Properties

Label 105.8.a.a.1.1
Level $105$
Weight $8$
Character 105.1
Self dual yes
Analytic conductor $32.800$
Analytic rank $1$
Dimension $1$
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] = [105,8,Mod(1,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, 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("105.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 105.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: \(32.8004276758\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 105.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -27.0000 q^{3} -124.000 q^{4} -125.000 q^{5} +54.0000 q^{6} +343.000 q^{7} +504.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -27.0000 q^{3} -124.000 q^{4} -125.000 q^{5} +54.0000 q^{6} +343.000 q^{7} +504.000 q^{8} +729.000 q^{9} +250.000 q^{10} +2724.00 q^{11} +3348.00 q^{12} +2874.00 q^{13} -686.000 q^{14} +3375.00 q^{15} +14864.0 q^{16} +9278.00 q^{17} -1458.00 q^{18} -4304.00 q^{19} +15500.0 q^{20} -9261.00 q^{21} -5448.00 q^{22} -41500.0 q^{23} -13608.0 q^{24} +15625.0 q^{25} -5748.00 q^{26} -19683.0 q^{27} -42532.0 q^{28} -35498.0 q^{29} -6750.00 q^{30} -52940.0 q^{31} -94240.0 q^{32} -73548.0 q^{33} -18556.0 q^{34} -42875.0 q^{35} -90396.0 q^{36} -84098.0 q^{37} +8608.00 q^{38} -77598.0 q^{39} -63000.0 q^{40} +180342. q^{41} +18522.0 q^{42} -33452.0 q^{43} -337776. q^{44} -91125.0 q^{45} +83000.0 q^{46} -136120. q^{47} -401328. q^{48} +117649. q^{49} -31250.0 q^{50} -250506. q^{51} -356376. q^{52} -1.27062e6 q^{53} +39366.0 q^{54} -340500. q^{55} +172872. q^{56} +116208. q^{57} +70996.0 q^{58} -1.55325e6 q^{59} -418500. q^{60} +213598. q^{61} +105880. q^{62} +250047. q^{63} -1.71411e6 q^{64} -359250. q^{65} +147096. q^{66} +487228. q^{67} -1.15047e6 q^{68} +1.12050e6 q^{69} +85750.0 q^{70} +1.08600e6 q^{71} +367416. q^{72} -5.92198e6 q^{73} +168196. q^{74} -421875. q^{75} +533696. q^{76} +934332. q^{77} +155196. q^{78} -5.42982e6 q^{79} -1.85800e6 q^{80} +531441. q^{81} -360684. q^{82} +6.93340e6 q^{83} +1.14836e6 q^{84} -1.15975e6 q^{85} +66904.0 q^{86} +958446. q^{87} +1.37290e6 q^{88} +262614. q^{89} +182250. q^{90} +985782. q^{91} +5.14600e6 q^{92} +1.42938e6 q^{93} +272240. q^{94} +538000. q^{95} +2.54448e6 q^{96} -522234. q^{97} -235298. q^{98} +1.98580e6 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.176777 −0.0883883 0.996086i \(-0.528172\pi\)
−0.0883883 + 0.996086i \(0.528172\pi\)
\(3\) −27.0000 −0.577350
\(4\) −124.000 −0.968750
\(5\) −125.000 −0.447214
\(6\) 54.0000 0.102062
\(7\) 343.000 0.377964
\(8\) 504.000 0.348029
\(9\) 729.000 0.333333
\(10\) 250.000 0.0790569
\(11\) 2724.00 0.617068 0.308534 0.951213i \(-0.400162\pi\)
0.308534 + 0.951213i \(0.400162\pi\)
\(12\) 3348.00 0.559308
\(13\) 2874.00 0.362815 0.181407 0.983408i \(-0.441935\pi\)
0.181407 + 0.983408i \(0.441935\pi\)
\(14\) −686.000 −0.0668153
\(15\) 3375.00 0.258199
\(16\) 14864.0 0.907227
\(17\) 9278.00 0.458019 0.229009 0.973424i \(-0.426451\pi\)
0.229009 + 0.973424i \(0.426451\pi\)
\(18\) −1458.00 −0.0589256
\(19\) −4304.00 −0.143958 −0.0719788 0.997406i \(-0.522931\pi\)
−0.0719788 + 0.997406i \(0.522931\pi\)
\(20\) 15500.0 0.433238
\(21\) −9261.00 −0.218218
\(22\) −5448.00 −0.109083
\(23\) −41500.0 −0.711215 −0.355607 0.934635i \(-0.615726\pi\)
−0.355607 + 0.934635i \(0.615726\pi\)
\(24\) −13608.0 −0.200935
\(25\) 15625.0 0.200000
\(26\) −5748.00 −0.0641372
\(27\) −19683.0 −0.192450
\(28\) −42532.0 −0.366153
\(29\) −35498.0 −0.270278 −0.135139 0.990827i \(-0.543148\pi\)
−0.135139 + 0.990827i \(0.543148\pi\)
\(30\) −6750.00 −0.0456435
\(31\) −52940.0 −0.319167 −0.159584 0.987184i \(-0.551015\pi\)
−0.159584 + 0.987184i \(0.551015\pi\)
\(32\) −94240.0 −0.508406
\(33\) −73548.0 −0.356264
\(34\) −18556.0 −0.0809670
\(35\) −42875.0 −0.169031
\(36\) −90396.0 −0.322917
\(37\) −84098.0 −0.272948 −0.136474 0.990644i \(-0.543577\pi\)
−0.136474 + 0.990644i \(0.543577\pi\)
\(38\) 8608.00 0.0254484
\(39\) −77598.0 −0.209471
\(40\) −63000.0 −0.155643
\(41\) 180342. 0.408652 0.204326 0.978903i \(-0.434500\pi\)
0.204326 + 0.978903i \(0.434500\pi\)
\(42\) 18522.0 0.0385758
\(43\) −33452.0 −0.0641627 −0.0320813 0.999485i \(-0.510214\pi\)
−0.0320813 + 0.999485i \(0.510214\pi\)
\(44\) −337776. −0.597784
\(45\) −91125.0 −0.149071
\(46\) 83000.0 0.125726
\(47\) −136120. −0.191240 −0.0956202 0.995418i \(-0.530483\pi\)
−0.0956202 + 0.995418i \(0.530483\pi\)
\(48\) −401328. −0.523788
\(49\) 117649. 0.142857
\(50\) −31250.0 −0.0353553
\(51\) −250506. −0.264437
\(52\) −356376. −0.351477
\(53\) −1.27062e6 −1.17233 −0.586166 0.810191i \(-0.699363\pi\)
−0.586166 + 0.810191i \(0.699363\pi\)
\(54\) 39366.0 0.0340207
\(55\) −340500. −0.275961
\(56\) 172872. 0.131543
\(57\) 116208. 0.0831140
\(58\) 70996.0 0.0477789
\(59\) −1.55325e6 −0.984600 −0.492300 0.870426i \(-0.663844\pi\)
−0.492300 + 0.870426i \(0.663844\pi\)
\(60\) −418500. −0.250130
\(61\) 213598. 0.120488 0.0602439 0.998184i \(-0.480812\pi\)
0.0602439 + 0.998184i \(0.480812\pi\)
\(62\) 105880. 0.0564213
\(63\) 250047. 0.125988
\(64\) −1.71411e6 −0.817352
\(65\) −359250. −0.162256
\(66\) 147096. 0.0629792
\(67\) 487228. 0.197911 0.0989556 0.995092i \(-0.468450\pi\)
0.0989556 + 0.995092i \(0.468450\pi\)
\(68\) −1.15047e6 −0.443706
\(69\) 1.12050e6 0.410620
\(70\) 85750.0 0.0298807
\(71\) 1.08600e6 0.360102 0.180051 0.983657i \(-0.442374\pi\)
0.180051 + 0.983657i \(0.442374\pi\)
