Properties

Label 384.8.a.i.1.1
Level $384$
Weight $8$
Character 384.1
Self dual yes
Analytic conductor $119.956$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(119.955849786\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 286x - 1680 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(19.3129\) 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)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} -368.444 q^{5} +1058.71 q^{7} +729.000 q^{9} +O(q^{10})\) \(q-27.0000 q^{3} -368.444 q^{5} +1058.71 q^{7} +729.000 q^{9} -4728.97 q^{11} -5522.34 q^{13} +9947.99 q^{15} +28487.5 q^{17} -12644.8 q^{19} -28585.1 q^{21} -16997.8 q^{23} +57626.0 q^{25} -19683.0 q^{27} +75684.6 q^{29} +69384.1 q^{31} +127682. q^{33} -390074. q^{35} +29223.4 q^{37} +149103. q^{39} +295928. q^{41} -16387.0 q^{43} -268596. q^{45} +68148.4 q^{47} +297318. q^{49} -769163. q^{51} +1.85505e6 q^{53} +1.74236e6 q^{55} +341409. q^{57} +2.75215e6 q^{59} -1.77198e6 q^{61} +771798. q^{63} +2.03467e6 q^{65} +2.76825e6 q^{67} +458941. q^{69} +4.66235e6 q^{71} -6.38405e6 q^{73} -1.55590e6 q^{75} -5.00659e6 q^{77} -7.32391e6 q^{79} +531441. q^{81} +3.05313e6 q^{83} -1.04961e7 q^{85} -2.04349e6 q^{87} -8.79581e6 q^{89} -5.84654e6 q^{91} -1.87337e6 q^{93} +4.65889e6 q^{95} +6.84797e6 q^{97} -3.44742e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 81 q^{3} - 308 q^{5} + 6 q^{7} + 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 81 q^{3} - 308 q^{5} + 6 q^{7} + 2187 q^{9} - 1468 q^{11} - 3070 q^{13} + 8316 q^{15} - 5126 q^{17} - 14168 q^{19} - 162 q^{21} + 59980 q^{23} + 31529 q^{25} - 59049 q^{27} - 71984 q^{29} + 177470 q^{31} + 39636 q^{33} + 84760 q^{35} + 184014 q^{37} + 82890 q^{39} + 657250 q^{41} + 120544 q^{43} - 224532 q^{45} + 665132 q^{47} + 1204647 q^{49} + 138402 q^{51} + 111888 q^{53} + 1854224 q^{55} + 382536 q^{57} + 1033308 q^{59} - 3584306 q^{61} + 4374 q^{63} + 1230472 q^{65} + 2344900 q^{67} - 1619460 q^{69} + 6311524 q^{71} - 2112174 q^{73} - 851283 q^{75} - 6670456 q^{77} + 2018830 q^{79} + 1594323 q^{81} + 7430484 q^{83} - 9821848 q^{85} + 1943568 q^{87} - 13300410 q^{89} - 10605052 q^{91} - 4791690 q^{93} + 303008 q^{95} + 2960414 q^{97} - 1070172 q^{99}+O(q^{100}) \) 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
\(3\) −27.0000 −0.577350
\(4\) 0 0
\(5\) −368.444 −1.31819 −0.659093 0.752062i \(-0.729059\pi\)
−0.659093 + 0.752062i \(0.729059\pi\)
\(6\) 0 0
\(7\) 1058.71 1.16663 0.583314 0.812246i \(-0.301756\pi\)
0.583314 + 0.812246i \(0.301756\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −4728.97 −1.07125 −0.535626 0.844455i \(-0.679924\pi\)
−0.535626 + 0.844455i \(0.679924\pi\)
\(12\) 0 0
\(13\) −5522.34 −0.697142 −0.348571 0.937282i \(-0.613333\pi\)
−0.348571 + 0.937282i \(0.613333\pi\)
\(14\) 0 0
\(15\) 9947.99 0.761055
\(16\) 0 0
\(17\) 28487.5 1.40632 0.703159 0.711033i \(-0.251772\pi\)
0.703159 + 0.711033i \(0.251772\pi\)
\(18\) 0 0
\(19\) −12644.8 −0.422935 −0.211467 0.977385i \(-0.567824\pi\)
−0.211467 + 0.977385i \(0.567824\pi\)
\(20\) 0 0
\(21\) −28585.1 −0.673554
\(22\) 0 0
\(23\) −16997.8 −0.291304 −0.145652 0.989336i \(-0.546528\pi\)
−0.145652 + 0.989336i \(0.546528\pi\)
\(24\) 0 0
\(25\) 57626.0 0.737612
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) 75684.6 0.576255 0.288128 0.957592i \(-0.406967\pi\)
0.288128 + 0.957592i \(0.406967\pi\)
\(30\) 0 0
\(31\) 69384.1 0.418306 0.209153 0.977883i \(-0.432929\pi\)
0.209153 + 0.977883i \(0.432929\pi\)
\(32\) 0 0
\(33\) 127682. 0.618488
\(34\) 0 0
\(35\) −390074. −1.53783
\(36\) 0 0
\(37\) 29223.4 0.0948472 0.0474236 0.998875i \(-0.484899\pi\)
0.0474236 + 0.998875i \(0.484899\pi\)
\(38\) 0 0
\(39\) 149103. 0.402495
\(40\) 0 0
\(41\) 295928. 0.670568 0.335284 0.942117i \(-0.391168\pi\)
0.335284 + 0.942117i \(0.391168\pi\)
\(42\) 0 0
\(43\) −16387.0 −0.0314310 −0.0157155 0.999877i \(-0.505003\pi\)
−0.0157155 + 0.999877i \(0.505003\pi\)
\(44\) 0 0
\(45\) −268596. −0.439395
\(46\) 0 0
\(47\) 68148.4 0.0957444 0.0478722 0.998853i \(-0.484756\pi\)
0.0478722 + 0.998853i \(0.484756\pi\)
\(48\) 0 0
\(49\) 297318. 0.361023
\(50\) 0 0
\(51\) −769163. −0.811938
\(52\) 0 0
\(53\) 1.85505e6 1.71155 0.855777 0.517346i \(-0.173080\pi\)
0.855777 + 0.517346i \(0.173080\pi\)
\(54\) 0 0
\(55\) 1.74236e6 1.41211
\(56\) 0 0
\(57\) 341409. 0.244181
\(58\) 0 0
\(59\) 2.75215e6 1.74458 0.872289 0.488991i \(-0.162635\pi\)
