Properties

Label 384.10.a.g.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 \(14.0249\) 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)
 
\(f(q)\) \(=\) \(q+81.0000 q^{3} -2687.31 q^{5} -950.516 q^{7} +6561.00 q^{9} +O(q^{10})\) \(q+81.0000 q^{3} -2687.31 q^{5} -950.516 q^{7} +6561.00 q^{9} -85357.5 q^{11} +124861. q^{13} -217672. q^{15} -21449.0 q^{17} +243033. q^{19} -76991.8 q^{21} +706695. q^{23} +5.26849e6 q^{25} +531441. q^{27} +3.90211e6 q^{29} -8.04528e6 q^{31} -6.91396e6 q^{33} +2.55433e6 q^{35} +1.67047e7 q^{37} +1.01137e7 q^{39} -3.31163e7 q^{41} +3.30635e7 q^{43} -1.76314e7 q^{45} +3.29633e7 q^{47} -3.94501e7 q^{49} -1.73737e6 q^{51} +5.61799e7 q^{53} +2.29382e8 q^{55} +1.96857e7 q^{57} +1.09356e7 q^{59} +3.77590e7 q^{61} -6.23634e6 q^{63} -3.35539e8 q^{65} -9.13783e7 q^{67} +5.72423e7 q^{69} -2.44278e8 q^{71} +1.30973e8 q^{73} +4.26747e8 q^{75} +8.11337e7 q^{77} +4.26074e8 q^{79} +4.30467e7 q^{81} +2.32850e8 q^{83} +5.76401e7 q^{85} +3.16071e8 q^{87} -2.89719e8 q^{89} -1.18682e8 q^{91} -6.51668e8 q^{93} -6.53103e8 q^{95} -2.74576e8 q^{97} -5.60031e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 324 q^{3} + 240 q^{5} - 4840 q^{7} + 26244 q^{9}+O(q^{10}) \) 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
\(5\) −2687.31 −1.92288 −0.961440 0.275016i \(-0.911317\pi\)
−0.961440 + 0.275016i \(0.911317\pi\)
\(6\) 0 0
\(7\) −950.516 −0.149630 −0.0748149 0.997197i \(-0.523837\pi\)
−0.0748149 + 0.997197i \(0.523837\pi\)
\(8\) 0 0
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) −85357.5 −1.75782 −0.878911 0.476986i \(-0.841729\pi\)
−0.878911 + 0.476986i \(0.841729\pi\)
\(12\) 0 0
\(13\) 124861. 1.21250 0.606249 0.795275i \(-0.292674\pi\)
0.606249 + 0.795275i \(0.292674\pi\)
\(14\) 0 0
\(15\) −217672. −1.11017
\(16\) 0 0
\(17\) −21449.0 −0.0622856 −0.0311428 0.999515i \(-0.509915\pi\)
−0.0311428 + 0.999515i \(0.509915\pi\)
\(18\) 0 0
\(19\) 243033. 0.427832 0.213916 0.976852i \(-0.431378\pi\)
0.213916 + 0.976852i \(0.431378\pi\)
\(20\) 0 0
\(21\) −76991.8 −0.0863888
\(22\) 0 0
\(23\) 706695. 0.526570 0.263285 0.964718i \(-0.415194\pi\)
0.263285 + 0.964718i \(0.415194\pi\)
\(24\) 0 0
\(25\) 5.26849e6 2.69747
\(26\) 0 0
\(27\) 531441. 0.192450
\(28\) 0 0
\(29\) 3.90211e6 1.02449 0.512246 0.858839i \(-0.328814\pi\)
0.512246 + 0.858839i \(0.328814\pi\)
\(30\) 0 0
\(31\) −8.04528e6 −1.56464 −0.782319 0.622878i \(-0.785963\pi\)
−0.782319 + 0.622878i \(0.785963\pi\)
\(32\) 0 0
\(33\) −6.91396e6 −1.01488
\(34\) 0 0
\(35\) 2.55433e6 0.287720
\(36\) 0 0
\(37\) 1.67047e7 1.46531 0.732656 0.680599i \(-0.238281\pi\)
0.732656 + 0.680599i \(0.238281\pi\)
\(38\) 0 0
\(39\) 1.01137e7 0.700036
\(40\) 0 0
\(41\) −3.31163e7 −1.83027 −0.915134 0.403149i \(-0.867915\pi\)
−0.915134 + 0.403149i \(0.867915\pi\)
\(42\) 0 0
\(43\) 3.30635e7 1.47483 0.737413 0.675442i \(-0.236047\pi\)
0.737413 + 0.675442i \(0.236047\pi\)
\(44\) 0 0
\(45\) −1.76314e7 −0.640960
\(46\) 0 0
\(47\) 3.29633e7 0.985351 0.492675 0.870213i \(-0.336019\pi\)
0.492675 + 0.870213i \(0.336019\pi\)
\(48\) 0 0
\(49\) −3.94501e7 −0.977611
\(50\) 0 0
\(51\) −1.73737e6 −0.0359606
\(52\) 0 0
\(53\) 5.61799e7 0.978001 0.489000 0.872284i \(-0.337362\pi\)
0.489000 + 0.872284i \(0.337362\pi\)
\(54\) 0 0
\(55\) 2.29382e8 3.38008
\(56\) 0 0
\(57\) 1.96857e7 0.247009
\(58\) 0 0
\(59\) 1.09356e7 0.117492 0.0587460 0.998273i \(-0.481290\pi\)
0.0587460 + 0.998273i \(0.481290\pi\)
\(60\) 0 0
\(61\) 3.77590e7 0.349170 0.174585 0.984642i \(-0.444142\pi\)
0.174585 + 0.984642i \(0.444142\pi\)
\(62\) 0 0
\(63\) −6.23634e6 −0.0498766
\(64\) 0 0
\(65\) −3.35539e8 −2.33149
\(66\) 0 0
\(67\) −9.13783e7 −0.553996 −0.276998 0.960871i \(-0.589340\pi\)
−0.276998 + 0.960871i \(0.589340\pi\)
\(68\) 0 0
\(69\) 5.72423e7 0.304016
\(70\) 0 0
\(71\) −2.44278e8 −1.14083 −0.570416 0.821356i \(-0.693218\pi\)
−0.570416 + 0.821356i \(0.693218\pi\)
\(72\) 0 0
\(73\) 1.30973e8 0.539795 0.269898 0.962889i \(-0.413010\pi\)
0.269898 + 0.962889i \(0.413010\pi\)
\(74\) 0 0
\(75\) 4.26747e8 1.55738
\(76\) 0 0
\(77\) 8.11337e7 0.263023
\(78\) 0 0
\(79\) 4.26074e8 1.23073 0.615366 0.788241i \(-0.289008\pi\)
0.615366 + 0.788241i \(0.289008\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 0 0
\(83\) 2.32850e8 0.538547 0.269274 0.963064i \(-0.413216\pi\)
