Properties

Label 384.10.a.f.1.2
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 1.2
Root \(45.8607\) 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)
 
\(f(q)\) \(=\) \(q+81.0000 q^{3} -250.262 q^{5} -1655.62 q^{7} +6561.00 q^{9} +O(q^{10})\) \(q+81.0000 q^{3} -250.262 q^{5} -1655.62 q^{7} +6561.00 q^{9} -848.983 q^{11} -69683.7 q^{13} -20271.3 q^{15} +282069. q^{17} -470922. q^{19} -134105. q^{21} +2.41851e6 q^{23} -1.89049e6 q^{25} +531441. q^{27} +3.90059e6 q^{29} -3.99099e6 q^{31} -68767.7 q^{33} +414339. q^{35} -5.47494e6 q^{37} -5.64438e6 q^{39} +1.25933e7 q^{41} -1.12591e7 q^{43} -1.64197e6 q^{45} +2.89424e7 q^{47} -3.76125e7 q^{49} +2.28476e7 q^{51} +9.61718e6 q^{53} +212469. q^{55} -3.81447e7 q^{57} +1.53642e7 q^{59} +2.89328e7 q^{61} -1.08625e7 q^{63} +1.74392e7 q^{65} -1.43864e8 q^{67} +1.95899e8 q^{69} -1.80186e8 q^{71} +2.57115e8 q^{73} -1.53130e8 q^{75} +1.40559e6 q^{77} -1.36452e8 q^{79} +4.30467e7 q^{81} -2.41915e7 q^{83} -7.05914e7 q^{85} +3.15948e8 q^{87} -8.66242e8 q^{89} +1.15370e8 q^{91} -3.23271e8 q^{93} +1.17854e8 q^{95} -6.24828e8 q^{97} -5.57018e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 324 q^{3} - 240 q^{5} + 4840 q^{7} + 26244 q^{9}+O(q^{10}) \) 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
\(5\) −250.262 −0.179073 −0.0895366 0.995984i \(-0.528539\pi\)
−0.0895366 + 0.995984i \(0.528539\pi\)
\(6\) 0 0
\(7\) −1655.62 −0.260627 −0.130313 0.991473i \(-0.541598\pi\)
−0.130313 + 0.991473i \(0.541598\pi\)
\(8\) 0 0
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) −848.983 −0.0174837 −0.00874183 0.999962i \(-0.502783\pi\)
−0.00874183 + 0.999962i \(0.502783\pi\)
\(12\) 0 0
\(13\) −69683.7 −0.676684 −0.338342 0.941023i \(-0.609866\pi\)
−0.338342 + 0.941023i \(0.609866\pi\)
\(14\) 0 0
\(15\) −20271.3 −0.103388
\(16\) 0 0
\(17\) 282069. 0.819098 0.409549 0.912288i \(-0.365686\pi\)
0.409549 + 0.912288i \(0.365686\pi\)
\(18\) 0 0
\(19\) −470922. −0.829006 −0.414503 0.910048i \(-0.636045\pi\)
−0.414503 + 0.910048i \(0.636045\pi\)
\(20\) 0 0
\(21\) −134105. −0.150473
\(22\) 0 0
\(23\) 2.41851e6 1.80207 0.901036 0.433745i \(-0.142808\pi\)
0.901036 + 0.433745i \(0.142808\pi\)
\(24\) 0 0
\(25\) −1.89049e6 −0.967933
\(26\) 0 0
\(27\) 531441. 0.192450
\(28\) 0 0
\(29\) 3.90059e6 1.02409 0.512047 0.858958i \(-0.328888\pi\)
0.512047 + 0.858958i \(0.328888\pi\)
\(30\) 0 0
\(31\) −3.99099e6 −0.776164 −0.388082 0.921625i \(-0.626862\pi\)
−0.388082 + 0.921625i \(0.626862\pi\)
\(32\) 0 0
\(33\) −68767.7 −0.0100942
\(34\) 0 0
\(35\) 414339. 0.0466713
\(36\) 0 0
\(37\) −5.47494e6 −0.480254 −0.240127 0.970741i \(-0.577189\pi\)
−0.240127 + 0.970741i \(0.577189\pi\)
\(38\) 0 0
\(39\) −5.64438e6 −0.390684
\(40\) 0 0
\(41\) 1.25933e7 0.696004 0.348002 0.937494i \(-0.386860\pi\)
0.348002 + 0.937494i \(0.386860\pi\)
\(42\) 0 0
\(43\) −1.12591e7 −0.502222 −0.251111 0.967958i \(-0.580796\pi\)
−0.251111 + 0.967958i \(0.580796\pi\)
\(44\) 0 0
\(45\) −1.64197e6 −0.0596911
\(46\) 0 0
\(47\) 2.89424e7 0.865155 0.432578 0.901597i \(-0.357604\pi\)
0.432578 + 0.901597i \(0.357604\pi\)
\(48\) 0 0
\(49\) −3.76125e7 −0.932074
\(50\) 0 0
\(51\) 2.28476e7 0.472907
\(52\) 0 0
\(53\) 9.61718e6 0.167420 0.0837098 0.996490i \(-0.473323\pi\)
0.0837098 + 0.996490i \(0.473323\pi\)
\(54\) 0 0
\(55\) 212469. 0.00313085
\(56\) 0 0
\(57\) −3.81447e7 −0.478627
\(58\) 0 0
\(59\) 1.53642e7 0.165073 0.0825364 0.996588i \(-0.473698\pi\)
0.0825364 + 0.996588i \(0.473698\pi\)
\(60\) 0 0
\(61\) 2.89328e7 0.267551 0.133776 0.991012i \(-0.457290\pi\)
0.133776 + 0.991012i \(0.457290\pi\)
\(62\) 0 0
\(63\) −1.08625e7 −0.0868756
\(64\) 0 0
\(65\) 1.74392e7 0.121176
\(66\) 0 0
\(67\) −1.43864e8 −0.872200 −0.436100 0.899898i \(-0.643641\pi\)
−0.436100 + 0.899898i \(0.643641\pi\)
\(68\) 0 0
\(69\) 1.95899e8 1.04043
\(70\) 0 0
\(71\) −1.80186e8 −0.841506 −0.420753 0.907175i \(-0.638234\pi\)
−0.420753 + 0.907175i \(0.638234\pi\)
\(72\) 0 0
\(73\) 2.57115e8 1.05968 0.529839 0.848098i \(-0.322252\pi\)
0.529839 + 0.848098i \(0.322252\pi\)
\(74\) 0 0
\(75\) −1.53130e8 −0.558836
\(76\) 0 0
\(77\) 1.40559e6 0.00455671
\(78\) 0 0
\(79\) −1.36452e8 −0.394148 −0.197074 0.980389i \(-0.563144\pi\)
−0.197074 + 0.980389i \(0.563144\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 0 0
\(83\) −2.41915e7 −0.0559514 −0.0279757 0.999609i \(-0.508906\pi\)