\(72\) 367416. 0.116010
\(73\) −5.92198e6 −1.78171 −0.890855 0.454289i \(-0.849893\pi\)
−0.890855 + 0.454289i \(0.849893\pi\)
\(74\) 168196. 0.0482508
\(75\) −421875. −0.115470
\(76\) 533696. 0.139459
\(77\) 934332. 0.233230
\(78\) 155196. 0.0370296
\(79\) −5.42982e6 −1.23906 −0.619528 0.784975i \(-0.712676\pi\)
−0.619528 + 0.784975i \(0.712676\pi\)
\(80\) −1.85800e6 −0.405724
\(81\) 531441. 0.111111
\(82\) −360684. −0.0722401
\(83\) 6.93340e6 1.33099 0.665493 0.746405i \(-0.268222\pi\)
0.665493 + 0.746405i \(0.268222\pi\)
\(84\) 1.14836e6 0.211399
\(85\) −1.15975e6 −0.204832
\(86\) 66904.0 0.0113425
\(87\) 958446. 0.156045
\(88\) 1.37290e6 0.214757
\(89\) 262614. 0.0394869 0.0197434 0.999805i \(-0.493715\pi\)
0.0197434 + 0.999805i \(0.493715\pi\)
\(90\) 182250. 0.0263523
\(91\) 985782. 0.137131
\(92\) 5.14600e6 0.688989
\(93\) 1.42938e6 0.184271
\(94\) 272240. 0.0338068
\(95\) 538000. 0.0643798
\(96\) 2.54448e6 0.293528
\(97\) −522234. −0.0580984 −0.0290492 0.999578i \(-0.509248\pi\)
−0.0290492 + 0.999578i \(0.509248\pi\)
\(98\) −235298. −0.0252538
\(99\) 1.98580e6 0.205689
\(100\) −1.93750e6 −0.193750
\(101\) −1.63961e6 −0.158349 −0.0791744 0.996861i \(-0.525228\pi\)
−0.0791744 + 0.996861i \(0.525228\pi\)
\(102\) 501012. 0.0467463
\(103\) −6.57506e6 −0.592883 −0.296442 0.955051i \(-0.595800\pi\)
−0.296442 + 0.955051i \(0.595800\pi\)
\(104\) 1.44850e6 0.126270
\(105\) 1.15762e6 0.0975900
\(106\) 2.54124e6 0.207241
\(107\) 1.21480e7 0.958654 0.479327 0.877636i \(-0.340881\pi\)
0.479327 + 0.877636i \(0.340881\pi\)
\(108\) 2.44069e6 0.186436
\(109\) −9.68452e6 −0.716284 −0.358142 0.933667i \(-0.616590\pi\)
−0.358142 + 0.933667i \(0.616590\pi\)
\(110\) 681000. 0.0487835
\(111\) 2.27065e6 0.157586
\(112\) 5.09835e6 0.342899
\(113\) 2.50296e7 1.63184 0.815922 0.578162i \(-0.196230\pi\)
0.815922 + 0.578162i \(0.196230\pi\)
\(114\) −232416. −0.0146926
\(115\) 5.18750e6 0.318065
\(116\) 4.40175e6 0.261832
\(117\) 2.09515e6 0.120938
\(118\) 3.10650e6 0.174054
\(119\) 3.18235e6 0.173115
\(120\) 1.70100e6 0.0898607
\(121\) −1.20670e7 −0.619228
\(122\) −427196. −0.0212994
\(123\) −4.86923e6 −0.235935
\(124\) 6.56456e6 0.309193
\(125\) −1.95312e6 −0.0894427
\(126\) −500094. −0.0222718
\(127\) 3.05960e6 0.132541 0.0662707 0.997802i \(-0.478890\pi\)
0.0662707 + 0.997802i \(0.478890\pi\)
\(128\) 1.54909e7 0.652894
\(129\) 903204. 0.0370443
\(130\) 718500. 0.0286830
\(131\) −3.56912e6 −0.138711 −0.0693555 0.997592i \(-0.522094\pi\)
−0.0693555 + 0.997592i \(0.522094\pi\)
\(132\) 9.11995e6 0.345131
\(133\) −1.47627e6 −0.0544109
\(134\) −974456. −0.0349861
\(135\) 2.46038e6 0.0860663
\(136\) 4.67611e6 0.159404
\(137\) 4.83736e6 0.160726 0.0803630 0.996766i \(-0.474392\pi\)
0.0803630 + 0.996766i \(0.474392\pi\)
\(138\) −2.24100e6 −0.0725880
\(139\) 2.48573e6 0.0785058 0.0392529 0.999229i \(-0.487502\pi\)
0.0392529 + 0.999229i \(0.487502\pi\)
\(140\) 5.31650e6 0.163749
\(141\) 3.67524e6 0.110413
\(142\) −2.17200e6 −0.0636577
\(143\) 7.82878e6 0.223881
\(144\) 1.08359e7 0.302409
\(145\) 4.43725e6 0.120872
\(146\) 1.18440e7 0.314965
\(147\) −3.17652e6 −0.0824786
\(148\) 1.04282e7 0.264418
\(149\) −3.99895e7 −0.990362 −0.495181 0.868790i \(-0.664898\pi\)
−0.495181 + 0.868790i \(0.664898\pi\)
\(150\) 843750. 0.0204124
\(151\) −5.37919e7 −1.27145 −0.635723 0.771918i \(-0.719298\pi\)
−0.635723 + 0.771918i \(0.719298\pi\)
\(152\) −2.16922e6 −0.0501014
\(153\) 6.76366e6 0.152673
\(154\) −1.86866e6 −0.0412296
\(155\) 6.61750e6 0.142736
\(156\) 9.62215e6 0.202925
\(157\) 9.64019e6 0.198809 0.0994046 0.995047i \(-0.468306\pi\)
0.0994046 + 0.995047i \(0.468306\pi\)
\(158\) 1.08596e7 0.219036
\(159\) 3.43068e7 0.676847
\(160\) 1.17800e7 0.227366
\(161\) −1.42345e7 −0.268814
\(162\) −1.06288e6 −0.0196419
\(163\) −7.67343e7 −1.38782 −0.693910 0.720062i \(-0.744113\pi\)
−0.693910 + 0.720062i \(0.744113\pi\)
\(164\) −2.23624e7 −0.395881
\(165\) 9.19350e6 0.159326
\(166\) −1.38668e7 −0.235287
\(167\) 1.03856e7 0.172554 0.0862770 0.996271i \(-0.472503\pi\)
0.0862770 + 0.996271i \(0.472503\pi\)
\(168\) −4.66754e6 −0.0759462
\(169\) −5.44886e7 −0.868365
\(170\) 2.31950e6 0.0362096
\(171\) −3.13762e6 −0.0479859
\(172\) 4.14805e6 0.0621576
\(173\) −5.64927e6 −0.0829528 −0.0414764 0.999139i \(-0.513206\pi\)
−0.0414764 + 0.999139i \(0.513206\pi\)
\(174\) −1.91689e6 −0.0275851
\(175\) 5.35938e6 0.0755929
\(176\) 4.04895e7 0.559820
\(177\) 4.19378e7 0.568459
\(178\) −525228. −0.00698036
\(179\) −1.04373e8 −1.36020 −0.680098 0.733121i \(-0.738063\pi\)
−0.680098 + 0.733121i \(0.738063\pi\)
\(180\) 1.12995e7 0.144413
\(181\) −1.42262e8 −1.78326 −0.891628 0.452769i \(-0.850436\pi\)
−0.891628 + 0.452769i \(0.850436\pi\)
\(182\) −1.97156e6 −0.0242416
\(183\) −5.76715e6 −0.0695636
\(184\) −2.09160e7 −0.247523
\(185\) 1.05122e7 0.122066
\(186\) −2.85876e6 −0.0325748
\(187\) 2.52733e7 0.282628
\(188\) 1.68789e7 0.185264
\(189\) −6.75127e6 −0.0727393
\(190\) −1.07600e6 −0.0113808
\(191\) −3.35533e7 −0.348433 −0.174216 0.984707i \(-0.555739\pi\)
−0.174216 + 0.984707i \(0.555739\pi\)
\(192\) 4.62810e7 0.471899
\(193\) −1.70599e8 −1.70815 −0.854073 0.520153i \(-0.825875\pi\)
−0.854073 + 0.520153i \(0.825875\pi\)
\(194\) 1.04447e6 0.0102704
\(195\) 9.69975e6 0.0936784
\(196\) −1.45885e7 −0.138393
\(197\) 6.92904e7 0.645716 0.322858 0.946447i \(-0.395356\pi\)
0.322858 + 0.946447i \(0.395356\pi\)
\(198\) −3.97159e6 −0.0363611
