Properties

Label 592.8.a.b.1.3
Level $592$
Weight $8$
Character 592.1
Self dual yes
Analytic conductor $184.932$
Analytic rank $0$
Dimension $4$
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] = [592,8,Mod(1,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 592.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: \(184.931935087\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 405x^{2} - 2998x - 4396 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-13.3612\) of defining polynomial
Character \(\chi\) \(=\) 592.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+36.0598 q^{3} -19.7362 q^{5} +50.9272 q^{7} -886.690 q^{9} +O(q^{10})\) \(q+36.0598 q^{3} -19.7362 q^{5} +50.9272 q^{7} -886.690 q^{9} -5175.62 q^{11} +4311.51 q^{13} -711.683 q^{15} +14063.5 q^{17} -26643.5 q^{19} +1836.43 q^{21} +77473.7 q^{23} -77735.5 q^{25} -110837. q^{27} -33124.8 q^{29} +172834. q^{31} -186632. q^{33} -1005.11 q^{35} +50653.0 q^{37} +155472. q^{39} -846738. q^{41} +307387. q^{43} +17499.9 q^{45} +322473. q^{47} -820949. q^{49} +507128. q^{51} +1.83562e6 q^{53} +102147. q^{55} -960758. q^{57} +797340. q^{59} -684801. q^{61} -45156.6 q^{63} -85092.7 q^{65} +3.79463e6 q^{67} +2.79369e6 q^{69} +342906. q^{71} +2.64684e6 q^{73} -2.80313e6 q^{75} -263580. q^{77} +2.44136e6 q^{79} -2.05756e6 q^{81} -5.61596e6 q^{83} -277560. q^{85} -1.19447e6 q^{87} +8.43578e6 q^{89} +219573. q^{91} +6.23237e6 q^{93} +525840. q^{95} -3.40422e6 q^{97} +4.58917e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 53 q^{3} + 111 q^{5} + 1666 q^{7} + 4609 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 53 q^{3} + 111 q^{5} + 1666 q^{7} + 4609 q^{9} + 4593 q^{11} + 7847 q^{13} - 18900 q^{15} + 23172 q^{17} - 23696 q^{19} + 69416 q^{21} - 24105 q^{23} - 138149 q^{25} + 433646 q^{27} - 140949 q^{29} + 664609 q^{31} - 240450 q^{33} + 248544 q^{35} + 202612 q^{37} + 2288827 q^{39} - 709737 q^{41} + 128962 q^{43} - 1755342 q^{45} + 445842 q^{47} - 1602774 q^{49} + 2883630 q^{51} - 975870 q^{53} + 644145 q^{55} + 3494630 q^{57} + 1812858 q^{59} - 2955031 q^{61} + 3362482 q^{63} + 666 q^{65} - 2737235 q^{67} - 1781673 q^{69} - 4958184 q^{71} - 931591 q^{73} - 4945810 q^{75} + 4352514 q^{77} - 5813561 q^{79} + 16394896 q^{81} - 2120460 q^{83} - 4845402 q^{85} - 7965333 q^{87} + 8833716 q^{89} + 18886274 q^{91} + 3024182 q^{93} + 3151794 q^{95} - 22666876 q^{97} + 17931894 q^{99}+O(q^{100}) \) 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
\(3\) 36.0598 0.771079 0.385540 0.922691i \(-0.374015\pi\)
0.385540 + 0.922691i \(0.374015\pi\)
\(4\) 0 0
\(5\) −19.7362 −0.0706103 −0.0353051 0.999377i \(-0.511240\pi\)
−0.0353051 + 0.999377i \(0.511240\pi\)
\(6\) 0 0
\(7\) 50.9272 0.0561186 0.0280593 0.999606i \(-0.491067\pi\)
0.0280593 + 0.999606i \(0.491067\pi\)
\(8\) 0 0
\(9\) −886.690 −0.405436
\(10\) 0 0
\(11\) −5175.62 −1.17243 −0.586217 0.810154i \(-0.699383\pi\)
−0.586217 + 0.810154i \(0.699383\pi\)
\(12\) 0 0
\(13\) 4311.51 0.544287 0.272143 0.962257i \(-0.412267\pi\)
0.272143 + 0.962257i \(0.412267\pi\)
\(14\) 0 0
\(15\) −711.683 −0.0544461
\(16\) 0 0
\(17\) 14063.5 0.694262 0.347131 0.937817i \(-0.387156\pi\)
0.347131 + 0.937817i \(0.387156\pi\)
\(18\) 0 0
\(19\) −26643.5 −0.891154 −0.445577 0.895244i \(-0.647001\pi\)
−0.445577 + 0.895244i \(0.647001\pi\)
\(20\) 0 0
\(21\) 1836.43 0.0432719
\(22\) 0 0
\(23\) 77473.7 1.32772 0.663861 0.747856i \(-0.268917\pi\)
0.663861 + 0.747856i \(0.268917\pi\)
\(24\) 0 0
\(25\) −77735.5 −0.995014
\(26\) 0 0
\(27\) −110837. −1.08370
\(28\) 0 0
\(29\) −33124.8 −0.252209 −0.126104 0.992017i \(-0.540247\pi\)
−0.126104 + 0.992017i \(0.540247\pi\)
\(30\) 0 0
\(31\) 172834. 1.04199 0.520995 0.853560i \(-0.325561\pi\)
0.520995 + 0.853560i \(0.325561\pi\)
\(32\) 0 0
\(33\) −186632. −0.904039
\(34\) 0 0
\(35\) −1005.11 −0.00396255
\(36\) 0 0
\(37\) 50653.0 0.164399
\(38\) 0 0
\(39\) 155472. 0.419688
\(40\) 0 0
\(41\) −846738. −1.91869 −0.959347 0.282229i \(-0.908926\pi\)
−0.959347 + 0.282229i \(0.908926\pi\)
\(42\) 0 0
\(43\) 307387. 0.589585 0.294792 0.955561i \(-0.404750\pi\)
0.294792 + 0.955561i \(0.404750\pi\)
\(44\) 0 0
\(45\) 17499.9 0.0286280
\(46\) 0 0
\(47\) 322473. 0.453056 0.226528 0.974005i \(-0.427263\pi\)
0.226528 + 0.974005i \(0.427263\pi\)
\(48\) 0 0
\(49\) −820949. −0.996851
\(50\) 0 0
\(51\) 507128. 0.535331
\(52\) 0 0
\(53\) 1.83562e6 1.69363 0.846814 0.531890i \(-0.178518\pi\)