0.872289 + 0.488991i \(0.162635\pi\)
\(60\) 0 0
\(61\) −1.77198e6 −0.999547 −0.499774 0.866156i \(-0.666583\pi\)
−0.499774 + 0.866156i \(0.666583\pi\)
\(62\) 0 0
\(63\) 771798. 0.388876
\(64\) 0 0
\(65\) 2.03467e6 0.918963
\(66\) 0 0
\(67\) 2.76825e6 1.12446 0.562230 0.826981i \(-0.309943\pi\)
0.562230 + 0.826981i \(0.309943\pi\)
\(68\) 0 0
\(69\) 458941. 0.168184
\(70\) 0 0
\(71\) 4.66235e6 1.54597 0.772985 0.634425i \(-0.218763\pi\)
0.772985 + 0.634425i \(0.218763\pi\)
\(72\) 0 0
\(73\) −6.38405e6 −1.92073 −0.960364 0.278748i \(-0.910081\pi\)
−0.960364 + 0.278748i \(0.910081\pi\)
\(74\) 0 0
\(75\) −1.55590e6 −0.425861
\(76\) 0 0
\(77\) −5.00659e6 −1.24975
\(78\) 0 0
\(79\) −7.32391e6 −1.67128 −0.835638 0.549281i \(-0.814902\pi\)
−0.835638 + 0.549281i \(0.814902\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 3.05313e6 0.586100 0.293050 0.956097i \(-0.405330\pi\)
0.293050 + 0.956097i \(0.405330\pi\)
\(84\) 0 0
\(85\) −1.04961e7 −1.85379
\(86\) 0 0
\(87\) −2.04349e6 −0.332701
\(88\) 0 0
\(89\) −8.79581e6 −1.32255 −0.661273 0.750145i \(-0.729984\pi\)
−0.661273 + 0.750145i \(0.729984\pi\)
\(90\) 0 0
\(91\) −5.84654e6 −0.813307
\(92\) 0 0
\(93\) −1.87337e6 −0.241509
\(94\) 0 0
\(95\) 4.65889e6 0.557506
\(96\) 0 0
\(97\) 6.84797e6 0.761835 0.380917 0.924609i \(-0.375608\pi\)
0.380917 + 0.924609i \(0.375608\pi\)
\(98\) 0 0
\(99\) −3.44742e6 −0.357084
\(100\) 0 0
\(101\) 1.01398e7 0.979274 0.489637 0.871926i \(-0.337129\pi\)
0.489637 + 0.871926i \(0.337129\pi\)
\(102\) 0 0
\(103\) −1.54107e7 −1.38961 −0.694803 0.719200i \(-0.744509\pi\)
−0.694803 + 0.719200i \(0.744509\pi\)
\(104\) 0 0
\(105\) 1.05320e7 0.887868
\(106\) 0 0
\(107\) −7.22867e6 −0.570447 −0.285224 0.958461i \(-0.592068\pi\)
−0.285224 + 0.958461i \(0.592068\pi\)
\(108\) 0 0
\(109\) −1.12169e7 −0.829625 −0.414812 0.909907i \(-0.636153\pi\)
−0.414812 + 0.909907i \(0.636153\pi\)
\(110\) 0 0
\(111\) −789031. −0.0547600
\(112\) 0 0
\(113\) −1.49419e7 −0.974162 −0.487081 0.873357i \(-0.661938\pi\)
−0.487081 + 0.873357i \(0.661938\pi\)
\(114\) 0 0
\(115\) 6.26275e6 0.383992
\(116\) 0 0
\(117\) −4.02579e6 −0.232381
\(118\) 0 0
\(119\) 3.01599e7 1.64065
\(120\) 0 0
\(121\) 2.87595e6 0.147582
\(122\) 0 0
\(123\) −7.99006e6 −0.387153
\(124\) 0 0
\(125\) 7.55274e6 0.345875
\(126\) 0 0
\(127\) 2.92110e7 1.26542 0.632709 0.774390i \(-0.281943\pi\)
0.632709 + 0.774390i \(0.281943\pi\)
\(128\) 0 0
\(129\) 442448. 0.0181467
\(130\) 0 0
\(131\) −1.38297e7 −0.537481 −0.268741 0.963213i \(-0.586607\pi\)
−0.268741 + 0.963213i \(0.586607\pi\)
\(132\) 0 0
\(133\) −1.33871e7 −0.493408
\(134\) 0 0
\(135\) 7.25208e6 0.253685
\(136\) 0 0
\(137\) −5.59943e7 −1.86047 −0.930233 0.366969i \(-0.880395\pi\)
−0.930233 + 0.366969i \(0.880395\pi\)
\(138\) 0 0
\(139\) −5.22397e7 −1.64987 −0.824934 0.565229i \(-0.808788\pi\)
−0.824934 + 0.565229i \(0.808788\pi\)
\(140\) 0 0
\(141\) −1.84001e6 −0.0552780
\(142\) 0 0
\(143\) 2.61150e7 0.746816
\(144\) 0 0
\(145\) −2.78855e7 −0.759611
\(146\) 0 0
\(147\) −8.02759e6 −0.208437
\(148\) 0 0
\(149\) 1.62977e7 0.403622 0.201811 0.979424i \(-0.435317\pi\)
0.201811 + 0.979424i \(0.435317\pi\)
\(150\) 0 0
\(151\) 1.71334e7 0.404971 0.202486 0.979285i \(-0.435098\pi\)
0.202486 + 0.979285i \(0.435098\pi\)
\(152\) 0 0
\(153\) 2.07674e7 0.468772
\(154\) 0 0
\(155\) −2.55641e7 −0.551404
\(156\) 0 0
\(157\) 8.35265e7 1.72257 0.861283 0.508126i \(-0.169662\pi\)
0.861283 + 0.508126i \(0.169662\pi\)
\(158\) 0 0
\(159\) −5.00864e7 −0.988166
\(160\) 0 0
\(161\) −1.79957e7 −0.339843
\(162\) 0 0
\(163\) −6.70834e7 −1.21327 −0.606637 0.794979i \(-0.707482\pi\)
−0.606637 + 0.794979i \(0.707482\pi\)
\(164\) 0 0
\(165\) −4.70437e7 −0.815282
\(166\) 0 0
\(167\) 5.88943e7 0.978510 0.489255 0.872141i \(-0.337269\pi\)
0.489255 + 0.872141i \(0.337269\pi\)
\(168\) 0 0
\(169\) −3.22523e7 −0.513992
\(170\) 0 0
\(171\) −9.21804e6 −0.140978
\(172\) 0 0
\(173\) −5.79040e7 −0.850250 −0.425125 0.905135i \(-0.639770\pi\)
−0.425125 + 0.905135i \(0.639770\pi\)
\(174\) 0 0
\(175\) 6.10090e7 0.860520
\(176\) 0 0
\(177\) −7.43081e7 −1.00723
\(178\) 0 0
\(179\) 4.73524e7 0.617102 0.308551 0.951208i \(-0.400156\pi\)
0.308551 + 0.951208i \(0.400156\pi\)
\(180\) 0 0
\(181\) −1.33152e8 −1.66907 −0.834534 0.550957i \(-0.814263\pi\)
−0.834534 + 0.550957i \(0.814263\pi\)
\(182\) 0 0
\(183\) 4.78433e7 0.577089
