Properties

Label 384.10.a.f.1.1
Level $384$
Weight $10$
Character 384.1
Self dual yes
Analytic conductor $197.774$
Analytic rank $1$
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] = [384,10,Mod(1,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 384.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: \(197.773761087\)
Analytic rank: \(1\)
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: \( x^{4} - 2070x^{2} - 13768x + 561570 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-34.7879\) of defining polynomial
Character \(\chi\) \(=\) 384.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+81.0000 q^{3} -2550.24 q^{5} +11484.8 q^{7} +6561.00 q^{9} +O(q^{10})\) \(q+81.0000 q^{3} -2550.24 q^{5} +11484.8 q^{7} +6561.00 q^{9} -17694.9 q^{11} -2878.66 q^{13} -206569. q^{15} -465200. q^{17} +409191. q^{19} +930265. q^{21} +374507. q^{23} +4.55058e6 q^{25} +531441. q^{27} -3.81849e6 q^{29} +2.57794e6 q^{31} -1.43328e6 q^{33} -2.92889e7 q^{35} -1.07373e7 q^{37} -233171. q^{39} +1.50461e7 q^{41} -1.21149e7 q^{43} -1.67321e7 q^{45} -4.65533e7 q^{47} +9.15461e7 q^{49} -3.76812e7 q^{51} +1.10710e8 q^{53} +4.51261e7 q^{55} +3.31445e7 q^{57} -9.19329e7 q^{59} +1.64770e8 q^{61} +7.53515e7 q^{63} +7.34126e6 q^{65} -2.68898e8 q^{67} +3.03350e7 q^{69} +3.57888e8 q^{71} -7.40257e7 q^{73} +3.68597e8 q^{75} -2.03221e8 q^{77} +9.17558e7 q^{79} +4.30467e7 q^{81} -1.86443e8 q^{83} +1.18637e9 q^{85} -3.09298e8 q^{87} -6.37166e8 q^{89} -3.30607e7 q^{91} +2.08813e8 q^{93} -1.04353e9 q^{95} -5.12367e7 q^{97} -1.16096e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 324 q^{3} - 240 q^{5} + 4840 q^{7} + 26244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 324 q^{3} - 240 q^{5} + 4840 q^{7} + 26244 q^{9} - 99664 q^{11} - 60840 q^{13} - 19440 q^{15} - 434952 q^{17} + 631776 q^{19} + 392040 q^{21} + 749392 q^{23} + 5991532 q^{25} + 2125764 q^{27} - 7908544 q^{29} + 11351240 q^{31} - 8072784 q^{33} - 25567008 q^{35} - 13592920 q^{37} - 4928040 q^{39} - 18838888 q^{41} + 14177920 q^{43} - 1574640 q^{45} - 37779120 q^{47} + 9409332 q^{49} - 35231112 q^{51} + 115336512 q^{53} - 184580544 q^{55} + 51173856 q^{57} - 115028080 q^{59} + 173228648 q^{61} + 31755240 q^{63} - 328077984 q^{65} - 231785104 q^{67} + 60700752 q^{69} + 197476208 q^{71} + 44629400 q^{73} + 485314092 q^{75} - 308117920 q^{77} - 355774584 q^{79} + 172186884 q^{81} - 607613328 q^{83} + 1087351392 q^{85} - 640592064 q^{87} - 1157146424 q^{89} - 847629840 q^{91} + 919450440 q^{93} - 329699328 q^{95} - 1599536472 q^{97} - 653895504 q^{99}+O(q^{100}) \) 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
\(3\) 81.0000 0.577350
\(4\) 0 0
\(5\) −2550.24 −1.82480 −0.912401 0.409298i \(-0.865774\pi\)
−0.912401 + 0.409298i \(0.865774\pi\)
\(6\) 0 0
\(7\) 11484.8 1.80793 0.903963 0.427610i \(-0.140644\pi\)
0.903963 + 0.427610i \(0.140644\pi\)
\(8\) 0 0
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) −17694.9 −0.364401 −0.182201 0.983261i \(-0.558322\pi\)
−0.182201 + 0.983261i \(0.558322\pi\)
\(12\) 0 0
\(13\) −2878.66 −0.0279541 −0.0139770 0.999902i \(-0.504449\pi\)
−0.0139770 + 0.999902i \(0.504449\pi\)
\(14\) 0 0
\(15\) −206569. −1.05355
\(16\) 0 0
\(17\) −465200. −1.35089 −0.675444 0.737411i \(-0.736048\pi\)
−0.675444 + 0.737411i \(0.736048\pi\)
\(18\) 0 0
\(19\) 409191. 0.720335 0.360168 0.932888i \(-0.382719\pi\)
0.360168 + 0.932888i \(0.382719\pi\)
\(20\) 0 0
\(21\) 930265. 1.04381
\(22\) 0 0
\(23\) 374507. 0.279051 0.139526 0.990218i \(-0.455442\pi\)
0.139526 + 0.990218i \(0.455442\pi\)
\(24\) 0 0
\(25\) 4.55058e6 2.32990
\(26\) 0 0
\(27\) 531441. 0.192450
\(28\) 0 0
\(29\) −3.81849e6 −1.00254 −0.501269 0.865291i \(-0.667133\pi\)
−0.501269 + 0.865291i \(0.667133\pi\)
\(30\) 0 0
\(31\) 2.57794e6 0.501355 0.250678 0.968071i \(-0.419347\pi\)
0.250678 + 0.968071i \(0.419347\pi\)
\(32\) 0 0
\(33\) −1.43328e6 −0.210387
\(34\) 0 0
\(35\) −2.92889e7 −3.29911
\(36\) 0 0
\(37\) −1.07373e7 −0.941858 −0.470929 0.882171i \(-0.656081\pi\)
−0.470929 + 0.882171i \(0.656081\pi\)
\(38\) 0 0
\(39\) −233171. −0.0161393
\(40\) 0 0
\(41\) 1.50461e7 0.831564 0.415782 0.909464i \(-0.363508\pi\)
0.415782 + 0.909464i \(0.363508\pi\)
\(42\) 0 0
\(43\) −1.21149e7 −0.540394 −0.270197 0.962805i \(-0.587089\pi\)
−0.270197 + 0.962805i \(0.587089\pi\)
\(44\) 0 0
\(45\) −1.67321e7 −0.608267
\(46\) 0 0
\(47\) −4.65533e7 −1.39159 −0.695793 0.718242i \(-0.744947\pi\)
−0.695793 + 0.718242i \(0.744947\pi\)
\(48\) 0 0