0.269274 + 0.963064i \(0.413216\pi\)
\(84\) 0 0
\(85\) 5.76401e7 0.119768
\(86\) 0 0
\(87\) 3.16071e8 0.591490
\(88\) 0 0
\(89\) −2.89719e8 −0.489465 −0.244732 0.969591i \(-0.578700\pi\)
−0.244732 + 0.969591i \(0.578700\pi\)
\(90\) 0 0
\(91\) −1.18682e8 −0.181426
\(92\) 0 0
\(93\) −6.51668e8 −0.903344
\(94\) 0 0
\(95\) −6.53103e8 −0.822670
\(96\) 0 0
\(97\) −2.74576e8 −0.314912 −0.157456 0.987526i \(-0.550329\pi\)
−0.157456 + 0.987526i \(0.550329\pi\)
\(98\) 0 0
\(99\) −5.60031e8 −0.585941
\(100\) 0 0
\(101\) 4.51816e8 0.432032 0.216016 0.976390i \(-0.430694\pi\)
0.216016 + 0.976390i \(0.430694\pi\)
\(102\) 0 0
\(103\) −8.90289e8 −0.779405 −0.389703 0.920941i \(-0.627422\pi\)
−0.389703 + 0.920941i \(0.627422\pi\)
\(104\) 0 0
\(105\) 2.06901e8 0.166115
\(106\) 0 0
\(107\) −2.56017e9 −1.88817 −0.944087 0.329696i \(-0.893054\pi\)
−0.944087 + 0.329696i \(0.893054\pi\)
\(108\) 0 0
\(109\) −1.33571e9 −0.906345 −0.453173 0.891423i \(-0.649708\pi\)
−0.453173 + 0.891423i \(0.649708\pi\)
\(110\) 0 0
\(111\) 1.35308e9 0.845998
\(112\) 0 0
\(113\) −2.87335e9 −1.65781 −0.828906 0.559388i \(-0.811036\pi\)
−0.828906 + 0.559388i \(0.811036\pi\)
\(114\) 0 0
\(115\) −1.89910e9 −1.01253
\(116\) 0 0
\(117\) 8.19211e8 0.404166
\(118\) 0 0
\(119\) 2.03876e7 0.00931978
\(120\) 0 0
\(121\) 4.92796e9 2.08994
\(122\) 0 0
\(123\) −2.68242e9 −1.05671
\(124\) 0 0
\(125\) −8.90939e9 −3.26402
\(126\) 0 0
\(127\) −5.49390e8 −0.187397 −0.0936987 0.995601i \(-0.529869\pi\)
−0.0936987 + 0.995601i \(0.529869\pi\)
\(128\) 0 0
\(129\) 2.67815e9 0.851492
\(130\) 0 0
\(131\) 3.50142e8 0.103878 0.0519390 0.998650i \(-0.483460\pi\)
0.0519390 + 0.998650i \(0.483460\pi\)
\(132\) 0 0
\(133\) −2.31007e8 −0.0640165
\(134\) 0 0
\(135\) −1.42814e9 −0.370058
\(136\) 0 0
\(137\) 2.45719e9 0.595930 0.297965 0.954577i \(-0.403692\pi\)
0.297965 + 0.954577i \(0.403692\pi\)
\(138\) 0 0
\(139\) −3.52695e9 −0.801368 −0.400684 0.916216i \(-0.631228\pi\)
−0.400684 + 0.916216i \(0.631228\pi\)
\(140\) 0 0
\(141\) 2.67003e9 0.568893
\(142\) 0 0
\(143\) −1.06578e10 −2.13135
\(144\) 0 0
\(145\) −1.04862e10 −1.96997
\(146\) 0 0
\(147\) −3.19546e9 −0.564424
\(148\) 0 0
\(149\) 4.64409e9 0.771903 0.385952 0.922519i \(-0.373873\pi\)
0.385952 + 0.922519i \(0.373873\pi\)
\(150\) 0 0
\(151\) 1.68753e9 0.264152 0.132076 0.991240i \(-0.457836\pi\)
0.132076 + 0.991240i \(0.457836\pi\)
\(152\) 0 0
\(153\) −1.40727e8 −0.0207619
\(154\) 0 0
\(155\) 2.16201e10 3.00861
\(156\) 0 0
\(157\) 5.63799e9 0.740587 0.370294 0.928915i \(-0.379257\pi\)
0.370294 + 0.928915i \(0.379257\pi\)
\(158\) 0 0
\(159\) 4.55057e9 0.564649
\(160\) 0 0
\(161\) −6.71725e8 −0.0787907
\(162\) 0 0
\(163\) −6.19678e9 −0.687577 −0.343789 0.939047i \(-0.611710\pi\)
−0.343789 + 0.939047i \(0.611710\pi\)
\(164\) 0 0
\(165\) 1.85799e10 1.95149
\(166\) 0 0
\(167\) 9.41917e9 0.937106 0.468553 0.883435i \(-0.344776\pi\)
0.468553 + 0.883435i \(0.344776\pi\)
\(168\) 0 0
\(169\) 4.98570e9 0.470149
\(170\) 0 0
\(171\) 1.59454e9 0.142611
\(172\) 0 0
\(173\) −1.18003e8 −0.0100158 −0.00500790 0.999987i \(-0.501594\pi\)
−0.00500790 + 0.999987i \(0.501594\pi\)
\(174\) 0 0
\(175\) −5.00778e9 −0.403621
\(176\) 0 0
\(177\) 8.85783e8 0.0678340
\(178\) 0 0
\(179\) 9.46326e9 0.688973 0.344486 0.938791i \(-0.388053\pi\)
0.344486 + 0.938791i \(0.388053\pi\)
\(180\) 0 0
\(181\) 1.75573e10 1.21592 0.607960 0.793967i \(-0.291988\pi\)
0.607960 + 0.793967i \(0.291988\pi\)
\(182\) 0 0
\(183\) 3.05848e9 0.201593
\(184\) 0 0
\(185\) −4.48905e10 −2.81762
\(186\) 0 0
\(187\) 1.83084e9 0.109487
\(188\) 0 0
\(189\) −5.05143e8 −0.0287963
\(190\) 0 0
\(191\) −1.11901e10 −0.608394 −0.304197 0.952609i \(-0.598388\pi\)
−0.304197 + 0.952609i \(0.598388\pi\)
\(192\) 0 0
\(193\) −2.78678e9 −0.144575 −0.0722877 0.997384i \(-0.523030\pi\)
−0.0722877 + 0.997384i \(0.523030\pi\)
\(194\) 0 0
\(195\) −2.71787e10 −1.34608
\(196\) 0 0
\(197\) −3.22790e10 −1.52694 −0.763470 0.645843i \(-0.776506\pi\)
−0.763470 + 0.645843i \(0.776506\pi\)
\(198\) 0 0
\(199\) 2.55007e10 1.15269 0.576345 0.817206i \(-0.304478\pi\)
0.576345 + 0.817206i \(0.304478\pi\)
\(200\) 0 0
\(201\) −7.40164e9 −0.319850
\(202\) 0 0
\(203\) −3.70902e9 −0.153294
\(204\) 0 0
\(205\) 8.89937e10 3.51939
\(206\) 0 0
\(207\) 4.63662e9 0.175523
\(208\) 0 0
\(209\) −2.07447e10 −0.752053
