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Properties

Label 161.4.e.b
Level $161$
Weight $4$
Character orbit 161.e
Analytic conductor $9.499$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [161,4,Mod(93,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.93");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 161.e (of order \(3\), 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: \(9.49930751092\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 18 q^{3} - 88 q^{4} + 20 q^{5} - 72 q^{6} - 44 q^{7} + 42 q^{8} - 126 q^{9} + 200 q^{10} - 20 q^{11} + 161 q^{12} - 392 q^{13} - 200 q^{14} + 40 q^{15} - 324 q^{16} + 242 q^{17} - 85 q^{18} + 128 q^{19}+ \cdots + 7140 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1 −2.74905 + 4.76150i −0.108392 − 0.187741i −11.1146 − 19.2510i 3.87133 − 6.70534i 1.19191 −3.00557 + 18.2748i 78.2332 13.4765 − 23.3420i 21.2850 + 36.8666i
93.2 −2.50376 + 4.33663i 3.33221 + 5.77156i −8.53760 − 14.7876i 6.94437 − 12.0280i −33.3722 −11.6755 − 14.3765i 45.4442 −8.70730 + 15.0815i 34.7740 + 60.2303i
93.3 −2.24392 + 3.88659i 4.45090 + 7.70919i −6.07039 − 10.5142i −1.04214 + 1.80505i −39.9500 17.8722 + 4.85657i 18.5832 −26.1211 + 45.2431i −4.67699 − 8.10078i
93.4 −2.08967 + 3.61941i −0.0901550 − 0.156153i −4.73340 − 8.19849i −6.23886 + 10.8060i 0.753575 1.04212 − 18.4909i 6.13025 13.4837 − 23.3545i −26.0742 − 45.1619i
93.5 −1.91224 + 3.31210i −3.77771 − 6.54318i −3.31333 − 5.73885i 9.23546 − 15.9963i 28.8955 17.3800 + 6.39809i −5.25234 −15.0422 + 26.0538i 35.3208 + 61.1775i
93.6 −1.68180 + 2.91297i −2.66618 − 4.61796i −1.65692 − 2.86987i 1.69468 − 2.93527i 17.9360 −18.5039 − 0.777724i −15.7624 −0.717036 + 1.24194i 5.70022 + 9.87308i
93.7 −1.19240 + 2.06529i −0.833094 − 1.44296i 1.15638 + 2.00290i −7.27589 + 12.6022i 3.97352 7.15265 + 17.0833i −24.5938 12.1119 − 20.9784i −17.3515 − 30.0537i
93.8 −1.09145 + 1.89045i 3.24856 + 5.62667i 1.61746 + 2.80153i −5.29818 + 9.17672i −14.1826 4.63307 + 17.9314i −24.5248 −7.60630 + 13.1745i −11.5654 − 20.0319i
93.9 −0.904891 + 1.56732i 4.25067 + 7.36238i 2.36235 + 4.09170i 10.8661 − 18.8206i −15.3856 −11.8667 + 14.2190i −23.0289 −22.6364 + 39.2075i 19.6653 + 34.0613i
93.10 −0.646697 + 1.12011i 0.748479 + 1.29640i 3.16357 + 5.47946i 6.70559 − 11.6144i −1.93616 8.99941 − 16.1867i −18.5306 12.3796 − 21.4420i 8.67296 + 15.0220i
93.11 0.248608 − 0.430602i −2.90835 − 5.03742i 3.87639 + 6.71410i 0.0115248 − 0.0199615i −2.89216 −12.3503 + 13.8011i 7.83254 −3.41704 + 5.91848i −0.00573031 − 0.00992519i
93.12 0.319148 − 0.552781i 2.55230 + 4.42071i 3.79629 + 6.57537i −2.96147 + 5.12941i 3.25824 −15.8891 + 9.51503i 9.95268 0.471554 − 0.816755i 1.89029 + 3.27409i
93.13 0.331592 − 0.574335i 0.441679 + 0.765010i 3.78009 + 6.54731i −0.460024 + 0.796786i 0.585829 −6.62409 − 17.2951i 10.3193 13.1098 − 22.7069i 0.305081 + 0.528416i
93.14 1.02001 − 1.76671i −2.44043 − 4.22696i 1.91915 + 3.32406i −2.20595 + 3.82082i −9.95709 18.4927 − 1.00908i 24.1504 1.58857 − 2.75148i 4.50019 + 7.79456i
93.15 1.11997 − 1.93985i −4.51048 − 7.81238i 1.49133 + 2.58306i −7.56441 + 13.1019i −20.2064 −11.3350 − 14.6464i 24.6005 −27.1888 + 47.0924i 16.9438 + 29.3476i
93.16 1.25818 − 2.17923i 3.59738 + 6.23085i 0.833966 + 1.44447i 6.85867 − 11.8796i 18.1046 7.12391 − 17.0953i 24.3280 −12.3823 + 21.4468i −17.2589 − 29.8932i
93.17 1.46890 − 2.54421i −1.64566 − 2.85037i −0.315335 − 0.546176i 9.62172 − 16.6653i −9.66926 15.8074 + 9.65013i 21.6496 8.08358 − 14.0012i −28.2667 − 48.9593i
93.18 1.56576 − 2.71198i 4.73453 + 8.20044i −0.903236 − 1.56445i −6.67323 + 11.5584i 29.6526 −18.0608 − 4.09965i 19.3952 −31.3315 + 54.2678i 20.8974 + 36.1954i
93.19 2.16079 − 3.74260i 2.14374 + 3.71307i −5.33806 − 9.24579i −4.70179 + 8.14373i 18.5287 16.6938 − 8.01969i −11.5651 4.30873 − 7.46295i 20.3192 + 35.1938i
93.20 2.26824 − 3.92871i −3.43053 − 5.94186i −6.28983 − 10.8943i 3.43720 − 5.95341i −31.1251 −17.2896 + 6.63846i −20.7755 −10.0371 + 17.3848i −15.5928 − 27.0075i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.4.e.b ✓ 44
7.c even 3 1 inner 161.4.e.b ✓ 44
7.c even 3 1 1127.4.a.j 22
7.d odd 6 1 1127.4.a.m 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.4.e.b ✓ 44 1.a even 1 1 trivial
161.4.e.b ✓ 44 7.c even 3 1 inner
1127.4.a.j 22 7.c even 3 1
1127.4.a.m 22 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{44} + 132 T_{2}^{42} - 14 T_{2}^{41} + 10025 T_{2}^{40} - 1811 T_{2}^{39} + 515489 T_{2}^{38} + \cdots + 94\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(161, [\chi])\). Copy content Toggle raw display