\(49\) 9.15461e7 2.26860
\(50\) 0 0
\(51\) −3.76812e7 −0.779936
\(52\) 0 0
\(53\) 1.10710e8 1.92728 0.963639 0.267209i \(-0.0861014\pi\)
0.963639 + 0.267209i \(0.0861014\pi\)
\(54\) 0 0
\(55\) 4.51261e7 0.664960
\(56\) 0 0
\(57\) 3.31445e7 0.415886
\(58\) 0 0
\(59\) −9.19329e7 −0.987726 −0.493863 0.869540i \(-0.664416\pi\)
−0.493863 + 0.869540i \(0.664416\pi\)
\(60\) 0 0
\(61\) 1.64770e8 1.52368 0.761838 0.647767i \(-0.224297\pi\)
0.761838 + 0.647767i \(0.224297\pi\)
\(62\) 0 0
\(63\) 7.53515e7 0.602642
\(64\) 0 0
\(65\) 7.34126e6 0.0510106
\(66\) 0 0
\(67\) −2.68898e8 −1.63024 −0.815119 0.579294i \(-0.803328\pi\)
−0.815119 + 0.579294i \(0.803328\pi\)
\(68\) 0 0
\(69\) 3.03350e7 0.161110
\(70\) 0 0
\(71\) 3.57888e8 1.67142 0.835708 0.549174i \(-0.185058\pi\)
0.835708 + 0.549174i \(0.185058\pi\)
\(72\) 0 0
\(73\) −7.40257e7 −0.305091 −0.152546 0.988296i \(-0.548747\pi\)
−0.152546 + 0.988296i \(0.548747\pi\)
\(74\) 0 0
\(75\) 3.68597e8 1.34517
\(76\) 0 0
\(77\) −2.03221e8 −0.658811
\(78\) 0 0
\(79\) 9.17558e7 0.265040 0.132520 0.991180i \(-0.457693\pi\)
0.132520 + 0.991180i \(0.457693\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 0 0
\(83\) −1.86443e8 −0.431215 −0.215608 0.976480i \(-0.569173\pi\)
−0.215608 + 0.976480i \(0.569173\pi\)
\(84\) 0 0
\(85\) 1.18637e9 2.46510
\(86\) 0 0
\(87\) −3.09298e8 −0.578816
\(88\) 0 0
\(89\) −6.37166e8 −1.07646 −0.538230 0.842798i \(-0.680907\pi\)
−0.538230 + 0.842798i \(0.680907\pi\)
\(90\) 0 0
\(91\) −3.30607e7 −0.0505389
\(92\) 0 0
\(93\) 2.08813e8 0.289458
\(94\) 0 0
\(95\) −1.04353e9 −1.31447
\(96\) 0 0
\(97\) −5.12367e7 −0.0587636 −0.0293818 0.999568i \(-0.509354\pi\)
−0.0293818 + 0.999568i \(0.509354\pi\)
\(98\) 0 0
\(99\) −1.16096e8 −0.121467
\(100\) 0 0
\(101\) −3.28576e8 −0.314188 −0.157094 0.987584i \(-0.550213\pi\)
−0.157094 + 0.987584i \(0.550213\pi\)
\(102\) 0 0
\(103\) 2.13814e9 1.87184 0.935918 0.352218i \(-0.114572\pi\)
0.935918 + 0.352218i \(0.114572\pi\)
\(104\) 0 0
\(105\) −2.37240e9 −1.90474
\(106\) 0 0
\(107\) 1.38054e9 1.01817 0.509087 0.860715i \(-0.329983\pi\)
0.509087 + 0.860715i \(0.329983\pi\)
\(108\) 0 0
\(109\) −1.42057e9 −0.963926 −0.481963 0.876192i \(-0.660076\pi\)
−0.481963 + 0.876192i \(0.660076\pi\)
\(110\) 0 0
\(111\) −8.69718e8 −0.543782
\(112\) 0 0
\(113\) 4.63697e8 0.267535 0.133768 0.991013i \(-0.457292\pi\)
0.133768 + 0.991013i \(0.457292\pi\)
\(114\) 0 0
\(115\) −9.55081e8 −0.509213
\(116\) 0 0
\(117\) −1.88869e7 −0.00931802
\(118\) 0 0
\(119\) −5.34271e9 −2.44231
\(120\) 0 0
\(121\) −2.04484e9 −0.867212
\(122\) 0 0
\(123\) 1.21873e9 0.480104
\(124\) 0 0
\(125\) −6.62414e9 −2.42680
\(126\) 0 0
\(127\) −3.20157e9 −1.09206 −0.546029 0.837766i \(-0.683861\pi\)
−0.546029 + 0.837766i \(0.683861\pi\)
\(128\) 0 0
\(129\) −9.81304e8 −0.311997
\(130\) 0 0
\(131\) −4.24459e9 −1.25926 −0.629630 0.776895i \(-0.716793\pi\)
−0.629630 + 0.776895i \(0.716793\pi\)
\(132\) 0 0
\(133\) 4.69946e9 1.30231
\(134\) 0 0
\(135\) −1.35530e9 −0.351183
\(136\) 0 0
\(137\) −1.87982e9 −0.455905 −0.227952 0.973672i \(-0.573203\pi\)
−0.227952 + 0.973672i \(0.573203\pi\)
\(138\) 0 0
\(139\) −3.99062e9 −0.906720 −0.453360 0.891327i \(-0.649775\pi\)
−0.453360 + 0.891327i \(0.649775\pi\)
\(140\) 0 0
\(141\) −3.77082e9 −0.803433
\(142\) 0 0
\(143\) 5.09374e7 0.0101865
\(144\) 0 0
\(145\) 9.73807e9 1.82943
\(146\) 0 0
\(147\) 7.41523e9 1.30978
\(148\) 0 0
\(149\) −8.67985e9 −1.44269 −0.721346 0.692574i \(-0.756476\pi\)
−0.721346 + 0.692574i \(0.756476\pi\)
\(150\) 0 0
\(151\) 4.40680e9 0.689806 0.344903 0.938638i \(-0.387912\pi\)
0.344903 + 0.938638i \(0.387912\pi\)
\(152\) 0 0
\(153\) −3.05218e9 −0.450296
\(154\) 0 0
\(155\) −6.57436e9 −0.914874
\(156\) 0 0
\(157\) 7.13694e9 0.937484 0.468742 0.883335i \(-0.344707\pi\)
0.468742 + 0.883335i \(0.344707\pi\)
\(158\) 0 0
\(159\) 8.96748e9 1.11271
\(160\) 0 0
\(161\) 4.30112e9 0.504504
\(162\) 0 0
\(163\) −9.33307e8 −0.103557 −0.0517786 0.998659i \(-0.516489\pi\)
−0.0517786 + 0.998659i \(0.516489\pi\)
\(164\) 0 0
\(165\) 3.65521e9 0.383915
\(166\) 0 0
\(167\) −1.62183e10 −1.61355 −0.806773 0.590862i \(-0.798788\pi\)
−0.806773 + 0.590862i \(0.798788\pi\)
\(168\) 0 0
\(169\) −1.05962e10 −0.999219
\(170\) 0 0
\(171\) 2.68470e9 0.240112
\(172\) 0 0
\(173\) 6.78681e9 0.576047 0.288024 0.957623i \(-0.407002\pi\)
0.288024 + 0.957623i \(0.407002\pi\)
\(174\) 0 0
\(175\) 5.22624e10 4.21229
\(176\) 0 0
\(177\) −7.44656e9 −0.570264
\(178\) 0 0
