Properties

Label 387.8.a.b.1.5
Level $387$
Weight $8$
Character 387.1
Self dual yes
Analytic conductor $120.893$
Analytic rank $0$
Dimension $11$
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] = [387,8,Mod(1,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("387.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 387.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: \(120.893004862\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + 209758592 x^{4} + 3410917248 x^{3} - 14180732672 x^{2} + \cdots + 238240894976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4}\cdot 7 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-5.14919\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.14919 q^{2} -118.083 q^{4} +39.2891 q^{5} +434.472 q^{7} +774.961 q^{8} +O(q^{10})\) \(q-3.14919 q^{2} -118.083 q^{4} +39.2891 q^{5} +434.472 q^{7} +774.961 q^{8} -123.729 q^{10} +6392.85 q^{11} +442.964 q^{13} -1368.23 q^{14} +12674.1 q^{16} +26318.9 q^{17} +7748.98 q^{19} -4639.35 q^{20} -20132.3 q^{22} +58609.7 q^{23} -76581.4 q^{25} -1394.98 q^{26} -51303.5 q^{28} -27040.0 q^{29} +100436. q^{31} -139108. q^{32} -82883.2 q^{34} +17070.0 q^{35} +358755. q^{37} -24403.0 q^{38} +30447.5 q^{40} +74304.4 q^{41} +79507.0 q^{43} -754884. q^{44} -184573. q^{46} -462198. q^{47} -634777. q^{49} +241169. q^{50} -52306.4 q^{52} -1.17404e6 q^{53} +251169. q^{55} +336699. q^{56} +85154.2 q^{58} -1.70392e6 q^{59} +2.64618e6 q^{61} -316293. q^{62} -1.18420e6 q^{64} +17403.6 q^{65} -2.11768e6 q^{67} -3.10780e6 q^{68} -53756.6 q^{70} +5.06201e6 q^{71} +899577. q^{73} -1.12979e6 q^{74} -915020. q^{76} +2.77751e6 q^{77} -4.05836e6 q^{79} +497952. q^{80} -233999. q^{82} +6.07269e6 q^{83} +1.03404e6 q^{85} -250383. q^{86} +4.95421e6 q^{88} +3.93741e6 q^{89} +192455. q^{91} -6.92079e6 q^{92} +1.45555e6 q^{94} +304450. q^{95} -1.24981e7 q^{97} +1.99904e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 24 q^{2} + 602 q^{4} + 752 q^{5} - 12 q^{7} + 3810 q^{8} - 1333 q^{10} - 1333 q^{11} - 17967 q^{13} + 22352 q^{14} - 34406 q^{16} + 63095 q^{17} - 54524 q^{19} + 280995 q^{20} - 289358 q^{22} + 138139 q^{23} + 3455 q^{25} + 132946 q^{26} - 12704 q^{28} + 308658 q^{29} - 209523 q^{31} + 644934 q^{32} + 762435 q^{34} + 578892 q^{35} - 298472 q^{37} + 369707 q^{38} + 2633173 q^{40} + 1346735 q^{41} + 874577 q^{43} - 3134292 q^{44} + 3588111 q^{46} - 499284 q^{47} + 2544563 q^{49} - 3049745 q^{50} + 983088 q^{52} + 2210495 q^{53} - 1855072 q^{55} + 469976 q^{56} + 4397067 q^{58} + 5824216 q^{59} - 4453034 q^{61} - 1002789 q^{62} + 4757538 q^{64} + 2162872 q^{65} - 6859513 q^{67} + 9397005 q^{68} + 845078 q^{70} + 10726554 q^{71} - 4456898 q^{73} - 1046637 q^{74} + 5861267 q^{76} + 17019816 q^{77} - 15541320 q^{79} + 15680911 q^{80} + 20233655 q^{82} + 11146767 q^{83} - 12471976 q^{85} + 1908168 q^{86} - 24463544 q^{88} + 13531356 q^{89} - 19746448 q^{91} + 26023161 q^{92} + 20288857 q^{94} + 12291624 q^{95} - 10999901 q^{97} - 29909168 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.14919 −0.278352 −0.139176 0.990268i \(-0.544445\pi\)
−0.139176 + 0.990268i \(0.544445\pi\)
\(3\) 0 0
\(4\) −118.083 −0.922520
\(5\) 39.2891 0.140565 0.0702824 0.997527i \(-0.477610\pi\)
0.0702824 + 0.997527i \(0.477610\pi\)
\(6\) 0 0
\(7\) 434.472 0.478760 0.239380 0.970926i \(-0.423056\pi\)
0.239380 + 0.970926i \(0.423056\pi\)
\(8\) 774.961 0.535137
\(9\) 0 0
\(10\) −123.729 −0.0391265
\(11\) 6392.85 1.44817 0.724086 0.689710i \(-0.242262\pi\)
0.724086 + 0.689710i \(0.242262\pi\)
\(12\) 0 0
\(13\) 442.964 0.0559200 0.0279600 0.999609i \(-0.491099\pi\)
0.0279600 + 0.999609i \(0.491099\pi\)
\(14\) −1368.23 −0.133264
\(15\) 0 0
\(16\) 12674.1 0.773564
\(17\) 26318.9 1.29926 0.649630 0.760250i \(-0.274924\pi\)
0.649630 + 0.760250i \(0.274924\pi\)
\(18\) 0 0
\(19\) 7748.98 0.259183 0.129592 0.991567i \(-0.458633\pi\)
0.129592 + 0.991567i \(0.458633\pi\)
\(20\) −4639.35 −0.129674
\(21\) 0 0
\(22\) −20132.3 −0.403101
\(23\) 58609.7 1.00444 0.502218 0.864741i \(-0.332517\pi\)
0.502218 + 0.864741i \(0.332517\pi\)
\(24\) 0 0
\(25\) −76581.4 −0.980242
\(26\) −1394.98 −0.0155654
\(27\) 0 0
\(28\) −51303.5 −0.441666
\(29\) −27040.0 −0.205880 −0.102940 0.994688i \(-0.532825\pi\)
−0.102940 + 0.994688i \(0.532825\pi\)
\(30\) 0 0
\(31\) 100436. 0.605514 0.302757 0.953068i \(-0.402093\pi\)
0.302757 + 0.953068i \(0.402093\pi\)
\(32\) −139108. −0.750460
\(33\) 0 0
\(34\) −82883.2 −0.361652
\(35\) 17070.0 0.0672969
\(36\) 0 0
\(37\) 358755. 1.16437 0.582186 0.813055i \(-0.302197\pi\)
0.582186 + 0.813055i \(0.302197\pi\)
\(38\) −24403.0 −0.0721441
\(39\) 0 0
\(40\) 30447.5 0.0752214
\(41\) 74304.4 0.168372 0.0841862 0.996450i \(-0.473171\pi\)
0.0841862 + 0.996450i \(0.473171\pi\)
\(42\) 0 0
\(43\) 79507.0 0.152499