0.846814 + 0.531890i \(0.178518\pi\)
\(54\) 0 0
\(55\) 102147. 0.0827859
\(56\) 0 0
\(57\) −960758. −0.687151
\(58\) 0 0
\(59\) 797340. 0.505430 0.252715 0.967541i \(-0.418676\pi\)
0.252715 + 0.967541i \(0.418676\pi\)
\(60\) 0 0
\(61\) −684801. −0.386287 −0.193144 0.981171i \(-0.561868\pi\)
−0.193144 + 0.981171i \(0.561868\pi\)
\(62\) 0 0
\(63\) −45156.6 −0.0227525
\(64\) 0 0
\(65\) −85092.7 −0.0384322
\(66\) 0 0
\(67\) 3.79463e6 1.54137 0.770687 0.637214i \(-0.219913\pi\)
0.770687 + 0.637214i \(0.219913\pi\)
\(68\) 0 0
\(69\) 2.79369e6 1.02378
\(70\) 0 0
\(71\) 342906. 0.113703 0.0568514 0.998383i \(-0.481894\pi\)
0.0568514 + 0.998383i \(0.481894\pi\)
\(72\) 0 0
\(73\) 2.64684e6 0.796337 0.398168 0.917312i \(-0.369646\pi\)
0.398168 + 0.917312i \(0.369646\pi\)
\(74\) 0 0
\(75\) −2.80313e6 −0.767235
\(76\) 0 0
\(77\) −263580. −0.0657953
\(78\) 0 0
\(79\) 2.44136e6 0.557104 0.278552 0.960421i \(-0.410146\pi\)
0.278552 + 0.960421i \(0.410146\pi\)
\(80\) 0 0
\(81\) −2.05756e6 −0.430185
\(82\) 0 0
\(83\) −5.61596e6 −1.07808 −0.539040 0.842280i \(-0.681213\pi\)
−0.539040 + 0.842280i \(0.681213\pi\)
\(84\) 0 0
\(85\) −277560. −0.0490220
\(86\) 0 0
\(87\) −1.19447e6 −0.194473
\(88\) 0 0
\(89\) 8.43578e6 1.26841 0.634206 0.773164i \(-0.281327\pi\)
0.634206 + 0.773164i \(0.281327\pi\)
\(90\) 0 0
\(91\) 219573. 0.0305446
\(92\) 0 0
\(93\) 6.23237e6 0.803457
\(94\) 0 0
\(95\) 525840. 0.0629246
\(96\) 0 0
\(97\) −3.40422e6 −0.378718 −0.189359 0.981908i \(-0.560641\pi\)
−0.189359 + 0.981908i \(0.560641\pi\)
\(98\) 0 0
\(99\) 4.58917e6 0.475347
\(100\) 0 0
\(101\) −1.62461e6 −0.156901 −0.0784504 0.996918i \(-0.524997\pi\)
−0.0784504 + 0.996918i \(0.524997\pi\)
\(102\) 0 0
\(103\) 7.21129e6 0.650253 0.325127 0.945671i \(-0.394593\pi\)
0.325127 + 0.945671i \(0.394593\pi\)
\(104\) 0 0
\(105\) −36244.0 −0.00305544
\(106\) 0 0
\(107\) 1.38534e7 1.09323 0.546616 0.837383i \(-0.315916\pi\)
0.546616 + 0.837383i \(0.315916\pi\)
\(108\) 0 0
\(109\) −5.56203e6 −0.411377 −0.205689 0.978617i \(-0.565943\pi\)
−0.205689 + 0.978617i \(0.565943\pi\)
\(110\) 0 0
\(111\) 1.82654e6 0.126765
\(112\) 0 0
\(113\) −2.13570e7 −1.39241 −0.696203 0.717845i \(-0.745129\pi\)
−0.696203 + 0.717845i \(0.745129\pi\)
\(114\) 0 0
\(115\) −1.52903e6 −0.0937508
\(116\) 0 0
\(117\) −3.82297e6 −0.220674
\(118\) 0 0
\(119\) 716216. 0.0389610
\(120\) 0 0
\(121\) 7.29990e6 0.374600
\(122\) 0 0
\(123\) −3.05332e7 −1.47947
\(124\) 0 0
\(125\) 3.07609e6 0.140869
\(126\) 0 0
\(127\) 2.51759e7 1.09062 0.545308 0.838236i \(-0.316413\pi\)
0.545308 + 0.838236i \(0.316413\pi\)
\(128\) 0 0
\(129\) 1.10843e7 0.454617
\(130\) 0 0
\(131\) 2.73598e7 1.06332 0.531659 0.846959i \(-0.321569\pi\)
0.531659 + 0.846959i \(0.321569\pi\)
\(132\) 0 0
\(133\) −1.35688e6 −0.0500103
\(134\) 0 0
\(135\) 2.18749e6 0.0765206
\(136\) 0 0
\(137\) 3.06260e7 1.01758 0.508789 0.860891i \(-0.330093\pi\)
0.508789 + 0.860891i \(0.330093\pi\)
\(138\) 0 0
\(139\) 3.33288e7 1.05261 0.526305 0.850296i \(-0.323577\pi\)
0.526305 + 0.850296i \(0.323577\pi\)
\(140\) 0 0
\(141\) 1.16283e7 0.349342
\(142\) 0 0
\(143\) −2.23148e7 −0.638140
\(144\) 0 0
\(145\) 653756. 0.0178085
\(146\) 0 0
\(147\) −2.96033e7 −0.768651
\(148\) 0 0
\(149\) 7.15248e6 0.177135 0.0885675 0.996070i \(-0.471771\pi\)
0.0885675 + 0.996070i \(0.471771\pi\)
\(150\) 0 0
\(151\) 4.28753e7 1.01342 0.506708 0.862118i \(-0.330862\pi\)
0.506708 + 0.862118i \(0.330862\pi\)
\(152\) 0 0
\(153\) −1.24700e7 −0.281479
\(154\) 0 0
\(155\) −3.41108e6 −0.0735752
\(156\) 0 0
\(157\) 6.89941e7 1.42286 0.711431 0.702756i \(-0.248047\pi\)
0.711431 + 0.702756i \(0.248047\pi\)
\(158\) 0 0
\(159\) 6.61922e7 1.30592
\(160\) 0 0
\(161\) 3.94552e6 0.0745098
\(162\) 0 0
\(163\) 2.76360e7 0.499826 0.249913 0.968268i \(-0.419598\pi\)
0.249913 + 0.968268i \(0.419598\pi\)
\(164\) 0 0
\(165\) 3.68340e6 0.0638345
\(166\) 0 0
\(167\) −4.33284e7 −0.719888 −0.359944 0.932974i \(-0.617204\pi\)
−0.359944 + 0.932974i \(0.617204\pi\)
\(168\) 0 0
\(169\) −4.41594e7 −0.703752
\(170\) 0 0
\(171\) 2.36245e7 0.361306
\(172\) 0 0
\(173\) 3.02184e7 0.443722 0.221861 0.975078i \(-0.428787\pi\)
0.221861 + 0.975078i \(0.428787\pi\)
\(174\) 0 0
\(175\) −3.95885e6 −0.0558388
\(176\) 0 0
\(177\) 2.87519e7 0.389727
\(178\) 0 0
\(179\) −1.27627e8 −1.66325 −0.831624 0.555340i \(-0.812588\pi\)
−0.831624 + 0.555340i \(0.812588\pi\)