\(179\) −6.04635e9 −0.440205 −0.220102 0.975477i \(-0.570639\pi\)
−0.220102 + 0.975477i \(0.570639\pi\)
\(180\) 0 0
\(181\) 1.73661e10 1.20268 0.601339 0.798994i \(-0.294634\pi\)
0.601339 + 0.798994i \(0.294634\pi\)
\(182\) 0 0
\(183\) 1.33463e10 0.879695
\(184\) 0 0
\(185\) 2.73825e10 1.71870
\(186\) 0 0
\(187\) 8.23165e9 0.492266
\(188\) 0 0
\(189\) 6.10347e9 0.347936
\(190\) 0 0
\(191\) 1.55901e10 0.847616 0.423808 0.905752i \(-0.360693\pi\)
0.423808 + 0.905752i \(0.360693\pi\)
\(192\) 0 0
\(193\) −2.58307e10 −1.34007 −0.670036 0.742329i \(-0.733721\pi\)
−0.670036 + 0.742329i \(0.733721\pi\)
\(194\) 0 0
\(195\) 5.94642e8 0.0294510
\(196\) 0 0
\(197\) −3.09030e10 −1.46185 −0.730924 0.682459i \(-0.760911\pi\)
−0.730924 + 0.682459i \(0.760911\pi\)
\(198\) 0 0
\(199\) 1.28574e10 0.581183 0.290592 0.956847i \(-0.406148\pi\)
0.290592 + 0.956847i \(0.406148\pi\)
\(200\) 0 0
\(201\) −2.17807e10 −0.941218
\(202\) 0 0
\(203\) −4.38545e10 −1.81252
\(204\) 0 0
\(205\) −3.83711e10 −1.51744
\(206\) 0 0
\(207\) 2.45714e9 0.0930171
\(208\) 0 0
\(209\) −7.24058e9 −0.262491
\(210\) 0 0
\(211\) −2.77349e10 −0.963288 −0.481644 0.876367i \(-0.659960\pi\)
−0.481644 + 0.876367i \(0.659960\pi\)
\(212\) 0 0
\(213\) 2.89889e10 0.964993
\(214\) 0 0
\(215\) 3.08958e10 0.986112
\(216\) 0 0
\(217\) 2.96070e10 0.906413
\(218\) 0 0
\(219\) −5.99608e9 −0.176145
\(220\) 0 0
\(221\) 1.33915e9 0.0377628
\(222\) 0 0
\(223\) −4.93029e10 −1.33506 −0.667530 0.744583i \(-0.732649\pi\)
−0.667530 + 0.744583i \(0.732649\pi\)
\(224\) 0 0
\(225\) 2.98564e10 0.776633
\(226\) 0 0
\(227\) −2.66250e10 −0.665538 −0.332769 0.943008i \(-0.607983\pi\)
−0.332769 + 0.943008i \(0.607983\pi\)
\(228\) 0 0
\(229\) −6.63083e10 −1.59334 −0.796670 0.604415i \(-0.793407\pi\)
−0.796670 + 0.604415i \(0.793407\pi\)
\(230\) 0 0
\(231\) −1.64609e10 −0.380365
\(232\) 0 0
\(233\) 5.24513e10 1.16588 0.582941 0.812514i \(-0.301902\pi\)
0.582941 + 0.812514i \(0.301902\pi\)
\(234\) 0 0
\(235\) 1.18722e11 2.53937
\(236\) 0 0
\(237\) 7.43222e9 0.153021
\(238\) 0 0
\(239\) −2.83125e10 −0.561291 −0.280646 0.959811i \(-0.590549\pi\)
−0.280646 + 0.959811i \(0.590549\pi\)
\(240\) 0 0
\(241\) −8.73233e10 −1.66745 −0.833726 0.552178i \(-0.813797\pi\)
−0.833726 + 0.552178i \(0.813797\pi\)
\(242\) 0 0
\(243\) 3.48678e9 0.0641500
\(244\) 0 0
\(245\) −2.33464e11 −4.13974
\(246\) 0 0
\(247\) −1.17792e9 −0.0201363
\(248\) 0 0
\(249\) −1.51019e10 −0.248962
\(250\) 0 0
\(251\) −3.44391e10 −0.547671 −0.273836 0.961776i \(-0.588292\pi\)
−0.273836 + 0.961776i \(0.588292\pi\)
\(252\) 0 0
\(253\) −6.62684e9 −0.101687
\(254\) 0 0
\(255\) 9.60960e10 1.42323
\(256\) 0 0
\(257\) −1.18147e11 −1.68937 −0.844685 0.535264i \(-0.820212\pi\)
−0.844685 + 0.535264i \(0.820212\pi\)
\(258\) 0 0
\(259\) −1.23315e11 −1.70281
\(260\) 0 0
\(261\) −2.50531e10 −0.334180
\(262\) 0 0
\(263\) 3.09809e9 0.0399294 0.0199647 0.999801i \(-0.493645\pi\)
0.0199647 + 0.999801i \(0.493645\pi\)
\(264\) 0 0
\(265\) −2.82336e11 −3.51690
\(266\) 0 0
\(267\) −5.16105e10 −0.621494
\(268\) 0 0
\(269\) 8.92323e10 1.03905 0.519526 0.854455i \(-0.326109\pi\)
0.519526 + 0.854455i \(0.326109\pi\)
\(270\) 0 0
\(271\) −1.33152e11 −1.49964 −0.749821 0.661641i \(-0.769860\pi\)
−0.749821 + 0.661641i \(0.769860\pi\)
\(272\) 0 0
\(273\) −2.67792e9 −0.0291786
\(274\) 0 0
\(275\) −8.05220e10 −0.849019
\(276\) 0 0
\(277\) −1.36458e11 −1.39264 −0.696322 0.717730i \(-0.745181\pi\)
−0.696322 + 0.717730i \(0.745181\pi\)
\(278\) 0 0
\(279\) 1.69139e10 0.167118
\(280\) 0 0
\(281\) 5.62851e10 0.538537 0.269268 0.963065i \(-0.413218\pi\)
0.269268 + 0.963065i \(0.413218\pi\)
\(282\) 0 0
\(283\) 1.30168e10 0.120633 0.0603163 0.998179i \(-0.480789\pi\)
0.0603163 + 0.998179i \(0.480789\pi\)
\(284\) 0 0
\(285\) −8.45262e10 −0.758909
\(286\) 0 0
\(287\) 1.72801e11 1.50341
\(288\) 0 0
\(289\) 9.78231e10 0.824900
\(290\) 0 0
\(291\) −4.15017e9 −0.0339272
\(292\) 0 0
\(293\) 2.06089e11 1.63362 0.816810 0.576907i \(-0.195741\pi\)
0.816810 + 0.576907i \(0.195741\pi\)
\(294\) 0 0
\(295\) 2.34451e11 1.80240
\(296\) 0 0
\(297\) −9.40377e9 −0.0701291
\(298\) 0 0
\(299\) −1.07808e9 −0.00780062
\(300\) 0 0
\(301\) −1.39136e11 −0.976993
\(302\) 0 0
\(303\) −2.66147e10 −0.181397
\(304\) 0 0
\(305\) −4.20202e11 −2.78041
\(306\) 0 0
\(307\) 6.28066e10 0.403536 0.201768 0.979433i \(-0.435331\pi\)
0.201768 + 0.979433i \(0.435331\pi\)
\(308\) 0 0
\(309\) 1.73189e11 1.08071
\(310\) 0 0
\(311\) −2.85384e10 −0.172985 −0.0864925 0.996253i \(-0.527566\pi\)