\(210\) 0 0
\(211\) −8.47230e9 −0.294259 −0.147130 0.989117i \(-0.547003\pi\)
−0.147130 + 0.989117i \(0.547003\pi\)
\(212\) 0 0
\(213\) −1.97865e10 −0.658659
\(214\) 0 0
\(215\) −8.88518e10 −2.83591
\(216\) 0 0
\(217\) 7.64717e9 0.234116
\(218\) 0 0
\(219\) 1.06088e10 0.311651
\(220\) 0 0
\(221\) −2.67814e9 −0.0755211
\(222\) 0 0
\(223\) −4.56578e10 −1.23636 −0.618178 0.786038i \(-0.712129\pi\)
−0.618178 + 0.786038i \(0.712129\pi\)
\(224\) 0 0
\(225\) 3.45665e10 0.899155
\(226\) 0 0
\(227\) 3.68244e10 0.920490 0.460245 0.887792i \(-0.347762\pi\)
0.460245 + 0.887792i \(0.347762\pi\)
\(228\) 0 0
\(229\) −2.88405e10 −0.693015 −0.346507 0.938047i \(-0.612632\pi\)
−0.346507 + 0.938047i \(0.612632\pi\)
\(230\) 0 0
\(231\) 6.57183e9 0.151856
\(232\) 0 0
\(233\) −3.88931e10 −0.864513 −0.432256 0.901751i \(-0.642282\pi\)
−0.432256 + 0.901751i \(0.642282\pi\)
\(234\) 0 0
\(235\) −8.85826e10 −1.89471
\(236\) 0 0
\(237\) 3.45120e10 0.710564
\(238\) 0 0
\(239\) −4.42661e10 −0.877568 −0.438784 0.898593i \(-0.644591\pi\)
−0.438784 + 0.898593i \(0.644591\pi\)
\(240\) 0 0
\(241\) 9.41867e10 1.79851 0.899255 0.437426i \(-0.144110\pi\)
0.899255 + 0.437426i \(0.144110\pi\)
\(242\) 0 0
\(243\) 3.48678e9 0.0641500
\(244\) 0 0
\(245\) 1.06015e11 1.87983
\(246\) 0 0
\(247\) 3.03452e10 0.518746
\(248\) 0 0
\(249\) 1.88608e10 0.310931
\(250\) 0 0
\(251\) −2.84495e10 −0.452421 −0.226211 0.974078i \(-0.572634\pi\)
−0.226211 + 0.974078i \(0.572634\pi\)
\(252\) 0 0
\(253\) −6.03217e10 −0.925617
\(254\) 0 0
\(255\) 4.66885e9 0.0691479
\(256\) 0 0
\(257\) −1.16667e10 −0.166820 −0.0834100 0.996515i \(-0.526581\pi\)
−0.0834100 + 0.996515i \(0.526581\pi\)
\(258\) 0 0
\(259\) −1.58781e10 −0.219254
\(260\) 0 0
\(261\) 2.56017e10 0.341497
\(262\) 0 0
\(263\) −1.37197e11 −1.76825 −0.884127 0.467247i \(-0.845246\pi\)
−0.884127 + 0.467247i \(0.845246\pi\)
\(264\) 0 0
\(265\) −1.50972e11 −1.88058
\(266\) 0 0
\(267\) −2.34672e10 −0.282593
\(268\) 0 0
\(269\) 5.76644e10 0.671463 0.335732 0.941958i \(-0.391016\pi\)
0.335732 + 0.941958i \(0.391016\pi\)
\(270\) 0 0
\(271\) −1.16504e11 −1.31213 −0.656067 0.754703i \(-0.727781\pi\)
−0.656067 + 0.754703i \(0.727781\pi\)
\(272\) 0 0
\(273\) −9.61325e9 −0.104746
\(274\) 0 0
\(275\) −4.49705e11 −4.74166
\(276\) 0 0
\(277\) −1.33146e10 −0.135884 −0.0679419 0.997689i \(-0.521643\pi\)
−0.0679419 + 0.997689i \(0.521643\pi\)
\(278\) 0 0
\(279\) −5.27851e10 −0.521546
\(280\) 0 0
\(281\) −3.34219e10 −0.319781 −0.159891 0.987135i \(-0.551114\pi\)
−0.159891 + 0.987135i \(0.551114\pi\)
\(282\) 0 0
\(283\) −5.11954e10 −0.474452 −0.237226 0.971455i \(-0.576238\pi\)
−0.237226 + 0.971455i \(0.576238\pi\)
\(284\) 0 0
\(285\) −5.29014e10 −0.474969
\(286\) 0 0
\(287\) 3.14776e10 0.273863
\(288\) 0 0
\(289\) −1.18128e11 −0.996121
\(290\) 0 0
\(291\) −2.22407e10 −0.181815
\(292\) 0 0
\(293\) −1.69283e11 −1.34187 −0.670934 0.741517i \(-0.734107\pi\)
−0.670934 + 0.741517i \(0.734107\pi\)
\(294\) 0 0
\(295\) −2.93873e10 −0.225923
\(296\) 0 0
\(297\) −4.53625e10 −0.338293
\(298\) 0 0
\(299\) 8.82384e10 0.638465
\(300\) 0 0
\(301\) −3.14274e10 −0.220678
\(302\) 0 0
\(303\) 3.65971e10 0.249434
\(304\) 0 0
\(305\) −1.01470e11 −0.671411
\(306\) 0 0
\(307\) 1.67510e11 1.07626 0.538130 0.842862i \(-0.319131\pi\)
0.538130 + 0.842862i \(0.319131\pi\)
\(308\) 0 0
\(309\) −7.21134e10 −0.449990
\(310\) 0 0
\(311\) 2.42940e11 1.47257 0.736287 0.676669i \(-0.236577\pi\)
0.736287 + 0.676669i \(0.236577\pi\)
\(312\) 0 0
\(313\) −2.59566e11 −1.52862 −0.764308 0.644851i \(-0.776919\pi\)
−0.764308 + 0.644851i \(0.776919\pi\)
\(314\) 0 0
\(315\) 1.67589e10 0.0959067
\(316\) 0 0
\(317\) 2.98973e11 1.66290 0.831448 0.555603i \(-0.187512\pi\)
0.831448 + 0.555603i \(0.187512\pi\)
\(318\) 0 0
\(319\) −3.33074e11 −1.80087
\(320\) 0 0
\(321\) −2.07374e11 −1.09014
\(322\) 0 0
\(323\) −5.21282e9 −0.0266478
\(324\) 0 0
\(325\) 6.57827e11 3.27067
\(326\) 0 0
\(327\) −1.08193e11 −0.523279
\(328\) 0 0
\(329\) −3.13322e10 −0.147438
\(330\) 0 0
\(331\) −3.64913e11 −1.67095 −0.835475 0.549528i \(-0.814807\pi\)
−0.835475 + 0.549528i \(0.814807\pi\)
\(332\) 0 0
\(333\) 1.09599e11 0.488437
\(334\) 0 0
\(335\) 2.45561e11 1.06527
\(336\) 0 0
\(337\) −1.45199e11 −0.613237 −0.306618 0.951833i \(-0.599197\pi\)
−0.306618 + 0.951833i \(0.599197\pi\)