−0.0279757 + 0.999609i \(0.508906\pi\)
\(84\) 0 0
\(85\) −7.05914e7 −0.146679
\(86\) 0 0
\(87\) 3.15948e8 0.591260
\(88\) 0 0
\(89\) −8.66242e8 −1.46347 −0.731735 0.681589i \(-0.761289\pi\)
−0.731735 + 0.681589i \(0.761289\pi\)
\(90\) 0 0
\(91\) 1.15370e8 0.176362
\(92\) 0 0
\(93\) −3.23271e8 −0.448118
\(94\) 0 0
\(95\) 1.17854e8 0.148453
\(96\) 0 0
\(97\) −6.24828e8 −0.716618 −0.358309 0.933603i \(-0.616647\pi\)
−0.358309 + 0.933603i \(0.616647\pi\)
\(98\) 0 0
\(99\) −5.57018e6 −0.00582788
\(100\) 0 0
\(101\) 4.63587e8 0.443288 0.221644 0.975128i \(-0.428858\pi\)
0.221644 + 0.975128i \(0.428858\pi\)
\(102\) 0 0
\(103\) −1.18683e9 −1.03901 −0.519507 0.854466i \(-0.673884\pi\)
−0.519507 + 0.854466i \(0.673884\pi\)
\(104\) 0 0
\(105\) 3.35615e7 0.0269457
\(106\) 0 0
\(107\) −1.61716e9 −1.19269 −0.596343 0.802730i \(-0.703380\pi\)
−0.596343 + 0.802730i \(0.703380\pi\)
\(108\) 0 0
\(109\) 2.78924e8 0.189264 0.0946318 0.995512i \(-0.469833\pi\)
0.0946318 + 0.995512i \(0.469833\pi\)
\(110\) 0 0
\(111\) −4.43470e8 −0.277275
\(112\) 0 0
\(113\) 7.83311e8 0.451940 0.225970 0.974134i \(-0.427445\pi\)
0.225970 + 0.974134i \(0.427445\pi\)
\(114\) 0 0
\(115\) −6.05261e8 −0.322703
\(116\) 0 0
\(117\) −4.57195e8 −0.225561
\(118\) 0 0
\(119\) −4.66999e8 −0.213479
\(120\) 0 0
\(121\) −2.35723e9 −0.999694
\(122\) 0 0
\(123\) 1.02006e9 0.401838
\(124\) 0 0
\(125\) 9.61913e8 0.352404
\(126\) 0 0
\(127\) 2.05194e9 0.699919 0.349960 0.936765i \(-0.386195\pi\)
0.349960 + 0.936765i \(0.386195\pi\)
\(128\) 0 0
\(129\) −9.11988e8 −0.289958
\(130\) 0 0
\(131\) −1.10603e9 −0.328131 −0.164065 0.986449i \(-0.552461\pi\)
−0.164065 + 0.986449i \(0.552461\pi\)
\(132\) 0 0
\(133\) 7.79667e8 0.216061
\(134\) 0 0
\(135\) −1.33000e8 −0.0344627
\(136\) 0 0
\(137\) 1.04415e9 0.253233 0.126617 0.991952i \(-0.459588\pi\)
0.126617 + 0.991952i \(0.459588\pi\)
\(138\) 0 0
\(139\) 2.93771e9 0.667487 0.333743 0.942664i \(-0.391688\pi\)
0.333743 + 0.942664i \(0.391688\pi\)
\(140\) 0 0
\(141\) 2.34433e9 0.499498
\(142\) 0 0
\(143\) 5.91603e7 0.0118309
\(144\) 0 0
\(145\) −9.76171e8 −0.183388
\(146\) 0 0
\(147\) −3.04662e9 −0.538133
\(148\) 0 0
\(149\) −3.75086e9 −0.623437 −0.311719 0.950174i \(-0.600905\pi\)
−0.311719 + 0.950174i \(0.600905\pi\)
\(150\) 0 0
\(151\) 2.48184e9 0.388487 0.194244 0.980953i \(-0.437775\pi\)
0.194244 + 0.980953i \(0.437775\pi\)
\(152\) 0 0
\(153\) 1.85066e9 0.273033
\(154\) 0 0
\(155\) 9.98796e8 0.138990
\(156\) 0 0
\(157\) −5.65770e9 −0.743175 −0.371588 0.928398i \(-0.621186\pi\)
−0.371588 + 0.928398i \(0.621186\pi\)
\(158\) 0 0
\(159\) 7.78991e8 0.0966597
\(160\) 0 0
\(161\) −4.00412e9 −0.469668
\(162\) 0 0
\(163\) −6.28827e9 −0.697729 −0.348865 0.937173i \(-0.613433\pi\)
−0.348865 + 0.937173i \(0.613433\pi\)
\(164\) 0 0
\(165\) 1.72100e7 0.00180760
\(166\) 0 0
\(167\) −1.43509e9 −0.142776 −0.0713882 0.997449i \(-0.522743\pi\)
−0.0713882 + 0.997449i \(0.522743\pi\)
\(168\) 0 0
\(169\) −5.74868e9 −0.542098
\(170\) 0 0
\(171\) −3.08972e9 −0.276335
\(172\) 0 0
\(173\) −1.06084e10 −0.900416 −0.450208 0.892924i \(-0.648650\pi\)
−0.450208 + 0.892924i \(0.648650\pi\)
\(174\) 0 0
\(175\) 3.12994e9 0.252269
\(176\) 0 0
\(177\) 1.24450e9 0.0953048
\(178\) 0 0
\(179\) 3.84540e8 0.0279964 0.0139982 0.999902i \(-0.495544\pi\)
0.0139982 + 0.999902i \(0.495544\pi\)
\(180\) 0 0
\(181\) −3.54919e9 −0.245797 −0.122898 0.992419i \(-0.539219\pi\)
−0.122898 + 0.992419i \(0.539219\pi\)
\(182\) 0 0
\(183\) 2.34356e9 0.154471
\(184\) 0 0
\(185\) 1.37017e9 0.0860007
\(186\) 0 0
\(187\) −2.39472e8 −0.0143208
\(188\) 0 0
\(189\) −8.79863e8 −0.0501576
\(190\) 0 0
\(191\) 2.80130e10 1.52304 0.761518 0.648144i \(-0.224455\pi\)
0.761518 + 0.648144i \(0.224455\pi\)
\(192\) 0 0
\(193\) 1.01773e10 0.527986 0.263993 0.964525i \(-0.414960\pi\)
0.263993 + 0.964525i \(0.414960\pi\)
\(194\) 0 0
\(195\) 1.41258e9 0.0699610
\(196\) 0 0
\(197\) −7.40263e9 −0.350177 −0.175089 0.984553i \(-0.556021\pi\)
−0.175089 + 0.984553i \(0.556021\pi\)
\(198\) 0 0
\(199\) 1.14131e10 0.515899 0.257950 0.966158i \(-0.416953\pi\)
0.257950 + 0.966158i \(0.416953\pi\)
\(200\) 0 0
\(201\) −1.16530e10 −0.503565
\(202\) 0 0
\(203\) −6.45789e9 −0.266906
\(204\) 0 0
\(205\) −3.15162e9 −0.124636
\(206\) 0 0
\(207\) 1.58678e10 0.600690
\(208\) 0 0