−0.0864925 + 0.996253i \(0.527566\pi\)
\(312\) 0 0
\(313\) 1.03679e10 0.0610578 0.0305289 0.999534i \(-0.490281\pi\)
0.0305289 + 0.999534i \(0.490281\pi\)
\(314\) 0 0
\(315\) −1.92164e11 −1.09970
\(316\) 0 0
\(317\) 1.10284e11 0.613400 0.306700 0.951806i \(-0.400775\pi\)
0.306700 + 0.951806i \(0.400775\pi\)
\(318\) 0 0
\(319\) 6.75677e10 0.365327
\(320\) 0 0
\(321\) 1.11824e11 0.587844
\(322\) 0 0
\(323\) −1.90356e11 −0.973093
\(324\) 0 0
\(325\) −1.30996e10 −0.0651301
\(326\) 0 0
\(327\) −1.15066e11 −0.556523
\(328\) 0 0
\(329\) −5.34654e11 −2.51589
\(330\) 0 0
\(331\) −2.32144e11 −1.06299 −0.531497 0.847060i \(-0.678370\pi\)
−0.531497 + 0.847060i \(0.678370\pi\)
\(332\) 0 0
\(333\) −7.04471e10 −0.313953
\(334\) 0 0
\(335\) 6.85753e11 2.97486
\(336\) 0 0
\(337\) 1.90284e11 0.803652 0.401826 0.915716i \(-0.368376\pi\)
0.401826 + 0.915716i \(0.368376\pi\)
\(338\) 0 0
\(339\) 3.75595e10 0.154462
\(340\) 0 0
\(341\) −4.56163e10 −0.182695
\(342\) 0 0
\(343\) 5.87933e11 2.29353
\(344\) 0 0
\(345\) −7.73615e10 −0.293994
\(346\) 0 0
\(347\) −2.80544e11 −1.03877 −0.519383 0.854541i \(-0.673838\pi\)
−0.519383 + 0.854541i \(0.673838\pi\)
\(348\) 0 0
\(349\) −5.24413e10 −0.189217 −0.0946083 0.995515i \(-0.530160\pi\)
−0.0946083 + 0.995515i \(0.530160\pi\)
\(350\) 0 0
\(351\) −1.52984e9 −0.00537976
\(352\) 0 0
\(353\) 5.36154e11 1.83782 0.918911 0.394466i \(-0.129070\pi\)
0.918911 + 0.394466i \(0.129070\pi\)
\(354\) 0 0
\(355\) −9.12699e11 −3.05000
\(356\) 0 0
\(357\) −4.32759e11 −1.41007
\(358\) 0 0
\(359\) 1.53244e11 0.486920 0.243460 0.969911i \(-0.421718\pi\)
0.243460 + 0.969911i \(0.421718\pi\)
\(360\) 0 0
\(361\) −1.55250e11 −0.481117
\(362\) 0 0
\(363\) −1.65632e11 −0.500685
\(364\) 0 0
\(365\) 1.88783e11 0.556731
\(366\) 0 0
\(367\) −3.74040e11 −1.07627 −0.538134 0.842859i \(-0.680871\pi\)
−0.538134 + 0.842859i \(0.680871\pi\)
\(368\) 0 0
\(369\) 9.87173e10 0.277188
\(370\) 0 0
\(371\) 1.27147e12 3.48437
\(372\) 0 0
\(373\) 2.01779e11 0.539742 0.269871 0.962896i \(-0.413019\pi\)
0.269871 + 0.962896i \(0.413019\pi\)
\(374\) 0 0
\(375\) −5.36555e11 −1.40111
\(376\) 0 0
\(377\) 1.09921e10 0.0280250
\(378\) 0 0
\(379\) −6.79938e11 −1.69275 −0.846375 0.532588i \(-0.821220\pi\)
−0.846375 + 0.532588i \(0.821220\pi\)
\(380\) 0 0
\(381\) −2.59327e11 −0.630500
\(382\) 0 0
\(383\) −4.62580e11 −1.09848 −0.549240 0.835664i \(-0.685083\pi\)
−0.549240 + 0.835664i \(0.685083\pi\)
\(384\) 0 0
\(385\) 5.18262e11 1.20220
\(386\) 0 0
\(387\) −7.94857e10 −0.180131
\(388\) 0 0
\(389\) −5.13044e11 −1.13601 −0.568004 0.823026i \(-0.692284\pi\)
−0.568004 + 0.823026i \(0.692284\pi\)
\(390\) 0 0
\(391\) −1.74220e11 −0.376967
\(392\) 0 0
\(393\) −3.43812e11 −0.727034
\(394\) 0 0
\(395\) −2.33999e11 −0.483646
\(396\) 0 0
\(397\) −7.14443e11 −1.44348 −0.721739 0.692165i \(-0.756657\pi\)
−0.721739 + 0.692165i \(0.756657\pi\)
\(398\) 0 0
\(399\) 3.80656e11 0.751891
\(400\) 0 0
\(401\) −2.80010e11 −0.540785 −0.270392 0.962750i \(-0.587153\pi\)
−0.270392 + 0.962750i \(0.587153\pi\)
\(402\) 0 0
\(403\) −7.42101e9 −0.0140149
\(404\) 0 0
\(405\) −1.09779e11 −0.202756
\(406\) 0 0
\(407\) 1.89994e11 0.343214
\(408\) 0 0
\(409\) 1.09398e11 0.193309 0.0966546 0.995318i \(-0.469186\pi\)
0.0966546 + 0.995318i \(0.469186\pi\)
\(410\) 0 0
\(411\) −1.52266e11 −0.263217
\(412\) 0 0
\(413\) −1.05583e12 −1.78574
\(414\) 0 0
\(415\) 4.75473e11 0.786882
\(416\) 0 0
\(417\) −3.23240e11 −0.523495
\(418\) 0 0
\(419\) 6.69404e10 0.106102 0.0530512 0.998592i \(-0.483105\pi\)
0.0530512 + 0.998592i \(0.483105\pi\)
\(420\) 0 0
\(421\) −4.27317e11 −0.662949 −0.331475 0.943464i \(-0.607546\pi\)
−0.331475 + 0.943464i \(0.607546\pi\)
\(422\) 0 0
\(423\) −3.05436e11 −0.463862
\(424\) 0 0
\(425\) −2.11693e12 −3.14743
\(426\) 0 0
\(427\) 1.89234e12 2.75469
\(428\) 0 0
\(429\) 4.12593e9 0.00588118
\(430\) 0 0
\(431\) 4.00618e11 0.559220 0.279610 0.960114i \(-0.409795\pi\)
0.279610 + 0.960114i \(0.409795\pi\)
\(432\) 0 0
\(433\) 3.69071e11 0.504562 0.252281 0.967654i \(-0.418819\pi\)
0.252281 + 0.967654i \(0.418819\pi\)
\(434\) 0 0
\(435\) 7.88783e11 1.05622
\(436\) 0 0
\(437\) 1.53245e11 0.201011
\(438\) 0 0
\(439\) 1.02355e12 1.31528 0.657640 0.753332i \(-0.271555\pi\)
0.657640 + 0.753332i \(0.271555\pi\)
\(440\) 0 0
\(441\) 6.00634e11 0.756199
\(442\) 0 0
\(443\) −2.64264e11 −0.326002 −0.163001 0.986626i \(-0.552117\pi\)
−0.163001 + 0.986626i \(0.552117\pi\)
\(444\) 0 0
\(445\) 1.62493e12 1.96432
\(446\) 0 0