\(199\) −6.71910e7 −0.604401 −0.302201 0.953244i \(-0.597721\pi\)
−0.302201 + 0.953244i \(0.597721\pi\)
\(200\) 7.87500e6 0.0696058
\(201\) −1.31552e7 −0.114264
\(202\) 3.27921e6 0.0279924
\(203\) −1.21758e7 −0.102156
\(204\) 3.10627e7 0.256174
\(205\) −2.25428e7 −0.182755
\(206\) 1.31501e7 0.104808
\(207\) −3.02535e7 −0.237072
\(208\) 4.27191e7 0.329155
\(209\) −1.17241e7 −0.0888316
\(210\) −2.31525e6 −0.0172516
\(211\) −5.14714e7 −0.377205 −0.188602 0.982054i \(-0.560396\pi\)
−0.188602 + 0.982054i \(0.560396\pi\)
\(212\) 1.57557e8 1.13570
\(213\) −2.93220e7 −0.207905
\(214\) −2.42960e7 −0.169468
\(215\) 4.18150e6 0.0286944
\(216\) −9.92023e6 −0.0669782
\(217\) −1.81584e7 −0.120634
\(218\) 1.93690e7 0.126622
\(219\) 1.59893e8 1.02867
\(220\) 4.22220e7 0.267337
\(221\) 2.66650e7 0.166176
\(222\) −4.54129e6 −0.0278576
\(223\) −2.49372e8 −1.50585 −0.752923 0.658109i \(-0.771357\pi\)
−0.752923 + 0.658109i \(0.771357\pi\)
\(224\) −3.23243e7 −0.192159
\(225\) 1.13906e7 0.0666667
\(226\) −5.00591e7 −0.288472
\(227\) 3.93928e7 0.223525 0.111762 0.993735i \(-0.464350\pi\)
0.111762 + 0.993735i \(0.464350\pi\)
\(228\) −1.44098e7 −0.0805167
\(229\) 1.25945e8 0.693035 0.346517 0.938044i \(-0.387364\pi\)
0.346517 + 0.938044i \(0.387364\pi\)
\(230\) −1.03750e7 −0.0562265
\(231\) −2.52270e7 −0.134655
\(232\) −1.78910e7 −0.0940647
\(233\) −1.55290e8 −0.804265 −0.402132 0.915582i \(-0.631731\pi\)
−0.402132 + 0.915582i \(0.631731\pi\)
\(234\) −4.19029e6 −0.0213791
\(235\) 1.70150e7 0.0855253
\(236\) 1.92603e8 0.953831
\(237\) 1.46605e8 0.715369
\(238\) −6.36471e6 −0.0306027
\(239\) −1.53561e8 −0.727593 −0.363797 0.931478i \(-0.618520\pi\)
−0.363797 + 0.931478i \(0.618520\pi\)
\(240\) 5.01660e7 0.234245
\(241\) 1.67184e8 0.769369 0.384685 0.923048i \(-0.374310\pi\)
0.384685 + 0.923048i \(0.374310\pi\)
\(242\) 2.41340e7 0.109465
\(243\) −1.43489e7 −0.0641500
\(244\) −2.64862e7 −0.116722
\(245\) −1.47061e7 −0.0638877
\(246\) 9.73847e6 0.0417078
\(247\) −1.23697e7 −0.0522300
\(248\) −2.66818e7 −0.111079
\(249\) −1.87202e8 −0.768445
\(250\) 3.90625e6 0.0158114
\(251\) −4.29028e7 −0.171249 −0.0856245 0.996327i \(-0.527289\pi\)
−0.0856245 + 0.996327i \(0.527289\pi\)
\(252\) −3.10058e7 −0.122051
\(253\) −1.13046e8 −0.438867
\(254\) −6.11920e6 −0.0234302
\(255\) 3.13132e7 0.118260
\(256\) 1.88424e8 0.701936
\(257\) 4.70837e8 1.73023 0.865116 0.501571i \(-0.167245\pi\)
0.865116 + 0.501571i \(0.167245\pi\)
\(258\) −1.80641e6 −0.00654858
\(259\) −2.88456e7 −0.103165
\(260\) 4.45470e7 0.157185
\(261\) −2.58780e7 −0.0900927
\(262\) 7.13823e6 0.0245209
\(263\) −1.02607e8 −0.347802 −0.173901 0.984763i \(-0.555637\pi\)
−0.173901 + 0.984763i \(0.555637\pi\)
\(264\) −3.70682e7 −0.123990
\(265\) 1.58828e8 0.524283
\(266\) 2.95254e6 0.00961857
\(267\) −7.09058e6 −0.0227978
\(268\) −6.04163e7 −0.191727
\(269\) −5.49478e8 −1.72114 −0.860571 0.509330i \(-0.829893\pi\)
−0.860571 + 0.509330i \(0.829893\pi\)
\(270\) −4.92075e6 −0.0152145
\(271\) 3.38205e8 1.03225 0.516127 0.856512i \(-0.327373\pi\)
0.516127 + 0.856512i \(0.327373\pi\)
\(272\) 1.37908e8 0.415527
\(273\) −2.66161e7 −0.0791727
\(274\) −9.67472e6 −0.0284126
\(275\) 4.25625e7 0.123414
\(276\) −1.38942e8 −0.397788
\(277\) 3.85029e8 1.08846 0.544232 0.838934i \(-0.316821\pi\)
0.544232 + 0.838934i \(0.316821\pi\)
\(278\) −4.97146e6 −0.0138780
\(279\) −3.85933e7 −0.106389
\(280\) −2.16090e7 −0.0588277
\(281\) 3.58731e8 0.964488 0.482244 0.876037i \(-0.339822\pi\)
0.482244 + 0.876037i \(0.339822\pi\)
\(282\) −7.35048e6 −0.0195184
\(283\) 5.81811e7 0.152591 0.0762955 0.997085i \(-0.475691\pi\)
0.0762955 + 0.997085i \(0.475691\pi\)
\(284\) −1.34664e8 −0.348849
\(285\) −1.45260e7 −0.0371697
\(286\) −1.56576e7 −0.0395770
\(287\) 6.18573e7 0.154456
\(288\) −6.87010e7 −0.169469
\(289\) −3.24257e8 −0.790219
\(290\) −8.87450e6 −0.0213674
\(291\) 1.41003e7 0.0335431
\(292\) 7.34325e8 1.72603
\(293\) 5.57054e8 1.29378 0.646890 0.762583i \(-0.276069\pi\)
0.646890 + 0.762583i \(0.276069\pi\)
\(294\) 6.35305e6 0.0145803
\(295\) 1.94156e8 0.440327
\(296\) −4.23854e7 −0.0949938
\(297\) −5.36165e7 −0.118755
\(298\) 7.99790e7 0.175073
\(299\) −1.19271e8 −0.258039
\(300\) 5.23125e7 0.111862
\(301\) −1.14740e7 −0.0242512
\(302\) 1.07584e8 0.224762
\(303\) 4.42694e7 0.0914227
\(304\) −6.39747e7 −0.130602
\(305\) −2.66998e7 −0.0538837
\(306\) −1.35273e7 −0.0269890
\(307\) 5.19798e8 1.02530 0.512649 0.858598i \(-0.328664\pi\)
0.512649 + 0.858598i \(0.328664\pi\)
\(308\) −1.15857e8 −0.225941
\(309\) 1.77527e8 0.342301
\(310\) −1.32350e7 −0.0252324
\(311\) −3.93391e8 −0.741589 −0.370794 0.928715i \(-0.620915\pi\)
−0.370794 + 0.928715i \(0.620915\pi\)
\(312\) −3.91094e7 −0.0729021
\(313\) 7.46643e8 1.37628 0.688142 0.725576i \(-0.258427\pi\)
0.688142 + 0.725576i \(0.258427\pi\)
\(314\) −1.92804e7 −0.0351448
\(315\) −3.12559e7 −0.0563436
\(316\) 6.73298e8 1.20034
\(317\) 1.90220e8 0.335389 0.167694 0.985839i \(-0.446368\pi\)
0.167694 + 0.985839i \(0.446368\pi\)
\(318\) −6.86136e7 −0.119651
\(319\) −9.66966e7 −0.166780
\(320\) 2.14264e8 0.365531
\(321\) −3.27996e8 −0.553479
\(322\) 2.84690e7 0.0475200
\(323\) −3.99325e7 −0.0659353
\(324\) −6.58987e7 −0.107639
\(325\) 4.49063e7 0.0725630
\(326\) 1.53469e8 0.245334
\(327\) 2.61482e8 0.413547
\(328\) 9.08924e7 0.142223
\(329\) −4.66892e7 −0.0722820
\(330\) −1.83870e7 −0.0281652
\(331\) −1.09993e9 −1.66712 −0.833559 0.552430i \(-0.813701\pi\)