\(180\) 0 0
\(181\) 4.76747e7 0.597603 0.298802 0.954315i \(-0.403413\pi\)
0.298802 + 0.954315i \(0.403413\pi\)
\(182\) 0 0
\(183\) −2.46938e7 −0.297858
\(184\) 0 0
\(185\) −999696. −0.0116083
\(186\) 0 0
\(187\) −7.27875e7 −0.813976
\(188\) 0 0
\(189\) −5.64460e6 −0.0608159
\(190\) 0 0
\(191\) −8.01658e7 −0.832477 −0.416239 0.909255i \(-0.636652\pi\)
−0.416239 + 0.909255i \(0.636652\pi\)
\(192\) 0 0
\(193\) 1.63480e6 0.0163687 0.00818436 0.999967i \(-0.497395\pi\)
0.00818436 + 0.999967i \(0.497395\pi\)
\(194\) 0 0
\(195\) −3.06843e6 −0.0296343
\(196\) 0 0
\(197\) −1.06595e8 −0.993356 −0.496678 0.867935i \(-0.665447\pi\)
−0.496678 + 0.867935i \(0.665447\pi\)
\(198\) 0 0
\(199\) −2.24394e7 −0.201848 −0.100924 0.994894i \(-0.532180\pi\)
−0.100924 + 0.994894i \(0.532180\pi\)
\(200\) 0 0
\(201\) 1.36834e8 1.18852
\(202\) 0 0
\(203\) −1.68695e6 −0.0141536
\(204\) 0 0
\(205\) 1.67114e7 0.135480
\(206\) 0 0
\(207\) −6.86951e7 −0.538307
\(208\) 0 0
\(209\) 1.37896e8 1.04482
\(210\) 0 0
\(211\) 2.68426e7 0.196714 0.0983571 0.995151i \(-0.468641\pi\)
0.0983571 + 0.995151i \(0.468641\pi\)
\(212\) 0 0
\(213\) 1.23651e7 0.0876739
\(214\) 0 0
\(215\) −6.06665e6 −0.0416307
\(216\) 0 0
\(217\) 8.80196e6 0.0584750
\(218\) 0 0
\(219\) 9.54444e7 0.614039
\(220\) 0 0
\(221\) 6.06351e7 0.377877
\(222\) 0 0
\(223\) 1.08818e7 0.0657102 0.0328551 0.999460i \(-0.489540\pi\)
0.0328551 + 0.999460i \(0.489540\pi\)
\(224\) 0 0
\(225\) 6.89272e7 0.403415
\(226\) 0 0
\(227\) 1.56786e8 0.889642 0.444821 0.895619i \(-0.353267\pi\)
0.444821 + 0.895619i \(0.353267\pi\)
\(228\) 0 0
\(229\) −1.95778e8 −1.07731 −0.538653 0.842528i \(-0.681067\pi\)
−0.538653 + 0.842528i \(0.681067\pi\)
\(230\) 0 0
\(231\) −9.50465e6 −0.0507334
\(232\) 0 0
\(233\) 1.45843e8 0.755337 0.377668 0.925941i \(-0.376726\pi\)
0.377668 + 0.925941i \(0.376726\pi\)
\(234\) 0 0
\(235\) −6.36439e6 −0.0319904
\(236\) 0 0
\(237\) 8.80348e7 0.429571
\(238\) 0 0
\(239\) 3.52653e8 1.67092 0.835458 0.549555i \(-0.185203\pi\)
0.835458 + 0.549555i \(0.185203\pi\)
\(240\) 0 0
\(241\) −3.09151e8 −1.42269 −0.711346 0.702842i \(-0.751914\pi\)
−0.711346 + 0.702842i \(0.751914\pi\)
\(242\) 0 0
\(243\) 1.68205e8 0.751997
\(244\) 0 0
\(245\) 1.62024e7 0.0703879
\(246\) 0 0
\(247\) −1.14874e8 −0.485043
\(248\) 0 0
\(249\) −2.02511e8 −0.831285
\(250\) 0 0
\(251\) 1.96899e8 0.785932 0.392966 0.919553i \(-0.371449\pi\)
0.392966 + 0.919553i \(0.371449\pi\)
\(252\) 0 0
\(253\) −4.00975e8 −1.55666
\(254\) 0 0
\(255\) −1.00088e7 −0.0377999
\(256\) 0 0
\(257\) 3.14559e8 1.15594 0.577972 0.816057i \(-0.303844\pi\)
0.577972 + 0.816057i \(0.303844\pi\)
\(258\) 0 0
\(259\) 2.57962e6 0.00922584
\(260\) 0 0
\(261\) 2.93714e7 0.102255
\(262\) 0 0
\(263\) −5.69257e7 −0.192958 −0.0964792 0.995335i \(-0.530758\pi\)
−0.0964792 + 0.995335i \(0.530758\pi\)
\(264\) 0 0
\(265\) −3.62282e7 −0.119587
\(266\) 0 0
\(267\) 3.04193e8 0.978046
\(268\) 0 0
\(269\) −3.56640e7 −0.111711 −0.0558557 0.998439i \(-0.517789\pi\)
−0.0558557 + 0.998439i \(0.517789\pi\)
\(270\) 0 0
\(271\) 5.25367e8 1.60350 0.801752 0.597657i \(-0.203901\pi\)
0.801752 + 0.597657i \(0.203901\pi\)
\(272\) 0 0
\(273\) 7.91777e6 0.0235523
\(274\) 0 0
\(275\) 4.02330e8 1.16659
\(276\) 0 0
\(277\) −6.45650e8 −1.82523 −0.912616 0.408818i \(-0.865941\pi\)
−0.912616 + 0.408818i \(0.865941\pi\)
\(278\) 0 0
\(279\) −1.53250e8 −0.422461
\(280\) 0 0
\(281\) 4.38689e8 1.17946 0.589731 0.807599i \(-0.299234\pi\)
0.589731 + 0.807599i \(0.299234\pi\)
\(282\) 0 0
\(283\) −1.91165e8 −0.501367 −0.250683 0.968069i \(-0.580655\pi\)
−0.250683 + 0.968069i \(0.580655\pi\)
\(284\) 0 0
\(285\) 1.89617e7 0.0485199
\(286\) 0 0
\(287\) −4.31220e7 −0.107674
\(288\) 0 0
\(289\) −2.12556e8 −0.518001
\(290\) 0 0
\(291\) −1.22755e8 −0.292022
\(292\) 0 0
\(293\) 6.78441e8 1.57571 0.787854 0.615862i \(-0.211192\pi\)
0.787854 + 0.615862i \(0.211192\pi\)
\(294\) 0 0
\(295\) −1.57364e7 −0.0356886
\(296\) 0 0
\(297\) 5.73649e8 1.27057
\(298\) 0 0
\(299\) 3.34029e8 0.722661
\(300\) 0 0
\(301\) 1.56544e7 0.0330867
\(302\) 0 0
\(303\) −5.85833e7 −0.120983
\(304\) 0 0
\(305\) 1.35154e7 0.0272758
\(306\) 0 0
\(307\) 4.40928e8 0.869728 0.434864 0.900496i \(-0.356797\pi\)
0.434864 + 0.900496i \(0.356797\pi\)
\(308\) 0 0
\(309\) 2.60038e8 0.501397
\(310\) 0 0
\(311\) 6.01820e8 1.13450 0.567251 0.823545i \(-0.308007\pi\)
0.567251 + 0.823545i \(0.308007\pi\)
\(312\) 0 0