\(447\) −7.03068e11 −0.832939
\(448\) 0 0
\(449\) −5.41761e11 −0.629070 −0.314535 0.949246i \(-0.601849\pi\)
−0.314535 + 0.949246i \(0.601849\pi\)
\(450\) 0 0
\(451\) −2.66238e11 −0.303023
\(452\) 0 0
\(453\) 3.56951e11 0.398260
\(454\) 0 0
\(455\) 8.43126e10 0.0922234
\(456\) 0 0
\(457\) −5.87461e10 −0.0630022 −0.0315011 0.999504i \(-0.510029\pi\)
−0.0315011 + 0.999504i \(0.510029\pi\)
\(458\) 0 0
\(459\) −2.47226e11 −0.259979
\(460\) 0 0
\(461\) 1.57233e12 1.62139 0.810697 0.585466i \(-0.199089\pi\)
0.810697 + 0.585466i \(0.199089\pi\)
\(462\) 0 0
\(463\) −9.14079e11 −0.924420 −0.462210 0.886771i \(-0.652943\pi\)
−0.462210 + 0.886771i \(0.652943\pi\)
\(464\) 0 0
\(465\) −5.32523e11 −0.528202
\(466\) 0 0
\(467\) 6.14555e11 0.597909 0.298954 0.954267i \(-0.403362\pi\)
0.298954 + 0.954267i \(0.403362\pi\)
\(468\) 0 0
\(469\) −3.08823e12 −2.94735
\(470\) 0 0
\(471\) 5.78092e11 0.541257
\(472\) 0 0
\(473\) 2.14371e11 0.196920
\(474\) 0 0
\(475\) 1.86206e12 1.67831
\(476\) 0 0
\(477\) 7.26366e11 0.642426
\(478\) 0 0
\(479\) −9.71639e11 −0.843325 −0.421663 0.906753i \(-0.638553\pi\)
−0.421663 + 0.906753i \(0.638553\pi\)
\(480\) 0 0
\(481\) 3.09089e10 0.0263288
\(482\) 0 0
\(483\) 3.48391e11 0.291276
\(484\) 0 0
\(485\) 1.30666e11 0.107232
\(486\) 0 0
\(487\) 2.35678e12 1.89862 0.949312 0.314335i \(-0.101782\pi\)
0.949312 + 0.314335i \(0.101782\pi\)
\(488\) 0 0
\(489\) −7.55979e10 −0.0597888
\(490\) 0 0
\(491\) 7.35095e11 0.570791 0.285395 0.958410i \(-0.407875\pi\)
0.285395 + 0.958410i \(0.407875\pi\)
\(492\) 0 0
\(493\) 1.77636e12 1.35432
\(494\) 0 0
\(495\) 2.96072e11 0.221653
\(496\) 0 0
\(497\) 4.11026e12 3.02180
\(498\) 0 0
\(499\) 2.92617e11 0.211275 0.105637 0.994405i \(-0.466312\pi\)
0.105637 + 0.994405i \(0.466312\pi\)
\(500\) 0 0
\(501\) −1.31368e12 −0.931581
\(502\) 0 0
\(503\) −2.02398e12 −1.40978 −0.704890 0.709317i \(-0.749004\pi\)
−0.704890 + 0.709317i \(0.749004\pi\)
\(504\) 0 0
\(505\) 8.37948e11 0.573332
\(506\) 0 0
\(507\) −8.58293e11 −0.576899
\(508\) 0 0
\(509\) 7.65483e11 0.505482 0.252741 0.967534i \(-0.418668\pi\)
0.252741 + 0.967534i \(0.418668\pi\)
\(510\) 0 0
\(511\) −8.50168e11 −0.551583
\(512\) 0 0
\(513\) 2.17461e11 0.138629
\(514\) 0 0
\(515\) −5.45275e12 −3.41573
\(516\) 0 0
\(517\) 8.23755e11 0.507096
\(518\) 0 0
\(519\) 5.49731e11 0.332581
\(520\) 0 0
\(521\) 2.50936e12 1.49208 0.746041 0.665900i \(-0.231952\pi\)
0.746041 + 0.665900i \(0.231952\pi\)
\(522\) 0 0
\(523\) 9.38072e10 0.0548250 0.0274125 0.999624i \(-0.491273\pi\)
0.0274125 + 0.999624i \(0.491273\pi\)
\(524\) 0 0
\(525\) 4.23325e12 2.43196
\(526\) 0 0
\(527\) −1.19926e12 −0.677275
\(528\) 0 0
\(529\) −1.66090e12 −0.922130
\(530\) 0 0
\(531\) −6.03171e11 −0.329242
\(532\) 0 0
\(533\) −4.33125e10 −0.0232456
\(534\) 0 0
\(535\) −3.52071e12 −1.85797
\(536\) 0 0
\(537\) −4.89755e11 −0.254152
\(538\) 0 0
\(539\) −1.61990e12 −0.826680
\(540\) 0 0
\(541\) 1.38883e12 0.697044 0.348522 0.937301i \(-0.386684\pi\)
0.348522 + 0.937301i \(0.386684\pi\)
\(542\) 0 0
\(543\) 1.40666e12 0.694366
\(544\) 0 0
\(545\) 3.62279e12 1.75897
\(546\) 0 0
\(547\) 2.50687e11 0.119726 0.0598631 0.998207i \(-0.480934\pi\)
0.0598631 + 0.998207i \(0.480934\pi\)
\(548\) 0 0
\(549\) 1.08105e12 0.507892
\(550\) 0 0
\(551\) −1.56249e12 −0.722164
\(552\) 0 0
\(553\) 1.05379e12 0.479173
\(554\) 0 0
\(555\) 2.21799e12 0.992294
\(556\) 0 0
\(557\) −2.07853e11 −0.0914973 −0.0457487 0.998953i \(-0.514567\pi\)
−0.0457487 + 0.998953i \(0.514567\pi\)
\(558\) 0 0
\(559\) 3.48746e10 0.0151062
\(560\) 0 0
\(561\) 6.66764e11 0.284210
\(562\) 0 0
\(563\) 1.85431e12 0.777850 0.388925 0.921269i \(-0.372847\pi\)
0.388925 + 0.921269i \(0.372847\pi\)
\(564\) 0 0
\(565\) −1.18254e12 −0.488199
\(566\) 0 0
\(567\) 4.94381e11 0.200881
\(568\) 0 0
\(569\) −1.99208e12 −0.796712 −0.398356 0.917231i \(-0.630419\pi\)
−0.398356 + 0.917231i \(0.630419\pi\)
\(570\) 0 0
\(571\) −1.49143e12 −0.587139 −0.293570 0.955938i \(-0.594843\pi\)
−0.293570 + 0.955938i \(0.594843\pi\)
\(572\) 0 0
\(573\) 1.26280e12 0.489372
\(574\) 0 0
\(575\) 1.70422e12 0.650162
\(576\) 0 0
\(577\) 1.82633e12 0.685942 0.342971 0.939346i \(-0.388567\pi\)
0.342971 + 0.939346i \(0.388567\pi\)
\(578\) 0 0
\(579\) −2.09229e12 −0.773691
\(580\) 0 0
\(581\) −2.14125e12 −0.779605
\(582\) 0 0
\(583\) −1.95899e12 −0.702303
\(584\) 0 0
\(585\) 4.81660e10 0.0170035
\(586\) 0 0
\(587\) −2.12345e12 −0.738193 −0.369096 0.929391i \(-0.620333\pi\)
−0.369096 + 0.929391i \(0.620333\pi\)