\(184\) 0 0
\(185\) −1.07672e7 −0.125026
\(186\) 0 0
\(187\) −1.34717e8 −1.50652
\(188\) 0 0
\(189\) −2.08385e7 −0.224518
\(190\) 0 0
\(191\) −9.13177e7 −0.948283 −0.474142 0.880449i \(-0.657242\pi\)
−0.474142 + 0.880449i \(0.657242\pi\)
\(192\) 0 0
\(193\) −1.06396e8 −1.06530 −0.532651 0.846335i \(-0.678804\pi\)
−0.532651 + 0.846335i \(0.678804\pi\)
\(194\) 0 0
\(195\) −5.49362e7 −0.530564
\(196\) 0 0
\(197\) −1.91262e8 −1.78236 −0.891182 0.453646i \(-0.850123\pi\)
−0.891182 + 0.453646i \(0.850123\pi\)
\(198\) 0 0
\(199\) 4.13489e7 0.371945 0.185972 0.982555i \(-0.440457\pi\)
0.185972 + 0.982555i \(0.440457\pi\)
\(200\) 0 0
\(201\) −7.47429e7 −0.649208
\(202\) 0 0
\(203\) 8.01279e7 0.672276
\(204\) 0 0
\(205\) −1.09033e8 −0.883933
\(206\) 0 0
\(207\) −1.23914e7 −0.0971012
\(208\) 0 0
\(209\) 5.97967e7 0.453070
\(210\) 0 0
\(211\) 5.48974e7 0.402312 0.201156 0.979559i \(-0.435530\pi\)
0.201156 + 0.979559i \(0.435530\pi\)
\(212\) 0 0
\(213\) −1.25884e8 −0.892566
\(214\) 0 0
\(215\) 6.03768e6 0.0414319
\(216\) 0 0
\(217\) 7.34574e7 0.488008
\(218\) 0 0
\(219\) 1.72369e8 1.10893
\(220\) 0 0
\(221\) −1.57318e8 −0.980404
\(222\) 0 0
\(223\) 1.22016e8 0.736799 0.368399 0.929668i \(-0.379906\pi\)
0.368399 + 0.929668i \(0.379906\pi\)
\(224\) 0 0
\(225\) 4.20093e7 0.245871
\(226\) 0 0
\(227\) −3.33351e8 −1.89152 −0.945762 0.324861i \(-0.894683\pi\)
−0.945762 + 0.324861i \(0.894683\pi\)
\(228\) 0 0
\(229\) −1.28881e7 −0.0709193 −0.0354597 0.999371i \(-0.511290\pi\)
−0.0354597 + 0.999371i \(0.511290\pi\)
\(230\) 0 0
\(231\) 1.35178e8 0.721546
\(232\) 0 0
\(233\) −3.67790e8 −1.90482 −0.952411 0.304817i \(-0.901405\pi\)
−0.952411 + 0.304817i \(0.901405\pi\)
\(234\) 0 0
\(235\) −2.51089e7 −0.126209
\(236\) 0 0
\(237\) 1.97746e8 0.964912
\(238\) 0 0
\(239\) 1.50083e8 0.711113 0.355556 0.934655i \(-0.384291\pi\)
0.355556 + 0.934655i \(0.384291\pi\)
\(240\) 0 0
\(241\) −7.57608e7 −0.348646 −0.174323 0.984689i \(-0.555774\pi\)
−0.174323 + 0.984689i \(0.555774\pi\)
\(242\) 0 0
\(243\) −1.43489e7 −0.0641500
\(244\) 0 0
\(245\) −1.09545e8 −0.475895
\(246\) 0 0
\(247\) 6.98287e7 0.294846
\(248\) 0 0
\(249\) −8.24345e7 −0.338385
\(250\) 0 0
\(251\) −2.46051e8 −0.982127 −0.491063 0.871124i \(-0.663392\pi\)
−0.491063 + 0.871124i \(0.663392\pi\)
\(252\) 0 0
\(253\) 8.03821e7 0.312060
\(254\) 0 0
\(255\) 2.83393e8 1.07028
\(256\) 0 0
\(257\) 3.21274e8 1.18062 0.590309 0.807177i \(-0.299006\pi\)
0.590309 + 0.807177i \(0.299006\pi\)
\(258\) 0 0
\(259\) 3.09390e7 0.110651
\(260\) 0 0
\(261\) 5.51741e7 0.192085
\(262\) 0 0
\(263\) 1.79444e8 0.608254 0.304127 0.952632i \(-0.401635\pi\)
0.304127 + 0.952632i \(0.401635\pi\)
\(264\) 0 0
\(265\) −6.83483e8 −2.25614
\(266\) 0 0
\(267\) 2.37487e8 0.763572
\(268\) 0 0
\(269\) 7.19069e7 0.225236 0.112618 0.993638i \(-0.464076\pi\)
0.112618 + 0.993638i \(0.464076\pi\)
\(270\) 0 0
\(271\) 5.37851e8 1.64161 0.820805 0.571209i \(-0.193525\pi\)
0.820805 + 0.571209i \(0.193525\pi\)
\(272\) 0 0
\(273\) 1.57857e8 0.469563
\(274\) 0 0
\(275\) −2.72511e8 −0.790169
\(276\) 0 0
\(277\) 2.21434e8 0.625986 0.312993 0.949755i \(-0.398668\pi\)
0.312993 + 0.949755i \(0.398668\pi\)
\(278\) 0 0
\(279\) 5.05810e7 0.139435
\(280\) 0 0
\(281\) 2.00421e8 0.538855 0.269428 0.963021i \(-0.413166\pi\)
0.269428 + 0.963021i \(0.413166\pi\)
\(282\) 0 0
\(283\) −5.60104e8 −1.46898 −0.734490 0.678619i \(-0.762579\pi\)
−0.734490 + 0.678619i \(0.762579\pi\)
\(284\) 0 0
\(285\) −1.25790e8 −0.321876
\(286\) 0 0
\(287\) 3.13301e8 0.782304
\(288\) 0 0
\(289\) 4.01200e8 0.977729
\(290\) 0 0
\(291\) −1.84895e8 −0.439845
\(292\) 0 0
\(293\) −4.99015e8 −1.15898 −0.579492 0.814978i \(-0.696749\pi\)
−0.579492 + 0.814978i \(0.696749\pi\)
\(294\) 0 0
\(295\) −1.01401e9 −2.29968
\(296\) 0 0
\(297\) 9.30802e7 0.206163
\(298\) 0 0
\(299\) 9.38678e7 0.203080
\(300\) 0 0
\(301\) −1.73490e7 −0.0366684
\(302\) 0 0
\(303\) −2.73774e8 −0.565384
\(304\) 0 0
\(305\) 6.52874e8 1.31759
\(306\) 0 0
\(307\) 8.21105e8 1.61963 0.809813 0.586689i \(-0.199569\pi\)
0.809813 + 0.586689i \(0.199569\pi\)
\(308\) 0 0
\(309\) 4.16089e8 0.802290
\(310\) 0 0
\(311\) −5.43818e7 −0.102516 −0.0512581 0.998685i \(-0.516323\pi\)
−0.0512581 + 0.998685i \(0.516323\pi\)
\(312\) 0 0
\(313\) −6.71990e8 −1.23868 −0.619338 0.785124i \(-0.712599\pi\)
−0.619338 + 0.785124i \(0.712599\pi\)
\(314\) 0 0
\(315\) −2.84364e8 −0.512611
\(316\) 0 0