\(338\) 0 0
\(339\) −2.32741e11 −0.957139
\(340\) 0 0
\(341\) 6.86725e11 2.75035
\(342\) 0 0
\(343\) 7.58547e10 0.295910
\(344\) 0 0
\(345\) −1.53827e11 −0.584585
\(346\) 0 0
\(347\) −1.28543e10 −0.0475955 −0.0237978 0.999717i \(-0.507576\pi\)
−0.0237978 + 0.999717i \(0.507576\pi\)
\(348\) 0 0
\(349\) −1.59013e11 −0.573744 −0.286872 0.957969i \(-0.592615\pi\)
−0.286872 + 0.957969i \(0.592615\pi\)
\(350\) 0 0
\(351\) 6.63561e10 0.233345
\(352\) 0 0
\(353\) 1.03715e11 0.355512 0.177756 0.984075i \(-0.443116\pi\)
0.177756 + 0.984075i \(0.443116\pi\)
\(354\) 0 0
\(355\) 6.56449e11 2.19368
\(356\) 0 0
\(357\) 1.65140e9 0.00538078
\(358\) 0 0
\(359\) 4.16136e10 0.132224 0.0661120 0.997812i \(-0.478941\pi\)
0.0661120 + 0.997812i \(0.478941\pi\)
\(360\) 0 0
\(361\) −2.63623e11 −0.816959
\(362\) 0 0
\(363\) 3.99165e11 1.20663
\(364\) 0 0
\(365\) −3.51965e11 −1.03796
\(366\) 0 0
\(367\) −3.78771e11 −1.08988 −0.544941 0.838474i \(-0.683448\pi\)
−0.544941 + 0.838474i \(0.683448\pi\)
\(368\) 0 0
\(369\) −2.17276e11 −0.610089
\(370\) 0 0
\(371\) −5.33999e10 −0.146338
\(372\) 0 0
\(373\) 7.90824e10 0.211539 0.105769 0.994391i \(-0.466269\pi\)
0.105769 + 0.994391i \(0.466269\pi\)
\(374\) 0 0
\(375\) −7.21661e11 −1.88448
\(376\) 0 0
\(377\) 4.87220e11 1.24219
\(378\) 0 0
\(379\) −5.43207e11 −1.35235 −0.676175 0.736741i \(-0.736364\pi\)
−0.676175 + 0.736741i \(0.736364\pi\)
\(380\) 0 0
\(381\) −4.45006e10 −0.108194
\(382\) 0 0
\(383\) 6.88749e11 1.63556 0.817780 0.575531i \(-0.195204\pi\)
0.817780 + 0.575531i \(0.195204\pi\)
\(384\) 0 0
\(385\) −2.18031e11 −0.505761
\(386\) 0 0
\(387\) 2.16930e11 0.491609
\(388\) 0 0
\(389\) 4.70395e11 1.04157 0.520786 0.853687i \(-0.325639\pi\)
0.520786 + 0.853687i \(0.325639\pi\)
\(390\) 0 0
\(391\) −1.51579e10 −0.0327977
\(392\) 0 0
\(393\) 2.83615e10 0.0599740
\(394\) 0 0
\(395\) −1.14499e12 −2.36655
\(396\) 0 0
\(397\) −5.27728e11 −1.06623 −0.533117 0.846042i \(-0.678979\pi\)
−0.533117 + 0.846042i \(0.678979\pi\)
\(398\) 0 0
\(399\) −1.87115e10 −0.0369599
\(400\) 0 0
\(401\) −1.78849e11 −0.345411 −0.172705 0.984974i \(-0.555251\pi\)
−0.172705 + 0.984974i \(0.555251\pi\)
\(402\) 0 0
\(403\) −1.00454e12 −1.89712
\(404\) 0 0
\(405\) −1.15680e11 −0.213653
\(406\) 0 0
\(407\) −1.42587e12 −2.57576
\(408\) 0 0
\(409\) 4.40686e11 0.778707 0.389353 0.921088i \(-0.372698\pi\)
0.389353 + 0.921088i \(0.372698\pi\)
\(410\) 0 0
\(411\) 1.99032e11 0.344061
\(412\) 0 0
\(413\) −1.03945e10 −0.0175803
\(414\) 0 0
\(415\) −6.25738e11 −1.03556
\(416\) 0 0
\(417\) −2.85683e11 −0.462670
\(418\) 0 0
\(419\) 3.08642e10 0.0489207 0.0244603 0.999701i \(-0.492213\pi\)
0.0244603 + 0.999701i \(0.492213\pi\)
\(420\) 0 0
\(421\) 2.60022e11 0.403405 0.201703 0.979447i \(-0.435353\pi\)
0.201703 + 0.979447i \(0.435353\pi\)
\(422\) 0 0
\(423\) 2.16272e11 0.328450
\(424\) 0 0
\(425\) −1.13004e11 −0.168013
\(426\) 0 0
\(427\) −3.58906e10 −0.0522462
\(428\) 0 0
\(429\) −8.63282e11 −1.23054
\(430\) 0 0
\(431\) −5.19190e11 −0.724734 −0.362367 0.932035i \(-0.618031\pi\)
−0.362367 + 0.932035i \(0.618031\pi\)
\(432\) 0 0
\(433\) 1.00829e12 1.37844 0.689222 0.724550i \(-0.257953\pi\)
0.689222 + 0.724550i \(0.257953\pi\)
\(434\) 0 0
\(435\) −8.49379e11 −1.13736
\(436\) 0 0
\(437\) 1.71750e11 0.225284
\(438\) 0 0
\(439\) 2.62358e10 0.0337135 0.0168568 0.999858i \(-0.494634\pi\)
0.0168568 + 0.999858i \(0.494634\pi\)
\(440\) 0 0
\(441\) −2.58832e11 −0.325870
\(442\) 0 0
\(443\) 6.82095e10 0.0841450 0.0420725 0.999115i \(-0.486604\pi\)
0.0420725 + 0.999115i \(0.486604\pi\)
\(444\) 0 0
\(445\) 7.78563e11 0.941182
\(446\) 0 0
\(447\) 3.76172e11 0.445658
\(448\) 0 0
\(449\) 1.23102e12 1.42941 0.714707 0.699424i \(-0.246560\pi\)
0.714707 + 0.699424i \(0.246560\pi\)
\(450\) 0 0
\(451\) 2.82673e12 3.21728
\(452\) 0 0
\(453\) 1.36690e11 0.152508
\(454\) 0 0
\(455\) 3.18935e11 0.348860
\(456\) 0 0
\(457\) 2.33313e11 0.250216 0.125108 0.992143i \(-0.460072\pi\)
0.125108 + 0.992143i \(0.460072\pi\)
\(458\) 0 0
\(459\) −1.13989e10 −0.0119869
\(460\) 0 0
\(461\) −7.98394e11 −0.823310 −0.411655 0.911340i \(-0.635049\pi\)
−0.411655 + 0.911340i \(0.635049\pi\)
\(462\) 0 0
\(463\) −1.46529e11 −0.148186 −0.0740931 0.997251i \(-0.523606\pi\)
−0.0740931 + 0.997251i \(0.523606\pi\)
\(464\) 0 0
\(465\) 1.75123e12 1.73702