\(44\) −754884. −1.33597
\(45\) 0 0
\(46\) −184573. −0.279587
\(47\) −462198. −0.649359 −0.324680 0.945824i \(-0.605257\pi\)
−0.324680 + 0.945824i \(0.605257\pi\)
\(48\) 0 0
\(49\) −634777. −0.770788
\(50\) 241169. 0.272852
\(51\) 0 0
\(52\) −52306.4 −0.0515873
\(53\) −1.17404e6 −1.08322 −0.541612 0.840629i \(-0.682186\pi\)
−0.541612 + 0.840629i \(0.682186\pi\)
\(54\) 0 0
\(55\) 251169. 0.203562
\(56\) 336699. 0.256202
\(57\) 0 0
\(58\) 85154.2 0.0573071
\(59\) −1.70392e6 −1.08011 −0.540055 0.841630i \(-0.681596\pi\)
−0.540055 + 0.841630i \(0.681596\pi\)
\(60\) 0 0
\(61\) 2.64618e6 1.49267 0.746336 0.665570i \(-0.231811\pi\)
0.746336 + 0.665570i \(0.231811\pi\)
\(62\) −316293. −0.168546
\(63\) 0 0
\(64\) −1.18420e6 −0.564672
\(65\) 17403.6 0.00786038
\(66\) 0 0
\(67\) −2.11768e6 −0.860199 −0.430099 0.902782i \(-0.641522\pi\)
−0.430099 + 0.902782i \(0.641522\pi\)
\(68\) −3.10780e6 −1.19859
\(69\) 0 0
\(70\) −53756.6 −0.0187322
\(71\) 5.06201e6 1.67849 0.839246 0.543752i \(-0.182997\pi\)
0.839246 + 0.543752i \(0.182997\pi\)
\(72\) 0 0
\(73\) 899577. 0.270650 0.135325 0.990801i \(-0.456792\pi\)
0.135325 + 0.990801i \(0.456792\pi\)
\(74\) −1.12979e6 −0.324105
\(75\) 0 0
\(76\) −915020. −0.239102
\(77\) 2.77751e6 0.693327
\(78\) 0 0
\(79\) −4.05836e6 −0.926095 −0.463047 0.886333i \(-0.653244\pi\)
−0.463047 + 0.886333i \(0.653244\pi\)
\(80\) 497952. 0.108736
\(81\) 0 0
\(82\) −233999. −0.0468668
\(83\) 6.07269e6 1.16576 0.582878 0.812559i \(-0.301926\pi\)
0.582878 + 0.812559i \(0.301926\pi\)
\(84\) 0 0
\(85\) 1.03404e6 0.182630
\(86\) −250383. −0.0424483
\(87\) 0 0
\(88\) 4.95421e6 0.774970
\(89\) 3.93741e6 0.592033 0.296017 0.955183i \(-0.404342\pi\)
0.296017 + 0.955183i \(0.404342\pi\)
\(90\) 0 0
\(91\) 192455. 0.0267723
\(92\) −6.92079e6 −0.926613
\(93\) 0 0
\(94\) 1.45555e6 0.180750
\(95\) 304450. 0.0364320
\(96\) 0 0
\(97\) −1.24981e7 −1.39041 −0.695203 0.718813i \(-0.744686\pi\)
−0.695203 + 0.718813i \(0.744686\pi\)
\(98\) 1.99904e6 0.214550
\(99\) 0 0
\(100\) 9.04293e6 0.904293
\(101\) −8.38183e6 −0.809495 −0.404747 0.914429i \(-0.632641\pi\)
−0.404747 + 0.914429i \(0.632641\pi\)
\(102\) 0 0
\(103\) 1.30426e7 1.17607 0.588036 0.808835i \(-0.299902\pi\)
0.588036 + 0.808835i \(0.299902\pi\)
\(104\) 343280. 0.0299248
\(105\) 0 0
\(106\) 3.69728e6 0.301517
\(107\) 8.79724e6 0.694230 0.347115 0.937823i \(-0.387161\pi\)
0.347115 + 0.937823i \(0.387161\pi\)
\(108\) 0 0
\(109\) −1.73854e7 −1.28585 −0.642927 0.765927i \(-0.722280\pi\)
−0.642927 + 0.765927i \(0.722280\pi\)
\(110\) −790979. −0.0566618
\(111\) 0 0
\(112\) 5.50652e6 0.370352
\(113\) 7.28232e6 0.474783 0.237391 0.971414i \(-0.423708\pi\)
0.237391 + 0.971414i \(0.423708\pi\)
\(114\) 0 0
\(115\) 2.30272e6 0.141188
\(116\) 3.19296e6 0.189929
\(117\) 0 0
\(118\) 5.36598e6 0.300650
\(119\) 1.14348e7 0.622035
\(120\) 0 0
\(121\) 2.13813e7 1.09720
\(122\) −8.33331e6 −0.415488
\(123\) 0 0
\(124\) −1.18598e7 −0.558599
\(125\) −6.07827e6 −0.278352
\(126\) 0 0
\(127\) 9.14216e6 0.396037 0.198018 0.980198i \(-0.436549\pi\)
0.198018 + 0.980198i \(0.436549\pi\)
\(128\) 2.15351e7 0.907637
\(129\) 0 0
\(130\) −54807.4 −0.00218795
\(131\) 3.19287e7 1.24089 0.620443 0.784251i \(-0.286953\pi\)
0.620443 + 0.784251i \(0.286953\pi\)
\(132\) 0 0
\(133\) 3.36671e6 0.124087
\(134\) 6.66899e6 0.239438
\(135\) 0 0
\(136\) 2.03961e7 0.695282
\(137\) −1.07371e7 −0.356750 −0.178375 0.983963i \(-0.557084\pi\)
−0.178375 + 0.983963i \(0.557084\pi\)
\(138\) 0 0
\(139\) −7.67775e6 −0.242483 −0.121242 0.992623i \(-0.538688\pi\)
−0.121242 + 0.992623i \(0.538688\pi\)
\(140\) −2.01567e6 −0.0620827
\(141\) 0 0
\(142\) −1.59413e7 −0.467211
\(143\) 2.83180e6 0.0809817
\(144\) 0 0
\(145\) −1.06238e6 −0.0289395
\(146\) −2.83294e6 −0.0753360
\(147\) 0 0
\(148\) −4.23627e7 −1.07416
\(149\) 5.77368e7 1.42988 0.714942 0.699184i \(-0.246453\pi\)
0.714942 + 0.699184i \(0.246453\pi\)
\(150\) 0 0
\(151\) −1.59611e7 −0.377263 −0.188632 0.982048i \(-0.560405\pi\)
−0.188632 + 0.982048i \(0.560405\pi\)
\(152\) 6.00516e6 0.138699
\(153\) 0 0
\(154\) −8.74691e6 −0.192989
\(155\) 3.94604e6 0.0851139
\(156\) 0 0
\(157\) 5.56041e7 1.14672 0.573361 0.819303i \(-0.305639\pi\)
0.573361 + 0.819303i \(0.305639\pi\)
\(158\) 1.27805e7 0.257780
\(159\) 0 0
\(160\) −5.46543e6 −0.105488
\(161\) 2.54643e7 0.480884
\(162\) 0 0
\(163\) −1.30748e7 −0.236472 −0.118236 0.992986i \(-0.537724\pi\)
−0.118236 + 0.992986i \(0.537724\pi\)
\(164\) −8.77405e6 −0.155327
\(165\) 0 0
\(166\) −1.91241e7 −0.324490
\(167\) −4.97944e7 −0.827319 −0.413660 0.910432i \(-0.635750\pi\)
−0.413660 + 0.910432i \(0.635750\pi\)
\(168\) 0 0
\(169\) −6.25523e7 −0.996873
\(170\) −3.25640e6 −0.0508355
\(171\) 0 0
\(172\) −9.38839e6 −0.140683
\(173\) −9.03870e7 −1.32722 −0.663612 0.748077i \(-0.730978\pi\)
−0.663612 + 0.748077i \(0.730978\pi\)