\(317\) −1.49707e8 −0.263958 −0.131979 0.991253i \(-0.542133\pi\)
−0.131979 + 0.991253i \(0.542133\pi\)
\(318\) 0 0
\(319\) −3.57910e8 −0.617315
\(320\) 0 0
\(321\) 1.95174e8 0.329348
\(322\) 0 0
\(323\) −3.60218e8 −0.594780
\(324\) 0 0
\(325\) −3.18230e8 −0.514221
\(326\) 0 0
\(327\) 3.02857e8 0.478984
\(328\) 0 0
\(329\) 7.21492e7 0.111698
\(330\) 0 0
\(331\) 2.29194e8 0.347380 0.173690 0.984800i \(-0.444431\pi\)
0.173690 + 0.984800i \(0.444431\pi\)
\(332\) 0 0
\(333\) 2.13038e7 0.0316157
\(334\) 0 0
\(335\) −1.01995e9 −1.48225
\(336\) 0 0
\(337\) −5.68268e8 −0.808814 −0.404407 0.914579i \(-0.632522\pi\)
−0.404407 + 0.914579i \(0.632522\pi\)
\(338\) 0 0
\(339\) 4.03431e8 0.562433
\(340\) 0 0
\(341\) −3.28115e8 −0.448111
\(342\) 0 0
\(343\) −5.57118e8 −0.745449
\(344\) 0 0
\(345\) −1.69094e8 −0.221698
\(346\) 0 0
\(347\) 2.99189e8 0.384408 0.192204 0.981355i \(-0.438436\pi\)
0.192204 + 0.981355i \(0.438436\pi\)
\(348\) 0 0
\(349\) 1.48423e9 1.86901 0.934503 0.355954i \(-0.115844\pi\)
0.934503 + 0.355954i \(0.115844\pi\)
\(350\) 0 0
\(351\) 1.08696e8 0.134165
\(352\) 0 0
\(353\) −6.05417e8 −0.732560 −0.366280 0.930505i \(-0.619369\pi\)
−0.366280 + 0.930505i \(0.619369\pi\)
\(354\) 0 0
\(355\) −1.71782e9 −2.03787
\(356\) 0 0
\(357\) −8.14318e8 −0.947230
\(358\) 0 0
\(359\) 2.74734e8 0.313388 0.156694 0.987647i \(-0.449916\pi\)
0.156694 + 0.987647i \(0.449916\pi\)
\(360\) 0 0
\(361\) −7.33982e8 −0.821126
\(362\) 0 0
\(363\) −7.76507e7 −0.0852064
\(364\) 0 0
\(365\) 2.35216e9 2.53188
\(366\) 0 0
\(367\) −8.66951e8 −0.915511 −0.457755 0.889078i \(-0.651346\pi\)
−0.457755 + 0.889078i \(0.651346\pi\)
\(368\) 0 0
\(369\) 2.15732e8 0.223523
\(370\) 0 0
\(371\) 1.96396e9 1.99675
\(372\) 0 0
\(373\) −5.09365e8 −0.508216 −0.254108 0.967176i \(-0.581782\pi\)
−0.254108 + 0.967176i \(0.581782\pi\)
\(374\) 0 0
\(375\) −2.03924e8 −0.199691
\(376\) 0 0
\(377\) −4.17956e8 −0.401732
\(378\) 0 0
\(379\) 9.85372e8 0.929743 0.464871 0.885378i \(-0.346101\pi\)
0.464871 + 0.885378i \(0.346101\pi\)
\(380\) 0 0
\(381\) −7.88698e8 −0.730590
\(382\) 0 0
\(383\) −3.47060e8 −0.315652 −0.157826 0.987467i \(-0.550449\pi\)
−0.157826 + 0.987467i \(0.550449\pi\)
\(384\) 0 0
\(385\) 1.84465e9 1.64741
\(386\) 0 0
\(387\) −1.19461e7 −0.0104770
\(388\) 0 0
\(389\) −8.19819e8 −0.706146 −0.353073 0.935596i \(-0.614863\pi\)
−0.353073 + 0.935596i \(0.614863\pi\)
\(390\) 0 0
\(391\) −4.84226e8 −0.409665
\(392\) 0 0
\(393\) 3.73402e8 0.310315
\(394\) 0 0
\(395\) 2.69845e9 2.20305
\(396\) 0 0
\(397\) 1.57699e9 1.26492 0.632460 0.774593i \(-0.282045\pi\)
0.632460 + 0.774593i \(0.282045\pi\)
\(398\) 0 0
\(399\) 3.61452e8 0.284869
\(400\) 0 0
\(401\) 5.99350e8 0.464168 0.232084 0.972696i \(-0.425446\pi\)
0.232084 + 0.972696i \(0.425446\pi\)
\(402\) 0 0
\(403\) −3.83162e8 −0.291619
\(404\) 0 0
\(405\) −1.95806e8 −0.146465
\(406\) 0 0
\(407\) −1.38196e8 −0.101605
\(408\) 0 0
\(409\) −4.08431e8 −0.295180 −0.147590 0.989049i \(-0.547152\pi\)
−0.147590 + 0.989049i \(0.547152\pi\)
\(410\) 0 0
\(411\) 1.51185e9 1.07414
\(412\) 0 0
\(413\) 2.91372e9 2.03527
\(414\) 0 0
\(415\) −1.12491e9 −0.772589
\(416\) 0 0
\(417\) 1.41047e9 0.952552
\(418\) 0 0
\(419\) 1.46270e9 0.971417 0.485708 0.874121i \(-0.338562\pi\)
0.485708 + 0.874121i \(0.338562\pi\)
\(420\) 0 0
\(421\) −1.76741e8 −0.115438 −0.0577191 0.998333i \(-0.518383\pi\)
−0.0577191 + 0.998333i \(0.518383\pi\)
\(422\) 0 0
\(423\) 4.96802e7 0.0319148
\(424\) 0 0
\(425\) 1.64162e9 1.03732
\(426\) 0 0
\(427\) −1.87600e9 −1.16610
\(428\) 0 0
\(429\) −7.05104e8 −0.431174
\(430\) 0 0
\(431\) 1.56757e9 0.943099 0.471549 0.881840i \(-0.343695\pi\)
0.471549 + 0.881840i \(0.343695\pi\)
\(432\) 0 0
\(433\) 2.75419e9 1.63037 0.815185 0.579201i \(-0.196635\pi\)
0.815185 + 0.579201i \(0.196635\pi\)
\(434\) 0 0
\(435\) 7.52910e8 0.438562
\(436\) 0 0
\(437\) 2.14934e8 0.123202
\(438\) 0 0
\(439\) 5.51524e8 0.311128 0.155564 0.987826i \(-0.450281\pi\)
0.155564 + 0.987826i \(0.450281\pi\)
\(440\) 0 0
\(441\) 2.16745e8 0.120341
\(442\) 0 0
\(443\) −1.80168e9 −0.984610 −0.492305 0.870423i \(-0.663845\pi\)
−0.492305 + 0.870423i \(0.663845\pi\)
\(444\) 0 0
\(445\) 3.24076e9 1.74336
\(446\) 0 0
\(447\) −4.40039e8 −0.233032
\(448\) 0 0
\(449\) 4.05710e8 0.211521 0.105760 0.994392i \(-0.466272\pi\)
0.105760 + 0.994392i \(0.466272\pi\)
\(450\) 0 0
\(451\) −1.39943e9 −0.718348
\(452\) 0 0