\(313\) 6.38537e8 1.17701 0.588506 0.808493i \(-0.299716\pi\)
0.588506 + 0.808493i \(0.299716\pi\)
\(314\) 0 0
\(315\) 891219. 0.00160656
\(316\) 0 0
\(317\) 6.62098e8 1.16739 0.583693 0.811974i \(-0.301607\pi\)
0.583693 + 0.811974i \(0.301607\pi\)
\(318\) 0 0
\(319\) 1.71441e8 0.295698
\(320\) 0 0
\(321\) 4.99550e8 0.842969
\(322\) 0 0
\(323\) −3.74701e8 −0.618694
\(324\) 0 0
\(325\) −3.35157e8 −0.541573
\(326\) 0 0
\(327\) −2.00566e8 −0.317205
\(328\) 0 0
\(329\) 1.64227e7 0.0254248
\(330\) 0 0
\(331\) 4.62478e8 0.700960 0.350480 0.936570i \(-0.386018\pi\)
0.350480 + 0.936570i \(0.386018\pi\)
\(332\) 0 0
\(333\) −4.49135e7 −0.0666533
\(334\) 0 0
\(335\) −7.48915e7 −0.108837
\(336\) 0 0
\(337\) 6.65983e8 0.947891 0.473946 0.880554i \(-0.342829\pi\)
0.473946 + 0.880554i \(0.342829\pi\)
\(338\) 0 0
\(339\) −7.70130e8 −1.07366
\(340\) 0 0
\(341\) −8.94524e8 −1.22166
\(342\) 0 0
\(343\) −8.37494e7 −0.112060
\(344\) 0 0
\(345\) −5.51367e7 −0.0722893
\(346\) 0 0
\(347\) −3.49930e8 −0.449601 −0.224801 0.974405i \(-0.572173\pi\)
−0.224801 + 0.974405i \(0.572173\pi\)
\(348\) 0 0
\(349\) 9.83350e8 1.23828 0.619140 0.785280i \(-0.287481\pi\)
0.619140 + 0.785280i \(0.287481\pi\)
\(350\) 0 0
\(351\) −4.77874e8 −0.589845
\(352\) 0 0
\(353\) −6.25026e8 −0.756287 −0.378144 0.925747i \(-0.623437\pi\)
−0.378144 + 0.925747i \(0.623437\pi\)
\(354\) 0 0
\(355\) −6.76766e6 −0.00802859
\(356\) 0 0
\(357\) 2.58266e7 0.0300420
\(358\) 0 0
\(359\) 6.83539e8 0.779709 0.389855 0.920876i \(-0.372525\pi\)
0.389855 + 0.920876i \(0.372525\pi\)
\(360\) 0 0
\(361\) −1.83998e8 −0.205844
\(362\) 0 0
\(363\) 2.63233e8 0.288847
\(364\) 0 0
\(365\) −5.22384e7 −0.0562296
\(366\) 0 0
\(367\) 5.50838e8 0.581691 0.290846 0.956770i \(-0.406063\pi\)
0.290846 + 0.956770i \(0.406063\pi\)
\(368\) 0 0
\(369\) 7.50794e8 0.777908
\(370\) 0 0
\(371\) 9.34831e7 0.0950440
\(372\) 0 0
\(373\) 1.03057e9 1.02824 0.514121 0.857717i \(-0.328118\pi\)
0.514121 + 0.857717i \(0.328118\pi\)
\(374\) 0 0
\(375\) 1.10923e8 0.108621
\(376\) 0 0
\(377\) −1.42818e8 −0.137274
\(378\) 0 0
\(379\) 1.53319e8 0.144663 0.0723317 0.997381i \(-0.476956\pi\)
0.0723317 + 0.997381i \(0.476956\pi\)
\(380\) 0 0
\(381\) 9.07839e8 0.840952
\(382\) 0 0
\(383\) 1.20085e9 1.09218 0.546089 0.837727i \(-0.316116\pi\)
0.546089 + 0.837727i \(0.316116\pi\)
\(384\) 0 0
\(385\) 5.20206e6 0.00464583
\(386\) 0 0
\(387\) −2.72557e8 −0.239039
\(388\) 0 0
\(389\) 9.04660e8 0.779223 0.389611 0.920979i \(-0.372609\pi\)
0.389611 + 0.920979i \(0.372609\pi\)
\(390\) 0 0
\(391\) 1.08955e9 0.921786
\(392\) 0 0
\(393\) 9.86589e8 0.819903
\(394\) 0 0
\(395\) −4.81830e7 −0.0393373
\(396\) 0 0
\(397\) 5.60100e8 0.449261 0.224630 0.974444i \(-0.427882\pi\)
0.224630 + 0.974444i \(0.427882\pi\)
\(398\) 0 0
\(399\) −4.89287e7 −0.0385619
\(400\) 0 0
\(401\) 1.75894e9 1.36222 0.681108 0.732183i \(-0.261498\pi\)
0.681108 + 0.732183i \(0.261498\pi\)
\(402\) 0 0
\(403\) 7.45176e8 0.567141
\(404\) 0 0
\(405\) 4.06084e7 0.0303755
\(406\) 0 0
\(407\) −2.62161e8 −0.192747
\(408\) 0 0
\(409\) 3.48105e8 0.251582 0.125791 0.992057i \(-0.459853\pi\)
0.125791 + 0.992057i \(0.459853\pi\)
\(410\) 0 0
\(411\) 1.10437e9 0.784633
\(412\) 0 0
\(413\) 4.06063e7 0.0283640
\(414\) 0 0
\(415\) 1.10838e8 0.0761235
\(416\) 0 0
\(417\) 1.20183e9 0.811647
\(418\) 0 0
\(419\) 9.10860e8 0.604927 0.302463 0.953161i \(-0.402191\pi\)
0.302463 + 0.953161i \(0.402191\pi\)
\(420\) 0 0
\(421\) −2.03192e9 −1.32715 −0.663574 0.748111i \(-0.730961\pi\)
−0.663574 + 0.748111i \(0.730961\pi\)
\(422\) 0 0
\(423\) −2.85934e8 −0.183685
\(424\) 0 0
\(425\) −1.09324e9 −0.690800
\(426\) 0 0
\(427\) −3.48750e7 −0.0216779
\(428\) 0 0
\(429\) −8.04666e8 −0.492057
\(430\) 0 0
\(431\) −1.21720e9 −0.732304 −0.366152 0.930555i \(-0.619325\pi\)
−0.366152 + 0.930555i \(0.619325\pi\)
\(432\) 0 0
\(433\) −1.40845e9 −0.833745 −0.416872 0.908965i \(-0.636874\pi\)
−0.416872 + 0.908965i \(0.636874\pi\)
\(434\) 0 0
\(435\) 2.35743e7 0.0137318
\(436\) 0 0
\(437\) −2.06417e9 −1.18320
\(438\) 0 0
\(439\) −1.96037e9 −1.10589 −0.552946 0.833217i \(-0.686496\pi\)
−0.552946 + 0.833217i \(0.686496\pi\)
\(440\) 0 0
\(441\) 7.27927e8 0.404160
\(442\) 0 0
\(443\) −2.51266e9 −1.37316 −0.686578 0.727056i \(-0.740888\pi\)
−0.686578 + 0.727056i \(0.740888\pi\)
\(444\) 0 0
\(445\) −1.66490e8 −0.0895629
\(446\) 0 0
\(447\) 2.57917e8 0.136585
\(448\) 0 0