\(174\) 0 0
\(175\) −3.32724e7 −0.469301
\(176\) 8.10234e7 1.12025
\(177\) 0 0
\(178\) −1.23997e7 −0.164794
\(179\) 1.32357e8 1.72489 0.862447 0.506147i \(-0.168931\pi\)
0.862447 + 0.506147i \(0.168931\pi\)
\(180\) 0 0
\(181\) −1.38366e7 −0.173441 −0.0867207 0.996233i \(-0.527639\pi\)
−0.0867207 + 0.996233i \(0.527639\pi\)
\(182\) −606079. −0.00745211
\(183\) 0 0
\(184\) 4.54203e7 0.537511
\(185\) 1.40951e7 0.163670
\(186\) 0 0
\(187\) 1.68253e8 1.88155
\(188\) 5.45775e7 0.599047
\(189\) 0 0
\(190\) −958772. −0.0101409
\(191\) −3.54814e7 −0.368455 −0.184227 0.982884i \(-0.558978\pi\)
−0.184227 + 0.982884i \(0.558978\pi\)
\(192\) 0 0
\(193\) 1.17213e8 1.17361 0.586807 0.809727i \(-0.300385\pi\)
0.586807 + 0.809727i \(0.300385\pi\)
\(194\) 3.93588e7 0.387022
\(195\) 0 0
\(196\) 7.49562e7 0.711068
\(197\) 3.73724e7 0.348272 0.174136 0.984722i \(-0.444287\pi\)
0.174136 + 0.984722i \(0.444287\pi\)
\(198\) 0 0
\(199\) 1.42971e8 1.28606 0.643031 0.765840i \(-0.277677\pi\)
0.643031 + 0.765840i \(0.277677\pi\)
\(200\) −5.93476e7 −0.524564
\(201\) 0 0
\(202\) 2.63960e7 0.225324
\(203\) −1.17481e7 −0.0985672
\(204\) 0 0
\(205\) 2.91935e6 0.0236672
\(206\) −4.10736e7 −0.327362
\(207\) 0 0
\(208\) 5.61416e6 0.0432577
\(209\) 4.95380e7 0.375342
\(210\) 0 0
\(211\) 1.59039e8 1.16551 0.582754 0.812648i \(-0.301975\pi\)
0.582754 + 0.812648i \(0.301975\pi\)
\(212\) 1.38634e8 0.999295
\(213\) 0 0
\(214\) −2.77042e7 −0.193240
\(215\) 3.12375e6 0.0214359
\(216\) 0 0
\(217\) 4.36367e7 0.289896
\(218\) 5.47499e7 0.357920
\(219\) 0 0
\(220\) −2.96587e7 −0.187790
\(221\) 1.16583e7 0.0726546
\(222\) 0 0
\(223\) −4.10300e7 −0.247762 −0.123881 0.992297i \(-0.539534\pi\)
−0.123881 + 0.992297i \(0.539534\pi\)
\(224\) −6.04385e7 −0.359291
\(225\) 0 0
\(226\) −2.29334e7 −0.132157
\(227\) −4.91817e7 −0.279070 −0.139535 0.990217i \(-0.544561\pi\)
−0.139535 + 0.990217i \(0.544561\pi\)
\(228\) 0 0
\(229\) −1.27086e8 −0.699315 −0.349657 0.936878i \(-0.613702\pi\)
−0.349657 + 0.936878i \(0.613702\pi\)
\(230\) −7.25171e6 −0.0393000
\(231\) 0 0
\(232\) −2.09550e7 −0.110174
\(233\) 7.98934e7 0.413776 0.206888 0.978365i \(-0.433666\pi\)
0.206888 + 0.978365i \(0.433666\pi\)
\(234\) 0 0
\(235\) −1.81593e7 −0.0912771
\(236\) 2.01204e8 0.996423
\(237\) 0 0
\(238\) −3.60104e7 −0.173144
\(239\) 1.85347e8 0.878199 0.439100 0.898438i \(-0.355298\pi\)
0.439100 + 0.898438i \(0.355298\pi\)
\(240\) 0 0
\(241\) 1.85587e7 0.0854061 0.0427031 0.999088i \(-0.486403\pi\)
0.0427031 + 0.999088i \(0.486403\pi\)
\(242\) −6.73338e7 −0.305407
\(243\) 0 0
\(244\) −3.12467e8 −1.37702
\(245\) −2.49398e7 −0.108346
\(246\) 0 0
\(247\) 3.43252e6 0.0144935
\(248\) 7.78341e7 0.324033
\(249\) 0 0
\(250\) 1.91416e7 0.0774799
\(251\) −1.86162e8 −0.743075 −0.371537 0.928418i \(-0.621169\pi\)
−0.371537 + 0.928418i \(0.621169\pi\)
\(252\) 0 0
\(253\) 3.74683e8 1.45460
\(254\) −2.87904e7 −0.110238
\(255\) 0 0
\(256\) 8.37598e7 0.312029
\(257\) 4.61094e8 1.69443 0.847214 0.531251i \(-0.178278\pi\)
0.847214 + 0.531251i \(0.178278\pi\)
\(258\) 0 0
\(259\) 1.55869e8 0.557456
\(260\) −2.05507e6 −0.00725136
\(261\) 0 0
\(262\) −1.00550e8 −0.345403
\(263\) 4.24659e8 1.43945 0.719723 0.694261i \(-0.244269\pi\)
0.719723 + 0.694261i \(0.244269\pi\)
\(264\) 0 0
\(265\) −4.61270e7 −0.152263
\(266\) −1.06024e7 −0.0345398
\(267\) 0 0
\(268\) 2.50061e8 0.793551
\(269\) 2.01202e7 0.0630230 0.0315115 0.999503i \(-0.489968\pi\)
0.0315115 + 0.999503i \(0.489968\pi\)
\(270\) 0 0
\(271\) 3.31931e8 1.01311 0.506554 0.862208i \(-0.330919\pi\)
0.506554 + 0.862208i \(0.330919\pi\)
\(272\) 3.33567e8 1.00506
\(273\) 0 0
\(274\) 3.38131e7 0.0993020
\(275\) −4.89573e8 −1.41956
\(276\) 0 0
\(277\) −6.27194e8 −1.77306 −0.886528 0.462674i \(-0.846890\pi\)
−0.886528 + 0.462674i \(0.846890\pi\)
\(278\) 2.41787e7 0.0674957
\(279\) 0 0
\(280\) 1.32286e7 0.0360130
\(281\) 5.46595e8 1.46958 0.734791 0.678294i \(-0.237280\pi\)
0.734791 + 0.678294i \(0.237280\pi\)
\(282\) 0 0
\(283\) 4.74316e8 1.24398 0.621992 0.783023i \(-0.286324\pi\)
0.621992 + 0.783023i \(0.286324\pi\)
\(284\) −5.97736e8 −1.54844
\(285\) 0 0
\(286\) −8.91789e6 −0.0225414
\(287\) 3.22831e7 0.0806100
\(288\) 0 0
\(289\) 2.82345e8 0.688078
\(290\) 3.34563e6 0.00805536
\(291\) 0 0
\(292\) −1.06224e8 −0.249680
\(293\) 2.61274e8 0.606820 0.303410 0.952860i \(-0.401875\pi\)
0.303410 + 0.952860i \(0.401875\pi\)
\(294\) 0 0
\(295\) −6.69455e7 −0.151825
\(296\) 2.78021e8 0.623099
\(297\) 0 0
\(298\) −1.81824e8 −0.398011
\(299\) 2.59620e7 0.0561680
\(300\) 0 0
\(301\) 3.45435e7 0.0730103
\(302\) 5.02647e7 0.105012
\(303\) 0 0
\(304\) 9.82111e7 0.200495
\(305\) 1.03966e8 0.209817
\(306\) 0 0
\(307\) 6.32337e8 1.24728 0.623640 0.781711i \(-0.285653\pi\)
0.623640 + 0.781711i \(0.285653\pi\)
\(308\) −3.27976e8 −0.639608