\(209\) 3.99805e8 0.0144941
\(210\) 0 0
\(211\) −1.50794e10 −0.523735 −0.261868 0.965104i \(-0.584338\pi\)
−0.261868 + 0.965104i \(0.584338\pi\)
\(212\) 0 0
\(213\) −1.45950e10 −0.485844
\(214\) 0 0
\(215\) 2.81773e9 0.0899345
\(216\) 0 0
\(217\) 6.60756e9 0.202289
\(218\) 0 0
\(219\) 2.08263e10 0.611806
\(220\) 0 0
\(221\) −1.96556e10 −0.554271
\(222\) 0 0
\(223\) 1.05027e10 0.284399 0.142200 0.989838i \(-0.454583\pi\)
0.142200 + 0.989838i \(0.454583\pi\)
\(224\) 0 0
\(225\) −1.24035e10 −0.322644
\(226\) 0 0
\(227\) −6.86346e10 −1.71564 −0.857821 0.513948i \(-0.828182\pi\)
−0.857821 + 0.513948i \(0.828182\pi\)
\(228\) 0 0
\(229\) −5.87491e10 −1.41170 −0.705848 0.708363i \(-0.749434\pi\)
−0.705848 + 0.708363i \(0.749434\pi\)
\(230\) 0 0
\(231\) 1.13853e8 0.00263082
\(232\) 0 0
\(233\) −3.66986e10 −0.815734 −0.407867 0.913041i \(-0.633727\pi\)
−0.407867 + 0.913041i \(0.633727\pi\)
\(234\) 0 0
\(235\) −7.24319e9 −0.154926
\(236\) 0 0
\(237\) −1.10526e10 −0.227561
\(238\) 0 0
\(239\) 2.46017e10 0.487724 0.243862 0.969810i \(-0.421586\pi\)
0.243862 + 0.969810i \(0.421586\pi\)
\(240\) 0 0
\(241\) −7.56409e10 −1.44438 −0.722188 0.691697i \(-0.756863\pi\)
−0.722188 + 0.691697i \(0.756863\pi\)
\(242\) 0 0
\(243\) 3.48678e9 0.0641500
\(244\) 0 0
\(245\) 9.41301e9 0.166909
\(246\) 0 0
\(247\) 3.28156e10 0.560975
\(248\) 0 0
\(249\) −1.95951e9 −0.0323036
\(250\) 0 0
\(251\) 2.92636e10 0.465367 0.232683 0.972553i \(-0.425249\pi\)
0.232683 + 0.972553i \(0.425249\pi\)
\(252\) 0 0
\(253\) −2.05327e9 −0.0315068
\(254\) 0 0
\(255\) −5.71790e9 −0.0846849
\(256\) 0 0
\(257\) −2.61646e10 −0.374123 −0.187062 0.982348i \(-0.559896\pi\)
−0.187062 + 0.982348i \(0.559896\pi\)
\(258\) 0 0
\(259\) 9.06440e9 0.125167
\(260\) 0 0
\(261\) 2.55918e10 0.341364
\(262\) 0 0
\(263\) 4.64847e9 0.0599114 0.0299557 0.999551i \(-0.490463\pi\)
0.0299557 + 0.999551i \(0.490463\pi\)
\(264\) 0 0
\(265\) −2.40682e9 −0.0299804
\(266\) 0 0
\(267\) −7.01656e10 −0.844935
\(268\) 0 0
\(269\) −5.15725e10 −0.600527 −0.300264 0.953856i \(-0.597075\pi\)
−0.300264 + 0.953856i \(0.597075\pi\)
\(270\) 0 0
\(271\) 1.43382e11 1.61485 0.807424 0.589971i \(-0.200861\pi\)
0.807424 + 0.589971i \(0.200861\pi\)
\(272\) 0 0
\(273\) 9.34494e9 0.101823
\(274\) 0 0
\(275\) 1.60500e9 0.0169230
\(276\) 0 0
\(277\) −7.25340e10 −0.740257 −0.370128 0.928981i \(-0.620686\pi\)
−0.370128 + 0.928981i \(0.620686\pi\)
\(278\) 0 0
\(279\) −2.61849e10 −0.258721
\(280\) 0 0
\(281\) −1.62285e11 −1.55275 −0.776374 0.630272i \(-0.782943\pi\)
−0.776374 + 0.630272i \(0.782943\pi\)
\(282\) 0 0
\(283\) −1.79317e11 −1.66182 −0.830908 0.556410i \(-0.812178\pi\)
−0.830908 + 0.556410i \(0.812178\pi\)
\(284\) 0 0
\(285\) 9.54618e9 0.0857093
\(286\) 0 0
\(287\) −2.08497e10 −0.181397
\(288\) 0 0
\(289\) −3.90247e10 −0.329078
\(290\) 0 0
\(291\) −5.06111e10 −0.413740
\(292\) 0 0
\(293\) −2.30277e11 −1.82535 −0.912676 0.408684i \(-0.865988\pi\)
−0.912676 + 0.408684i \(0.865988\pi\)
\(294\) 0 0
\(295\) −3.84508e9 −0.0295601
\(296\) 0 0
\(297\) −4.51185e8 −0.00336473
\(298\) 0 0
\(299\) −1.68530e11 −1.21943
\(300\) 0 0
\(301\) 1.86408e10 0.130893
\(302\) 0 0
\(303\) 3.75506e10 0.255932
\(304\) 0 0
\(305\) −7.24081e9 −0.0479113
\(306\) 0 0
\(307\) −2.81856e11 −1.81094 −0.905471 0.424409i \(-0.860482\pi\)
−0.905471 + 0.424409i \(0.860482\pi\)
\(308\) 0 0
\(309\) −9.61333e10 −0.599875
\(310\) 0 0
\(311\) 2.33240e11 1.41378 0.706889 0.707325i \(-0.250098\pi\)
0.706889 + 0.707325i \(0.250098\pi\)
\(312\) 0 0
\(313\) −3.29142e11 −1.93836 −0.969179 0.246360i \(-0.920766\pi\)
−0.969179 + 0.246360i \(0.920766\pi\)
\(314\) 0 0
\(315\) 2.71848e9 0.0155571
\(316\) 0 0
\(317\) 9.06182e10 0.504021 0.252011 0.967724i \(-0.418908\pi\)
0.252011 + 0.967724i \(0.418908\pi\)
\(318\) 0 0
\(319\) −3.31154e9 −0.0179049
\(320\) 0 0
\(321\) −1.30990e11 −0.688597
\(322\) 0 0
\(323\) −1.32833e11 −0.679037
\(324\) 0 0
\(325\) 1.31737e11 0.654985
\(326\) 0 0
\(327\) 2.25929e10 0.109271
\(328\) 0 0
\(329\) −4.79175e10 −0.225483
\(330\) 0 0
\(331\) 3.07245e11 1.40689 0.703443 0.710752i \(-0.251645\pi\)
0.703443 + 0.710752i \(0.251645\pi\)
\(332\) 0 0
\(333\) −3.59211e10 −0.160085
\(334\) 0 0
\(335\) 3.60038e10 0.156188
\(336\) 0 0
\(337\) 1.39132e11 0.587615 0.293807 0.955865i \(-0.405078\pi\)
0.293807 + 0.955865i \(0.405078\pi\)