\(449\) −1.67892e9 −0.875324 −0.437662 0.899140i \(-0.644193\pi\)
−0.437662 + 0.899140i \(0.644193\pi\)
\(450\) 0 0
\(451\) 4.38240e9 2.24954
\(452\) 0 0
\(453\) 1.54607e9 0.781424
\(454\) 0 0
\(455\) −4.33354e6 −0.00215676
\(456\) 0 0
\(457\) −6.46678e8 −0.316943 −0.158472 0.987364i \(-0.550657\pi\)
−0.158472 + 0.987364i \(0.550657\pi\)
\(458\) 0 0
\(459\) −1.55876e9 −0.752373
\(460\) 0 0
\(461\) 1.32962e9 0.632083 0.316041 0.948745i \(-0.397646\pi\)
0.316041 + 0.948745i \(0.397646\pi\)
\(462\) 0 0
\(463\) −3.34991e9 −1.56856 −0.784278 0.620409i \(-0.786967\pi\)
−0.784278 + 0.620409i \(0.786967\pi\)
\(464\) 0 0
\(465\) −1.23003e8 −0.0567323
\(466\) 0 0
\(467\) −6.13048e8 −0.278539 −0.139269 0.990255i \(-0.544475\pi\)
−0.139269 + 0.990255i \(0.544475\pi\)
\(468\) 0 0
\(469\) 1.93250e8 0.0864997
\(470\) 0 0
\(471\) 2.48791e9 1.09714
\(472\) 0 0
\(473\) −1.59092e9 −0.691249
\(474\) 0 0
\(475\) 2.07114e9 0.886711
\(476\) 0 0
\(477\) −1.62763e9 −0.686658
\(478\) 0 0
\(479\) 3.20691e9 1.33325 0.666627 0.745392i \(-0.267738\pi\)
0.666627 + 0.745392i \(0.267738\pi\)
\(480\) 0 0
\(481\) 2.18391e8 0.0894802
\(482\) 0 0
\(483\) 1.42275e8 0.0574530
\(484\) 0 0
\(485\) 6.71862e7 0.0267414
\(486\) 0 0
\(487\) −2.67973e9 −1.05133 −0.525665 0.850692i \(-0.676183\pi\)
−0.525665 + 0.850692i \(0.676183\pi\)
\(488\) 0 0
\(489\) 9.96550e8 0.385406
\(490\) 0 0
\(491\) −1.74917e9 −0.666879 −0.333439 0.942772i \(-0.608209\pi\)
−0.333439 + 0.942772i \(0.608209\pi\)
\(492\) 0 0
\(493\) −4.65851e8 −0.175099
\(494\) 0 0
\(495\) −9.05727e7 −0.0335644
\(496\) 0 0
\(497\) 1.74633e7 0.00638084
\(498\) 0 0
\(499\) 2.21759e9 0.798967 0.399484 0.916740i \(-0.369189\pi\)
0.399484 + 0.916740i \(0.369189\pi\)
\(500\) 0 0
\(501\) −1.56241e9 −0.555091
\(502\) 0 0
\(503\) −4.21833e9 −1.47792 −0.738962 0.673747i \(-0.764684\pi\)
−0.738962 + 0.673747i \(0.764684\pi\)
\(504\) 0 0
\(505\) 3.20637e7 0.0110788
\(506\) 0 0
\(507\) −1.59238e9 −0.542649
\(508\) 0 0
\(509\) −4.92334e9 −1.65481 −0.827404 0.561607i \(-0.810183\pi\)
−0.827404 + 0.561607i \(0.810183\pi\)
\(510\) 0 0
\(511\) 1.34796e8 0.0446893
\(512\) 0 0
\(513\) 2.95307e9 0.965747
\(514\) 0 0
\(515\) −1.42323e8 −0.0459146
\(516\) 0 0
\(517\) −1.66900e9 −0.531178
\(518\) 0 0
\(519\) 1.08967e9 0.342145
\(520\) 0 0
\(521\) 3.66580e9 1.13563 0.567814 0.823157i \(-0.307789\pi\)
0.567814 + 0.823157i \(0.307789\pi\)
\(522\) 0 0
\(523\) −5.07711e9 −1.55189 −0.775945 0.630800i \(-0.782727\pi\)
−0.775945 + 0.630800i \(0.782727\pi\)
\(524\) 0 0
\(525\) −1.42755e8 −0.0430561
\(526\) 0 0
\(527\) 2.43066e9 0.723414
\(528\) 0 0
\(529\) 2.59735e9 0.762843
\(530\) 0 0
\(531\) −7.06993e8 −0.204920
\(532\) 0 0
\(533\) −3.65072e9 −1.04432
\(534\) 0 0
\(535\) −2.73413e8 −0.0771934
\(536\) 0 0
\(537\) −4.60221e9 −1.28250
\(538\) 0 0
\(539\) 4.24892e9 1.16874
\(540\) 0 0
\(541\) 5.97271e9 1.62174 0.810870 0.585226i \(-0.198994\pi\)
0.810870 + 0.585226i \(0.198994\pi\)
\(542\) 0 0
\(543\) 1.71914e9 0.460800
\(544\) 0 0
\(545\) 1.09773e8 0.0290475
\(546\) 0 0
\(547\) 4.63928e9 1.21198 0.605990 0.795473i \(-0.292777\pi\)
0.605990 + 0.795473i \(0.292777\pi\)
\(548\) 0 0
\(549\) 6.07206e8 0.156615
\(550\) 0 0
\(551\) 8.82558e8 0.224757
\(552\) 0 0
\(553\) 1.24331e8 0.0312639
\(554\) 0 0
\(555\) −3.60489e7 −0.00895089
\(556\) 0 0
\(557\) −2.87674e9 −0.705354 −0.352677 0.935745i \(-0.614729\pi\)
−0.352677 + 0.935745i \(0.614729\pi\)
\(558\) 0 0
\(559\) 1.32530e9 0.320903
\(560\) 0 0
\(561\) −2.62471e9 −0.627640
\(562\) 0 0
\(563\) −5.48798e9 −1.29608 −0.648042 0.761604i \(-0.724412\pi\)
−0.648042 + 0.761604i \(0.724412\pi\)
\(564\) 0 0
\(565\) 4.21505e8 0.0983181
\(566\) 0 0
\(567\) −1.04786e8 −0.0241414
\(568\) 0 0
\(569\) 5.47367e9 1.24562 0.622810 0.782373i \(-0.285991\pi\)
0.622810 + 0.782373i \(0.285991\pi\)
\(570\) 0 0
\(571\) −1.73348e9 −0.389665 −0.194833 0.980837i \(-0.562416\pi\)
−0.194833 + 0.980837i \(0.562416\pi\)
\(572\) 0 0
\(573\) −2.89076e9 −0.641906
\(574\) 0 0
\(575\) −6.02246e9 −1.32110
\(576\) 0 0
\(577\) −2.24479e8 −0.0486475 −0.0243238 0.999704i \(-0.507743\pi\)
−0.0243238 + 0.999704i \(0.507743\pi\)
\(578\) 0 0
\(579\) 5.89507e7 0.0126216
\(580\) 0 0
\(581\) −2.86005e8 −0.0605003
\(582\) 0 0
\(583\) −9.50049e9 −1.98567
\(584\) 0 0
\(585\) 7.54508e7 0.0155818
\(586\) 0 0
\(587\) −2.44680e9 −0.499305 −0.249652 0.968336i \(-0.580316\pi\)
−0.249652 + 0.968336i \(0.580316\pi\)