\(309\) 0 0
\(310\) −1.24268e7 −0.0236916
\(311\) −9.11382e8 −1.71806 −0.859031 0.511923i \(-0.828933\pi\)
−0.859031 + 0.511923i \(0.828933\pi\)
\(312\) 0 0
\(313\) −4.76864e8 −0.879001 −0.439501 0.898242i \(-0.644845\pi\)
−0.439501 + 0.898242i \(0.644845\pi\)
\(314\) −1.75108e8 −0.319192
\(315\) 0 0
\(316\) 4.79221e8 0.854341
\(317\) −2.70951e8 −0.477731 −0.238865 0.971053i \(-0.576775\pi\)
−0.238865 + 0.971053i \(0.576775\pi\)
\(318\) 0 0
\(319\) −1.72863e8 −0.298150
\(320\) −4.65262e7 −0.0793730
\(321\) 0 0
\(322\) −8.01919e7 −0.133855
\(323\) 2.03945e8 0.336747
\(324\) 0 0
\(325\) −3.39228e7 −0.0548151
\(326\) 4.11751e7 0.0658223
\(327\) 0 0
\(328\) 5.75830e7 0.0901023
\(329\) −2.00812e8 −0.310888
\(330\) 0 0
\(331\) −4.79247e8 −0.726376 −0.363188 0.931716i \(-0.618312\pi\)
−0.363188 + 0.931716i \(0.618312\pi\)
\(332\) −7.17079e8 −1.07543
\(333\) 0 0
\(334\) 1.56812e8 0.230286
\(335\) −8.32017e7 −0.120914
\(336\) 0 0
\(337\) −4.14532e8 −0.590002 −0.295001 0.955497i \(-0.595320\pi\)
−0.295001 + 0.955497i \(0.595320\pi\)
\(338\) 1.96989e8 0.277481
\(339\) 0 0
\(340\) −1.22103e8 −0.168480
\(341\) 6.42073e8 0.876888
\(342\) 0 0
\(343\) −6.33599e8 −0.847784
\(344\) 6.16148e7 0.0816076
\(345\) 0 0
\(346\) 2.84646e8 0.369435
\(347\) 3.46330e7 0.0444976 0.0222488 0.999752i \(-0.492917\pi\)
0.0222488 + 0.999752i \(0.492917\pi\)
\(348\) 0 0
\(349\) −1.29687e9 −1.63308 −0.816541 0.577287i \(-0.804111\pi\)
−0.816541 + 0.577287i \(0.804111\pi\)
\(350\) 1.04781e8 0.130631
\(351\) 0 0
\(352\) −8.89297e8 −1.08679
\(353\) −2.44931e8 −0.296369 −0.148184 0.988960i \(-0.547343\pi\)
−0.148184 + 0.988960i \(0.547343\pi\)
\(354\) 0 0
\(355\) 1.98882e8 0.235937
\(356\) −4.64940e8 −0.546163
\(357\) 0 0
\(358\) −4.16819e8 −0.480128
\(359\) −1.89464e8 −0.216121 −0.108060 0.994144i \(-0.534464\pi\)
−0.108060 + 0.994144i \(0.534464\pi\)
\(360\) 0 0
\(361\) −8.33825e8 −0.932824
\(362\) 4.35740e7 0.0482777
\(363\) 0 0
\(364\) −2.27256e7 −0.0246980
\(365\) 3.53435e7 0.0380439
\(366\) 0 0
\(367\) −4.82269e8 −0.509282 −0.254641 0.967036i \(-0.581957\pi\)
−0.254641 + 0.967036i \(0.581957\pi\)
\(368\) 7.42824e8 0.776996
\(369\) 0 0
\(370\) −4.43883e7 −0.0455578
\(371\) −5.10088e8 −0.518604
\(372\) 0 0
\(373\) −1.14493e9 −1.14235 −0.571176 0.820828i \(-0.693512\pi\)
−0.571176 + 0.820828i \(0.693512\pi\)
\(374\) −5.29860e8 −0.523733
\(375\) 0 0
\(376\) −3.58185e8 −0.347496
\(377\) −1.19778e7 −0.0115128
\(378\) 0 0
\(379\) 1.22401e9 1.15491 0.577454 0.816423i \(-0.304046\pi\)
0.577454 + 0.816423i \(0.304046\pi\)
\(380\) −3.59503e7 −0.0336093
\(381\) 0 0
\(382\) 1.11738e8 0.102560
\(383\) −1.02057e9 −0.928210 −0.464105 0.885780i \(-0.653624\pi\)
−0.464105 + 0.885780i \(0.653624\pi\)
\(384\) 0 0
\(385\) 1.09126e8 0.0974574
\(386\) −3.69126e8 −0.326678
\(387\) 0 0
\(388\) 1.47580e9 1.28268
\(389\) 5.65923e7 0.0487454 0.0243727 0.999703i \(-0.492241\pi\)
0.0243727 + 0.999703i \(0.492241\pi\)
\(390\) 0 0
\(391\) 1.54254e9 1.30502
\(392\) −4.91928e8 −0.412477
\(393\) 0 0
\(394\) −1.17693e8 −0.0969422
\(395\) −1.59449e8 −0.130176
\(396\) 0 0
\(397\) −9.60632e8 −0.770532 −0.385266 0.922806i \(-0.625890\pi\)
−0.385266 + 0.922806i \(0.625890\pi\)
\(398\) −4.50243e8 −0.357978
\(399\) 0 0
\(400\) −9.70598e8 −0.758279
\(401\) −1.62607e9 −1.25931 −0.629656 0.776874i \(-0.716804\pi\)
−0.629656 + 0.776874i \(0.716804\pi\)
\(402\) 0 0
\(403\) 4.44896e7 0.0338603
\(404\) 9.89748e8 0.746775
\(405\) 0 0
\(406\) 3.69971e7 0.0274364
\(407\) 2.29347e9 1.68621
\(408\) 0 0
\(409\) −1.13080e9 −0.817247 −0.408623 0.912703i \(-0.633991\pi\)
−0.408623 + 0.912703i \(0.633991\pi\)
\(410\) −9.19359e6 −0.00658782
\(411\) 0 0
\(412\) −1.54010e9 −1.08495
\(413\) −7.40306e8 −0.517114
\(414\) 0 0
\(415\) 2.38590e8 0.163864
\(416\) −6.16199e7 −0.0419657
\(417\) 0 0
\(418\) −1.56005e8 −0.104477
\(419\) 5.90861e7 0.0392406 0.0196203 0.999808i \(-0.493754\pi\)
0.0196203 + 0.999808i \(0.493754\pi\)
\(420\) 0 0
\(421\) 2.62456e9 1.71423 0.857116 0.515124i \(-0.172254\pi\)
0.857116 + 0.515124i \(0.172254\pi\)
\(422\) −5.00845e8 −0.324421
\(423\) 0 0
\(424\) −9.09837e8 −0.579673
\(425\) −2.01554e9 −1.27359
\(426\) 0 0
\(427\) 1.14969e9 0.714632
\(428\) −1.03880e9 −0.640441
\(429\) 0 0
\(430\) −9.83730e6 −0.00596673
\(431\) 5.24926e8 0.315811 0.157906 0.987454i \(-0.449526\pi\)
0.157906 + 0.987454i \(0.449526\pi\)
\(432\) 0 0
\(433\) 2.25218e9 1.33320 0.666600 0.745415i \(-0.267749\pi\)
0.666600 + 0.745415i \(0.267749\pi\)
\(434\) −1.37420e8 −0.0806931
\(435\) 0 0
\(436\) 2.05291e9 1.18623
\(437\) 4.54166e8 0.260333
\(438\) 0 0
\(439\) 3.75255e8 0.211690 0.105845 0.994383i \(-0.466245\pi\)
0.105845 + 0.994383i \(0.466245\pi\)
\(440\) 1.94646e8 0.108933
\(441\) 0 0
\(442\) −3.67143e7 −0.0202235
\(443\) −5.78363e7 −0.0316073 −0.0158036 0.999875i \(-0.505031\pi\)