−0.833559 + 0.552430i \(0.813701\pi\)
\(332\) −8.59742e8 −1.28939
\(333\) −6.13074e7 −0.0909826
\(334\) −2.07712e7 −0.0305035
\(335\) −6.09035e7 −0.0885086
\(336\) −1.37656e8 −0.197973
\(337\) 4.94657e8 0.704043 0.352022 0.935992i \(-0.385494\pi\)
0.352022 + 0.935992i \(0.385494\pi\)
\(338\) 1.08977e8 0.153507
\(339\) −6.75798e8 −0.942146
\(340\) 1.43809e8 0.198431
\(341\) −1.44209e8 −0.196948
\(342\) 6.27523e6 0.00848278
\(343\) 4.03536e7 0.0539949
\(344\) −1.68598e7 −0.0223305
\(345\) −1.40062e8 −0.183635
\(346\) 1.12985e7 0.0146641
\(347\) −9.31091e8 −1.19630 −0.598148 0.801385i \(-0.704097\pi\)
−0.598148 + 0.801385i \(0.704097\pi\)
\(348\) −1.18847e8 −0.151169
\(349\) −4.56646e8 −0.575031 −0.287515 0.957776i \(-0.592829\pi\)
−0.287515 + 0.957776i \(0.592829\pi\)
\(350\) −1.07188e7 −0.0133631
\(351\) −5.65689e7 −0.0698237
\(352\) −2.56710e8 −0.313721
\(353\) −5.02682e8 −0.608250 −0.304125 0.952632i \(-0.598364\pi\)
−0.304125 + 0.952632i \(0.598364\pi\)
\(354\) −8.38756e7 −0.100490
\(355\) −1.35750e8 −0.161043
\(356\) −3.25641e7 −0.0382529
\(357\) −8.59236e7 −0.0999479
\(358\) 2.08746e8 0.240451
\(359\) −1.39665e9 −1.59316 −0.796578 0.604536i \(-0.793359\pi\)
−0.796578 + 0.604536i \(0.793359\pi\)
\(360\) −4.59270e7 −0.0518811
\(361\) −8.75347e8 −0.979276
\(362\) 2.84524e8 0.315238
\(363\) 3.25809e8 0.357511
\(364\) −1.22237e8 −0.132846
\(365\) 7.40247e8 0.796804
\(366\) 1.15343e7 0.0122972
\(367\) −2.70500e8 −0.285651 −0.142826 0.989748i \(-0.545619\pi\)
−0.142826 + 0.989748i \(0.545619\pi\)
\(368\) −6.16856e8 −0.645233
\(369\) 1.31469e8 0.136217
\(370\) −2.10245e7 −0.0215784
\(371\) −4.35823e8 −0.443100
\(372\) −1.77243e8 −0.178513
\(373\) −1.06422e9 −1.06182 −0.530912 0.847427i \(-0.678150\pi\)
−0.530912 + 0.847427i \(0.678150\pi\)
\(374\) −5.05465e7 −0.0499621
\(375\) 5.27344e7 0.0516398
\(376\) −6.86045e7 −0.0665572
\(377\) −1.02021e8 −0.0980609
\(378\) 1.35025e7 0.0128586
\(379\) −3.95456e8 −0.373131 −0.186565 0.982443i \(-0.559736\pi\)
−0.186565 + 0.982443i \(0.559736\pi\)
\(380\) −6.67120e7 −0.0623679
\(381\) −8.26092e7 −0.0765228
\(382\) 6.71067e7 0.0615948
\(383\) 1.64793e8 0.149880 0.0749400 0.997188i \(-0.476123\pi\)
0.0749400 + 0.997188i \(0.476123\pi\)
\(384\) −4.18255e8 −0.376949
\(385\) −1.16792e8 −0.104303
\(386\) 3.41197e8 0.301960
\(387\) −2.43865e7 −0.0213876
\(388\) 6.47570e7 0.0562828
\(389\) −1.71125e8 −0.147398 −0.0736989 0.997281i \(-0.523480\pi\)
−0.0736989 + 0.997281i \(0.523480\pi\)
\(390\) −1.93995e7 −0.0165602
\(391\) −3.85037e8 −0.325750
\(392\) 5.92951e7 0.0497184
\(393\) 9.63661e7 0.0800849
\(394\) −1.38581e8 −0.114147
\(395\) 6.78728e8 0.554123
\(396\) −2.46239e8 −0.199261
\(397\) 1.73552e9 1.39207 0.696037 0.718006i \(-0.254945\pi\)
0.696037 + 0.718006i \(0.254945\pi\)
\(398\) 1.34382e8 0.106844
\(399\) 3.98593e7 0.0314141
\(400\) 2.32250e8 0.181445
\(401\) 2.26343e8 0.175292 0.0876458 0.996152i \(-0.472066\pi\)
0.0876458 + 0.996152i \(0.472066\pi\)
\(402\) 2.63103e7 0.0201992
\(403\) −1.52150e8 −0.115799
\(404\) 2.03311e8 0.153400
\(405\) −6.64301e7 −0.0496904
\(406\) 2.43516e7 0.0180587
\(407\) −2.29083e8 −0.168427
\(408\) −1.26255e8 −0.0920318
\(409\) 1.14869e9 0.830176 0.415088 0.909781i \(-0.363751\pi\)
0.415088 + 0.909781i \(0.363751\pi\)
\(410\) 4.50855e7 0.0323068
\(411\) −1.30609e8 −0.0927952
\(412\) 8.15307e8 0.574356
\(413\) −5.32765e8 −0.372144
\(414\) 6.05070e7 0.0419087
\(415\) −8.66676e8 −0.595235
\(416\) −2.70846e8 −0.184457
\(417\) −6.71147e7 −0.0453254
\(418\) 2.34482e7 0.0157034
\(419\) 1.98365e9 1.31740 0.658699 0.752407i \(-0.271107\pi\)
0.658699 + 0.752407i \(0.271107\pi\)
\(420\) −1.43546e8 −0.0945403
\(421\) 2.66045e9 1.73767 0.868835 0.495103i \(-0.164870\pi\)
0.868835 + 0.495103i \(0.164870\pi\)
\(422\) 1.02943e8 0.0666810
\(423\) −9.92315e7 −0.0637468
\(424\) −6.40393e8 −0.408006
\(425\) 1.44969e8 0.0916037
\(426\) 5.86440e7 0.0367528
\(427\) 7.32641e7 0.0455401
\(428\) −1.50635e9 −0.928696
\(429\) −2.11377e8 −0.129258
\(430\) −8.36300e6 −0.00507251
\(431\) −1.50478e9 −0.905322 −0.452661 0.891683i \(-0.649525\pi\)
−0.452661 + 0.891683i \(0.649525\pi\)
\(432\) −2.92568e8 −0.174596
\(433\) 1.37803e9 0.815738 0.407869 0.913040i \(-0.366272\pi\)
0.407869 + 0.913040i \(0.366272\pi\)
\(434\) 3.63168e7 0.0213252
\(435\) −1.19806e8 −0.0697855
\(436\) 1.20088e9 0.693900
\(437\) 1.78616e8 0.102385
\(438\) −3.19787e8 −0.181845
\(439\) 2.15016e9 1.21296 0.606478 0.795100i \(-0.292582\pi\)
0.606478 + 0.795100i \(0.292582\pi\)
\(440\) −1.71612e8 −0.0960425
\(441\) 8.57661e7 0.0476190
\(442\) −5.33299e7 −0.0293760
\(443\) −1.01094e9 −0.552474 −0.276237 0.961090i \(-0.589087\pi\)
−0.276237 + 0.961090i \(0.589087\pi\)
\(444\) −2.81560e8 −0.152662
\(445\) −3.28268e7 −0.0176591
\(446\) 4.98744e8 0.266198
\(447\) 1.07972e9 0.571786
\(448\) −5.87940e8 −0.308930
\(449\) 1.82648e8 0.0952256 0.0476128 0.998866i \(-0.484839\pi\)
0.0476128 + 0.998866i \(0.484839\pi\)
\(450\) −2.27812e7 −0.0117851
\(451\) 4.91252e8 0.252166
\(452\) −3.10367e9 −1.58085
\(453\) 1.45238e9 0.734069
\(454\) −7.87855e7 −0.0395140
\(455\) −1.23223e8 −0.0613269
\(456\) 5.85688e7 0.0289261
\(457\) −1.51611e6 −0.000743060 0 −0.000371530 1.00000i \(-0.500118\pi\)
−0.000371530 1.00000i \(0.500118\pi\)
\(458\) −2.51889e8 −0.122512
\(459\) −1.82619e8 −0.0881457
\(460\) −6.43250e8 −0.308125
\(461\) −4.79940e8 −0.228157 −0.114079 0.993472i \(-0.536392\pi\)