\(588\) 0 0
\(589\) 1.05487e12 0.361144
\(590\) 0 0
\(591\) −2.50314e12 −0.843998
\(592\) 0 0
\(593\) 2.09754e12 0.696569 0.348284 0.937389i \(-0.386764\pi\)
0.348284 + 0.937389i \(0.386764\pi\)
\(594\) 0 0
\(595\) 1.36252e13 4.45672
\(596\) 0 0
\(597\) 1.04145e12 0.335546
\(598\) 0 0
\(599\) −2.89930e12 −0.920178 −0.460089 0.887873i \(-0.652182\pi\)
−0.460089 + 0.887873i \(0.652182\pi\)
\(600\) 0 0
\(601\) −4.36479e12 −1.36467 −0.682337 0.731038i \(-0.739036\pi\)
−0.682337 + 0.731038i \(0.739036\pi\)
\(602\) 0 0
\(603\) −1.76424e12 −0.543412
\(604\) 0 0
\(605\) 5.21483e12 1.58249
\(606\) 0 0
\(607\) 2.16703e12 0.647913 0.323956 0.946072i \(-0.394987\pi\)
0.323956 + 0.946072i \(0.394987\pi\)
\(608\) 0 0
\(609\) −3.55221e12 −1.04646
\(610\) 0 0
\(611\) 1.34011e11 0.0389005
\(612\) 0 0
\(613\) −1.83561e12 −0.525058 −0.262529 0.964924i \(-0.584557\pi\)
−0.262529 + 0.964924i \(0.584557\pi\)
\(614\) 0 0
\(615\) −3.10806e12 −0.876094
\(616\) 0 0
\(617\) −7.04193e11 −0.195618 −0.0978089 0.995205i \(-0.531183\pi\)
−0.0978089 + 0.995205i \(0.531183\pi\)
\(618\) 0 0
\(619\) 6.20869e12 1.69978 0.849889 0.526962i \(-0.176669\pi\)
0.849889 + 0.526962i \(0.176669\pi\)
\(620\) 0 0
\(621\) 1.99028e11 0.0537035
\(622\) 0 0
\(623\) −7.31770e12 −1.94616
\(624\) 0 0
\(625\) 8.00526e12 2.09853
\(626\) 0 0
\(627\) −5.86487e11 −0.151549
\(628\) 0 0
\(629\) 4.99497e12 1.27235
\(630\) 0 0
\(631\) −1.83096e12 −0.459776 −0.229888 0.973217i \(-0.573836\pi\)
−0.229888 + 0.973217i \(0.573836\pi\)
\(632\) 0 0
\(633\) −2.24653e12 −0.556155
\(634\) 0 0
\(635\) 8.16475e12 1.99279
\(636\) 0 0
\(637\) −2.63530e11 −0.0634165
\(638\) 0 0
\(639\) 2.34810e12 0.557139
\(640\) 0 0
\(641\) 9.57030e11 0.223905 0.111953 0.993714i \(-0.464290\pi\)
0.111953 + 0.993714i \(0.464290\pi\)
\(642\) 0 0
\(643\) 1.17043e12 0.270019 0.135010 0.990844i \(-0.456893\pi\)
0.135010 + 0.990844i \(0.456893\pi\)
\(644\) 0 0
\(645\) 2.50256e12 0.569332
\(646\) 0 0
\(647\) −7.13180e12 −1.60004 −0.800018 0.599976i \(-0.795177\pi\)
−0.800018 + 0.599976i \(0.795177\pi\)
\(648\) 0 0
\(649\) 1.62674e12 0.359929
\(650\) 0 0
\(651\) 2.39817e12 0.523318
\(652\) 0 0
\(653\) −1.45995e12 −0.314217 −0.157108 0.987581i \(-0.550217\pi\)
−0.157108 + 0.987581i \(0.550217\pi\)
\(654\) 0 0
\(655\) 1.08247e13 2.29790
\(656\) 0 0
\(657\) −4.85683e11 −0.101697
\(658\) 0 0
\(659\) 3.16319e12 0.653343 0.326672 0.945138i \(-0.394073\pi\)
0.326672 + 0.945138i \(0.394073\pi\)
\(660\) 0 0
\(661\) −5.94931e12 −1.21216 −0.606080 0.795404i \(-0.707259\pi\)
−0.606080 + 0.795404i \(0.707259\pi\)
\(662\) 0 0
\(663\) 1.08471e11 0.0218024
\(664\) 0 0
\(665\) −1.19847e13 −2.37646
\(666\) 0 0
\(667\) −1.43005e12 −0.279760
\(668\) 0 0
\(669\) −3.99354e12 −0.770798
\(670\) 0 0
\(671\) −2.91558e12 −0.555230
\(672\) 0 0
\(673\) 6.63233e12 1.24623 0.623115 0.782130i \(-0.285867\pi\)
0.623115 + 0.782130i \(0.285867\pi\)
\(674\) 0 0
\(675\) 2.41837e12 0.448389
\(676\) 0 0
\(677\) 2.68780e12 0.491754 0.245877 0.969301i \(-0.420924\pi\)
0.245877 + 0.969301i \(0.420924\pi\)
\(678\) 0 0
\(679\) −5.88441e11 −0.106240
\(680\) 0 0
\(681\) −2.15662e12 −0.384249
\(682\) 0 0
\(683\) −2.14691e11 −0.0377503 −0.0188751 0.999822i \(-0.506008\pi\)
−0.0188751 + 0.999822i \(0.506008\pi\)
\(684\) 0 0
\(685\) 4.79399e12 0.831935
\(686\) 0 0
\(687\) −5.37097e12 −0.919915
\(688\) 0 0
\(689\) −3.18695e11 −0.0538752
\(690\) 0 0
\(691\) −5.18271e12 −0.864780 −0.432390 0.901687i \(-0.642330\pi\)
−0.432390 + 0.901687i \(0.642330\pi\)
\(692\) 0 0
\(693\) −1.33333e12 −0.219604
\(694\) 0 0
\(695\) 1.01770e13 1.65458
\(696\) 0 0
\(697\) −6.99943e12 −1.12335
\(698\) 0 0
\(699\) 4.24856e12 0.673122
\(700\) 0 0
\(701\) 5.92911e12 0.927381 0.463691 0.885997i \(-0.346525\pi\)
0.463691 + 0.885997i \(0.346525\pi\)
\(702\) 0 0
\(703\) −4.39359e12 −0.678454
\(704\) 0 0
\(705\) 9.61648e12 1.46611
\(706\) 0 0
\(707\) −3.77362e12 −0.568030
\(708\) 0 0
\(709\) 4.63915e12 0.689493 0.344747 0.938696i \(-0.387965\pi\)
0.344747 + 0.938696i \(0.387965\pi\)
\(710\) 0 0
\(711\) 6.02010e11 0.0883467
\(712\) 0 0
\(713\) 9.65456e11 0.139904
\(714\) 0 0
\(715\) −1.29903e11 −0.0185883
\(716\) 0 0
\(717\) −2.29332e12 −0.324062
\(718\) 0 0
\(719\) 3.00203e12 0.418924 0.209462 0.977817i \(-0.432829\pi\)
0.209462 + 0.977817i \(0.432829\pi\)
\(720\) 0 0
\(721\) 2.45560e13 3.38414
\(722\) 0 0
\(723\) −7.07319e12 −0.962704
\(724\) 0 0
\(725\) −1.73764e13 −2.33581
\(726\) 0 0
\(727\) 9.19643e11 0.122100 0.0610498 0.998135i \(-0.480555\pi\)