−0.0158036 + 0.999875i \(0.505031\pi\)
\(444\) 0 0
\(445\) 1.54697e8 0.0832190
\(446\) 1.29211e8 0.0689650
\(447\) 0 0
\(448\) −5.14503e8 −0.270343
\(449\) −7.38749e8 −0.385154 −0.192577 0.981282i \(-0.561685\pi\)
−0.192577 + 0.981282i \(0.561685\pi\)
\(450\) 0 0
\(451\) 4.75016e8 0.243832
\(452\) −8.59915e8 −0.437997
\(453\) 0 0
\(454\) 1.54883e8 0.0776797
\(455\) 7.56139e6 0.00376324
\(456\) 0 0
\(457\) −1.98291e9 −0.971845 −0.485922 0.874002i \(-0.661516\pi\)
−0.485922 + 0.874002i \(0.661516\pi\)
\(458\) 4.00217e8 0.194656
\(459\) 0 0
\(460\) −2.71911e8 −0.130249
\(461\) −1.72573e9 −0.820389 −0.410195 0.911998i \(-0.634539\pi\)
−0.410195 + 0.911998i \(0.634539\pi\)
\(462\) 0 0
\(463\) 3.52996e9 1.65286 0.826429 0.563040i \(-0.190368\pi\)
0.826429 + 0.563040i \(0.190368\pi\)
\(464\) −3.42707e8 −0.159261
\(465\) 0 0
\(466\) −2.51600e8 −0.115175
\(467\) −3.16059e9 −1.43602 −0.718009 0.696034i \(-0.754946\pi\)
−0.718009 + 0.696034i \(0.754946\pi\)
\(468\) 0 0
\(469\) −9.20073e8 −0.411829
\(470\) 5.71871e7 0.0254071
\(471\) 0 0
\(472\) −1.32047e9 −0.578007
\(473\) 5.08276e8 0.220844
\(474\) 0 0
\(475\) −5.93428e8 −0.254062
\(476\) −1.35025e9 −0.573840
\(477\) 0 0
\(478\) −5.83693e8 −0.244448
\(479\) 1.20495e9 0.500950 0.250475 0.968123i \(-0.419413\pi\)
0.250475 + 0.968123i \(0.419413\pi\)
\(480\) 0 0
\(481\) 1.58916e8 0.0651117
\(482\) −5.84450e7 −0.0237730
\(483\) 0 0
\(484\) −2.52476e9 −1.01219
\(485\) −4.91037e8 −0.195442
\(486\) 0 0
\(487\) −1.76020e9 −0.690576 −0.345288 0.938497i \(-0.612219\pi\)
−0.345288 + 0.938497i \(0.612219\pi\)
\(488\) 2.05068e9 0.798784
\(489\) 0 0
\(490\) 7.85402e7 0.0301582
\(491\) 3.34075e9 1.27368 0.636839 0.770997i \(-0.280242\pi\)
0.636839 + 0.770997i \(0.280242\pi\)
\(492\) 0 0
\(493\) −7.11664e8 −0.267492
\(494\) −1.08097e7 −0.00403430
\(495\) 0 0
\(496\) 1.27293e9 0.468404
\(497\) 2.19930e9 0.803595
\(498\) 0 0
\(499\) −3.65908e9 −1.31832 −0.659159 0.752004i \(-0.729087\pi\)
−0.659159 + 0.752004i \(0.729087\pi\)
\(500\) 7.17738e8 0.256786
\(501\) 0 0
\(502\) 5.86259e8 0.206836
\(503\) 9.88140e7 0.0346203 0.0173101 0.999850i \(-0.494490\pi\)
0.0173101 + 0.999850i \(0.494490\pi\)
\(504\) 0 0
\(505\) −3.29314e8 −0.113786
\(506\) −1.17995e9 −0.404889
\(507\) 0 0
\(508\) −1.07953e9 −0.365352
\(509\) 3.29148e9 1.10632 0.553158 0.833076i \(-0.313422\pi\)
0.553158 + 0.833076i \(0.313422\pi\)
\(510\) 0 0
\(511\) 3.90841e8 0.129577
\(512\) −3.02027e9 −0.994491
\(513\) 0 0
\(514\) −1.45207e9 −0.471647
\(515\) 5.12431e8 0.165314
\(516\) 0 0
\(517\) −2.95476e9 −0.940384
\(518\) −4.90861e8 −0.155169
\(519\) 0 0
\(520\) 1.34871e7 0.00420638
\(521\) −1.48897e9 −0.461267 −0.230634 0.973041i \(-0.574080\pi\)
−0.230634 + 0.973041i \(0.574080\pi\)
\(522\) 0 0
\(523\) 5.83188e9 1.78259 0.891297 0.453420i \(-0.149796\pi\)
0.891297 + 0.453420i \(0.149796\pi\)
\(524\) −3.77023e9 −1.14474
\(525\) 0 0
\(526\) −1.33733e9 −0.400673
\(527\) 2.64337e9 0.786720
\(528\) 0 0
\(529\) 3.02763e7 0.00889216
\(530\) 1.45263e8 0.0423827
\(531\) 0 0
\(532\) −3.97550e8 −0.114472
\(533\) 3.29142e7 0.00941538
\(534\) 0 0
\(535\) 3.45635e8 0.0975843
\(536\) −1.64112e9 −0.460324
\(537\) 0 0
\(538\) −6.33623e7 −0.0175426
\(539\) −4.05803e9 −1.11623
\(540\) 0 0
\(541\) 3.86680e9 1.04993 0.524966 0.851123i \(-0.324078\pi\)
0.524966 + 0.851123i \(0.324078\pi\)
\(542\) −1.04532e9 −0.282001
\(543\) 0 0
\(544\) −3.66117e9 −0.975043
\(545\) −6.83056e8 −0.180746
\(546\) 0 0
\(547\) 3.69733e9 0.965901 0.482950 0.875648i \(-0.339565\pi\)
0.482950 + 0.875648i \(0.339565\pi\)
\(548\) 1.26786e9 0.329109
\(549\) 0 0
\(550\) 1.54176e9 0.395136
\(551\) −2.09533e8 −0.0533607
\(552\) 0 0
\(553\) −1.76324e9 −0.443378
\(554\) 1.97515e9 0.493534
\(555\) 0 0
\(556\) 9.06608e8 0.223696
\(557\) −2.06821e9 −0.507110 −0.253555 0.967321i \(-0.581600\pi\)
−0.253555 + 0.967321i \(0.581600\pi\)
\(558\) 0 0
\(559\) 3.52188e7 0.00852771
\(560\) 2.16346e8 0.0520584
\(561\) 0 0
\(562\) −1.72133e9 −0.409061
\(563\) −3.92675e9 −0.927372 −0.463686 0.886000i \(-0.653473\pi\)
−0.463686 + 0.886000i \(0.653473\pi\)
\(564\) 0 0
\(565\) 2.86115e8 0.0667377
\(566\) −1.49371e9 −0.346265
\(567\) 0 0
\(568\) 3.92286e9 0.898223
\(569\) 3.55278e9 0.808491 0.404245 0.914651i \(-0.367534\pi\)
0.404245 + 0.914651i \(0.367534\pi\)
\(570\) 0 0
\(571\) 1.16477e8 0.0261828 0.0130914 0.999914i \(-0.495833\pi\)
0.0130914 + 0.999914i \(0.495833\pi\)
\(572\) −3.34386e8 −0.0747072
\(573\) 0 0
\(574\) −1.01666e8 −0.0224380
\(575\) −4.48841e9 −0.984590
\(576\) 0 0
\(577\) 6.38024e9 1.38268 0.691340 0.722529i \(-0.257021\pi\)
0.691340 + 0.722529i \(0.257021\pi\)
\(578\) −8.89159e8 −0.191528
\(579\) 0 0
\(580\) 1.25448e8 0.0266973
\(581\) 2.63841e9 0.558118
\(582\) 0 0
\(583\) −7.50547e9 −1.56869
\(584\) 6.97137e8 0.144835