−0.114079 + 0.993472i \(0.536392\pi\)
\(462\) 5.04539e7 0.0238039
\(463\) −6.86812e8 −0.321591 −0.160796 0.986988i \(-0.551406\pi\)
−0.160796 + 0.986988i \(0.551406\pi\)
\(464\) −5.27642e8 −0.245203
\(465\) −1.78672e8 −0.0824086
\(466\) 3.10581e8 0.142175
\(467\) 1.69004e9 0.767869 0.383934 0.923360i \(-0.374569\pi\)
0.383934 + 0.923360i \(0.374569\pi\)
\(468\) −2.59798e8 −0.117159
\(469\) 1.67119e8 0.0748034
\(470\) −3.40300e7 −0.0151189
\(471\) −2.60285e8 −0.114783
\(472\) −7.82839e8 −0.342670
\(473\) −9.11232e7 −0.0395927
\(474\) −2.93210e8 −0.126461
\(475\) −6.72500e7 −0.0287915
\(476\) −3.94612e8 −0.167705
\(477\) −9.26283e8 −0.390778
\(478\) 3.07122e8 0.128622
\(479\) 2.39843e9 0.997132 0.498566 0.866852i \(-0.333860\pi\)
0.498566 + 0.866852i \(0.333860\pi\)
\(480\) −3.18060e8 −0.131270
\(481\) −2.41698e8 −0.0990295
\(482\) −3.34368e8 −0.136007
\(483\) 3.84332e8 0.155200
\(484\) 1.49631e9 0.599877
\(485\) 6.52792e7 0.0259824
\(486\) 2.86978e7 0.0113402
\(487\) −6.76814e8 −0.265533 −0.132766 0.991147i \(-0.542386\pi\)
−0.132766 + 0.991147i \(0.542386\pi\)
\(488\) 1.07653e8 0.0419332
\(489\) 2.07183e9 0.801258
\(490\) 2.94122e7 0.0112938
\(491\) 3.97038e9 1.51372 0.756862 0.653574i \(-0.226731\pi\)
0.756862 + 0.653574i \(0.226731\pi\)
\(492\) 6.03785e8 0.228562
\(493\) −3.29350e8 −0.123792
\(494\) 2.47394e7 0.00923304
\(495\) −2.48224e8 −0.0919870
\(496\) −7.86900e8 −0.289557
\(497\) 3.72498e8 0.136106
\(498\) 3.74404e8 0.135843
\(499\) 1.59757e9 0.575584 0.287792 0.957693i \(-0.407079\pi\)
0.287792 + 0.957693i \(0.407079\pi\)
\(500\) 2.42188e8 0.0866476
\(501\) −2.80412e8 −0.0996241
\(502\) 8.58056e7 0.0302728
\(503\) 1.72479e9 0.604295 0.302148 0.953261i \(-0.402296\pi\)
0.302148 + 0.953261i \(0.402296\pi\)
\(504\) 1.26024e8 0.0438475
\(505\) 2.04951e8 0.0708157
\(506\) 2.26092e8 0.0775815
\(507\) 1.47119e9 0.501351
\(508\) −3.79390e8 −0.128400
\(509\) 5.83626e9 1.96165 0.980827 0.194880i \(-0.0624316\pi\)
0.980827 + 0.194880i \(0.0624316\pi\)
\(510\) −6.26265e7 −0.0209056
\(511\) −2.03124e9 −0.673423
\(512\) −2.35969e9 −0.776980
\(513\) 8.47156e7 0.0277047
\(514\) −9.41674e8 −0.305865
\(515\) 8.21882e8 0.265145
\(516\) −1.11997e8 −0.0358867
\(517\) −3.70791e8 −0.118008
\(518\) 5.76912e7 0.0182371
\(519\) 1.52530e8 0.0478928
\(520\) −1.81062e8 −0.0564697
\(521\) 4.99457e9 1.54727 0.773634 0.633632i \(-0.218437\pi\)
0.773634 + 0.633632i \(0.218437\pi\)
\(522\) 5.17561e7 0.0159263
\(523\) 1.03723e9 0.317045 0.158522 0.987355i \(-0.449327\pi\)
0.158522 + 0.987355i \(0.449327\pi\)
\(524\) 4.42570e8 0.134376
\(525\) −1.44703e8 −0.0436436
\(526\) 2.05214e8 0.0614834
\(527\) −4.91177e8 −0.146184
\(528\) −1.09322e9 −0.323212
\(529\) −1.68258e9 −0.494174
\(530\) −3.17655e8 −0.0926810
\(531\) −1.13232e9 −0.328200
\(532\) 1.83058e8 0.0527105
\(533\) 5.18303e8 0.148265
\(534\) 1.41812e7 0.00403011
\(535\) −1.51850e9 −0.428723
\(536\) 2.45563e8 0.0688789
\(537\) 2.81807e9 0.785310
\(538\) 1.09896e9 0.304258
\(539\) 3.20476e8 0.0881525
\(540\) −3.05086e8 −0.0833767
\(541\) −2.48336e9 −0.674295 −0.337147 0.941452i \(-0.609462\pi\)
−0.337147 + 0.941452i \(0.609462\pi\)
\(542\) −6.76409e8 −0.182479
\(543\) 3.84107e9 1.02956
\(544\) −8.74359e8 −0.232859
\(545\) 1.21057e9 0.320332
\(546\) 5.32322e7 0.0139959
\(547\) −3.74874e9 −0.979331 −0.489665 0.871910i \(-0.662881\pi\)
−0.489665 + 0.871910i \(0.662881\pi\)
\(548\) −5.99832e8 −0.155703
\(549\) 1.55713e8 0.0401626
\(550\) −8.51250e7 −0.0218166
\(551\) 1.52783e8 0.0389086
\(552\) 5.64732e8 0.142908
\(553\) −1.86243e9 −0.468319
\(554\) −7.70058e8 −0.192415
\(555\) −2.83831e8 −0.0704748
\(556\) −3.08230e8 −0.0760525
\(557\) −2.18032e9 −0.534596 −0.267298 0.963614i \(-0.586131\pi\)
−0.267298 + 0.963614i \(0.586131\pi\)
\(558\) 7.71865e7 0.0188071
\(559\) −9.61410e7 −0.0232792
\(560\) −6.37294e8 −0.153349
\(561\) −6.82378e8 −0.163176
\(562\) −7.17462e8 −0.170499
\(563\) 2.56548e9 0.605883 0.302941 0.953009i \(-0.402031\pi\)
0.302941 + 0.953009i \(0.402031\pi\)
\(564\) −4.55730e8 −0.106962
\(565\) −3.12869e9 −0.729783
\(566\) −1.16362e8 −0.0269745
\(567\) 1.82284e8 0.0419961
\(568\) 5.47344e8 0.125326
\(569\) 1.65916e8 0.0377568 0.0188784 0.999822i \(-0.493990\pi\)
0.0188784 + 0.999822i \(0.493990\pi\)
\(570\) 2.90520e7 0.00657074
\(571\) −5.11928e9 −1.15075 −0.575377 0.817888i \(-0.695145\pi\)
−0.575377 + 0.817888i \(0.695145\pi\)
\(572\) −9.70768e8 −0.216885
\(573\) 9.05940e8 0.201168
\(574\) −1.23715e8 −0.0273042
\(575\) −6.48438e8 −0.142243
\(576\) −1.24959e9 −0.272451
\(577\) −6.90669e9 −1.49677 −0.748384 0.663265i \(-0.769170\pi\)
−0.748384 + 0.663265i \(0.769170\pi\)
\(578\) 6.48515e8 0.139692
\(579\) 4.60616e9 0.986199
\(580\) −5.50219e8 −0.117095
\(581\) 2.37816e9 0.503065
\(582\) −2.82006e7 −0.00592964
\(583\) −3.46117e9 −0.723408
\(584\) −2.98468e9 −0.620087
\(585\) −2.61893e8 −0.0540852
\(586\) −1.11411e9 −0.228710
\(587\) −4.92686e9 −1.00540 −0.502698 0.864462i \(-0.667659\pi\)
−0.502698 + 0.864462i \(0.667659\pi\)
\(588\) 3.93889e8 0.0799012
\(589\) 2.27854e8 0.0459465
\(590\) −3.88313e8 −0.0778395
\(591\) −1.87084e9 −0.372804
\(592\) −1.25003e9 −0.247626
\(593\) 2.48990e9 0.490332 0.245166 0.969481i \(-0.421158\pi\)
0.245166 + 0.969481i \(0.421158\pi\)
\(594\) 1.07233e8 0.0209931
\(595\) −3.97794e8 −0.0774193
\(596\) 4.95870e9 0.959413
\(597\) 1.81416e9 0.348951
\(598\) 2.38542e8 0.0456153