\(338\) 0 0
\(339\) 6.34482e10 0.260928
\(340\) 0 0
\(341\) 3.38829e9 0.0135702
\(342\) 0 0
\(343\) 1.29082e11 0.503550
\(344\) 0 0
\(345\) −4.90262e10 −0.186313
\(346\) 0 0
\(347\) −2.79900e11 −1.03638 −0.518192 0.855264i \(-0.673395\pi\)
−0.518192 + 0.855264i \(0.673395\pi\)
\(348\) 0 0
\(349\) −4.32395e11 −1.56015 −0.780075 0.625686i \(-0.784819\pi\)
−0.780075 + 0.625686i \(0.784819\pi\)
\(350\) 0 0
\(351\) −3.70328e10 −0.130228
\(352\) 0 0
\(353\) −1.11287e11 −0.381467 −0.190734 0.981642i \(-0.561087\pi\)
−0.190734 + 0.981642i \(0.561087\pi\)
\(354\) 0 0
\(355\) 4.50937e10 0.150691
\(356\) 0 0
\(357\) −3.78269e10 −0.123252
\(358\) 0 0
\(359\) 1.49311e10 0.0474423 0.0237212 0.999719i \(-0.492449\pi\)
0.0237212 + 0.999719i \(0.492449\pi\)
\(360\) 0 0
\(361\) −1.00920e11 −0.312749
\(362\) 0 0
\(363\) −1.90935e11 −0.577174
\(364\) 0 0
\(365\) −6.43462e10 −0.189760
\(366\) 0 0
\(367\) −9.39582e10 −0.270357 −0.135178 0.990821i \(-0.543161\pi\)
−0.135178 + 0.990821i \(0.543161\pi\)
\(368\) 0 0
\(369\) 8.26245e10 0.232001
\(370\) 0 0
\(371\) −1.59224e10 −0.0436340
\(372\) 0 0
\(373\) −6.00502e11 −1.60629 −0.803146 0.595783i \(-0.796842\pi\)
−0.803146 + 0.595783i \(0.796842\pi\)
\(374\) 0 0
\(375\) 7.79150e10 0.203461
\(376\) 0 0
\(377\) −2.71808e11 −0.692988
\(378\) 0 0
\(379\) −5.20120e11 −1.29487 −0.647437 0.762119i \(-0.724159\pi\)
−0.647437 + 0.762119i \(0.724159\pi\)
\(380\) 0 0
\(381\) 1.66207e11 0.404099
\(382\) 0 0
\(383\) 4.99135e11 1.18529 0.592643 0.805465i \(-0.298084\pi\)
0.592643 + 0.805465i \(0.298084\pi\)
\(384\) 0 0
\(385\) −3.51767e8 −0.000815984 0
\(386\) 0 0
\(387\) −7.38710e10 −0.167407
\(388\) 0 0
\(389\) 1.73874e11 0.385001 0.192500 0.981297i \(-0.438340\pi\)
0.192500 + 0.981297i \(0.438340\pi\)
\(390\) 0 0
\(391\) 6.82187e11 1.47607
\(392\) 0 0
\(393\) −8.95886e10 −0.189446
\(394\) 0 0
\(395\) 3.41489e10 0.0705813
\(396\) 0 0
\(397\) 4.03254e11 0.814745 0.407373 0.913262i \(-0.366445\pi\)
0.407373 + 0.913262i \(0.366445\pi\)
\(398\) 0 0
\(399\) 6.31530e10 0.124743
\(400\) 0 0
\(401\) −3.41251e11 −0.659059 −0.329529 0.944145i \(-0.606890\pi\)
−0.329529 + 0.944145i \(0.606890\pi\)
\(402\) 0 0
\(403\) 2.78107e11 0.525218
\(404\) 0 0
\(405\) −1.07730e10 −0.0198970
\(406\) 0 0
\(407\) 4.64813e9 0.00839660
\(408\) 0 0
\(409\) 2.45414e11 0.433655 0.216828 0.976210i \(-0.430429\pi\)
0.216828 + 0.976210i \(0.430429\pi\)
\(410\) 0 0
\(411\) 8.45762e10 0.146204
\(412\) 0 0
\(413\) −2.54372e10 −0.0430224
\(414\) 0 0
\(415\) 6.05422e9 0.0100194
\(416\) 0 0
\(417\) 2.37955e11 0.385374
\(418\) 0 0
\(419\) 3.19132e11 0.505833 0.252917 0.967488i \(-0.418610\pi\)
0.252917 + 0.967488i \(0.418610\pi\)
\(420\) 0 0
\(421\) −1.08208e12 −1.67877 −0.839386 0.543535i \(-0.817085\pi\)
−0.839386 + 0.543535i \(0.817085\pi\)
\(422\) 0 0
\(423\) 1.89891e11 0.288385
\(424\) 0 0
\(425\) −5.33251e11 −0.792832
\(426\) 0 0
\(427\) −4.79017e10 −0.0697310
\(428\) 0 0
\(429\) 4.79199e9 0.00683058
\(430\) 0 0
\(431\) −6.02827e11 −0.841482 −0.420741 0.907181i \(-0.638230\pi\)
−0.420741 + 0.907181i \(0.638230\pi\)
\(432\) 0 0
\(433\) 3.98033e11 0.544157 0.272078 0.962275i \(-0.412289\pi\)
0.272078 + 0.962275i \(0.412289\pi\)
\(434\) 0 0
\(435\) −7.90699e10 −0.105879
\(436\) 0 0
\(437\) −1.13893e12 −1.49393
\(438\) 0 0
\(439\) 1.67851e11 0.215692 0.107846 0.994168i \(-0.465605\pi\)
0.107846 + 0.994168i \(0.465605\pi\)
\(440\) 0 0
\(441\) −2.46776e11 −0.310691
\(442\) 0 0
\(443\) 2.78026e11 0.342980 0.171490 0.985186i \(-0.445142\pi\)
0.171490 + 0.985186i \(0.445142\pi\)
\(444\) 0 0
\(445\) 2.16788e11 0.262068
\(446\) 0 0
\(447\) −3.03820e11 −0.359942
\(448\) 0 0
\(449\) 1.66356e12 1.93165 0.965826 0.259191i \(-0.0834560\pi\)
0.965826 + 0.259191i \(0.0834560\pi\)
\(450\) 0 0
\(451\) −1.06915e10 −0.0121687
\(452\) 0 0
\(453\) 2.01029e11 0.224293
\(454\) 0 0
\(455\) −2.88727e10 −0.0315817
\(456\) 0 0
\(457\) −6.38333e11 −0.684580 −0.342290 0.939594i \(-0.611203\pi\)
−0.342290 + 0.939594i \(0.611203\pi\)
\(458\) 0 0
\(459\) 1.49903e11 0.157636
\(460\) 0 0
\(461\) 2.76291e11 0.284913 0.142456 0.989801i \(-0.454500\pi\)
0.142456 + 0.989801i \(0.454500\pi\)
\(462\) 0 0
\(463\) 1.55833e11 0.157596 0.0787980 0.996891i \(-0.474892\pi\)
0.0787980 + 0.996891i \(0.474892\pi\)
\(464\) 0 0
\(465\) 8.09025e10 0.0802460