\(466\) 0 0
\(467\) −1.53277e12 −1.49125 −0.745627 0.666364i \(-0.767850\pi\)
−0.745627 + 0.666364i \(0.767850\pi\)
\(468\) 0 0
\(469\) 8.68566e10 0.0828943
\(470\) 0 0
\(471\) 4.56677e11 0.427578
\(472\) 0 0
\(473\) −2.82222e12 −2.59248
\(474\) 0 0
\(475\) 1.28041e12 1.15406
\(476\) 0 0
\(477\) 3.68596e11 0.326000
\(478\) 0 0
\(479\) −7.94475e11 −0.689558 −0.344779 0.938684i \(-0.612046\pi\)
−0.344779 + 0.938684i \(0.612046\pi\)
\(480\) 0 0
\(481\) 2.08576e12 1.77669
\(482\) 0 0
\(483\) −5.44097e10 −0.0454898
\(484\) 0 0
\(485\) 7.37870e11 0.605539
\(486\) 0 0
\(487\) −1.34532e12 −1.08379 −0.541893 0.840447i \(-0.682292\pi\)
−0.541893 + 0.840447i \(0.682292\pi\)
\(488\) 0 0
\(489\) −5.01939e11 −0.396973
\(490\) 0 0
\(491\) 1.75188e12 1.36031 0.680154 0.733069i \(-0.261913\pi\)
0.680154 + 0.733069i \(0.261913\pi\)
\(492\) 0 0
\(493\) −8.36964e10 −0.0638110
\(494\) 0 0
\(495\) 1.50497e12 1.12669
\(496\) 0 0
\(497\) 2.32190e11 0.170702
\(498\) 0 0
\(499\) −7.66101e11 −0.553138 −0.276569 0.960994i \(-0.589197\pi\)
−0.276569 + 0.960994i \(0.589197\pi\)
\(500\) 0 0
\(501\) 7.62953e11 0.541038
\(502\) 0 0
\(503\) 2.02872e12 1.41308 0.706540 0.707673i \(-0.250255\pi\)
0.706540 + 0.707673i \(0.250255\pi\)
\(504\) 0 0
\(505\) −1.21417e12 −0.830745
\(506\) 0 0
\(507\) 4.03842e11 0.271441
\(508\) 0 0
\(509\) −3.08049e11 −0.203418 −0.101709 0.994814i \(-0.532431\pi\)
−0.101709 + 0.994814i \(0.532431\pi\)
\(510\) 0 0
\(511\) −1.24492e11 −0.0807695
\(512\) 0 0
\(513\) 1.29158e11 0.0823364
\(514\) 0 0
\(515\) 2.39248e12 1.49870
\(516\) 0 0
\(517\) −2.81367e12 −1.73207
\(518\) 0 0
\(519\) −9.55824e9 −0.00578262
\(520\) 0 0
\(521\) −2.45259e12 −1.45833 −0.729165 0.684338i \(-0.760091\pi\)
−0.729165 + 0.684338i \(0.760091\pi\)
\(522\) 0 0
\(523\) −2.41650e12 −1.41231 −0.706154 0.708059i \(-0.749571\pi\)
−0.706154 + 0.708059i \(0.749571\pi\)
\(524\) 0 0
\(525\) −4.05630e11 −0.233031
\(526\) 0 0
\(527\) 1.72564e11 0.0974543
\(528\) 0 0
\(529\) −1.30174e12 −0.722724
\(530\) 0 0
\(531\) 7.17484e10 0.0391640
\(532\) 0 0
\(533\) −4.13493e12 −2.21920
\(534\) 0 0
\(535\) 6.87996e12 3.63073
\(536\) 0 0
\(537\) 7.66524e11 0.397779
\(538\) 0 0
\(539\) 3.36737e12 1.71847
\(540\) 0 0
\(541\) 1.14349e12 0.573911 0.286955 0.957944i \(-0.407357\pi\)
0.286955 + 0.957944i \(0.407357\pi\)
\(542\) 0 0
\(543\) 1.42215e12 0.702012
\(544\) 0 0
\(545\) 3.58947e12 1.74279
\(546\) 0 0
\(547\) 1.20917e12 0.577492 0.288746 0.957406i \(-0.406762\pi\)
0.288746 + 0.957406i \(0.406762\pi\)
\(548\) 0 0
\(549\) 2.47737e11 0.116390
\(550\) 0 0
\(551\) 9.48340e11 0.438310
\(552\) 0 0
\(553\) −4.04991e11 −0.184154
\(554\) 0 0
\(555\) −3.63613e12 −1.62675
\(556\) 0 0
\(557\) −2.18500e12 −0.961838 −0.480919 0.876765i \(-0.659697\pi\)
−0.480919 + 0.876765i \(0.659697\pi\)
\(558\) 0 0
\(559\) 4.12833e12 1.78822
\(560\) 0 0
\(561\) 1.48298e11 0.0632123
\(562\) 0 0
\(563\) −1.68753e11 −0.0707886 −0.0353943 0.999373i \(-0.511269\pi\)
−0.0353943 + 0.999373i \(0.511269\pi\)
\(564\) 0 0
\(565\) 7.72157e12 3.18777
\(566\) 0 0
\(567\) −4.09166e10 −0.0166255
\(568\) 0 0
\(569\) −3.00268e12 −1.20089 −0.600446 0.799665i \(-0.705010\pi\)
−0.600446 + 0.799665i \(0.705010\pi\)
\(570\) 0 0
\(571\) 4.51453e12 1.77726 0.888628 0.458628i \(-0.151659\pi\)
0.888628 + 0.458628i \(0.151659\pi\)
\(572\) 0 0
\(573\) −9.06401e11 −0.351257
\(574\) 0 0
\(575\) 3.72321e12 1.42041
\(576\) 0 0
\(577\) 3.10024e12 1.16441 0.582203 0.813044i \(-0.302191\pi\)
0.582203 + 0.813044i \(0.302191\pi\)
\(578\) 0 0
\(579\) −2.25729e11 −0.0834706
\(580\) 0 0
\(581\) −2.21327e11 −0.0805828
\(582\) 0 0
\(583\) −4.79537e12 −1.71915
\(584\) 0 0
\(585\) −2.20147e12 −0.777162
\(586\) 0 0
\(587\) 2.50051e11 0.0869275 0.0434637 0.999055i \(-0.486161\pi\)
0.0434637 + 0.999055i \(0.486161\pi\)
\(588\) 0 0
\(589\) −1.95527e12 −0.669402
\(590\) 0 0
\(591\) −2.61460e12 −0.881580
\(592\) 0 0
\(593\) −1.58185e12 −0.525314 −0.262657 0.964889i \(-0.584599\pi\)
−0.262657 + 0.964889i \(0.584599\pi\)
\(594\) 0 0
\(595\) −5.47878e10 −0.0179208
\(596\) 0 0
\(597\) 2.06555e12 0.665506
\(598\) 0 0
\(599\) −4.55230e12 −1.44481 −0.722404 0.691471i \(-0.756963\pi\)
−0.722404 + 0.691471i \(0.756963\pi\)
\(600\) 0 0
\(601\) 5.54016e12 1.73216 0.866079 0.499908i \(-0.166633\pi\)
0.866079 + 0.499908i \(0.166633\pi\)