\(599\) −1.79975e9 −0.342151 −0.171076 0.985258i \(-0.554724\pi\)
−0.171076 + 0.985258i \(0.554724\pi\)
\(600\) −2.12625e8 −0.0401869
\(601\) −9.64250e9 −1.81188 −0.905939 0.423409i \(-0.860834\pi\)
−0.905939 + 0.423409i \(0.860834\pi\)
\(602\) 2.29481e7 0.00428705
\(603\) 3.55189e8 0.0659704
\(604\) 6.67020e9 1.23171
\(605\) 1.50837e9 0.276927
\(606\) −8.85387e7 −0.0161614
\(607\) −7.59240e9 −1.37790 −0.688951 0.724808i \(-0.741929\pi\)
−0.688951 + 0.724808i \(0.741929\pi\)
\(608\) 4.05609e8 0.0731889
\(609\) 3.28747e8 0.0589795
\(610\) 5.33995e7 0.00952539
\(611\) −3.91209e8 −0.0693848
\(612\) −8.38694e8 −0.147902
\(613\) 6.21966e9 1.09057 0.545287 0.838249i \(-0.316421\pi\)
0.545287 + 0.838249i \(0.316421\pi\)
\(614\) −1.03960e9 −0.181249
\(615\) 6.08654e8 0.105513
\(616\) 4.70903e8 0.0811707
\(617\) −1.01931e10 −1.74706 −0.873529 0.486772i \(-0.838174\pi\)
−0.873529 + 0.486772i \(0.838174\pi\)
\(618\) −3.55053e8 −0.0605109
\(619\) −1.86525e9 −0.316097 −0.158048 0.987431i \(-0.550520\pi\)
−0.158048 + 0.987431i \(0.550520\pi\)
\(620\) −8.20570e8 −0.138275
\(621\) 8.16844e8 0.136873
\(622\) 7.86782e8 0.131096
\(623\) 9.00766e7 0.0149246
\(624\) −1.15342e9 −0.190038
\(625\) 2.44141e8 0.0400000
\(626\) −1.49329e9 −0.243295
\(627\) 3.16551e8 0.0512869
\(628\) −1.19538e9 −0.192596
\(629\) −7.80261e8 −0.125015
\(630\) 6.25118e7 0.00996024
\(631\) 3.03048e9 0.480185 0.240092 0.970750i \(-0.422822\pi\)
0.240092 + 0.970750i \(0.422822\pi\)
\(632\) −2.73663e9 −0.431228
\(633\) 1.38973e9 0.217779
\(634\) −3.80440e8 −0.0592889
\(635\) −3.82450e8 −0.0592743
\(636\) −4.25404e9 −0.655695
\(637\) 3.38123e8 0.0518307
\(638\) 1.93393e8 0.0294828
\(639\) 7.91694e8 0.120034
\(640\) −1.93637e9 −0.291983
\(641\) 1.08550e10 1.62789 0.813945 0.580942i \(-0.197316\pi\)
0.813945 + 0.580942i \(0.197316\pi\)
\(642\) 6.55992e8 0.0978422
\(643\) 9.35593e8 0.138787 0.0693934 0.997589i \(-0.477894\pi\)
0.0693934 + 0.997589i \(0.477894\pi\)
\(644\) 1.76508e9 0.260413
\(645\) −1.12900e8 −0.0165667
\(646\) 7.98650e7 0.0116558
\(647\) −8.82519e9 −1.28103 −0.640514 0.767946i \(-0.721279\pi\)
−0.640514 + 0.767946i \(0.721279\pi\)
\(648\) 2.67846e8 0.0386699
\(649\) −4.23106e9 −0.607565
\(650\) −8.98125e7 −0.0128274
\(651\) 4.90277e8 0.0696480
\(652\) 9.51506e9 1.34445
\(653\) 2.99850e9 0.421413 0.210707 0.977549i \(-0.432424\pi\)
0.210707 + 0.977549i \(0.432424\pi\)
\(654\) −5.22964e8 −0.0731055
\(655\) 4.46140e8 0.0620335
\(656\) 2.68060e9 0.370740
\(657\) −4.31712e9 −0.593903
\(658\) 9.33783e7 0.0127778
\(659\) −3.93595e9 −0.535736 −0.267868 0.963456i \(-0.586319\pi\)
−0.267868 + 0.963456i \(0.586319\pi\)
\(660\) −1.13999e9 −0.154347
\(661\) −6.26839e8 −0.0844211 −0.0422106 0.999109i \(-0.513440\pi\)
−0.0422106 + 0.999109i \(0.513440\pi\)
\(662\) 2.19986e9 0.294708
\(663\) −7.19954e8 −0.0959417
\(664\) 3.49444e9 0.463222
\(665\) 1.84534e8 0.0243333
\(666\) 1.22615e8 0.0160836
\(667\) 1.47317e9 0.192226
\(668\) −1.28782e9 −0.167162
\(669\) 6.73304e9 0.869400
\(670\) 1.21807e8 0.0156463
\(671\) 5.81841e8 0.0743491
\(672\) 8.72757e8 0.110943
\(673\) 4.74113e9 0.599555 0.299777 0.954009i \(-0.403088\pi\)
0.299777 + 0.954009i \(0.403088\pi\)
\(674\) −9.89313e8 −0.124458
\(675\) −3.07547e8 −0.0384900
\(676\) 6.75659e9 0.841229
\(677\) −8.34660e9 −1.03383 −0.516915 0.856037i \(-0.672920\pi\)
−0.516915 + 0.856037i \(0.672920\pi\)
\(678\) 1.35160e9 0.166549
\(679\) −1.79126e8 −0.0219591
\(680\) −5.84514e8 −0.0712876
\(681\) −1.06360e9 −0.129052
\(682\) 2.88417e8 0.0348157
\(683\) 1.00158e10 1.20286 0.601428 0.798927i \(-0.294599\pi\)
0.601428 + 0.798927i \(0.294599\pi\)
\(684\) 3.89064e8 0.0464863
\(685\) −6.04670e8 −0.0718789
\(686\) −8.07072e7 −0.00954504
\(687\) −3.40050e9 −0.400124
\(688\) −4.97231e8 −0.0582101
\(689\) −3.65177e9 −0.425340
\(690\) 2.80125e8 0.0324624
\(691\) −1.16744e10 −1.34605 −0.673024 0.739621i \(-0.735005\pi\)
−0.673024 + 0.739621i \(0.735005\pi\)
\(692\) 7.00509e8 0.0803605
\(693\) 6.81128e8 0.0777432
\(694\) 1.86218e9 0.211477
\(695\) −3.10716e8 −0.0351089
\(696\) 4.83057e8 0.0543083
\(697\) 1.67321e9 0.187170
\(698\) 9.13293e8 0.101652
\(699\) 4.19284e9 0.464342
\(700\) −6.64562e8 −0.0732306
\(701\) 7.05584e9 0.773634 0.386817 0.922157i \(-0.373575\pi\)
0.386817 + 0.922157i \(0.373575\pi\)
\(702\) 1.13138e8 0.0123432
\(703\) 3.61958e8 0.0392929
\(704\) −4.66924e9 −0.504362
\(705\) −4.59405e8 −0.0493780
\(706\) 1.00536e9 0.107524
\(707\) −5.62385e8 −0.0598502
\(708\) −5.20029e9 −0.550695
\(709\) 1.96477e8 0.0207038 0.0103519 0.999946i \(-0.496705\pi\)
0.0103519 + 0.999946i \(0.496705\pi\)
\(710\) 2.71500e8 0.0284686
\(711\) −3.95834e9 −0.413019
\(712\) 1.32357e8 0.0137426
\(713\) 2.19701e9 0.226996
\(714\) 1.71847e8 0.0176685
\(715\) −9.78597e8 −0.100123
\(716\) 1.29422e10 1.31769
\(717\) 4.14615e9 0.420076
\(718\) 2.79331e9 0.281633
\(719\) −8.96572e9 −0.899569 −0.449784 0.893137i \(-0.648499\pi\)
−0.449784 + 0.893137i \(0.648499\pi\)
\(720\) −1.35448e9 −0.135241
\(721\) −2.25524e9 −0.224089
\(722\) 1.75069e9 0.173113
\(723\) −4.51397e9 −0.444196
\(724\) 1.76405e10 1.72753
\(725\) −5.54656e8 −0.0540556
\(726\) −6.51618e8 −0.0631997
\(727\) −4.44082e9 −0.428640 −0.214320 0.976763i \(-0.568754\pi\)
−0.214320 + 0.976763i \(0.568754\pi\)
\(728\) 4.96834e8 0.0477256
\(729\) 3.87420e8 0.0370370
\(730\) −1.48049e9 −0.140856
\(731\) −3.10368e8 −0.0293877
\(732\) 7.15126e8 0.0673898