\(585\) 0 0
\(586\) −8.22803e8 −0.168910
\(587\) 3.91131e9 0.798157 0.399079 0.916917i \(-0.369330\pi\)
0.399079 + 0.916917i \(0.369330\pi\)
\(588\) 0 0
\(589\) 7.78278e8 0.156939
\(590\) 2.10824e8 0.0422609
\(591\) 0 0
\(592\) 4.54689e9 0.900716
\(593\) 9.01946e9 1.77619 0.888094 0.459661i \(-0.152029\pi\)
0.888094 + 0.459661i \(0.152029\pi\)
\(594\) 0 0
\(595\) 4.49263e8 0.0874362
\(596\) −6.81771e9 −1.31910
\(597\) 0 0
\(598\) −8.17593e7 −0.0156345
\(599\) −3.06365e9 −0.582432 −0.291216 0.956657i \(-0.594060\pi\)
−0.291216 + 0.956657i \(0.594060\pi\)
\(600\) 0 0
\(601\) 4.13537e9 0.777059 0.388529 0.921436i \(-0.372983\pi\)
0.388529 + 0.921436i \(0.372983\pi\)
\(602\) −1.08784e8 −0.0203225
\(603\) 0 0
\(604\) 1.88473e9 0.348033
\(605\) 8.40052e8 0.154228
\(606\) 0 0
\(607\) −3.50688e9 −0.636445 −0.318222 0.948016i \(-0.603086\pi\)
−0.318222 + 0.948016i \(0.603086\pi\)
\(608\) −1.07795e9 −0.194507
\(609\) 0 0
\(610\) −3.27408e8 −0.0584030
\(611\) −2.04737e8 −0.0363122
\(612\) 0 0
\(613\) −2.51080e9 −0.440251 −0.220125 0.975472i \(-0.570647\pi\)
−0.220125 + 0.975472i \(0.570647\pi\)
\(614\) −1.99135e9 −0.347183
\(615\) 0 0
\(616\) 2.15246e9 0.371025
\(617\) 8.91772e9 1.52847 0.764233 0.644941i \(-0.223118\pi\)
0.764233 + 0.644941i \(0.223118\pi\)
\(618\) 0 0
\(619\) −1.01712e10 −1.72368 −0.861838 0.507183i \(-0.830687\pi\)
−0.861838 + 0.507183i \(0.830687\pi\)
\(620\) −4.65959e8 −0.0785193
\(621\) 0 0
\(622\) 2.87012e9 0.478226
\(623\) 1.71069e9 0.283442
\(624\) 0 0
\(625\) 5.74411e9 0.941115
\(626\) 1.50174e9 0.244672
\(627\) 0 0
\(628\) −6.56588e9 −1.05787
\(629\) 9.44203e9 1.51282
\(630\) 0 0
\(631\) −8.57149e9 −1.35817 −0.679084 0.734061i \(-0.737623\pi\)
−0.679084 + 0.734061i \(0.737623\pi\)
\(632\) −3.14507e9 −0.495588
\(633\) 0 0
\(634\) 8.53276e8 0.132977
\(635\) 3.59187e8 0.0556688
\(636\) 0 0
\(637\) −2.81184e8 −0.0431025
\(638\) 5.44378e8 0.0829905
\(639\) 0 0
\(640\) 8.46094e8 0.127582
\(641\) −3.31058e9 −0.496479 −0.248240 0.968699i \(-0.579852\pi\)
−0.248240 + 0.968699i \(0.579852\pi\)
\(642\) 0 0
\(643\) −3.42873e9 −0.508622 −0.254311 0.967122i \(-0.581849\pi\)
−0.254311 + 0.967122i \(0.581849\pi\)
\(644\) −3.00689e9 −0.443626
\(645\) 0 0
\(646\) −6.42260e8 −0.0937340
\(647\) −2.83989e8 −0.0412228 −0.0206114 0.999788i \(-0.506561\pi\)
−0.0206114 + 0.999788i \(0.506561\pi\)
\(648\) 0 0
\(649\) −1.08929e10 −1.56418
\(650\) 1.06829e8 0.0152579
\(651\) 0 0
\(652\) 1.54391e9 0.218150
\(653\) −3.48386e8 −0.0489626 −0.0244813 0.999700i \(-0.507793\pi\)
−0.0244813 + 0.999700i \(0.507793\pi\)
\(654\) 0 0
\(655\) 1.25445e9 0.174425
\(656\) 9.41739e8 0.130247
\(657\) 0 0
\(658\) 6.32395e8 0.0865361
\(659\) 9.43143e9 1.28375 0.641873 0.766811i \(-0.278158\pi\)
0.641873 + 0.766811i \(0.278158\pi\)
\(660\) 0 0
\(661\) 1.05208e10 1.41692 0.708458 0.705753i \(-0.249391\pi\)
0.708458 + 0.705753i \(0.249391\pi\)
\(662\) 1.50924e9 0.202188
\(663\) 0 0
\(664\) 4.70610e9 0.623840
\(665\) 1.32275e8 0.0174422
\(666\) 0 0
\(667\) −1.58481e9 −0.206793
\(668\) 5.87986e9 0.763219
\(669\) 0 0
\(670\) 2.62018e8 0.0336565
\(671\) 1.69166e10 2.16164
\(672\) 0 0
\(673\) −2.80231e8 −0.0354376 −0.0177188 0.999843i \(-0.505640\pi\)
−0.0177188 + 0.999843i \(0.505640\pi\)
\(674\) 1.30544e9 0.164228
\(675\) 0 0
\(676\) 7.38634e9 0.919636
\(677\) 3.28856e9 0.407329 0.203664 0.979041i \(-0.434715\pi\)
0.203664 + 0.979041i \(0.434715\pi\)
\(678\) 0 0
\(679\) −5.43006e9 −0.665672
\(680\) 8.01344e8 0.0977322
\(681\) 0 0
\(682\) −2.02201e9 −0.244083
\(683\) 2.54212e9 0.305298 0.152649 0.988280i \(-0.451220\pi\)
0.152649 + 0.988280i \(0.451220\pi\)
\(684\) 0 0
\(685\) −4.21849e8 −0.0501465
\(686\) 1.99532e9 0.235982
\(687\) 0 0
\(688\) 1.00768e9 0.117967
\(689\) −5.20058e8 −0.0605738
\(690\) 0 0
\(691\) 1.50328e10 1.73328 0.866638 0.498937i \(-0.166276\pi\)
0.866638 + 0.498937i \(0.166276\pi\)
\(692\) 1.06731e10 1.22439
\(693\) 0 0
\(694\) −1.09066e8 −0.0123860
\(695\) −3.01651e8 −0.0340846
\(696\) 0 0
\(697\) 1.95561e9 0.218760
\(698\) 4.08410e9 0.454571
\(699\) 0 0
\(700\) 3.92890e9 0.432940
\(701\) 3.96542e9 0.434787 0.217393 0.976084i \(-0.430245\pi\)
0.217393 + 0.976084i \(0.430245\pi\)
\(702\) 0 0
\(703\) 2.77999e9 0.301786
\(704\) −7.57043e9 −0.817742
\(705\) 0 0
\(706\) 7.71335e8 0.0824947
\(707\) −3.64167e9 −0.387554
\(708\) 0 0
\(709\) 8.82480e9 0.929914 0.464957 0.885333i \(-0.346070\pi\)
0.464957 + 0.885333i \(0.346070\pi\)
\(710\) −6.26317e8 −0.0656734
\(711\) 0 0
\(712\) 3.05134e9 0.316819
\(713\) 5.88654e9 0.608200
\(714\) 0 0
\(715\) 1.11259e8 0.0113832
\(716\) −1.56291e10 −1.59125
\(717\) 0 0
\(718\) 5.96659e8 0.0601576
\(719\) 1.63954e10 1.64501 0.822507 0.568755i \(-0.192575\pi\)
0.822507 + 0.568755i \(0.192575\pi\)
\(720\) 0 0
\(721\) 5.66664e9 0.563056
\(722\) 2.62587e9 0.259653