\(588\) 0 0
\(589\) −4.60490e9 −0.928574
\(590\) 0 0
\(591\) −3.84380e9 −0.765956
\(592\) 0 0
\(593\) −1.85980e9 −0.366248 −0.183124 0.983090i \(-0.558621\pi\)
−0.183124 + 0.983090i \(0.558621\pi\)
\(594\) 0 0
\(595\) −1.41354e7 −0.00275105
\(596\) 0 0
\(597\) −8.09161e8 −0.155641
\(598\) 0 0
\(599\) 1.26622e8 0.0240722 0.0120361 0.999928i \(-0.496169\pi\)
0.0120361 + 0.999928i \(0.496169\pi\)
\(600\) 0 0
\(601\) −1.07228e9 −0.201488 −0.100744 0.994912i \(-0.532122\pi\)
−0.100744 + 0.994912i \(0.532122\pi\)
\(602\) 0 0
\(603\) −3.36466e9 −0.624929
\(604\) 0 0
\(605\) −1.44072e8 −0.0264506
\(606\) 0 0
\(607\) 5.73945e9 1.04162 0.520811 0.853672i \(-0.325630\pi\)
0.520811 + 0.853672i \(0.325630\pi\)
\(608\) 0 0
\(609\) −6.08312e7 −0.0109135
\(610\) 0 0
\(611\) 1.39035e9 0.246592
\(612\) 0 0
\(613\) 8.64474e8 0.151579 0.0757897 0.997124i \(-0.475852\pi\)
0.0757897 + 0.997124i \(0.475852\pi\)
\(614\) 0 0
\(615\) 6.02609e8 0.104465
\(616\) 0 0
\(617\) −8.52653e9 −1.46142 −0.730708 0.682690i \(-0.760810\pi\)
−0.730708 + 0.682690i \(0.760810\pi\)
\(618\) 0 0
\(619\) −3.14258e9 −0.532561 −0.266281 0.963896i \(-0.585795\pi\)
−0.266281 + 0.963896i \(0.585795\pi\)
\(620\) 0 0
\(621\) −8.58693e9 −1.43886
\(622\) 0 0
\(623\) 4.29611e8 0.0711815
\(624\) 0 0
\(625\) 6.01237e9 0.985067
\(626\) 0 0
\(627\) 4.97252e9 0.805639
\(628\) 0 0
\(629\) 7.12360e8 0.114136
\(630\) 0 0
\(631\) −4.61616e9 −0.731439 −0.365719 0.930725i \(-0.619177\pi\)
−0.365719 + 0.930725i \(0.619177\pi\)
\(632\) 0 0
\(633\) 9.67939e8 0.151682
\(634\) 0 0
\(635\) −4.96876e8 −0.0770087
\(636\) 0 0
\(637\) −3.53953e9 −0.542573
\(638\) 0 0
\(639\) −3.04051e8 −0.0460993
\(640\) 0 0
\(641\) −1.00510e10 −1.50732 −0.753658 0.657266i \(-0.771713\pi\)
−0.753658 + 0.657266i \(0.771713\pi\)
\(642\) 0 0
\(643\) 3.42743e9 0.508429 0.254214 0.967148i \(-0.418183\pi\)
0.254214 + 0.967148i \(0.418183\pi\)
\(644\) 0 0
\(645\) −2.18762e8 −0.0321006
\(646\) 0 0
\(647\) 1.28014e9 0.185821 0.0929103 0.995674i \(-0.470383\pi\)
0.0929103 + 0.995674i \(0.470383\pi\)
\(648\) 0 0
\(649\) −4.12673e9 −0.592584
\(650\) 0 0
\(651\) 3.17397e8 0.0450889
\(652\) 0 0
\(653\) 6.73100e9 0.945983 0.472992 0.881067i \(-0.343174\pi\)
0.472992 + 0.881067i \(0.343174\pi\)
\(654\) 0 0
\(655\) −5.39977e8 −0.0750812
\(656\) 0 0
\(657\) −2.34692e9 −0.322864
\(658\) 0 0
\(659\) 8.07982e9 1.09977 0.549886 0.835240i \(-0.314671\pi\)
0.549886 + 0.835240i \(0.314671\pi\)
\(660\) 0 0
\(661\) −1.58791e8 −0.0213856 −0.0106928 0.999943i \(-0.503404\pi\)
−0.0106928 + 0.999943i \(0.503404\pi\)
\(662\) 0 0
\(663\) 2.18649e9 0.291374
\(664\) 0 0
\(665\) 2.67796e7 0.00353124
\(666\) 0 0
\(667\) −2.56630e9 −0.334863
\(668\) 0 0
\(669\) 3.92395e8 0.0506678
\(670\) 0 0
\(671\) 3.54427e9 0.452896
\(672\) 0 0
\(673\) −7.20617e8 −0.0911280 −0.0455640 0.998961i \(-0.514509\pi\)
−0.0455640 + 0.998961i \(0.514509\pi\)
\(674\) 0 0
\(675\) 8.61594e9 1.07830
\(676\) 0 0
\(677\) −8.42691e9 −1.04378 −0.521889 0.853013i \(-0.674773\pi\)
−0.521889 + 0.853013i \(0.674773\pi\)
\(678\) 0 0
\(679\) −1.73367e8 −0.0212531
\(680\) 0 0
\(681\) 5.65366e9 0.685985
\(682\) 0 0
\(683\) −5.36376e9 −0.644164 −0.322082 0.946712i \(-0.604383\pi\)
−0.322082 + 0.946712i \(0.604383\pi\)
\(684\) 0 0
\(685\) −6.04439e8 −0.0718515
\(686\) 0 0
\(687\) −7.05971e9 −0.830688
\(688\) 0 0
\(689\) 7.91431e9 0.921819
\(690\) 0 0
\(691\) −1.48364e10 −1.71063 −0.855313 0.518112i \(-0.826635\pi\)
−0.855313 + 0.518112i \(0.826635\pi\)
\(692\) 0 0
\(693\) 2.33714e8 0.0266758
\(694\) 0 0
\(695\) −6.57783e8 −0.0743252
\(696\) 0 0
\(697\) −1.19081e10 −1.33208
\(698\) 0 0
\(699\) 5.25908e9 0.582425
\(700\) 0 0
\(701\) 6.52624e9 0.715566 0.357783 0.933805i \(-0.383533\pi\)
0.357783 + 0.933805i \(0.383533\pi\)
\(702\) 0 0
\(703\) −1.34957e9 −0.146505
\(704\) 0 0
\(705\) −2.29499e8 −0.0246671
\(706\) 0 0
\(707\) −8.27371e7 −0.00880506
\(708\) 0 0
\(709\) 1.69533e10 1.78646 0.893228 0.449603i \(-0.148435\pi\)
0.893228 + 0.449603i \(0.148435\pi\)
\(710\) 0 0
\(711\) −2.16472e9 −0.225870
\(712\) 0 0
\(713\) 1.33901e10 1.38347
\(714\) 0 0
\(715\) 4.40408e8 0.0450592
\(716\) 0 0
\(717\) 1.27166e10 1.28841
\(718\) 0 0
\(719\) 4.30200e9 0.431637 0.215819 0.976433i \(-0.430758\pi\)
0.215819 + 0.976433i \(0.430758\pi\)
\(720\) 0 0
\(721\) 3.67251e8 0.0364913
\(722\) 0 0
\(723\) −1.11479e10 −1.09701
\(724\) 0 0
\(725\) 2.57497e9 0.250951