\(602\) 0 0
\(603\) −5.99533e11 −0.184665
\(604\) 0 0
\(605\) −1.32429e13 −4.01870
\(606\) 0 0
\(607\) −2.77998e12 −0.831175 −0.415587 0.909553i \(-0.636424\pi\)
−0.415587 + 0.909553i \(0.636424\pi\)
\(608\) 0 0
\(609\) −3.00430e11 −0.0885046
\(610\) 0 0
\(611\) 4.11583e12 1.19474
\(612\) 0 0
\(613\) 1.59888e11 0.0457346 0.0228673 0.999739i \(-0.492720\pi\)
0.0228673 + 0.999739i \(0.492720\pi\)
\(614\) 0 0
\(615\) 7.20849e12 2.03192
\(616\) 0 0
\(617\) 6.35346e12 1.76493 0.882464 0.470380i \(-0.155883\pi\)
0.882464 + 0.470380i \(0.155883\pi\)
\(618\) 0 0
\(619\) 1.67188e12 0.457716 0.228858 0.973460i \(-0.426501\pi\)
0.228858 + 0.973460i \(0.426501\pi\)
\(620\) 0 0
\(621\) 3.75566e11 0.101339
\(622\) 0 0
\(623\) 2.75382e11 0.0732386
\(624\) 0 0
\(625\) 1.36522e13 3.57885
\(626\) 0 0
\(627\) −1.68032e12 −0.434198
\(628\) 0 0
\(629\) −3.58299e11 −0.0912678
\(630\) 0 0
\(631\) −3.69214e12 −0.927141 −0.463571 0.886060i \(-0.653432\pi\)
−0.463571 + 0.886060i \(0.653432\pi\)
\(632\) 0 0
\(633\) −6.86256e11 −0.169891
\(634\) 0 0
\(635\) 1.47638e12 0.360343
\(636\) 0 0
\(637\) −4.92577e12 −1.18535
\(638\) 0 0
\(639\) −1.60271e12 −0.380277
\(640\) 0 0
\(641\) −2.43986e12 −0.570826 −0.285413 0.958405i \(-0.592131\pi\)
−0.285413 + 0.958405i \(0.592131\pi\)
\(642\) 0 0
\(643\) −1.91171e12 −0.441034 −0.220517 0.975383i \(-0.570774\pi\)
−0.220517 + 0.975383i \(0.570774\pi\)
\(644\) 0 0
\(645\) −7.19699e12 −1.63732
\(646\) 0 0
\(647\) 6.33401e12 1.42105 0.710525 0.703672i \(-0.248458\pi\)
0.710525 + 0.703672i \(0.248458\pi\)
\(648\) 0 0
\(649\) −9.33435e11 −0.206530
\(650\) 0 0
\(651\) 6.19421e11 0.135167
\(652\) 0 0
\(653\) −2.21499e12 −0.476718 −0.238359 0.971177i \(-0.576610\pi\)
−0.238359 + 0.971177i \(0.576610\pi\)
\(654\) 0 0
\(655\) −9.40939e11 −0.199745
\(656\) 0 0
\(657\) 8.59314e11 0.179932
\(658\) 0 0
\(659\) −3.59753e12 −0.743054 −0.371527 0.928422i \(-0.621166\pi\)
−0.371527 + 0.928422i \(0.621166\pi\)
\(660\) 0 0
\(661\) 2.41730e12 0.492521 0.246260 0.969204i \(-0.420798\pi\)
0.246260 + 0.969204i \(0.420798\pi\)
\(662\) 0 0
\(663\) −2.16929e11 −0.0436021
\(664\) 0 0
\(665\) 6.20785e11 0.123096
\(666\) 0 0
\(667\) 2.75760e12 0.539467
\(668\) 0 0
\(669\) −3.69828e12 −0.713810
\(670\) 0 0
\(671\) −3.22302e12 −0.613778
\(672\) 0 0
\(673\) −1.32596e12 −0.249152 −0.124576 0.992210i \(-0.539757\pi\)
−0.124576 + 0.992210i \(0.539757\pi\)
\(674\) 0 0
\(675\) 2.79989e12 0.519127
\(676\) 0 0
\(677\) 2.49781e12 0.456995 0.228497 0.973545i \(-0.426619\pi\)
0.228497 + 0.973545i \(0.426619\pi\)
\(678\) 0 0
\(679\) 2.60989e11 0.0471203
\(680\) 0 0
\(681\) 2.98277e12 0.531445
\(682\) 0 0
\(683\) −1.02091e13 −1.79512 −0.897558 0.440895i \(-0.854661\pi\)
−0.897558 + 0.440895i \(0.854661\pi\)
\(684\) 0 0
\(685\) −6.60321e12 −1.14590
\(686\) 0 0
\(687\) −2.33608e12 −0.400112
\(688\) 0 0
\(689\) 7.01466e12 1.18582
\(690\) 0 0
\(691\) −5.33760e12 −0.890625 −0.445313 0.895375i \(-0.646907\pi\)
−0.445313 + 0.895375i \(0.646907\pi\)
\(692\) 0 0
\(693\) 5.32318e11 0.0876742
\(694\) 0 0
\(695\) 9.47798e12 1.54093
\(696\) 0 0
\(697\) 7.10313e11 0.113999
\(698\) 0 0
\(699\) −3.15034e12 −0.499127
\(700\) 0 0
\(701\) 6.96175e12 1.08890 0.544449 0.838794i \(-0.316739\pi\)
0.544449 + 0.838794i \(0.316739\pi\)
\(702\) 0 0
\(703\) 4.05978e12 0.626908
\(704\) 0 0
\(705\) −7.17519e12 −1.09391
\(706\) 0 0
\(707\) −4.29459e11 −0.0646449
\(708\) 0 0
\(709\) 1.14423e13 1.70061 0.850307 0.526287i \(-0.176416\pi\)
0.850307 + 0.526287i \(0.176416\pi\)
\(710\) 0 0
\(711\) 2.79547e12 0.410244
\(712\) 0 0
\(713\) −5.68556e12 −0.823892
\(714\) 0 0
\(715\) 2.86408e13 4.09834
\(716\) 0 0
\(717\) −3.58555e12 −0.506664
\(718\) 0 0
\(719\) −5.43048e12 −0.757806 −0.378903 0.925436i \(-0.623699\pi\)
−0.378903 + 0.925436i \(0.623699\pi\)
\(720\) 0 0
\(721\) 8.46234e11 0.116622
\(722\) 0 0
\(723\) 7.62912e12 1.03837
\(724\) 0 0
\(725\) 2.05582e13 2.76353
\(726\) 0 0
\(727\) −6.18258e12 −0.820852 −0.410426 0.911894i \(-0.634620\pi\)
−0.410426 + 0.911894i \(0.634620\pi\)
\(728\) 0 0
\(729\) 2.82430e11 0.0370370
\(730\) 0 0
\(731\) −7.09180e11 −0.0918604
\(732\) 0 0
\(733\) −1.14558e13 −1.46574 −0.732870 0.680368i \(-0.761820\pi\)
−0.732870 + 0.680368i \(0.761820\pi\)
\(734\) 0 0
\(735\) 8.58718e12 1.08532
\(736\) 0 0
\(737\) 7.79983e12 0.973826
