Properties

Label 99.8.g.b
Level $99$
Weight $8$
Character orbit 99.g
Analytic conductor $30.926$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,8,Mod(32,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.32");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 99.g (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(30.9261175229\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(80\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 126 q^{3} - 5378 q^{4} + 1728 q^{5} + 2490 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 126 q^{3} - 5378 q^{4} + 1728 q^{5} + 2490 q^{9} - 9408 q^{11} + 21906 q^{12} - 6 q^{14} + 54138 q^{15} - 278786 q^{16} + 140922 q^{20} + 35511 q^{22} + 140628 q^{23} + 935196 q^{25} + 391626 q^{27} + 60260 q^{31} - 288312 q^{33} - 177714 q^{34} + 925488 q^{36} - 1276732 q^{37} + 888852 q^{38} + 1782594 q^{42} + 881562 q^{45} + 2958090 q^{47} + 7466664 q^{48} + 9882514 q^{49} + 745436 q^{55} - 2569764 q^{56} - 691746 q^{58} + 12473748 q^{59} - 301842 q^{60} + 44106232 q^{64} - 11217708 q^{66} + 14336 q^{67} - 388500 q^{69} + 5740056 q^{70} - 36174936 q^{75} + 6403392 q^{77} - 20423958 q^{78} - 7870254 q^{81} - 3789672 q^{82} - 3411414 q^{86} + 12330003 q^{88} + 13655508 q^{91} - 39357234 q^{92} + 23243640 q^{93} + 33974312 q^{97} - 12557952 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −11.1004 19.2264i −37.9674 + 27.3034i −182.436 + 315.989i 403.826 + 233.149i 946.398 + 426.898i −544.859 + 314.574i 5258.74 696.044 2073.28i 10352.2i
32.2 −10.9145 18.9044i 19.0635 + 42.7034i −174.251 + 301.812i −50.9103 29.3931i 599.215 826.469i 369.937 213.583i 4813.32 −1460.17 + 1628.15i 1283.24i
32.3 −10.6254 18.4037i −18.8318 42.8061i −161.797 + 280.241i −170.697 98.5517i −587.695 + 801.406i −1048.79 + 605.518i 4156.53 −1477.72 + 1612.24i 4188.60i
32.4 −10.3142 17.8647i −32.7089 33.4234i −148.766 + 257.670i 158.720 + 91.6372i −259.734 + 929.073i 1129.80 652.288i 3497.17 −47.2531 + 2186.49i 3780.66i
32.5 −10.2303 17.7194i 46.3518 6.20590i −145.318 + 251.697i −400.321 231.125i −584.157 757.836i 34.4910 19.9134i 3327.61 2109.97 575.309i 9457.91i
32.6 −10.0424 17.3940i 20.7995 41.8853i −137.700 + 238.504i 16.8339 + 9.71907i −937.430 + 58.8446i 647.365 373.757i 2960.52 −1321.76 1742.39i 390.412i
32.7 −9.98017 17.2862i 45.8450 9.23254i −135.208 + 234.187i 164.830 + 95.1644i −617.136 700.341i −1091.68 + 630.280i 2842.66 2016.52 846.531i 3799.03i
32.8 −9.77690 16.9341i −45.9977 + 8.43870i −127.176 + 220.275i −347.525 200.644i 592.617 + 696.425i −609.165 + 351.701i 2470.64 2044.58 776.322i 7846.70i
32.9 −9.57900 16.5913i −33.1843 + 32.9515i −119.515 + 207.005i −220.726 127.437i 864.582 + 234.928i 1258.64 726.678i 2127.09 15.3931 2186.95i 4882.86i
32.10 −8.88768 15.3939i 30.6308 + 35.3377i −93.9816 + 162.781i 174.032 + 100.477i 271.748 785.598i −782.137 + 451.567i 1065.87 −310.506 + 2164.85i 3572.03i
32.11 −8.88000 15.3806i 45.3135 + 11.5622i −93.7087 + 162.308i 275.259 + 158.921i −224.551 799.622i 1305.79 753.901i 1055.25 1919.63 + 1047.84i 5644.87i
32.12 −8.83322 15.2996i 19.9399 42.3013i −92.0515 + 159.438i 394.120 + 227.545i −823.326 + 68.5842i −358.591 + 207.033i 991.141 −1391.80 1686.97i 8039.83i
32.13 −8.37654 14.5086i −6.54719 + 46.3048i −76.3327 + 132.212i −145.108 83.7783i 726.660 292.883i −87.4072 + 50.4646i 413.220 −2101.27 606.333i 2807.09i
32.14 −8.26723 14.3193i 13.1032 44.8922i −72.6941 + 125.910i −289.726 167.273i −751.149 + 183.506i 245.896 141.968i 287.505 −1843.61 1176.46i 5531.55i
32.15 −7.84085 13.5808i −46.7654 0.00852324i −58.9579 + 102.118i 219.976 + 127.003i 366.565 + 635.176i 156.676 90.4568i −158.138 2187.00 + 0.797185i 3983.26i
32.16 −7.83042 13.5627i −41.3361 21.8707i −58.6309 + 101.552i 17.6188 + 10.1722i 27.0537 + 731.885i −402.625 + 232.456i −168.169 1230.35 + 1808.10i 318.611i
32.17 −7.47708 12.9507i −23.8681 + 40.2158i −47.8133 + 82.8151i 100.829 + 58.2136i 699.286 + 8.41095i −1316.64 + 760.165i −484.116 −1047.63 1919.75i 1741.07i
32.18 −7.05474 12.2192i −8.49600 + 45.9872i −35.5388 + 61.5551i 425.574 + 245.705i 621.862 220.614i 1040.43 600.695i −803.145 −2042.64 781.414i 6933.54i
32.19 −6.89273 11.9386i 38.0471 + 27.1922i −31.0195 + 53.7274i −387.432 223.684i 62.3867 641.657i −707.122 + 408.257i −909.302 708.171 + 2069.17i 6167.18i
32.20 −6.59267 11.4188i −31.0750 34.9477i −22.9267 + 39.7102i −451.295 260.555i −194.196 + 585.240i 1470.13 848.780i −1083.13 −255.690 + 2172.00i 6871.03i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
11.b odd 2 1 inner
99.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.8.g.b 160
9.d odd 6 1 inner 99.8.g.b 160
11.b odd 2 1 inner 99.8.g.b 160
99.g even 6 1 inner 99.8.g.b 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.8.g.b 160 1.a even 1 1 trivial
99.8.g.b 160 9.d odd 6 1 inner
99.8.g.b 160 11.b odd 2 1 inner
99.8.g.b 160 99.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{160} + 7809 T_{2}^{158} + 31731585 T_{2}^{156} + 88448970114 T_{2}^{154} + 188883079962540 T_{2}^{152} + \cdots + 11\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display