\(466\) 0 0
\(467\) −1.11029e11 −0.108022 −0.0540108 0.998540i \(-0.517201\pi\)
−0.0540108 + 0.998540i \(0.517201\pi\)
\(468\) 0 0
\(469\) 2.38184e11 0.227319
\(470\) 0 0
\(471\) −4.58273e11 −0.429072
\(472\) 0 0
\(473\) 9.55879e9 0.00878068
\(474\) 0 0
\(475\) 8.90275e11 0.802422
\(476\) 0 0
\(477\) 6.30983e10 0.0558065
\(478\) 0 0
\(479\) 6.41651e11 0.556915 0.278458 0.960449i \(-0.410177\pi\)
0.278458 + 0.960449i \(0.410177\pi\)
\(480\) 0 0
\(481\) 3.81514e11 0.324981
\(482\) 0 0
\(483\) −3.24334e11 −0.271163
\(484\) 0 0
\(485\) 1.56371e11 0.128327
\(486\) 0 0
\(487\) −5.00844e11 −0.403480 −0.201740 0.979439i \(-0.564660\pi\)
−0.201740 + 0.979439i \(0.564660\pi\)
\(488\) 0 0
\(489\) −5.09350e11 −0.402834
\(490\) 0 0
\(491\) −1.02945e12 −0.799353 −0.399677 0.916656i \(-0.630878\pi\)
−0.399677 + 0.916656i \(0.630878\pi\)
\(492\) 0 0
\(493\) 1.10024e12 0.838833
\(494\) 0 0
\(495\) 1.39401e9 0.00104362
\(496\) 0 0
\(497\) 2.98318e11 0.219319
\(498\) 0 0
\(499\) 1.93280e12 1.39552 0.697758 0.716333i \(-0.254181\pi\)
0.697758 + 0.716333i \(0.254181\pi\)
\(500\) 0 0
\(501\) −1.16243e11 −0.0824320
\(502\) 0 0
\(503\) −6.06855e11 −0.422697 −0.211348 0.977411i \(-0.567785\pi\)
−0.211348 + 0.977411i \(0.567785\pi\)
\(504\) 0 0
\(505\) −1.16019e11 −0.0793809
\(506\) 0 0
\(507\) −4.65643e11 −0.312981
\(508\) 0 0
\(509\) 2.82186e12 1.86340 0.931698 0.363235i \(-0.118328\pi\)
0.931698 + 0.363235i \(0.118328\pi\)
\(510\) 0 0
\(511\) −4.25684e11 −0.276181
\(512\) 0 0
\(513\) −2.50267e11 −0.159542
\(514\) 0 0
\(515\) 2.97019e11 0.186060
\(516\) 0 0
\(517\) −2.45716e10 −0.0151261
\(518\) 0 0
\(519\) −8.59281e11 −0.519855
\(520\) 0 0
\(521\) 3.19165e11 0.189778 0.0948888 0.995488i \(-0.469750\pi\)
0.0948888 + 0.995488i \(0.469750\pi\)
\(522\) 0 0
\(523\) 7.03090e11 0.410916 0.205458 0.978666i \(-0.434132\pi\)
0.205458 + 0.978666i \(0.434132\pi\)
\(524\) 0 0
\(525\) 2.53525e11 0.145648
\(526\) 0 0
\(527\) −1.12574e12 −0.635754
\(528\) 0 0
\(529\) 4.04802e12 2.24746
\(530\) 0 0
\(531\) 1.00804e11 0.0550243
\(532\) 0 0
\(533\) −8.77546e11 −0.470975
\(534\) 0 0
\(535\) 4.04714e11 0.213578
\(536\) 0 0
\(537\) 3.11477e10 0.0161637
\(538\) 0 0
\(539\) 3.19324e10 0.0162961
\(540\) 0 0
\(541\) 2.63274e11 0.132136 0.0660679 0.997815i \(-0.478955\pi\)
0.0660679 + 0.997815i \(0.478955\pi\)
\(542\) 0 0
\(543\) −2.87485e11 −0.141911
\(544\) 0 0
\(545\) −6.98043e10 −0.0338920
\(546\) 0 0
\(547\) −9.43116e11 −0.450425 −0.225212 0.974310i \(-0.572308\pi\)
−0.225212 + 0.974310i \(0.572308\pi\)
\(548\) 0 0
\(549\) 1.89828e11 0.0891837
\(550\) 0 0
\(551\) −1.83687e12 −0.848979
\(552\) 0 0
\(553\) 2.25913e11 0.102725
\(554\) 0 0
\(555\) 1.10984e11 0.0496525
\(556\) 0 0
\(557\) 3.73948e12 1.64613 0.823063 0.567950i \(-0.192263\pi\)
0.823063 + 0.567950i \(0.192263\pi\)
\(558\) 0 0
\(559\) 7.84576e11 0.339846
\(560\) 0 0
\(561\) −1.93973e10 −0.00826813
\(562\) 0 0
\(563\) −3.81433e11 −0.160004 −0.0800019 0.996795i \(-0.525493\pi\)
−0.0800019 + 0.996795i \(0.525493\pi\)
\(564\) 0 0
\(565\) −1.96033e11 −0.0809304
\(566\) 0 0
\(567\) −7.12689e10 −0.0289585
\(568\) 0 0
\(569\) −3.58954e12 −1.43560 −0.717800 0.696249i \(-0.754851\pi\)
−0.717800 + 0.696249i \(0.754851\pi\)
\(570\) 0 0
\(571\) 3.19723e12 1.25867 0.629334 0.777135i \(-0.283328\pi\)
0.629334 + 0.777135i \(0.283328\pi\)
\(572\) 0 0
\(573\) 2.26906e12 0.879325
\(574\) 0 0
\(575\) −4.57217e12 −1.74428
\(576\) 0 0
\(577\) 1.67840e12 0.630381 0.315191 0.949028i \(-0.397932\pi\)
0.315191 + 0.949028i \(0.397932\pi\)
\(578\) 0 0
\(579\) 8.24357e11 0.304833
\(580\) 0 0
\(581\) 4.00518e10 0.0145824
\(582\) 0 0
\(583\) −8.16482e9 −0.00292711
\(584\) 0 0
\(585\) 1.14419e11 0.0403920
\(586\) 0 0
\(587\) 4.78170e12 1.66231 0.831153 0.556043i \(-0.187681\pi\)
0.831153 + 0.556043i \(0.187681\pi\)
\(588\) 0 0
\(589\) 1.87945e12 0.643445
\(590\) 0 0
\(591\) −5.99613e11 −0.202175
\(592\) 0 0
\(593\) 1.42074e12 0.471811 0.235906 0.971776i \(-0.424194\pi\)
0.235906 + 0.971776i \(0.424194\pi\)
\(594\) 0 0
\(595\) 1.16872e11 0.0382284
\(596\) 0 0
\(597\) 9.24461e11 0.297854
\(598\) 0 0
\(599\) −3.72919e12 −1.18357 −0.591784 0.806096i \(-0.701576\pi\)
−0.591784 + 0.806096i \(0.701576\pi\)
\(600\) 0 0
\(601\) 3.32876e11 0.104075 0.0520375 0.998645i \(-0.483428\pi\)
0.0520375 + 0.998645i \(0.483428\pi\)