\(733\) −8.97113e9 −0.841363 −0.420681 0.907208i \(-0.638209\pi\)
−0.420681 + 0.907208i \(0.638209\pi\)
\(734\) 5.41000e8 0.0504965
\(735\) 3.97065e8 0.0368856
\(736\) 3.91096e9 0.361586
\(737\) 1.32721e9 0.122125
\(738\) −2.62939e8 −0.0240800
\(739\) 1.27011e10 1.15767 0.578835 0.815445i \(-0.303508\pi\)
0.578835 + 0.815445i \(0.303508\pi\)
\(740\) −1.30352e9 −0.118251
\(741\) 3.33982e8 0.0301550
\(742\) 8.71647e8 0.0783298
\(743\) −1.54200e10 −1.37919 −0.689593 0.724197i \(-0.742211\pi\)
−0.689593 + 0.724197i \(0.742211\pi\)
\(744\) 7.20408e8 0.0641317
\(745\) 4.99869e9 0.442903
\(746\) 2.12845e9 0.187706
\(747\) 5.05445e9 0.443662
\(748\) −3.13389e9 −0.273796
\(749\) 4.16677e9 0.362337
\(750\) −1.05469e8 −0.00912871
\(751\) 4.99561e9 0.430376 0.215188 0.976573i \(-0.430964\pi\)
0.215188 + 0.976573i \(0.430964\pi\)
\(752\) −2.02329e9 −0.173498
\(753\) 1.15838e9 0.0988706
\(754\) 2.04043e8 0.0173349
\(755\) 6.72399e9 0.568608
\(756\) 8.37157e8 0.0704662
\(757\) 1.13781e10 0.953313 0.476657 0.879090i \(-0.341848\pi\)
0.476657 + 0.879090i \(0.341848\pi\)
\(758\) 7.90912e8 0.0659608
\(759\) 3.05224e9 0.253380
\(760\) 2.71152e8 0.0224060
\(761\) 9.31372e9 0.766084 0.383042 0.923731i \(-0.374876\pi\)
0.383042 + 0.923731i \(0.374876\pi\)
\(762\) 1.65218e8 0.0135275
\(763\) −3.32179e9 −0.270730
\(764\) 4.16061e9 0.337544
\(765\) −8.45458e8 −0.0682774
\(766\) −3.29586e8 −0.0264953
\(767\) −4.46405e9 −0.357228
\(768\) −5.08746e9 −0.405263
\(769\) −5.75726e9 −0.456535 −0.228267 0.973598i \(-0.573306\pi\)
−0.228267 + 0.973598i \(0.573306\pi\)
\(770\) 2.33583e8 0.0184384
\(771\) −1.27126e10 −0.998950
\(772\) 2.11542e10 1.65477
\(773\) 1.45658e10 1.13424 0.567121 0.823634i \(-0.308057\pi\)
0.567121 + 0.823634i \(0.308057\pi\)
\(774\) 4.87730e7 0.00378082
\(775\) −8.27188e8 −0.0638334
\(776\) −2.63206e8 −0.0202199
\(777\) 7.78832e8 0.0595621
\(778\) 3.42251e8 0.0260565
\(779\) −7.76192e8 −0.0588285
\(780\) −1.20277e9 −0.0907509
\(781\) 2.95826e9 0.222207
\(782\) 7.70074e8 0.0575849
\(783\) 6.98707e8 0.0520150
\(784\) 1.74873e9 0.129604
\(785\) −1.20502e9 −0.0889102
\(786\) −1.92732e8 −0.0141571
\(787\) 7.81273e9 0.571336 0.285668 0.958329i \(-0.407785\pi\)
0.285668 + 0.958329i \(0.407785\pi\)
\(788\) −8.59201e9 −0.625537
\(789\) 2.77039e9 0.200804
\(790\) −1.35746e9 −0.0979560
\(791\) 8.58514e9 0.616779
\(792\) 1.00084e9 0.0715858
\(793\) 6.13881e8 0.0437147
\(794\) −3.47103e9 −0.246086
\(795\) −4.28835e9 −0.302695
\(796\) 8.33168e9 0.585514
\(797\) 5.72212e9 0.400362 0.200181 0.979759i \(-0.435847\pi\)
0.200181 + 0.979759i \(0.435847\pi\)
\(798\) −7.97187e7 −0.00555329
\(799\) −1.26292e9 −0.0875916
\(800\) −1.47250e9 −0.101681
\(801\) 1.91446e8 0.0131623
\(802\) −4.52685e8 −0.0309875
\(803\) −1.61315e10 −1.09943
\(804\) 1.63124e9 0.110693
\(805\) 1.77931e9 0.120217
\(806\) 3.04299e8 0.0204705
\(807\) 1.48359e10 0.993702
\(808\) −8.26361e8 −0.0551100
\(809\) −9.16070e9 −0.608287 −0.304144 0.952626i \(-0.598370\pi\)
−0.304144 + 0.952626i \(0.598370\pi\)
\(810\) 1.32860e8 0.00878410
\(811\) −2.04091e10 −1.34354 −0.671771 0.740759i \(-0.734466\pi\)
−0.671771 + 0.740759i \(0.734466\pi\)
\(812\) 1.50980e9 0.0989632
\(813\) −9.13152e9 −0.595973
\(814\) 4.58166e8 0.0297740
\(815\) 9.59179e9 0.620652
\(816\) −3.72352e9 −0.239904
\(817\) 1.43977e8 0.00923671
\(818\) −2.29738e9 −0.146756
\(819\) 7.18635e8 0.0457104
\(820\) 2.79530e9 0.177044
\(821\) 6.26680e9 0.395225 0.197613 0.980280i \(-0.436681\pi\)
0.197613 + 0.980280i \(0.436681\pi\)
\(822\) 2.61217e8 0.0164040
\(823\) 4.37792e9 0.273759 0.136879 0.990588i \(-0.456293\pi\)
0.136879 + 0.990588i \(0.456293\pi\)
\(824\) −3.31383e9 −0.206341
\(825\) −1.14919e9 −0.0712528
\(826\) 1.06553e9 0.0657864
\(827\) −1.61665e10 −0.993912 −0.496956 0.867776i \(-0.665549\pi\)
−0.496956 + 0.867776i \(0.665549\pi\)
\(828\) 3.75143e9 0.229663
\(829\) 2.76523e10 1.68574 0.842868 0.538120i \(-0.180865\pi\)
0.842868 + 0.538120i \(0.180865\pi\)
\(830\) 1.73335e9 0.105224
\(831\) −1.03958e10 −0.628425
\(832\) −4.92636e9 −0.296548
\(833\) 1.09155e9 0.0654312
\(834\) 1.34229e8 0.00801247
\(835\) −1.29820e9 −0.0771685
\(836\) 1.45379e9 0.0860556
\(837\) 1.04202e9 0.0614237
\(838\) −3.96731e9 −0.232885
\(839\) −1.72137e10 −1.00625 −0.503126 0.864213i \(-0.667817\pi\)
−0.503126 + 0.864213i \(0.667817\pi\)
\(840\) 5.83443e8 0.0339642
\(841\) −1.59898e10 −0.926950
\(842\) −5.32089e9 −0.307179
\(843\) −9.68574e9 −0.556847
\(844\) 6.38245e9 0.365417
\(845\) 6.81108e9 0.388345
\(846\) 1.98463e8 0.0112689
\(847\) −4.13898e9 −0.234046
\(848\) −1.88865e10 −1.06357
\(849\) −1.57089e9 −0.0880985
\(850\) −2.89938e8 −0.0161934
\(851\) 3.49007e9 0.194124
\(852\) 3.63593e9 0.201408
\(853\) 6.71439e9 0.370412 0.185206 0.982700i \(-0.440705\pi\)
0.185206 + 0.982700i \(0.440705\pi\)
\(854\) −1.46528e8 −0.00805042
\(855\) 3.92202e8 0.0214599
\(856\) 6.12260e9 0.333639
\(857\) −9.89153e9 −0.536822 −0.268411 0.963304i \(-0.586499\pi\)
−0.268411 + 0.963304i \(0.586499\pi\)
\(858\) 4.22754e8 0.0228498
\(859\) −3.35625e10 −1.80667 −0.903335 0.428937i \(-0.858888\pi\)
−0.903335 + 0.428937i \(0.858888\pi\)
\(860\) −5.18506e8 −0.0277977
\(861\) −1.67015e9 −0.0891751
\(862\) 3.00956e9 0.160040
\(863\) 9.99427e9 0.529314 0.264657 0.964343i \(-0.414741\pi\)
0.264657 + 0.964343i \(0.414741\pi\)
\(864\) 1.85493e9 0.0978427
\(865\) 7.06159e8 0.0370976
\(866\) −2.75606e9 −0.144203
\(867\) 8.75495e9 0.456233
\(868\) 2.25164e9 0.116864