0.0610498 + 0.998135i \(0.480555\pi\)
\(728\) 0 0
\(729\) 2.82430e11 0.0370370
\(730\) 0 0
\(731\) 5.63584e12 0.730012
\(732\) 0 0
\(733\) −1.03255e13 −1.32113 −0.660564 0.750769i \(-0.729683\pi\)
−0.660564 + 0.750769i \(0.729683\pi\)
\(734\) 0 0
\(735\) −1.89106e13 −2.39008
\(736\) 0 0
\(737\) 4.75811e12 0.594061
\(738\) 0 0
\(739\) 3.25490e12 0.401456 0.200728 0.979647i \(-0.435669\pi\)
0.200728 + 0.979647i \(0.435669\pi\)
\(740\) 0 0
\(741\) −9.54116e10 −0.0116257
\(742\) 0 0
\(743\) 3.84135e12 0.462418 0.231209 0.972904i \(-0.425732\pi\)
0.231209 + 0.972904i \(0.425732\pi\)
\(744\) 0 0
\(745\) 2.21357e13 2.63263
\(746\) 0 0
\(747\) −1.22325e12 −0.143738
\(748\) 0 0
\(749\) 1.58552e13 1.84078
\(750\) 0 0
\(751\) 1.42668e13 1.63662 0.818311 0.574776i \(-0.194911\pi\)
0.818311 + 0.574776i \(0.194911\pi\)
\(752\) 0 0
\(753\) −2.78957e12 −0.316198
\(754\) 0 0
\(755\) −1.12384e13 −1.25876
\(756\) 0 0
\(757\) −2.85396e12 −0.315876 −0.157938 0.987449i \(-0.550485\pi\)
−0.157938 + 0.987449i \(0.550485\pi\)
\(758\) 0 0
\(759\) −5.36774e11 −0.0587089
\(760\) 0 0
\(761\) −2.19569e12 −0.237323 −0.118662 0.992935i \(-0.537860\pi\)
−0.118662 + 0.992935i \(0.537860\pi\)
\(762\) 0 0
\(763\) −1.63149e13 −1.74271
\(764\) 0 0
\(765\) 7.78378e12 0.821701
\(766\) 0 0
\(767\) 2.64643e11 0.0276110
\(768\) 0 0
\(769\) 5.45095e12 0.562087 0.281044 0.959695i \(-0.409319\pi\)
0.281044 + 0.959695i \(0.409319\pi\)
\(770\) 0 0
\(771\) −9.56993e12 −0.975358
\(772\) 0 0
\(773\) 6.19087e12 0.623654 0.311827 0.950139i \(-0.399059\pi\)
0.311827 + 0.950139i \(0.399059\pi\)
\(774\) 0 0
\(775\) 1.17311e13 1.16811
\(776\) 0 0
\(777\) −9.98850e12 −0.983118
\(778\) 0 0
\(779\) 6.15672e12 0.599005
\(780\) 0 0
\(781\) −6.33278e12 −0.609066
\(782\) 0 0
\(783\) −2.02930e12 −0.192939
\(784\) 0 0
\(785\) −1.82009e13 −1.71072
\(786\) 0 0
\(787\) 1.77687e12 0.165109 0.0825543 0.996587i \(-0.473692\pi\)
0.0825543 + 0.996587i \(0.473692\pi\)
\(788\) 0 0
\(789\) 2.50945e11 0.0230533
\(790\) 0 0
\(791\) 5.32545e12 0.483684
\(792\) 0 0
\(793\) −4.74315e11 −0.0425929
\(794\) 0 0
\(795\) −2.28692e13 −2.03048
\(796\) 0 0
\(797\) 9.53311e12 0.836898 0.418449 0.908240i \(-0.362574\pi\)
0.418449 + 0.908240i \(0.362574\pi\)
\(798\) 0 0
\(799\) 2.16566e13 1.87988
\(800\) 0 0
\(801\) −4.18045e12 −0.358820
\(802\) 0 0
\(803\) 1.30988e12 0.111176
\(804\) 0 0
\(805\) −1.09689e13 −0.920620
\(806\) 0 0
\(807\) 7.22782e12 0.599897
\(808\) 0 0
\(809\) −4.71178e12 −0.386738 −0.193369 0.981126i \(-0.561941\pi\)
−0.193369 + 0.981126i \(0.561941\pi\)
\(810\) 0 0
\(811\) −9.33679e12 −0.757885 −0.378943 0.925420i \(-0.623712\pi\)
−0.378943 + 0.925420i \(0.623712\pi\)
\(812\) 0 0
\(813\) −1.07854e13 −0.865818
\(814\) 0 0
\(815\) 2.38015e12 0.188971
\(816\) 0 0
\(817\) −4.95729e12 −0.389265
\(818\) 0 0
\(819\) −2.16911e11 −0.0168463
\(820\) 0 0
\(821\) −9.62688e11 −0.0739505 −0.0369753 0.999316i \(-0.511772\pi\)
−0.0369753 + 0.999316i \(0.511772\pi\)
\(822\) 0 0
\(823\) 2.35265e12 0.178755 0.0893774 0.995998i \(-0.471512\pi\)
0.0893774 + 0.995998i \(0.471512\pi\)
\(824\) 0 0
\(825\) −6.52228e12 −0.490181
\(826\) 0 0
\(827\) 2.49308e13 1.85337 0.926685 0.375838i \(-0.122645\pi\)
0.926685 + 0.375838i \(0.122645\pi\)
\(828\) 0 0
\(829\) −4.92371e12 −0.362073 −0.181037 0.983476i \(-0.557945\pi\)
−0.181037 + 0.983476i \(0.557945\pi\)
\(830\) 0 0
\(831\) −1.10531e13 −0.804043
\(832\) 0 0
\(833\) −4.25872e13 −3.06462
\(834\) 0 0
\(835\) 4.13605e13 2.94440
\(836\) 0 0
\(837\) 1.37002e12 0.0964858
\(838\) 0 0
\(839\) 1.58161e13 1.10197 0.550987 0.834514i \(-0.314252\pi\)
0.550987 + 0.834514i \(0.314252\pi\)
\(840\) 0 0
\(841\) 7.37525e10 0.00508388
\(842\) 0 0
\(843\) 4.55910e12 0.310924
\(844\) 0 0
\(845\) 2.70229e13 1.82338
\(846\) 0 0
\(847\) −2.34845e13 −1.56785
\(848\) 0 0
\(849\) 1.05436e12 0.0696473
\(850\) 0 0
\(851\) −4.02117e12 −0.262827
\(852\) 0 0
\(853\) −2.37094e13 −1.53338 −0.766690 0.642017i \(-0.778098\pi\)
−0.766690 + 0.642017i \(0.778098\pi\)
\(854\) 0 0
\(855\) −6.84663e12 −0.438156
\(856\) 0 0
\(857\) 1.88590e13 1.19427 0.597137 0.802139i \(-0.296305\pi\)
0.597137 + 0.802139i \(0.296305\pi\)
\(858\) 0 0
\(859\) 2.73506e13 1.71395 0.856974 0.515360i \(-0.172342\pi\)
0.856974 + 0.515360i \(0.172342\pi\)
\(860\) 0 0
\(861\) 1.39968e13 0.867992
\(862\) 0 0
\(863\) −5.40502e12 −0.331703 −0.165851 0.986151i \(-0.553037\pi\)
−0.165851 + 0.986151i \(0.553037\pi\)
\(864\) 0 0
\(865\) −1.73080e13 −1.05117