\(602\) 0 0
\(603\) −9.43893e11 −0.290733
\(604\) 0 0
\(605\) 5.89925e11 0.179018
\(606\) 0 0
\(607\) 1.38715e12 0.414739 0.207369 0.978263i \(-0.433510\pi\)
0.207369 + 0.978263i \(0.433510\pi\)
\(608\) 0 0
\(609\) −5.23089e11 −0.154098
\(610\) 0 0
\(611\) −2.01681e12 −0.585437
\(612\) 0 0
\(613\) −1.54790e12 −0.442763 −0.221382 0.975187i \(-0.571057\pi\)
−0.221382 + 0.975187i \(0.571057\pi\)
\(614\) 0 0
\(615\) −2.55282e11 −0.0719584
\(616\) 0 0
\(617\) 1.17696e12 0.326947 0.163473 0.986548i \(-0.447730\pi\)
0.163473 + 0.986548i \(0.447730\pi\)
\(618\) 0 0
\(619\) −1.72922e12 −0.473415 −0.236708 0.971581i \(-0.576068\pi\)
−0.236708 + 0.971581i \(0.576068\pi\)
\(620\) 0 0
\(621\) 1.28529e12 0.346809
\(622\) 0 0
\(623\) 1.43417e12 0.381420
\(624\) 0 0
\(625\) 3.45164e12 0.904827
\(626\) 0 0
\(627\) 3.23842e10 0.00836815
\(628\) 0 0
\(629\) −1.54431e12 −0.393376
\(630\) 0 0
\(631\) −5.73742e12 −1.44074 −0.720368 0.693592i \(-0.756027\pi\)
−0.720368 + 0.693592i \(0.756027\pi\)
\(632\) 0 0
\(633\) −1.22143e12 −0.302379
\(634\) 0 0
\(635\) −5.13524e11 −0.125337
\(636\) 0 0
\(637\) 2.62098e12 0.630720
\(638\) 0 0
\(639\) −1.18220e12 −0.280502
\(640\) 0 0
\(641\) 7.57753e12 1.77283 0.886414 0.462894i \(-0.153189\pi\)
0.886414 + 0.462894i \(0.153189\pi\)
\(642\) 0 0
\(643\) 4.80446e12 1.10840 0.554199 0.832384i \(-0.313025\pi\)
0.554199 + 0.832384i \(0.313025\pi\)
\(644\) 0 0
\(645\) 2.28236e11 0.0519237
\(646\) 0 0
\(647\) 1.43505e12 0.321958 0.160979 0.986958i \(-0.448535\pi\)
0.160979 + 0.986958i \(0.448535\pi\)
\(648\) 0 0
\(649\) −1.30439e10 −0.00288608
\(650\) 0 0
\(651\) 5.35213e11 0.116792
\(652\) 0 0
\(653\) 6.17024e12 1.32798 0.663991 0.747740i \(-0.268861\pi\)
0.663991 + 0.747740i \(0.268861\pi\)
\(654\) 0 0
\(655\) 2.76798e11 0.0587594
\(656\) 0 0
\(657\) 1.68693e12 0.353226
\(658\) 0 0
\(659\) 2.01866e11 0.0416945 0.0208472 0.999783i \(-0.493364\pi\)
0.0208472 + 0.999783i \(0.493364\pi\)
\(660\) 0 0
\(661\) −8.37300e12 −1.70598 −0.852991 0.521925i \(-0.825214\pi\)
−0.852991 + 0.521925i \(0.825214\pi\)
\(662\) 0 0
\(663\) −1.59211e12 −0.320008
\(664\) 0 0
\(665\) −1.95121e11 −0.0386908
\(666\) 0 0
\(667\) 9.43360e12 1.84549
\(668\) 0 0
\(669\) 8.50717e11 0.164198
\(670\) 0 0
\(671\) −2.45635e10 −0.00467777
\(672\) 0 0
\(673\) 9.11671e12 1.71305 0.856525 0.516105i \(-0.172619\pi\)
0.856525 + 0.516105i \(0.172619\pi\)
\(674\) 0 0
\(675\) −1.00469e12 −0.186279
\(676\) 0 0
\(677\) 8.62986e12 1.57890 0.789450 0.613815i \(-0.210366\pi\)
0.789450 + 0.613815i \(0.210366\pi\)
\(678\) 0 0
\(679\) 1.03448e12 0.186770
\(680\) 0 0
\(681\) −5.55940e12 −0.990527
\(682\) 0 0
\(683\) 2.73461e12 0.480842 0.240421 0.970669i \(-0.422715\pi\)
0.240421 + 0.970669i \(0.422715\pi\)
\(684\) 0 0
\(685\) −2.61312e11 −0.0453473
\(686\) 0 0
\(687\) −4.75867e12 −0.815043
\(688\) 0 0
\(689\) −6.70161e11 −0.113290
\(690\) 0 0
\(691\) 7.23661e12 1.20749 0.603746 0.797177i \(-0.293674\pi\)
0.603746 + 0.797177i \(0.293674\pi\)
\(692\) 0 0
\(693\) 9.22209e9 0.00151890
\(694\) 0 0
\(695\) −7.35199e11 −0.119529
\(696\) 0 0
\(697\) 3.55218e12 0.570095
\(698\) 0 0
\(699\) −2.97259e12 −0.470964
\(700\) 0 0
\(701\) −2.35635e12 −0.368561 −0.184280 0.982874i \(-0.558995\pi\)
−0.184280 + 0.982874i \(0.558995\pi\)
\(702\) 0 0
\(703\) 2.57827e12 0.398134
\(704\) 0 0
\(705\) −5.86699e11 −0.0894466
\(706\) 0 0
\(707\) −7.67524e11 −0.115533
\(708\) 0 0
\(709\) −1.84172e12 −0.273725 −0.136862 0.990590i \(-0.543702\pi\)
−0.136862 + 0.990590i \(0.543702\pi\)
\(710\) 0 0
\(711\) −8.95264e11 −0.131383
\(712\) 0 0
\(713\) −9.65225e12 −1.39870
\(714\) 0 0
\(715\) −1.48056e10 −0.00211860
\(716\) 0 0
\(717\) 1.99274e12 0.281588
\(718\) 0 0
\(719\) 1.00065e12 0.139637 0.0698186 0.997560i \(-0.477758\pi\)
0.0698186 + 0.997560i \(0.477758\pi\)
\(720\) 0 0
\(721\) 1.96494e12 0.270795
\(722\) 0 0
\(723\) −6.12691e12 −0.833910
\(724\) 0 0
\(725\) −7.37404e12 −0.991253
\(726\) 0 0
\(727\) 1.46409e13 1.94385 0.971925 0.235291i \(-0.0756044\pi\)
0.971925 + 0.235291i \(0.0756044\pi\)
\(728\) 0 0
\(729\) 2.82430e11 0.0370370
\(730\) 0 0
\(731\) −3.17585e12 −0.411369
\(732\) 0 0
\(733\) 1.30354e12 0.166785 0.0833925 0.996517i \(-0.473424\pi\)
0.0833925 + 0.996517i \(0.473424\pi\)
\(734\) 0 0
\(735\) 7.62453e11 0.0963652
\(736\) 0 0
\(737\) 1.22138e11 0.0152492