\(738\) 0 0
\(739\) 1.13658e13 1.40184 0.700920 0.713240i \(-0.252773\pi\)
0.700920 + 0.713240i \(0.252773\pi\)
\(740\) 0 0
\(741\) 2.45796e12 0.299498
\(742\) 0 0
\(743\) 1.42813e13 1.71917 0.859584 0.510995i \(-0.170723\pi\)
0.859584 + 0.510995i \(0.170723\pi\)
\(744\) 0 0
\(745\) −1.24801e13 −1.48428
\(746\) 0 0
\(747\) 1.52773e12 0.179516
\(748\) 0 0
\(749\) 2.43348e12 0.282527
\(750\) 0 0
\(751\) 2.50025e12 0.286816 0.143408 0.989664i \(-0.454194\pi\)
0.143408 + 0.989664i \(0.454194\pi\)
\(752\) 0 0
\(753\) −2.30441e12 −0.261205
\(754\) 0 0
\(755\) −4.53490e12 −0.507933
\(756\) 0 0
\(757\) 1.03072e13 1.14080 0.570402 0.821366i \(-0.306787\pi\)
0.570402 + 0.821366i \(0.306787\pi\)
\(758\) 0 0
\(759\) −4.88606e12 −0.534405
\(760\) 0 0
\(761\) −8.02116e12 −0.866974 −0.433487 0.901160i \(-0.642717\pi\)
−0.433487 + 0.901160i \(0.642717\pi\)
\(762\) 0 0
\(763\) 1.26962e12 0.135616
\(764\) 0 0
\(765\) 3.78177e11 0.0399226
\(766\) 0 0
\(767\) 1.36543e12 0.142459
\(768\) 0 0
\(769\) 1.09614e13 1.13031 0.565154 0.824985i \(-0.308817\pi\)
0.565154 + 0.824985i \(0.308817\pi\)
\(770\) 0 0
\(771\) −9.45001e11 −0.0963136
\(772\) 0 0
\(773\) −7.65013e12 −0.770657 −0.385329 0.922779i \(-0.625912\pi\)
−0.385329 + 0.922779i \(0.625912\pi\)
\(774\) 0 0
\(775\) −4.23865e13 −4.22055
\(776\) 0 0
\(777\) −1.28612e12 −0.126587
\(778\) 0 0
\(779\) −8.04835e12 −0.783048
\(780\) 0 0
\(781\) 2.08509e13 2.00538
\(782\) 0 0
\(783\) 2.07374e12 0.197163
\(784\) 0 0
\(785\) −1.51510e13 −1.42406
\(786\) 0 0
\(787\) 2.07846e13 1.93133 0.965663 0.259797i \(-0.0836556\pi\)
0.965663 + 0.259797i \(0.0836556\pi\)
\(788\) 0 0
\(789\) −1.11130e13 −1.02090
\(790\) 0 0
\(791\) 2.73116e12 0.248058
\(792\) 0 0
\(793\) 4.71462e12 0.423367
\(794\) 0 0
\(795\) −1.22288e13 −1.08575
\(796\) 0 0
\(797\) 5.87904e12 0.516112 0.258056 0.966130i \(-0.416918\pi\)
0.258056 + 0.966130i \(0.416918\pi\)
\(798\) 0 0
\(799\) −7.07032e11 −0.0613731
\(800\) 0 0
\(801\) −1.90084e12 −0.163155
\(802\) 0 0
\(803\) −1.11795e13 −0.948863
\(804\) 0 0
\(805\) 1.80513e12 0.151505
\(806\) 0 0
\(807\) 4.67081e12 0.387669
\(808\) 0 0
\(809\) 1.17127e13 0.961363 0.480682 0.876895i \(-0.340389\pi\)
0.480682 + 0.876895i \(0.340389\pi\)
\(810\) 0 0
\(811\) −2.27415e12 −0.184597 −0.0922987 0.995731i \(-0.529421\pi\)
−0.0922987 + 0.995731i \(0.529421\pi\)
\(812\) 0 0
\(813\) −9.43681e12 −0.757561
\(814\) 0 0
\(815\) 1.66526e13 1.32213
\(816\) 0 0
\(817\) 8.03552e12 0.630979
\(818\) 0 0
\(819\) −7.78673e11 −0.0604753
\(820\) 0 0
\(821\) 8.39279e12 0.644707 0.322353 0.946619i \(-0.395526\pi\)
0.322353 + 0.946619i \(0.395526\pi\)
\(822\) 0 0
\(823\) 3.70938e12 0.281839 0.140920 0.990021i \(-0.454994\pi\)
0.140920 + 0.990021i \(0.454994\pi\)
\(824\) 0 0
\(825\) −3.64261e13 −2.73760
\(826\) 0 0
\(827\) −2.53080e13 −1.88141 −0.940706 0.339224i \(-0.889836\pi\)
−0.940706 + 0.339224i \(0.889836\pi\)
\(828\) 0 0
\(829\) 1.59061e13 1.16968 0.584842 0.811147i \(-0.301156\pi\)
0.584842 + 0.811147i \(0.301156\pi\)
\(830\) 0 0
\(831\) −1.07848e12 −0.0784525
\(832\) 0 0
\(833\) 8.46167e11 0.0608911
\(834\) 0 0
\(835\) −2.53122e13 −1.80194
\(836\) 0 0
\(837\) −4.27559e12 −0.301115
\(838\) 0 0
\(839\) 2.57050e12 0.179097 0.0895487 0.995982i \(-0.471458\pi\)
0.0895487 + 0.995982i \(0.471458\pi\)
\(840\) 0 0
\(841\) 7.19297e11 0.0495822
\(842\) 0 0
\(843\) −2.70718e12 −0.184626
\(844\) 0 0
\(845\) −1.33981e13 −0.904041
\(846\) 0 0
\(847\) −4.68411e12 −0.312717
\(848\) 0 0
\(849\) −4.14683e12 −0.273925
\(850\) 0 0
\(851\) 1.18051e13 0.771590
\(852\) 0 0
\(853\) −6.41468e12 −0.414863 −0.207431 0.978250i \(-0.566510\pi\)
−0.207431 + 0.978250i \(0.566510\pi\)
\(854\) 0 0
\(855\) −4.28501e12 −0.274223
\(856\) 0 0
\(857\) −1.31074e13 −0.830046 −0.415023 0.909811i \(-0.636226\pi\)
−0.415023 + 0.909811i \(0.636226\pi\)
\(858\) 0 0
\(859\) −1.06234e13 −0.665723 −0.332861 0.942976i \(-0.608014\pi\)
−0.332861 + 0.942976i \(0.608014\pi\)
\(860\) 0 0
\(861\) 2.54969e12 0.158115
\(862\) 0 0
\(863\) 4.71438e12 0.289318 0.144659 0.989482i \(-0.453791\pi\)
0.144659 + 0.989482i \(0.453791\pi\)
\(864\) 0 0
\(865\) 3.17110e11 0.0192592
\(866\) 0 0
\(867\) −9.56835e12 −0.575110
\(868\) 0 0
\(869\) −3.63687e13 −2.16341
\(870\) 0 0
\(871\) −1.14096e13 −0.671719
\(872\) 0 0
\(873\) −1.80149e12 −0.104971