\(726\) 0 0
\(727\) −1.60836e10 −1.55243 −0.776217 0.630465i \(-0.782864\pi\)
−0.776217 + 0.630465i \(0.782864\pi\)
\(728\) 0 0
\(729\) 1.05653e10 1.01003
\(730\) 0 0
\(731\) 4.32295e9 0.409326
\(732\) 0 0
\(733\) 1.61253e9 0.151232 0.0756162 0.997137i \(-0.475908\pi\)
0.0756162 + 0.997137i \(0.475908\pi\)
\(734\) 0 0
\(735\) 5.84256e8 0.0542747
\(736\) 0 0
\(737\) −1.96396e10 −1.80716
\(738\) 0 0
\(739\) −5.21771e9 −0.475581 −0.237790 0.971317i \(-0.576423\pi\)
−0.237790 + 0.971317i \(0.576423\pi\)
\(740\) 0 0
\(741\) −4.14232e9 −0.374007
\(742\) 0 0
\(743\) −3.55715e9 −0.318157 −0.159078 0.987266i \(-0.550852\pi\)
−0.159078 + 0.987266i \(0.550852\pi\)
\(744\) 0 0
\(745\) −1.41163e8 −0.0125076
\(746\) 0 0
\(747\) 4.97961e9 0.437093
\(748\) 0 0
\(749\) 7.05514e8 0.0613507
\(750\) 0 0
\(751\) −2.02483e10 −1.74441 −0.872205 0.489140i \(-0.837311\pi\)
−0.872205 + 0.489140i \(0.837311\pi\)
\(752\) 0 0
\(753\) 7.10014e9 0.606016
\(754\) 0 0
\(755\) −8.46194e8 −0.0715576
\(756\) 0 0
\(757\) 1.87776e10 1.57327 0.786636 0.617417i \(-0.211821\pi\)
0.786636 + 0.617417i \(0.211821\pi\)
\(758\) 0 0
\(759\) −1.44591e10 −1.20031
\(760\) 0 0
\(761\) −9.48144e9 −0.779880 −0.389940 0.920840i \(-0.627504\pi\)
−0.389940 + 0.920840i \(0.627504\pi\)
\(762\) 0 0
\(763\) −2.83259e8 −0.0230859
\(764\) 0 0
\(765\) 2.46110e8 0.0198753
\(766\) 0 0
\(767\) 3.43774e9 0.275099
\(768\) 0 0
\(769\) −7.67008e9 −0.608216 −0.304108 0.952638i \(-0.598358\pi\)
−0.304108 + 0.952638i \(0.598358\pi\)
\(770\) 0 0
\(771\) 1.13430e10 0.891324
\(772\) 0 0
\(773\) −7.33496e9 −0.571175 −0.285588 0.958353i \(-0.592189\pi\)
−0.285588 + 0.958353i \(0.592189\pi\)
\(774\) 0 0
\(775\) −1.34353e10 −1.03680
\(776\) 0 0
\(777\) 9.30205e7 0.00711386
\(778\) 0 0
\(779\) 2.25600e10 1.70985
\(780\) 0 0
\(781\) −1.77475e9 −0.133309
\(782\) 0 0
\(783\) 3.67144e9 0.273319
\(784\) 0 0
\(785\) −1.36168e9 −0.100469
\(786\) 0 0
\(787\) 3.05525e9 0.223427 0.111713 0.993740i \(-0.464366\pi\)
0.111713 + 0.993740i \(0.464366\pi\)
\(788\) 0 0
\(789\) −2.05273e9 −0.148786
\(790\) 0 0
\(791\) −1.08765e9 −0.0781398
\(792\) 0 0
\(793\) −2.95253e9 −0.210251
\(794\) 0 0
\(795\) −1.30638e9 −0.0922115
\(796\) 0 0
\(797\) −1.30056e10 −0.909971 −0.454985 0.890499i \(-0.650355\pi\)
−0.454985 + 0.890499i \(0.650355\pi\)
\(798\) 0 0
\(799\) 4.53512e9 0.314539
\(800\) 0 0
\(801\) −7.47992e9 −0.514260
\(802\) 0 0
\(803\) −1.36990e10 −0.933652
\(804\) 0 0
\(805\) −7.78695e7 −0.00526116
\(806\) 0 0
\(807\) −1.28604e9 −0.0861383
\(808\) 0 0
\(809\) −2.37212e10 −1.57513 −0.787565 0.616231i \(-0.788659\pi\)
−0.787565 + 0.616231i \(0.788659\pi\)
\(810\) 0 0
\(811\) 1.28296e10 0.844577 0.422288 0.906462i \(-0.361227\pi\)
0.422288 + 0.906462i \(0.361227\pi\)
\(812\) 0 0
\(813\) 1.89446e10 1.23643
\(814\) 0 0
\(815\) −5.45429e8 −0.0352929
\(816\) 0 0
\(817\) −8.18986e9 −0.525411
\(818\) 0 0
\(819\) −1.94693e8 −0.0123839
\(820\) 0 0
\(821\) −7.10512e9 −0.448096 −0.224048 0.974578i \(-0.571927\pi\)
−0.224048 + 0.974578i \(0.571927\pi\)
\(822\) 0 0
\(823\) −3.31606e9 −0.207359 −0.103680 0.994611i \(-0.533062\pi\)
−0.103680 + 0.994611i \(0.533062\pi\)
\(824\) 0 0
\(825\) 1.45079e10 0.899532
\(826\) 0 0
\(827\) 2.02601e10 1.24558 0.622791 0.782388i \(-0.285999\pi\)
0.622791 + 0.782388i \(0.285999\pi\)
\(828\) 0 0
\(829\) 3.28928e9 0.200521 0.100261 0.994961i \(-0.468032\pi\)
0.100261 + 0.994961i \(0.468032\pi\)
\(830\) 0 0
\(831\) −2.32820e10 −1.40740
\(832\) 0 0
\(833\) −1.15454e10 −0.692075
\(834\) 0 0
\(835\) 8.55136e8 0.0508315
\(836\) 0 0
\(837\) −1.91564e10 −1.12921
\(838\) 0 0
\(839\) 1.33201e10 0.778646 0.389323 0.921101i \(-0.372709\pi\)
0.389323 + 0.921101i \(0.372709\pi\)
\(840\) 0 0
\(841\) −1.61526e10 −0.936391
\(842\) 0 0
\(843\) 1.58190e10 0.909460
\(844\) 0 0
\(845\) 8.71537e8 0.0496921
\(846\) 0 0
\(847\) 3.71764e8 0.0210220
\(848\) 0 0
\(849\) −6.89337e9 −0.386593
\(850\) 0 0
\(851\) 3.92427e9 0.218276
\(852\) 0 0
\(853\) −2.73074e10 −1.50646 −0.753231 0.657756i \(-0.771506\pi\)
−0.753231 + 0.657756i \(0.771506\pi\)
\(854\) 0 0
\(855\) −4.66257e8 −0.0255119
\(856\) 0 0
\(857\) 1.77629e10 0.964008 0.482004 0.876169i \(-0.339909\pi\)
0.482004 + 0.876169i \(0.339909\pi\)
\(858\) 0 0
\(859\) −3.55499e10 −1.91365 −0.956825 0.290664i \(-0.906124\pi\)
−0.956825 + 0.290664i \(0.906124\pi\)
\(860\) 0 0
\(861\) −1.55497e9 −0.0830255
\(862\) 0 0