\(453\) −4.62602e8 −0.233810
\(454\) 0 0
\(455\) 2.15412e9 1.07209
\(456\) 0 0
\(457\) 1.68101e9 0.823877 0.411939 0.911212i \(-0.364852\pi\)
0.411939 + 0.911212i \(0.364852\pi\)
\(458\) 0 0
\(459\) −5.60720e8 −0.270646
\(460\) 0 0
\(461\) −2.11400e9 −1.00497 −0.502484 0.864586i \(-0.667580\pi\)
−0.502484 + 0.864586i \(0.667580\pi\)
\(462\) 0 0
\(463\) −2.49558e9 −1.16852 −0.584261 0.811566i \(-0.698616\pi\)
−0.584261 + 0.811566i \(0.698616\pi\)
\(464\) 0 0
\(465\) 6.90232e8 0.318353
\(466\) 0 0
\(467\) 2.80333e9 1.27369 0.636847 0.770990i \(-0.280238\pi\)
0.636847 + 0.770990i \(0.280238\pi\)
\(468\) 0 0
\(469\) 2.93077e9 1.31183
\(470\) 0 0
\(471\) −2.25522e9 −0.994524
\(472\) 0 0
\(473\) 7.74934e7 0.0336706
\(474\) 0 0
\(475\) −7.28667e8 −0.311962
\(476\) 0 0
\(477\) 1.35233e9 0.570518
\(478\) 0 0
\(479\) 1.17285e8 0.0487605 0.0243803 0.999703i \(-0.492239\pi\)
0.0243803 + 0.999703i \(0.492239\pi\)
\(480\) 0 0
\(481\) −1.61381e8 −0.0661220
\(482\) 0 0
\(483\) 4.85884e8 0.196209
\(484\) 0 0
\(485\) −2.52309e9 −1.00424
\(486\) 0 0
\(487\) −1.37486e9 −0.539393 −0.269697 0.962945i \(-0.586923\pi\)
−0.269697 + 0.962945i \(0.586923\pi\)
\(488\) 0 0
\(489\) 1.81125e9 0.700484
\(490\) 0 0
\(491\) −2.33074e8 −0.0888606 −0.0444303 0.999012i \(-0.514147\pi\)
−0.0444303 + 0.999012i \(0.514147\pi\)
\(492\) 0 0
\(493\) 2.15607e9 0.810397
\(494\) 0 0
\(495\) 1.27018e9 0.470703
\(496\) 0 0
\(497\) 4.93607e9 1.80357
\(498\) 0 0
\(499\) 8.97141e7 0.0323228 0.0161614 0.999869i \(-0.494855\pi\)
0.0161614 + 0.999869i \(0.494855\pi\)
\(500\) 0 0
\(501\) −1.59014e9 −0.564943
\(502\) 0 0
\(503\) −2.47433e9 −0.866901 −0.433450 0.901177i \(-0.642704\pi\)
−0.433450 + 0.901177i \(0.642704\pi\)
\(504\) 0 0
\(505\) −3.73594e9 −1.29086
\(506\) 0 0
\(507\) 8.70811e8 0.296754
\(508\) 0 0
\(509\) 1.68523e8 0.0566430 0.0283215 0.999599i \(-0.490984\pi\)
0.0283215 + 0.999599i \(0.490984\pi\)
\(510\) 0 0
\(511\) −6.75884e9 −2.24078
\(512\) 0 0
\(513\) 2.48887e8 0.0813938
\(514\) 0 0
\(515\) 5.67798e9 1.83176
\(516\) 0 0
\(517\) −3.22272e8 −0.102566
\(518\) 0 0
\(519\) 1.56341e9 0.490892
\(520\) 0 0
\(521\) −2.41709e9 −0.748792 −0.374396 0.927269i \(-0.622150\pi\)
−0.374396 + 0.927269i \(0.622150\pi\)
\(522\) 0 0
\(523\) 9.40312e8 0.287419 0.143710 0.989620i \(-0.454097\pi\)
0.143710 + 0.989620i \(0.454097\pi\)
\(524\) 0 0
\(525\) −1.64724e9 −0.496821
\(526\) 0 0
\(527\) 1.97658e9 0.588271
\(528\) 0 0
\(529\) −3.11590e9 −0.915142
\(530\) 0 0
\(531\) 2.00632e9 0.581526
\(532\) 0 0
\(533\) −1.63422e9 −0.467482
\(534\) 0 0
\(535\) 2.66336e9 0.751955
\(536\) 0 0
\(537\) −1.27852e9 −0.356284
\(538\) 0 0
\(539\) −1.40601e9 −0.386747
\(540\) 0 0
\(541\) 2.65125e8 0.0719879 0.0359939 0.999352i \(-0.488540\pi\)
0.0359939 + 0.999352i \(0.488540\pi\)
\(542\) 0 0
\(543\) 3.59511e9 0.963636
\(544\) 0 0
\(545\) 4.13281e9 1.09360
\(546\) 0 0
\(547\) 6.41977e8 0.167712 0.0838559 0.996478i \(-0.473276\pi\)
0.0838559 + 0.996478i \(0.473276\pi\)
\(548\) 0 0
\(549\) −1.29177e9 −0.333182
\(550\) 0 0
\(551\) −9.57015e8 −0.243718
\(552\) 0 0
\(553\) −7.75388e9 −1.94976
\(554\) 0 0
\(555\) 2.90714e8 0.0721839
\(556\) 0 0
\(557\) −2.92485e9 −0.717152 −0.358576 0.933501i \(-0.616738\pi\)
−0.358576 + 0.933501i \(0.616738\pi\)
\(558\) 0 0
\(559\) 9.04944e7 0.0219119
\(560\) 0 0
\(561\) 3.63735e9 0.869790
\(562\) 0 0
\(563\) −5.62455e9 −1.32834 −0.664169 0.747583i \(-0.731214\pi\)
−0.664169 + 0.747583i \(0.731214\pi\)
\(564\) 0 0
\(565\) 5.50525e9 1.28413
\(566\) 0 0
\(567\) 5.62640e8 0.129625
\(568\) 0 0
\(569\) −8.48615e9 −1.93116 −0.965579 0.260111i \(-0.916241\pi\)
−0.965579 + 0.260111i \(0.916241\pi\)
\(570\) 0 0
\(571\) 5.42789e9 1.22013 0.610063 0.792353i \(-0.291144\pi\)
0.610063 + 0.792353i \(0.291144\pi\)
\(572\) 0 0
\(573\) 2.46558e9 0.547492
\(574\) 0 0
\(575\) −9.79516e8 −0.214869
\(576\) 0 0
\(577\) −8.80138e9 −1.90737 −0.953686 0.300803i \(-0.902745\pi\)
−0.953686 + 0.300803i \(0.902745\pi\)
\(578\) 0 0
\(579\) 2.87268e9 0.615053
\(580\) 0 0
\(581\) 3.23237e9 0.683761
\(582\) 0 0
\(583\) −8.77248e9 −1.83351
\(584\) 0 0
\(585\) 1.48328e9 0.306321
\(586\) 0 0
\(587\) 3.50790e8 0.0715836 0.0357918 0.999359i \(-0.488605\pi\)
0.0357918 + 0.999359i \(0.488605\pi\)
\(588\) 0 0
\(589\) −8.77345e8 −0.176916
\(590\) 0 0
\(591\) 5.16407e9 1.02905
\(592\) 0 0
\(593\) 6.34103e9 1.24873 0.624364 0.781133i \(-0.285358\pi\)