\(738\) 0 0
\(739\) 5.15412e12 0.635704 0.317852 0.948140i \(-0.397039\pi\)
0.317852 + 0.948140i \(0.397039\pi\)
\(740\) 0 0
\(741\) 2.65806e12 0.323879
\(742\) 0 0
\(743\) −8.22504e12 −0.990121 −0.495060 0.868859i \(-0.664854\pi\)
−0.495060 + 0.868859i \(0.664854\pi\)
\(744\) 0 0
\(745\) 9.38700e11 0.111641
\(746\) 0 0
\(747\) −1.58720e11 −0.0186505
\(748\) 0 0
\(749\) 2.67740e12 0.310846
\(750\) 0 0
\(751\) 2.09125e12 0.239898 0.119949 0.992780i \(-0.461727\pi\)
0.119949 + 0.992780i \(0.461727\pi\)
\(752\) 0 0
\(753\) 2.37035e12 0.268680
\(754\) 0 0
\(755\) −6.21110e11 −0.0695677
\(756\) 0 0
\(757\) −1.40301e13 −1.55285 −0.776427 0.630208i \(-0.782970\pi\)
−0.776427 + 0.630208i \(0.782970\pi\)
\(758\) 0 0
\(759\) −1.66315e11 −0.0181905
\(760\) 0 0
\(761\) 4.88984e11 0.0528523 0.0264262 0.999651i \(-0.491587\pi\)
0.0264262 + 0.999651i \(0.491587\pi\)
\(762\) 0 0
\(763\) −4.61792e11 −0.0493272
\(764\) 0 0
\(765\) −4.63150e11 −0.0488928
\(766\) 0 0
\(767\) −1.07063e12 −0.111702
\(768\) 0 0
\(769\) 5.51278e12 0.568463 0.284232 0.958756i \(-0.408262\pi\)
0.284232 + 0.958756i \(0.408262\pi\)
\(770\) 0 0
\(771\) −2.11933e12 −0.216000
\(772\) 0 0
\(773\) −7.93968e12 −0.799826 −0.399913 0.916553i \(-0.630960\pi\)
−0.399913 + 0.916553i \(0.630960\pi\)
\(774\) 0 0
\(775\) 7.54495e12 0.751275
\(776\) 0 0
\(777\) 7.34217e11 0.0722653
\(778\) 0 0
\(779\) −5.93045e12 −0.576991
\(780\) 0 0
\(781\) 1.52975e11 0.0147126
\(782\) 0 0
\(783\) 2.07293e12 0.197087
\(784\) 0 0
\(785\) 1.41591e12 0.133083
\(786\) 0 0
\(787\) 2.12131e12 0.197114 0.0985569 0.995131i \(-0.468577\pi\)
0.0985569 + 0.995131i \(0.468577\pi\)
\(788\) 0 0
\(789\) 3.76526e11 0.0345899
\(790\) 0 0
\(791\) −1.29686e12 −0.117788
\(792\) 0 0
\(793\) −2.01615e12 −0.181048
\(794\) 0 0
\(795\) −1.94952e11 −0.0173092
\(796\) 0 0
\(797\) 1.56756e13 1.37613 0.688067 0.725647i \(-0.258459\pi\)
0.688067 + 0.725647i \(0.258459\pi\)
\(798\) 0 0
\(799\) 8.16376e12 0.708647
\(800\) 0 0
\(801\) −5.68341e12 −0.487824
\(802\) 0 0
\(803\) −2.18286e11 −0.0185271
\(804\) 0 0
\(805\) 1.00208e12 0.0841050
\(806\) 0 0
\(807\) −4.17737e12 −0.346715
\(808\) 0 0
\(809\) −1.33817e13 −1.09836 −0.549178 0.835705i \(-0.685059\pi\)
−0.549178 + 0.835705i \(0.685059\pi\)
\(810\) 0 0
\(811\) 7.33572e12 0.595455 0.297727 0.954651i \(-0.403771\pi\)
0.297727 + 0.954651i \(0.403771\pi\)
\(812\) 0 0
\(813\) 1.16139e13 0.932333
\(814\) 0 0
\(815\) 1.57372e12 0.124945
\(816\) 0 0
\(817\) 5.30216e12 0.416345
\(818\) 0 0
\(819\) 7.56940e11 0.0587874
\(820\) 0 0
\(821\) 6.64810e12 0.510685 0.255343 0.966851i \(-0.417812\pi\)
0.255343 + 0.966851i \(0.417812\pi\)
\(822\) 0 0
\(823\) 4.61148e12 0.350381 0.175191 0.984535i \(-0.443946\pi\)
0.175191 + 0.984535i \(0.443946\pi\)
\(824\) 0 0
\(825\) 1.30005e11 0.00977050
\(826\) 0 0
\(827\) −1.64749e13 −1.22475 −0.612377 0.790566i \(-0.709786\pi\)
−0.612377 + 0.790566i \(0.709786\pi\)
\(828\) 0 0
\(829\) −2.33872e13 −1.71982 −0.859910 0.510446i \(-0.829480\pi\)
−0.859910 + 0.510446i \(0.829480\pi\)
\(830\) 0 0
\(831\) −5.87525e12 −0.427388
\(832\) 0 0
\(833\) −1.06093e13 −0.763460
\(834\) 0 0
\(835\) 3.59150e11 0.0255674
\(836\) 0 0
\(837\) −2.12098e12 −0.149373
\(838\) 0 0
\(839\) 3.19129e12 0.222350 0.111175 0.993801i \(-0.464539\pi\)
0.111175 + 0.993801i \(0.464539\pi\)
\(840\) 0 0
\(841\) 7.07463e11 0.0487665
\(842\) 0 0
\(843\) −1.31451e13 −0.896480
\(844\) 0 0
\(845\) 1.43868e12 0.0970753
\(846\) 0 0
\(847\) 3.90267e12 0.260547
\(848\) 0 0
\(849\) −1.45247e13 −0.959450
\(850\) 0 0
\(851\) −1.32412e13 −0.865453
\(852\) 0 0
\(853\) −2.01905e13 −1.30580 −0.652899 0.757445i \(-0.726448\pi\)
−0.652899 + 0.757445i \(0.726448\pi\)
\(854\) 0 0
\(855\) 7.73241e11 0.0494843
\(856\) 0 0
\(857\) −1.40886e13 −0.892185 −0.446093 0.894987i \(-0.647185\pi\)
−0.446093 + 0.894987i \(0.647185\pi\)
\(858\) 0 0
\(859\) −2.11678e13 −1.32650 −0.663250 0.748398i \(-0.730823\pi\)
−0.663250 + 0.748398i \(0.730823\pi\)
\(860\) 0 0
\(861\) −1.68882e12 −0.104730
\(862\) 0 0
\(863\) −7.28437e12 −0.447037 −0.223519 0.974700i \(-0.571754\pi\)
−0.223519 + 0.974700i \(0.571754\pi\)
\(864\) 0 0
\(865\) 2.65489e12 0.161240
\(866\) 0 0
\(867\) −3.16100e12 −0.189993
\(868\) 0 0
\(869\) 1.15846e11 0.00689115
\(870\) 0 0
\(871\) 1.00250e13 0.590204
\(872\) 0 0
\(873\) −4.09950e12 −0.238873
