.391492\pi\)
\(954\) 0 0
\(955\) −2.70708e31 −1.38825
\(956\) 0 0
\(957\) −2.54010e31 −1.27431
\(958\) 0 0
\(959\) 1.31247e30 0.0644160
\(960\) 0 0
\(961\) −2.04282e31 −0.980923
\(962\) 0 0
\(963\) −4.46377e30 −0.209713
\(964\) 0 0
\(965\) −1.09646e31 −0.504031
\(966\) 0 0
\(967\) −2.48789e31 −1.11906 −0.559531 0.828810i \(-0.689019\pi\)
−0.559531 + 0.828810i \(0.689019\pi\)
\(968\) 0 0
\(969\) 4.41264e31 1.94222
\(970\) 0 0
\(971\) −2.93374e31 −1.26363 −0.631814 0.775120i \(-0.717689\pi\)
−0.631814 + 0.775120i \(0.717689\pi\)
\(972\) 0 0
\(973\) 1.77203e31 0.746939
\(974\) 0 0
\(975\) −8.98446e30 −0.370633
\(976\) 0 0
\(977\) −1.41713e31 −0.572159 −0.286080 0.958206i \(-0.592352\pi\)
−0.286080 + 0.958206i \(0.592352\pi\)
\(978\) 0 0
\(979\) 2.50493e31 0.989867
\(980\) 0 0
\(981\) 7.74239e30 0.299468
\(982\) 0 0
\(983\) 3.05921e31 1.15824 0.579119 0.815243i \(-0.303397\pi\)
0.579119 + 0.815243i \(0.303397\pi\)
\(984\) 0 0
\(985\) 2.40017e31 0.889532
\(986\) 0 0
\(987\) −1.52365e31 −0.552784
\(988\) 0 0
\(989\) −1.36264e31 −0.483971
\(990\) 0 0
\(991\) −4.24783e31 −1.47704 −0.738522 0.674229i \(-0.764476\pi\)
−0.738522 + 0.674229i \(0.764476\pi\)
\(992\) 0 0
\(993\) −5.07095e31 −1.72632
\(994\) 0 0
\(995\) 1.90843e31 0.636113
\(996\) 0 0
\(997\) −1.55461e31 −0.507369 −0.253684 0.967287i \(-0.581642\pi\)
−0.253684 + 0.967287i \(0.581642\pi\)
\(998\) 0 0
\(999\) −1.60463e31 −0.512788
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 16.22.a.d.1.2 2
4.3 odd 2 4.22.a.a.1.1 2
8.3 odd 2 64.22.a.i.1.2 2
8.5 even 2 64.22.a.j.1.1 2
12.11 even 2 36.22.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.22.a.a.1.1 2 4.3 odd 2
16.22.a.d.1.2 2 1.1 even 1 trivial
36.22.a.c.1.2 2 12.11 even 2
64.22.a.i.1.2 2 8.3 odd 2
64.22.a.j.1.1 2 8.5 even 2