0.624364 + 0.781133i \(0.285358\pi\)
\(594\) 0 0
\(595\) −1.11122e10 −2.16268
\(596\) 0 0
\(597\) −1.11642e9 −0.214742
\(598\) 0 0
\(599\) 2.85447e9 0.542664 0.271332 0.962486i \(-0.412536\pi\)
0.271332 + 0.962486i \(0.412536\pi\)
\(600\) 0 0
\(601\) 3.22420e9 0.605844 0.302922 0.953015i \(-0.402038\pi\)
0.302922 + 0.953015i \(0.402038\pi\)
\(602\) 0 0
\(603\) 2.01806e9 0.374820
\(604\) 0 0
\(605\) −1.05963e9 −0.194540
\(606\) 0 0
\(607\) 8.25547e8 0.149824 0.0749120 0.997190i \(-0.476132\pi\)
0.0749120 + 0.997190i \(0.476132\pi\)
\(608\) 0 0
\(609\) −2.16345e9 −0.388139
\(610\) 0 0
\(611\) −3.76339e8 −0.0667475
\(612\) 0 0
\(613\) −1.26909e9 −0.222526 −0.111263 0.993791i \(-0.535490\pi\)
−0.111263 + 0.993791i \(0.535490\pi\)
\(614\) 0 0
\(615\) 2.94389e9 0.510339
\(616\) 0 0
\(617\) 1.08847e10 1.86560 0.932802 0.360389i \(-0.117356\pi\)
0.932802 + 0.360389i \(0.117356\pi\)
\(618\) 0 0
\(619\) −5.60725e9 −0.950239 −0.475119 0.879921i \(-0.657595\pi\)
−0.475119 + 0.879921i \(0.657595\pi\)
\(620\) 0 0
\(621\) 3.34568e8 0.0560614
\(622\) 0 0
\(623\) −9.31219e9 −1.54292
\(624\) 0 0
\(625\) −7.28479e9 −1.19354
\(626\) 0 0
\(627\) −1.61451e9 −0.261580
\(628\) 0 0
\(629\) 8.32501e8 0.133385
\(630\) 0 0
\(631\) −3.03140e9 −0.480330 −0.240165 0.970732i \(-0.577202\pi\)
−0.240165 + 0.970732i \(0.577202\pi\)
\(632\) 0 0
\(633\) −1.48223e9 −0.232275
\(634\) 0 0
\(635\) −1.07626e10 −1.66806
\(636\) 0 0
\(637\) −1.64189e9 −0.251685
\(638\) 0 0
\(639\) 3.39886e9 0.515323
\(640\) 0 0
\(641\) 1.88737e9 0.283045 0.141522 0.989935i \(-0.454800\pi\)
0.141522 + 0.989935i \(0.454800\pi\)
\(642\) 0 0
\(643\) −1.23016e10 −1.82483 −0.912417 0.409263i \(-0.865786\pi\)
−0.912417 + 0.409263i \(0.865786\pi\)
\(644\) 0 0
\(645\) −1.63017e8 −0.0239207
\(646\) 0 0
\(647\) 5.68136e9 0.824683 0.412342 0.911029i \(-0.364711\pi\)
0.412342 + 0.911029i \(0.364711\pi\)
\(648\) 0 0
\(649\) −1.30148e10 −1.86888
\(650\) 0 0
\(651\) −1.98335e9 −0.281751
\(652\) 0 0
\(653\) −9.38850e9 −1.31947 −0.659736 0.751497i \(-0.729332\pi\)
−0.659736 + 0.751497i \(0.729332\pi\)
\(654\) 0 0
\(655\) 5.09547e9 0.708500
\(656\) 0 0
\(657\) −4.65397e9 −0.640243
\(658\) 0 0
\(659\) −3.88229e9 −0.528432 −0.264216 0.964463i \(-0.585113\pi\)
−0.264216 + 0.964463i \(0.585113\pi\)
\(660\) 0 0
\(661\) −3.30387e9 −0.444957 −0.222478 0.974938i \(-0.571415\pi\)
−0.222478 + 0.974938i \(0.571415\pi\)
\(662\) 0 0
\(663\) 4.24758e9 0.566036
\(664\) 0 0
\(665\) 4.93240e9 0.650403
\(666\) 0 0
\(667\) −1.28647e9 −0.167865
\(668\) 0 0
\(669\) −3.29442e9 −0.425391
\(670\) 0 0
\(671\) 8.37961e9 1.07077
\(672\) 0 0
\(673\) 3.44280e9 0.435371 0.217685 0.976019i \(-0.430149\pi\)
0.217685 + 0.976019i \(0.430149\pi\)
\(674\) 0 0
\(675\) −1.13425e9 −0.141954
\(676\) 0 0
\(677\) −1.59211e10 −1.97202 −0.986012 0.166677i \(-0.946696\pi\)
−0.986012 + 0.166677i \(0.946696\pi\)
\(678\) 0 0
\(679\) 7.24999e9 0.888778
\(680\) 0 0
\(681\) 9.00049e9 1.09207
\(682\) 0 0
\(683\) 7.17757e9 0.861995 0.430997 0.902353i \(-0.358162\pi\)
0.430997 + 0.902353i \(0.358162\pi\)
\(684\) 0 0
\(685\) 2.06308e10 2.45244
\(686\) 0 0
\(687\) 3.47979e8 0.0409453
\(688\) 0 0
\(689\) −1.02442e10 −1.19320
\(690\) 0 0
\(691\) −1.18199e10 −1.36282 −0.681411 0.731901i \(-0.738633\pi\)
−0.681411 + 0.731901i \(0.738633\pi\)
\(692\) 0 0
\(693\) −3.64981e9 −0.416585
\(694\) 0 0
\(695\) 1.92474e10 2.17483
\(696\) 0 0
\(697\) 8.43026e9 0.943032
\(698\) 0 0
\(699\) 9.93033e9 1.09975
\(700\) 0 0
\(701\) −8.79765e9 −0.964614 −0.482307 0.876002i \(-0.660201\pi\)
−0.482307 + 0.876002i \(0.660201\pi\)
\(702\) 0 0
\(703\) −3.69523e8 −0.0401142
\(704\) 0 0
\(705\) 6.77940e8 0.0728667
\(706\) 0 0
\(707\) 1.07351e10 1.14245
\(708\) 0 0
\(709\) −4.09495e9 −0.431506 −0.215753 0.976448i \(-0.569221\pi\)
−0.215753 + 0.976448i \(0.569221\pi\)
\(710\) 0 0
\(711\) −5.33913e9 −0.557092
\(712\) 0 0
\(713\) −1.17938e9 −0.121854
\(714\) 0 0
\(715\) −9.62190e9 −0.984441
\(716\) 0 0
\(717\) −4.05224e9 −0.410561
\(718\) 0 0
\(719\) 1.16329e10 1.16718 0.583589 0.812049i \(-0.301648\pi\)
0.583589 + 0.812049i \(0.301648\pi\)
\(720\) 0 0
\(721\) −1.63154e10 −1.62116
\(722\) 0 0
\(723\) 2.04554e9 0.201291
\(724\) 0 0
\(725\) 4.36140e9 0.425053
\(726\) 0 0
\(727\) −1.69867e10 −1.63960 −0.819799 0.572651i \(-0.805915\pi\)
−0.819799 + 0.572651i \(0.805915\pi\)
\(728\) 0 0
\(729\) 3.87420e8 0.0370370
\(730\) 0 0