\(723\) 0 0
\(724\) 1.63386e9 0.160003
\(725\) 2.07076e9 0.201812
\(726\) 0 0
\(727\) −4.74182e9 −0.457693 −0.228847 0.973462i \(-0.573495\pi\)
−0.228847 + 0.973462i \(0.573495\pi\)
\(728\) 1.49145e8 0.0143268
\(729\) 0 0
\(730\) −1.11304e8 −0.0105896
\(731\) 2.09254e9 0.198135
\(732\) 0 0
\(733\) −1.56780e10 −1.47037 −0.735184 0.677868i \(-0.762904\pi\)
−0.735184 + 0.677868i \(0.762904\pi\)
\(734\) 1.51876e9 0.141760
\(735\) 0 0
\(736\) −8.15309e9 −0.753789
\(737\) −1.35380e10 −1.24572
\(738\) 0 0
\(739\) 2.09854e10 1.91277 0.956383 0.292116i \(-0.0943594\pi\)
0.956383 + 0.292116i \(0.0943594\pi\)
\(740\) −1.66439e9 −0.150989
\(741\) 0 0
\(742\) 1.60636e9 0.144354
\(743\) −1.57915e10 −1.41242 −0.706209 0.708004i \(-0.749596\pi\)
−0.706209 + 0.708004i \(0.749596\pi\)
\(744\) 0 0
\(745\) 2.26842e9 0.200991
\(746\) 3.60562e9 0.317976
\(747\) 0 0
\(748\) −1.98677e10 −1.73577
\(749\) 3.82215e9 0.332370
\(750\) 0 0
\(751\) 1.46937e10 1.26588 0.632939 0.774201i \(-0.281848\pi\)
0.632939 + 0.774201i \(0.281848\pi\)
\(752\) −5.85792e9 −0.502321
\(753\) 0 0
\(754\) 3.77203e7 0.00320461
\(755\) −6.27098e8 −0.0530299
\(756\) 0 0
\(757\) 8.22699e9 0.689295 0.344648 0.938732i \(-0.387998\pi\)
0.344648 + 0.938732i \(0.387998\pi\)
\(758\) −3.85464e9 −0.321471
\(759\) 0 0
\(760\) 2.35937e8 0.0194961
\(761\) −2.23002e10 −1.83427 −0.917133 0.398581i \(-0.869503\pi\)
−0.917133 + 0.398581i \(0.869503\pi\)
\(762\) 0 0
\(763\) −7.55346e9 −0.615616
\(764\) 4.18974e9 0.339907
\(765\) 0 0
\(766\) 3.21397e9 0.258369
\(767\) −7.54777e8 −0.0603997
\(768\) 0 0
\(769\) 1.62540e10 1.28890 0.644448 0.764648i \(-0.277087\pi\)
0.644448 + 0.764648i \(0.277087\pi\)
\(770\) −3.43658e8 −0.0271274
\(771\) 0 0
\(772\) −1.38408e10 −1.08268
\(773\) 1.29767e10 1.01050 0.505248 0.862974i \(-0.331401\pi\)
0.505248 + 0.862974i \(0.331401\pi\)
\(774\) 0 0
\(775\) −7.69154e9 −0.593550
\(776\) −9.68552e9 −0.744058
\(777\) 0 0
\(778\) −1.78220e8 −0.0135684
\(779\) 5.75783e8 0.0436393
\(780\) 0 0
\(781\) 3.23607e10 2.43074
\(782\) −4.85776e9 −0.363256
\(783\) 0 0
\(784\) −8.04521e9 −0.596254
\(785\) 2.18463e9 0.161189
\(786\) 0 0
\(787\) −1.18344e10 −0.865436 −0.432718 0.901529i \(-0.642445\pi\)
−0.432718 + 0.901529i \(0.642445\pi\)
\(788\) −4.41302e9 −0.321288
\(789\) 0 0
\(790\) 5.02136e8 0.0362348
\(791\) 3.16396e9 0.227307
\(792\) 0 0
\(793\) 1.17216e9 0.0834701
\(794\) 3.02522e9 0.214479
\(795\) 0 0
\(796\) −1.68824e10 −1.18642
\(797\) 5.74935e8 0.0402267 0.0201134 0.999798i \(-0.493597\pi\)
0.0201134 + 0.999798i \(0.493597\pi\)
\(798\) 0 0
\(799\) −1.21645e10 −0.843687
\(800\) 1.06531e10 0.735632
\(801\) 0 0
\(802\) 5.12080e9 0.350532
\(803\) 5.75086e9 0.391948
\(804\) 0 0
\(805\) 1.00047e9 0.0675954
\(806\) −1.40106e8 −0.00942508
\(807\) 0 0
\(808\) −6.49559e9 −0.433191
\(809\) −1.85697e10 −1.23306 −0.616532 0.787330i \(-0.711463\pi\)
−0.616532 + 0.787330i \(0.711463\pi\)
\(810\) 0 0
\(811\) −1.35102e10 −0.889384 −0.444692 0.895683i \(-0.646687\pi\)
−0.444692 + 0.895683i \(0.646687\pi\)
\(812\) 1.38725e9 0.0909303
\(813\) 0 0
\(814\) −7.22256e9 −0.469360
\(815\) −5.13697e8 −0.0332396
\(816\) 0 0
\(817\) 6.16098e8 0.0395251
\(818\) 3.56110e9 0.227482
\(819\) 0 0
\(820\) −3.44724e8 −0.0218335
\(821\) 2.18113e10 1.37557 0.687783 0.725917i \(-0.258584\pi\)
0.687783 + 0.725917i \(0.258584\pi\)
\(822\) 0 0
\(823\) 2.25550e10 1.41040 0.705202 0.709006i \(-0.250856\pi\)
0.705202 + 0.709006i \(0.250856\pi\)
\(824\) 1.01075e10 0.629359
\(825\) 0 0
\(826\) 2.33137e9 0.143940
\(827\) 2.31862e10 1.42548 0.712739 0.701430i \(-0.247455\pi\)
0.712739 + 0.701430i \(0.247455\pi\)
\(828\) 0 0
\(829\) 2.89637e10 1.76568 0.882842 0.469671i \(-0.155627\pi\)
0.882842 + 0.469671i \(0.155627\pi\)
\(830\) −7.51366e8 −0.0456119
\(831\) 0 0
\(832\) −5.24559e8 −0.0315764
\(833\) −1.67066e10 −1.00145
\(834\) 0 0
\(835\) −1.95638e9 −0.116292
\(836\) −5.84958e9 −0.346260
\(837\) 0 0
\(838\) −1.86073e8 −0.0109227
\(839\) 9.53603e9 0.557444 0.278722 0.960372i \(-0.410089\pi\)
0.278722 + 0.960372i \(0.410089\pi\)
\(840\) 0 0
\(841\) −1.65187e10 −0.957613
\(842\) −8.26525e9 −0.477160
\(843\) 0 0
\(844\) −1.87798e10 −1.07521
\(845\) −2.45762e9 −0.140125
\(846\) 0 0
\(847\) 9.28957e9 0.525296
\(848\) −1.48799e10 −0.837942
\(849\) 0 0
\(850\) 6.34731e9 0.354506
\(851\) 2.10265e10 1.16954
\(852\) 0 0
\(853\) 2.23351e10 1.23216 0.616078 0.787685i \(-0.288721\pi\)
0.616078 + 0.787685i \(0.288721\pi\)
\(854\) −3.62059e9 −0.198919
\(855\) 0 0
\(856\) 6.81752e9 0.371508
\(857\) 2.98724e10 1.62120 0.810602 0.585598i \(-0.199140\pi\)
0.810602 + 0.585598i \(0.199140\pi\)
\(858\) 0 0
\(859\) −1.94294e10 −1.04588 −0.522941 0.852369i \(-0.675165\pi\)
−0.522941 + 0.852369i \(0.675165\pi\)
\(860\) −3.68861e8 −0.0197751
\(861\) 0 0
\(862\) −1.65309e9 −0.0879067