\(874\) 0 0
\(875\) −1.59256e12 −0.0918459
\(876\) 0 0
\(877\) −2.59566e13 −1.48166 −0.740831 0.671692i \(-0.765568\pi\)
−0.740831 + 0.671692i \(0.765568\pi\)
\(878\) 0 0
\(879\) −1.86525e13 −1.05387
\(880\) 0 0
\(881\) −9.85083e12 −0.550911 −0.275456 0.961314i \(-0.588829\pi\)
−0.275456 + 0.961314i \(0.588829\pi\)
\(882\) 0 0
\(883\) 7.71297e11 0.0426971 0.0213486 0.999772i \(-0.493204\pi\)
0.0213486 + 0.999772i \(0.493204\pi\)
\(884\) 0 0
\(885\) −3.11451e11 −0.0170665
\(886\) 0 0
\(887\) 9.94743e12 0.539579 0.269789 0.962919i \(-0.413046\pi\)
0.269789 + 0.962919i \(0.413046\pi\)
\(888\) 0 0
\(889\) −3.39723e12 −0.182418
\(890\) 0 0
\(891\) −3.65460e10 −0.00194263
\(892\) 0 0
\(893\) −1.36296e13 −0.717219
\(894\) 0 0
\(895\) −9.62359e10 −0.00501341
\(896\) 0 0
\(897\) −1.36510e13 −0.704040
\(898\) 0 0
\(899\) −1.55672e13 −0.794864
\(900\) 0 0
\(901\) 2.71271e12 0.137133
\(902\) 0 0
\(903\) 1.50990e12 0.0755708
\(904\) 0 0
\(905\) 8.88230e11 0.0440156
\(906\) 0 0
\(907\) 1.51839e13 0.744988 0.372494 0.928035i \(-0.378503\pi\)
0.372494 + 0.928035i \(0.378503\pi\)
\(908\) 0 0
\(909\) 3.04160e12 0.147763
\(910\) 0 0
\(911\) 1.30119e13 0.625903 0.312951 0.949769i \(-0.398682\pi\)
0.312951 + 0.949769i \(0.398682\pi\)
\(912\) 0 0
\(913\) 2.05382e10 0.000978235 0
\(914\) 0 0
\(915\) −5.86505e11 −0.0276616
\(916\) 0 0
\(917\) 1.83117e12 0.0855197
\(918\) 0 0
\(919\) 2.88036e13 1.33207 0.666034 0.745921i \(-0.267990\pi\)
0.666034 + 0.745921i \(0.267990\pi\)
\(920\) 0 0
\(921\) −2.28303e13 −1.04555
\(922\) 0 0
\(923\) 1.25560e13 0.569434
\(924\) 0 0
\(925\) 1.03503e13 0.464854
\(926\) 0 0
\(927\) −7.78680e12 −0.346338
\(928\) 0 0
\(929\) −2.00577e13 −0.883510 −0.441755 0.897136i \(-0.645644\pi\)
−0.441755 + 0.897136i \(0.645644\pi\)
\(930\) 0 0
\(931\) 1.77126e13 0.772695
\(932\) 0 0
\(933\) 1.88924e13 0.816245
\(934\) 0 0
\(935\) 5.99309e10 0.00256448
\(936\) 0 0
\(937\) 3.18883e13 1.35146 0.675731 0.737149i \(-0.263828\pi\)
0.675731 + 0.737149i \(0.263828\pi\)
\(938\) 0 0
\(939\) −2.66605e13 −1.11911
\(940\) 0 0
\(941\) −3.39219e12 −0.141035 −0.0705175 0.997511i \(-0.522465\pi\)
−0.0705175 + 0.997511i \(0.522465\pi\)
\(942\) 0 0
\(943\) 3.04569e13 1.25425
\(944\) 0 0
\(945\) 2.20197e11 0.00898189
\(946\) 0 0
\(947\) −3.55975e13 −1.43828 −0.719142 0.694863i \(-0.755465\pi\)
−0.719142 + 0.694863i \(0.755465\pi\)
\(948\) 0 0
\(949\) −1.79167e13 −0.717068
\(950\) 0 0
\(951\) 7.34007e12 0.290997
\(952\) 0 0
\(953\) −2.43100e13 −0.954702 −0.477351 0.878713i \(-0.658403\pi\)
−0.477351 + 0.878713i \(0.658403\pi\)
\(954\) 0 0
\(955\) −7.01061e12 −0.272735
\(956\) 0 0
\(957\) −2.68235e11 −0.0103374
\(958\) 0 0
\(959\) −1.72872e12 −0.0659994
\(960\) 0 0
\(961\) −1.05116e13 −0.397569
\(962\) 0 0
\(963\) −1.06102e13 −0.397562
\(964\) 0 0
\(965\) −2.54698e12 −0.0945482
\(966\) 0 0
\(967\) 1.53573e13 0.564801 0.282401 0.959297i \(-0.408869\pi\)
0.282401 + 0.959297i \(0.408869\pi\)
\(968\) 0 0
\(969\) −1.07594e13 −0.392042
\(970\) 0 0
\(971\) −3.16083e13 −1.14108 −0.570539 0.821271i \(-0.693266\pi\)
−0.570539 + 0.821271i \(0.693266\pi\)
\(972\) 0 0
\(973\) −4.86373e12 −0.173965
\(974\) 0 0
\(975\) 1.06707e13 0.378156
\(976\) 0 0
\(977\) −1.05660e13 −0.371008 −0.185504 0.982643i \(-0.559392\pi\)
−0.185504 + 0.982643i \(0.559392\pi\)
\(978\) 0 0
\(979\) 7.35425e11 0.0255868
\(980\) 0 0
\(981\) 1.83002e12 0.0630879
\(982\) 0 0
\(983\) −4.36692e13 −1.49171 −0.745856 0.666108i \(-0.767959\pi\)
−0.745856 + 0.666108i \(0.767959\pi\)
\(984\) 0 0
\(985\) 1.85260e12 0.0627074
\(986\) 0 0
\(987\) −3.88132e12 −0.130182
\(988\) 0 0
\(989\) −2.72302e13 −0.905040
\(990\) 0 0
\(991\) 3.20991e13 1.05721 0.528605 0.848868i \(-0.322715\pi\)
0.528605 + 0.848868i \(0.322715\pi\)
\(992\) 0 0
\(993\) 2.48868e13 0.812266
\(994\) 0 0
\(995\) −2.85627e12 −0.0923837
\(996\) 0 0
\(997\) −3.79644e13 −1.21688 −0.608441 0.793599i \(-0.708205\pi\)
−0.608441 + 0.793599i \(0.708205\pi\)
\(998\) 0 0
\(999\) −2.90961e12 −0.0924250
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.1.2 yes 4
4.3 odd 2 384.10.a.b.1.2 4
8.3 odd 2 384.10.a.g.1.3 yes 4
8.5 even 2 384.10.a.c.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.a.b.1.2 4 4.3 odd 2
384.10.a.c.1.3 yes 4 8.5 even 2
384.10.a.f.1.2 yes 4 1.1 even 1 trivial
384.10.a.g.1.3 yes 4 8.3 odd 2