\(731\) −4.66824e8 −0.0442020
\(732\) 0 0
\(733\) −3.28036e9 −0.307650 −0.153825 0.988098i \(-0.549159\pi\)
−0.153825 + 0.988098i \(0.549159\pi\)
\(734\) 0 0
\(735\) 2.95772e9 0.274758
\(736\) 0 0
\(737\) −1.30910e10 −1.20458
\(738\) 0 0
\(739\) 1.76555e10 1.60926 0.804628 0.593779i \(-0.202365\pi\)
0.804628 + 0.593779i \(0.202365\pi\)
\(740\) 0 0
\(741\) −1.88538e9 −0.170229
\(742\) 0 0
\(743\) 8.33458e9 0.745458 0.372729 0.927940i \(-0.378422\pi\)
0.372729 + 0.927940i \(0.378422\pi\)
\(744\) 0 0
\(745\) −6.00480e9 −0.532049
\(746\) 0 0
\(747\) 2.22573e9 0.195367
\(748\) 0 0
\(749\) −7.65305e9 −0.665500
\(750\) 0 0
\(751\) 1.15928e10 0.998729 0.499365 0.866392i \(-0.333567\pi\)
0.499365 + 0.866392i \(0.333567\pi\)
\(752\) 0 0
\(753\) 6.64338e9 0.567031
\(754\) 0 0
\(755\) −6.31270e9 −0.533827
\(756\) 0 0
\(757\) 2.07716e10 1.74034 0.870170 0.492751i \(-0.164009\pi\)
0.870170 + 0.492751i \(0.164009\pi\)
\(758\) 0 0
\(759\) −2.17032e9 −0.180168
\(760\) 0 0
\(761\) 6.33390e9 0.520985 0.260492 0.965476i \(-0.416115\pi\)
0.260492 + 0.965476i \(0.416115\pi\)
\(762\) 0 0
\(763\) −1.18755e10 −0.967864
\(764\) 0 0
\(765\) −7.65162e9 −0.617929
\(766\) 0 0
\(767\) −1.51983e10 −1.21622
\(768\) 0 0
\(769\) −6.52316e9 −0.517268 −0.258634 0.965975i \(-0.583272\pi\)
−0.258634 + 0.965975i \(0.583272\pi\)
\(770\) 0 0
\(771\) −8.67439e9 −0.681630
\(772\) 0 0
\(773\) −6.77839e9 −0.527835 −0.263918 0.964545i \(-0.585015\pi\)
−0.263918 + 0.964545i \(0.585015\pi\)
\(774\) 0 0
\(775\) 3.99832e9 0.308547
\(776\) 0 0
\(777\) −8.35353e8 −0.0638846
\(778\) 0 0
\(779\) −3.74195e9 −0.283607
\(780\) 0 0
\(781\) −2.20481e10 −1.65612
\(782\) 0 0
\(783\) −1.48970e9 −0.110900
\(784\) 0 0
\(785\) −3.07748e10 −2.27066
\(786\) 0 0
\(787\) 2.82358e9 0.206485 0.103242 0.994656i \(-0.467078\pi\)
0.103242 + 0.994656i \(0.467078\pi\)
\(788\) 0 0
\(789\) −4.84500e9 −0.351175
\(790\) 0 0
\(791\) −1.58191e10 −1.13649
\(792\) 0 0
\(793\) 9.78546e9 0.696827
\(794\) 0 0
\(795\) 1.84540e10 1.30259
\(796\) 0 0
\(797\) −5.30653e9 −0.371284 −0.185642 0.982617i \(-0.559436\pi\)
−0.185642 + 0.982617i \(0.559436\pi\)
\(798\) 0 0
\(799\) 1.94138e9 0.134647
\(800\) 0 0
\(801\) −6.41215e9 −0.440849
\(802\) 0 0
\(803\) 3.01899e10 2.05759
\(804\) 0 0
\(805\) 6.63041e9 0.447976
\(806\) 0 0
\(807\) −1.94149e9 −0.130040
\(808\) 0 0
\(809\) −6.64448e9 −0.441206 −0.220603 0.975364i \(-0.570803\pi\)
−0.220603 + 0.975364i \(0.570803\pi\)
\(810\) 0 0
\(811\) −1.83502e10 −1.20800 −0.604000 0.796984i \(-0.706427\pi\)
−0.604000 + 0.796984i \(0.706427\pi\)
\(812\) 0 0
\(813\) −1.45220e10 −0.947784
\(814\) 0 0
\(815\) 2.47165e10 1.59932
\(816\) 0 0
\(817\) 2.07209e8 0.0132933
\(818\) 0 0
\(819\) −4.26213e9 −0.271102
\(820\) 0 0
\(821\) 1.10009e10 0.693788 0.346894 0.937904i \(-0.387236\pi\)
0.346894 + 0.937904i \(0.387236\pi\)
\(822\) 0 0
\(823\) −1.35279e10 −0.845926 −0.422963 0.906147i \(-0.639010\pi\)
−0.422963 + 0.906147i \(0.639010\pi\)
\(824\) 0 0
\(825\) 7.35780e9 0.456204
\(826\) 0 0
\(827\) −1.77028e9 −0.108836 −0.0544181 0.998518i \(-0.517330\pi\)
−0.0544181 + 0.998518i \(0.517330\pi\)
\(828\) 0 0
\(829\) −8.54339e9 −0.520822 −0.260411 0.965498i \(-0.583858\pi\)
−0.260411 + 0.965498i \(0.583858\pi\)
\(830\) 0 0
\(831\) −5.97871e9 −0.361413
\(832\) 0 0
\(833\) 8.46986e9 0.507713
\(834\) 0 0
\(835\) −2.16992e10 −1.28986
\(836\) 0 0
\(837\) −1.36569e9 −0.0805030
\(838\) 0 0
\(839\) 1.23734e10 0.723307 0.361653 0.932313i \(-0.382212\pi\)
0.361653 + 0.932313i \(0.382212\pi\)
\(840\) 0 0
\(841\) −1.15217e10 −0.667930
\(842\) 0 0
\(843\) −5.41138e9 −0.311108
\(844\) 0 0
\(845\) 1.18832e10 0.677537
\(846\) 0 0
\(847\) 3.04479e9 0.172173
\(848\) 0 0
\(849\) 1.51228e10 0.848117
\(850\) 0 0
\(851\) −4.96734e8 −0.0276293
\(852\) 0 0
\(853\) −1.67180e10 −0.922282 −0.461141 0.887327i \(-0.652560\pi\)
−0.461141 + 0.887327i \(0.652560\pi\)
\(854\) 0 0
\(855\) 3.39633e9 0.185835
\(856\) 0 0
\(857\) 3.33011e9 0.180728 0.0903642 0.995909i \(-0.471197\pi\)
0.0903642 + 0.995909i \(0.471197\pi\)
\(858\) 0 0
\(859\) 1.10846e10 0.596682 0.298341 0.954459i \(-0.403567\pi\)
0.298341 + 0.954459i \(0.403567\pi\)
\(860\) 0 0
\(861\) −8.45914e9 −0.451664
\(862\) 0 0
\(863\) −2.94912e10 −1.56191 −0.780954 0.624589i \(-0.785266\pi\)
−0.780954 + 0.624589i \(0.785266\pi\)
\(864\) 0 0
\(865\) 2.13344e10 1.12079
\(866\) 0 0
\(867\) −1.08324e10 −0.564492