\(863\) 1.54322e10 0.817317 0.408658 0.912687i \(-0.365997\pi\)
0.408658 + 0.912687i \(0.365997\pi\)
\(864\) 0 0
\(865\) −5.96396e8 −0.0313313
\(866\) 0 0
\(867\) −7.66472e9 −0.399420
\(868\) 0 0
\(869\) −1.26355e10 −0.653167
\(870\) 0 0
\(871\) 1.63606e10 0.838950
\(872\) 0 0
\(873\) 3.01848e9 0.153546
\(874\) 0 0
\(875\) 1.56657e8 0.00790534
\(876\) 0 0
\(877\) 2.45939e10 1.23120 0.615599 0.788059i \(-0.288914\pi\)
0.615599 + 0.788059i \(0.288914\pi\)
\(878\) 0 0
\(879\) 2.44645e10 1.21500
\(880\) 0 0
\(881\) −3.24805e10 −1.60032 −0.800159 0.599788i \(-0.795252\pi\)
−0.800159 + 0.599788i \(0.795252\pi\)
\(882\) 0 0
\(883\) 1.24274e10 0.607460 0.303730 0.952758i \(-0.401768\pi\)
0.303730 + 0.952758i \(0.401768\pi\)
\(884\) 0 0
\(885\) −5.67453e8 −0.0275187
\(886\) 0 0
\(887\) 2.06132e10 0.991774 0.495887 0.868387i \(-0.334843\pi\)
0.495887 + 0.868387i \(0.334843\pi\)
\(888\) 0 0
\(889\) 1.28214e9 0.0612039
\(890\) 0 0
\(891\) 1.06492e10 0.504363
\(892\) 0 0
\(893\) −8.59181e9 −0.403742
\(894\) 0 0
\(895\) 2.51887e9 0.117442
\(896\) 0 0
\(897\) 1.20450e10 0.557229
\(898\) 0 0
\(899\) −5.72509e9 −0.262799
\(900\) 0 0
\(901\) 2.58153e10 1.17582
\(902\) 0 0
\(903\) 5.64494e8 0.0255125
\(904\) 0 0
\(905\) −9.40916e8 −0.0421969
\(906\) 0 0
\(907\) 2.15584e10 0.959380 0.479690 0.877438i \(-0.340749\pi\)
0.479690 + 0.877438i \(0.340749\pi\)
\(908\) 0 0
\(909\) 1.44053e9 0.0636133
\(910\) 0 0
\(911\) −2.47542e10 −1.08476 −0.542381 0.840132i \(-0.682477\pi\)
−0.542381 + 0.840132i \(0.682477\pi\)
\(912\) 0 0
\(913\) 2.90661e10 1.26398
\(914\) 0 0
\(915\) 4.87361e8 0.0210318
\(916\) 0 0
\(917\) 1.39336e9 0.0596719
\(918\) 0 0
\(919\) −3.82049e10 −1.62373 −0.811866 0.583843i \(-0.801548\pi\)
−0.811866 + 0.583843i \(0.801548\pi\)
\(920\) 0 0
\(921\) 1.58998e10 0.670629
\(922\) 0 0
\(923\) 1.47844e9 0.0618869
\(924\) 0 0
\(925\) −3.93754e9 −0.163579
\(926\) 0 0
\(927\) −6.39417e9 −0.263636
\(928\) 0 0
\(929\) −1.51912e10 −0.621636 −0.310818 0.950469i \(-0.600603\pi\)
−0.310818 + 0.950469i \(0.600603\pi\)
\(930\) 0 0
\(931\) 2.18729e10 0.888348
\(932\) 0 0
\(933\) 2.17015e10 0.874791
\(934\) 0 0
\(935\) 1.43655e9 0.0574750
\(936\) 0 0
\(937\) −2.93618e10 −1.16599 −0.582995 0.812476i \(-0.698119\pi\)
−0.582995 + 0.812476i \(0.698119\pi\)
\(938\) 0 0
\(939\) 2.30255e10 0.907570
\(940\) 0 0
\(941\) −1.88409e10 −0.737119 −0.368559 0.929604i \(-0.620149\pi\)
−0.368559 + 0.929604i \(0.620149\pi\)
\(942\) 0 0
\(943\) −6.55999e10 −2.54749
\(944\) 0 0
\(945\) 1.11403e8 0.00429423
\(946\) 0 0
\(947\) 1.73098e10 0.662319 0.331159 0.943575i \(-0.392560\pi\)
0.331159 + 0.943575i \(0.392560\pi\)
\(948\) 0 0
\(949\) 1.14119e10 0.433436
\(950\) 0 0
\(951\) 2.38751e10 0.900148
\(952\) 0 0
\(953\) 1.22640e10 0.458993 0.229497 0.973309i \(-0.426292\pi\)
0.229497 + 0.973309i \(0.426292\pi\)
\(954\) 0 0
\(955\) 1.58217e9 0.0587815
\(956\) 0 0
\(957\) 6.18214e9 0.228007
\(958\) 0 0
\(959\) 1.55969e9 0.0571050
\(960\) 0 0
\(961\) 2.35903e9 0.0857436
\(962\) 0 0
\(963\) −1.22836e10 −0.443236
\(964\) 0 0
\(965\) −3.22647e7 −0.00115580
\(966\) 0 0
\(967\) 2.65054e10 0.942632 0.471316 0.881964i \(-0.343779\pi\)
0.471316 + 0.881964i \(0.343779\pi\)
\(968\) 0 0
\(969\) −1.35116e10 −0.477062
\(970\) 0 0
\(971\) 4.44819e10 1.55925 0.779626 0.626246i \(-0.215409\pi\)
0.779626 + 0.626246i \(0.215409\pi\)
\(972\) 0 0
\(973\) 1.69734e9 0.0590710
\(974\) 0 0
\(975\) −1.20857e10 −0.417596
\(976\) 0 0
\(977\) 2.66074e10 0.912792 0.456396 0.889777i \(-0.349140\pi\)
0.456396 + 0.889777i \(0.349140\pi\)
\(978\) 0 0
\(979\) −4.36604e10 −1.48713
\(980\) 0 0
\(981\) 4.93179e9 0.166787
\(982\) 0 0
\(983\) −3.98658e10 −1.33864 −0.669319 0.742975i \(-0.733414\pi\)
−0.669319 + 0.742975i \(0.733414\pi\)
\(984\) 0 0
\(985\) 2.10378e9 0.0701411
\(986\) 0 0
\(987\) 5.92199e8 0.0196046
\(988\) 0 0
\(989\) 2.38144e10 0.782804
\(990\) 0 0
\(991\) 1.96883e10 0.642613 0.321306 0.946975i \(-0.395878\pi\)
0.321306 + 0.946975i \(0.395878\pi\)
\(992\) 0 0
\(993\) 1.66769e10 0.540496
\(994\) 0 0
\(995\) 4.42868e8 0.0142526
\(996\) 0 0
\(997\) 7.78796e9 0.248880 0.124440 0.992227i \(-0.460287\pi\)
0.124440 + 0.992227i \(0.460287\pi\)
\(998\) 0 0
\(999\) −5.61421e9 −0.178160
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 592.8.a.b.1.3 4
4.3 odd 2 74.8.a.a.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.8.a.a.1.2 4 4.3 odd 2
592.8.a.b.1.3 4 1.1 even 1 trivial