\(874\) 0 0
\(875\) 8.46852e12 0.488395
\(876\) 0 0
\(877\) −4.59207e12 −0.262126 −0.131063 0.991374i \(-0.541839\pi\)
−0.131063 + 0.991374i \(0.541839\pi\)
\(878\) 0 0
\(879\) −1.37119e13 −0.774728
\(880\) 0 0
\(881\) 1.82236e13 1.01916 0.509580 0.860423i \(-0.329801\pi\)
0.509580 + 0.860423i \(0.329801\pi\)
\(882\) 0 0
\(883\) 3.47414e12 0.192320 0.0961598 0.995366i \(-0.469344\pi\)
0.0961598 + 0.995366i \(0.469344\pi\)
\(884\) 0 0
\(885\) −2.38037e12 −0.130437
\(886\) 0 0
\(887\) −7.73676e12 −0.419665 −0.209833 0.977737i \(-0.567292\pi\)
−0.209833 + 0.977737i \(0.567292\pi\)
\(888\) 0 0
\(889\) 5.22204e11 0.0280403
\(890\) 0 0
\(891\) −3.67436e12 −0.195314
\(892\) 0 0
\(893\) 8.01117e12 0.421565
\(894\) 0 0
\(895\) −2.54307e13 −1.32481
\(896\) 0 0
\(897\) 7.14731e12 0.368618
\(898\) 0 0
\(899\) −3.13936e13 −1.60296
\(900\) 0 0
\(901\) −1.20500e12 −0.0609153
\(902\) 0 0
\(903\) −2.54562e12 −0.127409
\(904\) 0 0
\(905\) −4.71820e13 −2.33807
\(906\) 0 0
\(907\) −2.47553e13 −1.21461 −0.607303 0.794470i \(-0.707749\pi\)
−0.607303 + 0.794470i \(0.707749\pi\)
\(908\) 0 0
\(909\) 2.96437e12 0.144011
\(910\) 0 0
\(911\) −3.49811e13 −1.68268 −0.841339 0.540508i \(-0.818232\pi\)
−0.841339 + 0.540508i \(0.818232\pi\)
\(912\) 0 0
\(913\) −1.98755e13 −0.946670
\(914\) 0 0
\(915\) −8.21907e12 −0.387639
\(916\) 0 0
\(917\) −3.32816e11 −0.0155432
\(918\) 0 0
\(919\) −1.70135e13 −0.786816 −0.393408 0.919364i \(-0.628704\pi\)
−0.393408 + 0.919364i \(0.628704\pi\)
\(920\) 0 0
\(921\) 1.35683e13 0.621379
\(922\) 0 0
\(923\) −3.05007e13 −1.38325
\(924\) 0 0
\(925\) 8.80083e13 3.95263
\(926\) 0 0
\(927\) −5.84118e12 −0.259802
\(928\) 0 0
\(929\) 7.63520e12 0.336318 0.168159 0.985760i \(-0.446218\pi\)
0.168159 + 0.985760i \(0.446218\pi\)
\(930\) 0 0
\(931\) −9.58767e12 −0.418254
\(932\) 0 0
\(933\) 1.96781e13 0.850191
\(934\) 0 0
\(935\) −4.92002e12 −0.210530
\(936\) 0 0
\(937\) −3.23096e12 −0.136932 −0.0684658 0.997653i \(-0.521810\pi\)
−0.0684658 + 0.997653i \(0.521810\pi\)
\(938\) 0 0
\(939\) −2.10249e13 −0.882547
\(940\) 0 0
\(941\) −3.68554e13 −1.53232 −0.766158 0.642652i \(-0.777834\pi\)
−0.766158 + 0.642652i \(0.777834\pi\)
\(942\) 0 0
\(943\) −2.34031e13 −0.963765
\(944\) 0 0
\(945\) 1.35747e12 0.0553718
\(946\) 0 0
\(947\) 2.16984e13 0.876703 0.438352 0.898804i \(-0.355563\pi\)
0.438352 + 0.898804i \(0.355563\pi\)
\(948\) 0 0
\(949\) 1.63534e13 0.654500
\(950\) 0 0
\(951\) 2.42168e13 0.960073
\(952\) 0 0
\(953\) −9.33996e12 −0.366798 −0.183399 0.983039i \(-0.558710\pi\)
−0.183399 + 0.983039i \(0.558710\pi\)
\(954\) 0 0
\(955\) 3.00713e13 1.16987
\(956\) 0 0
\(957\) −2.69790e13 −1.03973
\(958\) 0 0
\(959\) −2.33560e12 −0.0891690
\(960\) 0 0
\(961\) 3.82869e13 1.44809
\(962\) 0 0
\(963\) −1.67973e13 −0.629392
\(964\) 0 0
\(965\) 7.48892e12 0.278001
\(966\) 0 0
\(967\) 6.17891e12 0.227244 0.113622 0.993524i \(-0.463755\pi\)
0.113622 + 0.993524i \(0.463755\pi\)
\(968\) 0 0
\(969\) −4.22238e11 −0.0153851
\(970\) 0 0
\(971\) 4.66682e13 1.68475 0.842373 0.538894i \(-0.181158\pi\)
0.842373 + 0.538894i \(0.181158\pi\)
\(972\) 0 0
\(973\) 3.35242e12 0.119909
\(974\) 0 0
\(975\) 5.32840e13 1.88832
\(976\) 0 0
\(977\) −4.66780e13 −1.63903 −0.819515 0.573058i \(-0.805757\pi\)
−0.819515 + 0.573058i \(0.805757\pi\)
\(978\) 0 0
\(979\) 2.47297e13 0.860392
\(980\) 0 0
\(981\) −8.76361e12 −0.302115
\(982\) 0 0
\(983\) −3.93231e13 −1.34325 −0.671625 0.740891i \(-0.734404\pi\)
−0.671625 + 0.740891i \(0.734404\pi\)
\(984\) 0 0
\(985\) 8.67436e13 2.93612
\(986\) 0 0
\(987\) −2.53791e12 −0.0851233
\(988\) 0 0
\(989\) 2.33658e13 0.776600
\(990\) 0 0
\(991\) −1.82257e13 −0.600280 −0.300140 0.953895i \(-0.597033\pi\)
−0.300140 + 0.953895i \(0.597033\pi\)
\(992\) 0 0
\(993\) −2.95580e13 −0.964724
\(994\) 0 0
\(995\) −6.85281e13 −2.21648
\(996\) 0 0
\(997\) 3.16514e13 1.01453 0.507264 0.861791i \(-0.330657\pi\)
0.507264 + 0.861791i \(0.330657\pi\)
\(998\) 0 0
\(999\) 8.87754e12 0.281999
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.g.1.1 yes 4
4.3 odd 2 384.10.a.c.1.1 yes 4
8.3 odd 2 384.10.a.f.1.4 yes 4
8.5 even 2 384.10.a.b.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.a.b.1.4 4 8.5 even 2
384.10.a.c.1.1 yes 4 4.3 odd 2
384.10.a.f.1.4 yes 4 8.3 odd 2
384.10.a.g.1.1 yes 4 1.1 even 1 trivial