\(863\) −2.20900e10 −1.16992 −0.584962 0.811060i \(-0.698891\pi\)
−0.584962 + 0.811060i \(0.698891\pi\)
\(864\) 0 0
\(865\) −3.55122e9 −0.186561
\(866\) −7.09255e9 −0.371099
\(867\) 0 0
\(868\) −5.15273e9 −0.267435
\(869\) −2.59445e10 −1.34114
\(870\) 0 0
\(871\) −9.38057e8 −0.0481023
\(872\) −1.34730e10 −0.688108
\(873\) 0 0
\(874\) −1.43025e9 −0.0724642
\(875\) −2.64083e9 −0.133264
\(876\) 0 0
\(877\) −3.02430e10 −1.51400 −0.757001 0.653413i \(-0.773336\pi\)
−0.757001 + 0.653413i \(0.773336\pi\)
\(878\) −1.18175e9 −0.0589243
\(879\) 0 0
\(880\) 3.18333e9 0.157468
\(881\) 2.28024e10 1.12348 0.561739 0.827314i \(-0.310132\pi\)
0.561739 + 0.827314i \(0.310132\pi\)
\(882\) 0 0
\(883\) −3.29419e10 −1.61023 −0.805113 0.593122i \(-0.797895\pi\)
−0.805113 + 0.593122i \(0.797895\pi\)
\(884\) −1.37664e9 −0.0670253
\(885\) 0 0
\(886\) 1.82138e8 0.00879795
\(887\) −2.72521e9 −0.131119 −0.0655597 0.997849i \(-0.520883\pi\)
−0.0655597 + 0.997849i \(0.520883\pi\)
\(888\) 0 0
\(889\) 3.97201e9 0.189607
\(890\) −4.87171e8 −0.0231642
\(891\) 0 0
\(892\) 4.84493e9 0.228565
\(893\) −3.58156e9 −0.168303
\(894\) 0 0
\(895\) 5.20020e9 0.242459
\(896\) 9.35640e9 0.434541
\(897\) 0 0
\(898\) 2.32646e9 0.107208
\(899\) −2.71580e9 −0.124663
\(900\) 0 0
\(901\) −3.08995e10 −1.40739
\(902\) −1.49592e9 −0.0678711
\(903\) 0 0
\(904\) 5.64351e9 0.254074
\(905\) −5.43625e8 −0.0243798
\(906\) 0 0
\(907\) 6.55130e9 0.291542 0.145771 0.989318i \(-0.453434\pi\)
0.145771 + 0.989318i \(0.453434\pi\)
\(908\) 5.80751e9 0.257448
\(909\) 0 0
\(910\) −2.38123e7 −0.00104750
\(911\) 2.40723e10 1.05488 0.527440 0.849592i \(-0.323152\pi\)
0.527440 + 0.849592i \(0.323152\pi\)
\(912\) 0 0
\(913\) 3.88218e10 1.68821
\(914\) 6.24457e9 0.270515
\(915\) 0 0
\(916\) 1.50066e10 0.645132
\(917\) 1.38721e10 0.594087
\(918\) 0 0
\(919\) 4.57146e10 1.94290 0.971450 0.237244i \(-0.0762441\pi\)
0.971450 + 0.237244i \(0.0762441\pi\)
\(920\) 1.78452e9 0.0755551
\(921\) 0 0
\(922\) 5.43466e9 0.228357
\(923\) 2.24229e9 0.0938612
\(924\) 0 0
\(925\) −2.74739e10 −1.14137
\(926\) −1.11165e10 −0.460076
\(927\) 0 0
\(928\) 3.76149e9 0.154505
\(929\) 3.25677e10 1.33270 0.666349 0.745640i \(-0.267856\pi\)
0.666349 + 0.745640i \(0.267856\pi\)
\(930\) 0 0
\(931\) −4.91888e9 −0.199775
\(932\) −9.43402e9 −0.381717
\(933\) 0 0
\(934\) 9.95332e9 0.399718
\(935\) 6.61048e9 0.264480
\(936\) 0 0
\(937\) −3.92369e10 −1.55814 −0.779069 0.626938i \(-0.784308\pi\)
−0.779069 + 0.626938i \(0.784308\pi\)
\(938\) 2.89748e9 0.114633
\(939\) 0 0
\(940\) 2.14430e9 0.0842049
\(941\) −3.07063e10 −1.20133 −0.600667 0.799499i \(-0.705098\pi\)
−0.600667 + 0.799499i \(0.705098\pi\)
\(942\) 0 0
\(943\) 4.35496e9 0.169119
\(944\) −2.15956e10 −0.835534
\(945\) 0 0
\(946\) −1.60066e9 −0.0614723
\(947\) −4.74031e10 −1.81377 −0.906883 0.421382i \(-0.861545\pi\)
−0.906883 + 0.421382i \(0.861545\pi\)
\(948\) 0 0
\(949\) 3.98480e8 0.0151348
\(950\) 1.86882e9 0.0707187
\(951\) 0 0
\(952\) 8.86153e9 0.332874
\(953\) −4.93561e10 −1.84721 −0.923604 0.383347i \(-0.874771\pi\)
−0.923604 + 0.383347i \(0.874771\pi\)
\(954\) 0 0
\(955\) −1.39403e9 −0.0517918
\(956\) −2.18863e10 −0.810157
\(957\) 0 0
\(958\) −3.79462e9 −0.139440
\(959\) −4.66495e9 −0.170798
\(960\) 0 0
\(961\) −1.74252e10 −0.633353
\(962\) −5.00456e8 −0.0181239
\(963\) 0 0
\(964\) −2.19147e9 −0.0787889
\(965\) 4.60519e9 0.164969
\(966\) 0 0
\(967\) −6.84804e9 −0.243542 −0.121771 0.992558i \(-0.538857\pi\)
−0.121771 + 0.992558i \(0.538857\pi\)
\(968\) 1.65697e10 0.587152
\(969\) 0 0
\(970\) 1.54637e9 0.0544017
\(971\) −4.01610e8 −0.0140779 −0.00703894 0.999975i \(-0.502241\pi\)
−0.00703894 + 0.999975i \(0.502241\pi\)
\(972\) 0 0
\(973\) −3.33576e9 −0.116091
\(974\) 5.54322e9 0.192223
\(975\) 0 0
\(976\) 3.35378e10 1.15468
\(977\) −3.71330e10 −1.27388 −0.636940 0.770913i \(-0.719800\pi\)
−0.636940 + 0.770913i \(0.719800\pi\)
\(978\) 0 0
\(979\) 2.51713e10 0.857365
\(980\) 2.94496e9 0.0999511
\(981\) 0 0
\(982\) −1.05207e10 −0.354530
\(983\) −1.12625e10 −0.378179 −0.189089 0.981960i \(-0.560554\pi\)
−0.189089 + 0.981960i \(0.560554\pi\)
\(984\) 0 0
\(985\) 1.46832e9 0.0489548
\(986\) 2.24116e9 0.0744568
\(987\) 0 0
\(988\) −4.05321e8 −0.0133706
\(989\) 4.65988e9 0.153175
\(990\) 0 0
\(991\) 4.74895e10 1.55003 0.775015 0.631943i \(-0.217743\pi\)
0.775015 + 0.631943i \(0.217743\pi\)
\(992\) −1.39715e10 −0.454414
\(993\) 0 0
\(994\) −6.92602e9 −0.223682
\(995\) 5.61719e9 0.180775
\(996\) 0 0
\(997\) 1.12122e10 0.358310 0.179155 0.983821i \(-0.442664\pi\)
0.179155 + 0.983821i \(0.442664\pi\)
\(998\) 1.15231e10 0.366956
\(999\) 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 387.8.a.b.1.5 11
3.2 odd 2 43.8.a.a.1.7 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.a.1.7 11 3.2 odd 2
387.8.a.b.1.5 11 1.1 even 1 trivial