\(866\) 0 0
\(867\) 7.92367e12 0.476256
\(868\) 0 0
\(869\) −1.62361e12 −0.0965810
\(870\) 0 0
\(871\) 7.74065e11 0.0455718
\(872\) 0 0
\(873\) −3.36164e11 −0.0195879
\(874\) 0 0
\(875\) −7.60766e13 −4.38748
\(876\) 0 0
\(877\) 5.41805e12 0.309275 0.154638 0.987971i \(-0.450579\pi\)
0.154638 + 0.987971i \(0.450579\pi\)
\(878\) 0 0
\(879\) 1.66932e13 0.943171
\(880\) 0 0
\(881\) −7.92850e12 −0.443404 −0.221702 0.975114i \(-0.571161\pi\)
−0.221702 + 0.975114i \(0.571161\pi\)
\(882\) 0 0
\(883\) −2.68918e13 −1.48866 −0.744331 0.667811i \(-0.767231\pi\)
−0.744331 + 0.667811i \(0.767231\pi\)
\(884\) 0 0
\(885\) 1.89905e13 1.04062
\(886\) 0 0
\(887\) −2.82440e13 −1.53204 −0.766020 0.642817i \(-0.777766\pi\)
−0.766020 + 0.642817i \(0.777766\pi\)
\(888\) 0 0
\(889\) −3.67692e13 −1.97436
\(890\) 0 0
\(891\) −7.61706e11 −0.0404890
\(892\) 0 0
\(893\) −1.90492e13 −1.00241
\(894\) 0 0
\(895\) 1.54196e13 0.803286
\(896\) 0 0
\(897\) −8.73242e10 −0.00450369
\(898\) 0 0
\(899\) −9.84386e12 −0.502628
\(900\) 0 0
\(901\) −5.15021e13 −2.60354
\(902\) 0 0
\(903\) −1.12700e13 −0.564067
\(904\) 0 0
\(905\) −4.42877e13 −2.19465
\(906\) 0 0
\(907\) 1.28245e13 0.629226 0.314613 0.949220i \(-0.398125\pi\)
0.314613 + 0.949220i \(0.398125\pi\)
\(908\) 0 0
\(909\) −2.15579e12 −0.104729
\(910\) 0 0
\(911\) 9.36953e12 0.450697 0.225349 0.974278i \(-0.427648\pi\)
0.225349 + 0.974278i \(0.427648\pi\)
\(912\) 0 0
\(913\) 3.29908e12 0.157135
\(914\) 0 0
\(915\) −3.40363e13 −1.60527
\(916\) 0 0
\(917\) −4.87481e13 −2.27665
\(918\) 0 0
\(919\) 9.04580e12 0.418338 0.209169 0.977880i \(-0.432924\pi\)
0.209169 + 0.977880i \(0.432924\pi\)
\(920\) 0 0
\(921\) 5.08734e12 0.232982
\(922\) 0 0
\(923\) −1.03024e12 −0.0467229
\(924\) 0 0
\(925\) −4.88608e13 −2.19443
\(926\) 0 0
\(927\) 1.40283e13 0.623945
\(928\) 0 0
\(929\) 1.59777e13 0.703789 0.351895 0.936040i \(-0.385538\pi\)
0.351895 + 0.936040i \(0.385538\pi\)
\(930\) 0 0
\(931\) 3.74598e13 1.63415
\(932\) 0 0
\(933\) −2.31161e12 −0.0998729
\(934\) 0 0
\(935\) −2.09927e13 −0.898287
\(936\) 0 0
\(937\) −5.20573e12 −0.220624 −0.110312 0.993897i \(-0.535185\pi\)
−0.110312 + 0.993897i \(0.535185\pi\)
\(938\) 0 0
\(939\) 8.39800e11 0.0352517
\(940\) 0 0
\(941\) −1.35801e13 −0.564612 −0.282306 0.959324i \(-0.591099\pi\)
−0.282306 + 0.959324i \(0.591099\pi\)
\(942\) 0 0
\(943\) 5.63485e12 0.232049
\(944\) 0 0
\(945\) −1.55653e13 −0.634913
\(946\) 0 0
\(947\) −1.11344e13 −0.449874 −0.224937 0.974373i \(-0.572218\pi\)
−0.224937 + 0.974373i \(0.572218\pi\)
\(948\) 0 0
\(949\) 2.13095e11 0.00852854
\(950\) 0 0
\(951\) 8.93297e12 0.354147
\(952\) 0 0
\(953\) 3.70642e13 1.45558 0.727791 0.685799i \(-0.240547\pi\)
0.727791 + 0.685799i \(0.240547\pi\)
\(954\) 0 0
\(955\) −3.97585e13 −1.54673
\(956\) 0 0
\(957\) 5.47299e12 0.210921
\(958\) 0 0
\(959\) −2.15893e13 −0.824242
\(960\) 0 0
\(961\) −1.97938e13 −0.748643
\(962\) 0 0
\(963\) 9.05773e12 0.339392
\(964\) 0 0
\(965\) 6.58744e13 2.44537
\(966\) 0 0
\(967\) 2.56420e13 0.943047 0.471524 0.881853i \(-0.343704\pi\)
0.471524 + 0.881853i \(0.343704\pi\)
\(968\) 0 0
\(969\) −1.54188e13 −0.561815
\(970\) 0 0
\(971\) −4.47900e13 −1.61694 −0.808471 0.588536i \(-0.799704\pi\)
−0.808471 + 0.588536i \(0.799704\pi\)
\(972\) 0 0
\(973\) −4.58313e13 −1.63928
\(974\) 0 0
\(975\) −1.06107e12 −0.0376029
\(976\) 0 0
\(977\) −4.70660e12 −0.165265 −0.0826326 0.996580i \(-0.526333\pi\)
−0.0826326 + 0.996580i \(0.526333\pi\)
\(978\) 0 0
\(979\) 1.12746e13 0.392263
\(980\) 0 0
\(981\) −9.32036e12 −0.321309
\(982\) 0 0
\(983\) −4.45816e12 −0.152288 −0.0761438 0.997097i \(-0.524261\pi\)
−0.0761438 + 0.997097i \(0.524261\pi\)
\(984\) 0 0
\(985\) 7.88099e13 2.66758
\(986\) 0 0
\(987\) −4.33069e13 −1.45255
\(988\) 0 0
\(989\) −4.53710e12 −0.150798
\(990\) 0 0
\(991\) 1.02988e13 0.339201 0.169600 0.985513i \(-0.445752\pi\)
0.169600 + 0.985513i \(0.445752\pi\)
\(992\) 0 0
\(993\) −1.88036e13 −0.613720
\(994\) 0 0
\(995\) −3.27893e13 −1.06054
\(996\) 0 0
\(997\) 4.40551e13 1.41211 0.706054 0.708158i \(-0.250474\pi\)
0.706054 + 0.708158i \(0.250474\pi\)
\(998\) 0 0
\(999\) −5.70622e12 −0.181261
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 384.10.a.f.1.1 yes 4
4.3 odd 2 384.10.a.b.1.1 4
8.3 odd 2 384.10.a.g.1.4 yes 4
8.5 even 2 384.10.a.c.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.a.b.1.1 4 4.3 odd 2
384.10.a.c.1.4 yes 4 8.5 even 2
384.10.a.f.1.1 yes 4 1.1 even 1 trivial
384.10.a.g.1.4 yes 4 8.3 odd 2