\(869\) −1.47908e10 −0.764581
\(870\) 2.39611e8 0.0123365
\(871\) 1.40029e9 0.0718051
\(872\) −4.88100e9 −0.249288
\(873\) −3.80709e8 −0.0193661
\(874\) −3.57232e8 −0.0180992
\(875\) −6.69922e8 −0.0338062
\(876\) −1.98268e10 −0.996524
\(877\) 1.24239e8 0.00621953 0.00310977 0.999995i \(-0.499010\pi\)
0.00310977 + 0.999995i \(0.499010\pi\)
\(878\) −4.30032e9 −0.214422
\(879\) −1.50405e10 −0.746964
\(880\) −5.06119e9 −0.250359
\(881\) 2.45438e10 1.20928 0.604640 0.796499i \(-0.293317\pi\)
0.604640 + 0.796499i \(0.293317\pi\)
\(882\) −1.71532e8 −0.00841794
\(883\) 9.35492e9 0.457275 0.228638 0.973512i \(-0.426573\pi\)
0.228638 + 0.973512i \(0.426573\pi\)
\(884\) −3.30646e9 −0.160983
\(885\) −5.24223e9 −0.254223
\(886\) 2.02188e9 0.0976646
\(887\) −3.18320e10 −1.53155 −0.765776 0.643107i \(-0.777645\pi\)
−0.765776 + 0.643107i \(0.777645\pi\)
\(888\) 1.14441e9 0.0548447
\(889\) 1.04944e9 0.0500959
\(890\) 6.56535e7 0.00312171
\(891\) 1.44765e9 0.0685631
\(892\) 3.09221e10 1.45879
\(893\) 5.85860e8 0.0275305
\(894\) −2.15943e9 −0.101078
\(895\) 1.30466e10 0.608298
\(896\) 5.31339e9 0.246771
\(897\) 3.22032e9 0.148979
\(898\) −3.65297e8 −0.0168337
\(899\) 1.87926e9 0.0862639
\(900\) −1.41244e9 −0.0645833
\(901\) −1.17888e10 −0.536950
\(902\) −9.82503e8 −0.0445770
\(903\) 3.09799e8 0.0140014
\(904\) 1.26149e10 0.567929
\(905\) 1.77827e10 0.797496
\(906\) −2.90476e9 −0.129766
\(907\) 2.00342e10 0.891552 0.445776 0.895144i \(-0.352928\pi\)
0.445776 + 0.895144i \(0.352928\pi\)
\(908\) −4.88470e9 −0.216540
\(909\) −1.19527e9 −0.0527829
\(910\) 2.46446e8 0.0108412
\(911\) −2.82239e10 −1.23681 −0.618405 0.785859i \(-0.712221\pi\)
−0.618405 + 0.785859i \(0.712221\pi\)
\(912\) 1.72732e9 0.0754032
\(913\) 1.88866e10 0.821308
\(914\) 3.03222e6 0.000131356 0
\(915\) 7.20893e8 0.0311098
\(916\) −1.56171e10 −0.671377
\(917\) −1.22421e9 −0.0524279
\(918\) 3.65238e8 0.0155821
\(919\) −3.40984e10 −1.44921 −0.724603 0.689167i \(-0.757977\pi\)
−0.724603 + 0.689167i \(0.757977\pi\)
\(920\) 2.61450e9 0.110696
\(921\) −1.40345e10 −0.591956
\(922\) 9.59881e8 0.0403329
\(923\) 3.12116e9 0.130650
\(924\) 3.12814e9 0.130447
\(925\) −1.31403e9 −0.0545896
\(926\) 1.37362e9 0.0568499
\(927\) −4.79322e9 −0.197628
\(928\) 3.34533e9 0.137411
\(929\) −2.18397e10 −0.893698 −0.446849 0.894609i \(-0.647454\pi\)
−0.446849 + 0.894609i \(0.647454\pi\)
\(930\) 3.57345e8 0.0145679
\(931\) −5.06361e8 −0.0205654
\(932\) 1.92560e10 0.779131
\(933\) 1.06216e10 0.428157
\(934\) −3.38007e9 −0.135741
\(935\) −3.15916e9 −0.126395
\(936\) 1.05595e9 0.0420900
\(937\) −1.29359e10 −0.513699 −0.256849 0.966451i \(-0.582684\pi\)
−0.256849 + 0.966451i \(0.582684\pi\)
\(938\) −3.34238e8 −0.0132235
\(939\) −2.01594e10 −0.794598
\(940\) −2.10986e9 −0.0828526
\(941\) −1.70996e10 −0.668995 −0.334497 0.942397i \(-0.608567\pi\)
−0.334497 + 0.942397i \(0.608567\pi\)
\(942\) 5.20570e8 0.0202909
\(943\) −7.48419e9 −0.290639
\(944\) −2.30875e10 −0.893255
\(945\) 8.43909e8 0.0325300
\(946\) 1.82246e8 0.00699907
\(947\) −1.95022e8 −0.00746207 −0.00373104 0.999993i \(-0.501188\pi\)
−0.00373104 + 0.999993i \(0.501188\pi\)
\(948\) −1.81791e10 −0.693014
\(949\) −1.70198e10 −0.646430
\(950\) 1.34500e8 0.00508967
\(951\) −5.13594e9 −0.193637
\(952\) 1.60391e9 0.0602490
\(953\) −1.97958e10 −0.740878 −0.370439 0.928857i \(-0.620793\pi\)
−0.370439 + 0.928857i \(0.620793\pi\)
\(954\) 1.85257e9 0.0690804
\(955\) 4.19417e9 0.155824
\(956\) 1.90416e10 0.704856
\(957\) 2.61081e9 0.0962904
\(958\) −4.79686e9 −0.176270
\(959\) 1.65921e9 0.0607487
\(960\) −5.78513e9 −0.211039
\(961\) −2.47100e10 −0.898132
\(962\) 4.83395e8 0.0175061
\(963\) 8.85590e9 0.319551
\(964\) −2.07308e10 −0.745327
\(965\) 2.13248e10 0.763906
\(966\) −7.68663e8 −0.0274357
\(967\) 4.18825e10 1.48950 0.744748 0.667346i \(-0.232570\pi\)
0.744748 + 0.667346i \(0.232570\pi\)
\(968\) −6.08177e9 −0.215509
\(969\) 1.07818e9 0.0380677
\(970\) −1.30559e8 −0.00459308
\(971\) −1.82766e9 −0.0640661 −0.0320331 0.999487i \(-0.510198\pi\)
−0.0320331 + 0.999487i \(0.510198\pi\)
\(972\) 1.77926e9 0.0621453
\(973\) 8.52605e8 0.0296724
\(974\) 1.35363e9 0.0469400
\(975\) −1.21247e9 −0.0418942
\(976\) 3.17492e9 0.109310
\(977\) 4.04873e10 1.38895 0.694477 0.719515i \(-0.255636\pi\)
0.694477 + 0.719515i \(0.255636\pi\)
\(978\) −4.14365e9 −0.141644
\(979\) 7.15361e8 0.0243661
\(980\) 1.82356e9 0.0618912
\(981\) −7.06002e9 −0.238761
\(982\) −7.94076e9 −0.267591
\(983\) 1.79913e9 0.0604124 0.0302062 0.999544i \(-0.490384\pi\)
0.0302062 + 0.999544i \(0.490384\pi\)
\(984\) −2.45409e9 −0.0821123
\(985\) −8.66130e9 −0.288773
\(986\) 6.58701e8 0.0218836
\(987\) 1.26061e9 0.0417321
\(988\) 1.53384e9 0.0505978
\(989\) 1.38826e9 0.0456334
\(990\) 4.96449e8 0.0162612
\(991\) −1.80926e10 −0.590532 −0.295266 0.955415i \(-0.595408\pi\)
−0.295266 + 0.955415i \(0.595408\pi\)
\(992\) 4.98907e9 0.162266
\(993\) 2.96981e10 0.962511
\(994\) −7.44996e8 −0.0240603
\(995\) 8.39888e9 0.270296
\(996\) 2.32130e10 0.744431
\(997\) 4.79210e10 1.53141 0.765707 0.643189i \(-0.222389\pi\)
0.765707 + 0.643189i \(0.222389\pi\)
\(998\) −3.19514e9 −0.101750
\(999\) 1.65530e9 0.0525288
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 105.8.a.a.1.1 1
3.2 odd 2 315.8.a.b.1.1 1
5.4 even 2 525.8.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.8.a.a.1.1 1 1.1 even 1 trivial
315.8.a.b.1.1 1 3.2 odd 2
525.8.a.c.1.1 1 5.4 even 2