\(868\) 0 0
\(869\) 3.46345e10 1.79036
\(870\) 0 0
\(871\) −1.52872e10 −0.783909
\(872\) 0 0
\(873\) 4.99217e9 0.253945
\(874\) 0 0
\(875\) 7.99615e9 0.403508
\(876\) 0 0
\(877\) −3.29461e9 −0.164932 −0.0824660 0.996594i \(-0.526280\pi\)
−0.0824660 + 0.996594i \(0.526280\pi\)
\(878\) 0 0
\(879\) 1.34734e10 0.669139
\(880\) 0 0
\(881\) 1.93619e10 0.953966 0.476983 0.878913i \(-0.341730\pi\)
0.476983 + 0.878913i \(0.341730\pi\)
\(882\) 0 0
\(883\) −2.16110e9 −0.105636 −0.0528181 0.998604i \(-0.516820\pi\)
−0.0528181 + 0.998604i \(0.516820\pi\)
\(884\) 0 0
\(885\) 2.73784e10 1.32772
\(886\) 0 0
\(887\) 3.57751e10 1.72127 0.860633 0.509226i \(-0.170068\pi\)
0.860633 + 0.509226i \(0.170068\pi\)
\(888\) 0 0
\(889\) 3.09259e10 1.47627
\(890\) 0 0
\(891\) −2.51317e9 −0.119028
\(892\) 0 0
\(893\) −8.61721e8 −0.0404936
\(894\) 0 0
\(895\) −1.74467e10 −0.813454
\(896\) 0 0
\(897\) −2.53443e9 −0.117248
\(898\) 0 0
\(899\) 5.25131e9 0.241051
\(900\) 0 0
\(901\) 5.28458e10 2.40699
\(902\) 0 0
\(903\) 4.68423e8 0.0211705
\(904\) 0 0
\(905\) 4.90592e10 2.20014
\(906\) 0 0
\(907\) −1.96026e10 −0.872347 −0.436173 0.899863i \(-0.643667\pi\)
−0.436173 + 0.899863i \(0.643667\pi\)
\(908\) 0 0
\(909\) 7.39191e9 0.326425
\(910\) 0 0
\(911\) −3.00474e10 −1.31672 −0.658359 0.752704i \(-0.728749\pi\)
−0.658359 + 0.752704i \(0.728749\pi\)
\(912\) 0 0
\(913\) −1.44381e10 −0.627861
\(914\) 0 0
\(915\) −1.76276e10 −0.760710
\(916\) 0 0
\(917\) −1.46416e10 −0.627041
\(918\) 0 0
\(919\) −2.48018e10 −1.05409 −0.527046 0.849837i \(-0.676701\pi\)
−0.527046 + 0.849837i \(0.676701\pi\)
\(920\) 0 0
\(921\) −2.21698e10 −0.935091
\(922\) 0 0
\(923\) −2.57471e10 −1.07776
\(924\) 0 0
\(925\) 1.68403e9 0.0699604
\(926\) 0 0
\(927\) −1.12344e10 −0.463202
\(928\) 0 0
\(929\) −2.43504e10 −0.996438 −0.498219 0.867051i \(-0.666012\pi\)
−0.498219 + 0.867051i \(0.666012\pi\)
\(930\) 0 0
\(931\) −3.75952e9 −0.152689
\(932\) 0 0
\(933\) 1.46831e9 0.0591877
\(934\) 0 0
\(935\) 4.96355e10 1.98587
\(936\) 0 0
\(937\) −4.85512e10 −1.92802 −0.964011 0.265864i \(-0.914343\pi\)
−0.964011 + 0.265864i \(0.914343\pi\)
\(938\) 0 0
\(939\) 1.81437e10 0.715150
\(940\) 0 0
\(941\) 2.87097e10 1.12322 0.561610 0.827402i \(-0.310182\pi\)
0.561610 + 0.827402i \(0.310182\pi\)
\(942\) 0 0
\(943\) −5.03014e9 −0.195339
\(944\) 0 0
\(945\) 7.67783e9 0.295956
\(946\) 0 0
\(947\) 4.85339e10 1.85703 0.928517 0.371290i \(-0.121084\pi\)
0.928517 + 0.371290i \(0.121084\pi\)
\(948\) 0 0
\(949\) 3.52549e10 1.33902
\(950\) 0 0
\(951\) 4.04209e9 0.152396
\(952\) 0 0
\(953\) −5.62731e9 −0.210608 −0.105304 0.994440i \(-0.533582\pi\)
−0.105304 + 0.994440i \(0.533582\pi\)
\(954\) 0 0
\(955\) 3.36454e10 1.25001
\(956\) 0 0
\(957\) 9.66357e9 0.356407
\(958\) 0 0
\(959\) −5.92816e10 −2.17047
\(960\) 0 0
\(961\) −2.26985e10 −0.825020
\(962\) 0 0
\(963\) −5.26970e9 −0.190149
\(964\) 0 0
\(965\) 3.92008e10 1.40427
\(966\) 0 0
\(967\) −1.43610e10 −0.510732 −0.255366 0.966844i \(-0.582196\pi\)
−0.255366 + 0.966844i \(0.582196\pi\)
\(968\) 0 0
\(969\) 9.72589e9 0.343397
\(970\) 0 0
\(971\) −3.71664e9 −0.130281 −0.0651407 0.997876i \(-0.520750\pi\)
−0.0651407 + 0.997876i \(0.520750\pi\)
\(972\) 0 0
\(973\) −5.53066e10 −1.92478
\(974\) 0 0
\(975\) 8.59222e9 0.296886
\(976\) 0 0
\(977\) 1.35573e10 0.465096 0.232548 0.972585i \(-0.425294\pi\)
0.232548 + 0.972585i \(0.425294\pi\)
\(978\) 0 0
\(979\) 4.15951e10 1.41678
\(980\) 0 0
\(981\) −8.17715e9 −0.276542
\(982\) 0 0
\(983\) −7.75077e9 −0.260260 −0.130130 0.991497i \(-0.541539\pi\)
−0.130130 + 0.991497i \(0.541539\pi\)
\(984\) 0 0
\(985\) 7.04693e10 2.34949
\(986\) 0 0
\(987\) −1.94803e9 −0.0644890
\(988\) 0 0
\(989\) 2.78543e8 0.00915597
\(990\) 0 0
\(991\) 4.43046e10 1.44608 0.723039 0.690808i \(-0.242745\pi\)
0.723039 + 0.690808i \(0.242745\pi\)
\(992\) 0 0
\(993\) −6.18824e9 −0.200560
\(994\) 0 0
\(995\) −1.52348e10 −0.490292
\(996\) 0 0
\(997\) −3.71271e9 −0.118647 −0.0593236 0.998239i \(-0.518894\pi\)
−0.0593236 + 0.998239i \(0.518894\pi\)
\(998\) 0 0
\(999\) −5.75204e8 −0.0182533
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 384.8.a.i.1.1 3
4.3 odd 2 384.8.a.k.1.1 yes 3
8.3 odd 2 384.8.a.j.1.3 yes 3
8.5 even 2 384.8.a.l.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.8.a.i.1.1 3 1.1 even 1 trivial
384.8.a.j.1.3 yes 3 8.3 odd 2
384.8.a.k.1.1 yes 3 4.3 odd 2
384.8.a.l.1.3 yes 3 8.5 even 2