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Properties

Label 161.4.e.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 6 q^{3} - 88 q^{4} - 20 q^{5} + 24 q^{6} - 12 q^{7} + 42 q^{8} - 238 q^{9} - 182 q^{10} + 28 q^{11} - 127 q^{12} + 440 q^{13} + 16 q^{14} + 40 q^{15} - 436 q^{16} - 294 q^{17} + 155 q^{18} - 252 q^{19}+ \cdots - 5764 q^{99}+O(q^{100}) \) 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 −2.59283 + 4.49091i −3.54145 − 6.13398i −9.44551 − 16.3601i 2.19895 − 3.80869i 36.7295 −4.74084 − 17.9032i 56.4771 −11.5838 + 20.0637i 11.4030 + 19.7505i
93.2 −2.51222 + 4.35130i −0.509513 − 0.882503i −8.62252 − 14.9346i −5.35622 + 9.27725i 5.12004 18.4112 + 2.00676i 46.4512 12.9808 − 22.4834i −26.9120 − 46.6130i
93.3 −2.44746 + 4.23912i 3.42021 + 5.92398i −7.98009 − 13.8219i −9.57184 + 16.5789i −33.4833 −17.5280 + 5.98075i 38.9644 −9.89567 + 17.1398i −46.8533 − 81.1523i
93.4 −2.21363 + 3.83412i −4.64135 − 8.03905i −5.80032 − 10.0465i −8.69922 + 15.0675i 41.0969 −7.29013 + 17.0251i 15.9410 −29.5843 + 51.2414i −38.5137 − 66.7078i
93.5 −1.86723 + 3.23414i 1.89237 + 3.27768i −2.97311 − 5.14958i 5.27939 − 9.14417i −14.1340 18.0686 − 4.06535i −7.66976 6.33786 − 10.9775i 19.7157 + 34.1486i
93.6 −1.82844 + 3.16696i −0.0618046 − 0.107049i −2.68642 − 4.65302i 4.27095 − 7.39750i 0.452025 −9.80694 + 15.7106i −9.60722 13.4924 − 23.3695i 15.6184 + 27.0518i
93.7 −1.29395 + 2.24118i 2.49222 + 4.31665i 0.651400 + 1.12826i −1.97487 + 3.42057i −12.8992 −16.1811 − 9.00953i −24.0747 1.07767 − 1.86659i −5.11074 − 8.85207i
93.8 −0.702984 + 1.21760i 4.90053 + 8.48797i 3.01163 + 5.21629i −2.29818 + 3.98056i −13.7800 4.61323 − 17.9365i −19.7162 −34.5305 + 59.8085i −3.23116 − 5.59654i
93.9 −0.428463 + 0.742119i −2.51902 − 4.36308i 3.63284 + 6.29226i 8.85981 − 15.3456i 4.31723 −14.5237 − 11.4918i −13.0815 0.809047 − 1.40131i 7.59219 + 13.1501i
93.10 −0.263205 + 0.455884i −1.93918 − 3.35876i 3.86145 + 6.68822i −10.0597 + 17.4239i 2.04161 −16.7081 − 7.98998i −8.27668 5.97916 − 10.3562i −5.29553 − 9.17212i
93.11 −0.224025 + 0.388023i 0.0803360 + 0.139146i 3.89963 + 6.75435i 2.97104 − 5.14599i −0.0719892 13.9349 + 12.1992i −7.07886 13.4871 − 23.3603i 1.33117 + 2.30566i
93.12 −0.125710 + 0.217736i −4.73599 − 8.20297i 3.96839 + 6.87346i 2.14049 − 3.70743i 2.38144 1.43047 + 18.4649i −4.00682 −31.3591 + 54.3156i 0.538160 + 0.932121i
93.13 0.408313 − 0.707219i 2.18838 + 3.79038i 3.66656 + 6.35067i −10.2780 + 17.8020i 3.57417 17.0320 − 7.27397i 12.5214 3.92202 − 6.79314i 8.39327 + 14.5376i
93.14 0.674796 − 1.16878i 3.89262 + 6.74221i 3.08930 + 5.35082i 3.28419 − 5.68839i 10.5069 14.9503 + 10.9311i 19.1353 −16.8050 + 29.1070i −4.43232 − 7.67701i
93.15 1.15757 − 2.00498i −4.37132 − 7.57134i 1.32004 + 2.28638i 7.70847 − 13.3515i −20.2405 2.31770 − 18.3747i 24.6334 −24.7168 + 42.8108i −17.8463 − 30.9106i
93.16 1.16398 − 2.01608i 0.912712 + 1.58086i 1.29028 + 2.23483i 7.25165 − 12.5602i 4.24953 −16.6871 + 8.03367i 24.6312 11.8339 − 20.4969i −16.8816 − 29.2398i
93.17 1.60577 − 2.78127i −1.32439 − 2.29391i −1.15697 − 2.00393i −7.10907 + 12.3133i −8.50665 −2.29977 + 18.3769i 18.2610 9.99197 − 17.3066i 22.8310 + 39.5445i
93.18 1.81191 − 3.13831i −0.112864 − 0.195487i −2.56601 − 4.44446i −1.55669 + 2.69626i −0.818000 −8.28058 − 16.5660i 10.3930 13.4745 − 23.3386i 5.64114 + 9.77074i
93.19 2.08672 − 3.61431i 3.80998 + 6.59908i −4.70881 − 8.15590i −1.49553 + 2.59033i 31.8015 1.97882 + 18.4142i −5.91636 −15.5319 + 26.9021i 6.24150 + 10.8106i
93.20 2.42180 − 4.19468i −4.52353 − 7.83499i −7.73021 − 13.3891i −7.08114 + 12.2649i −43.8204 17.4156 + 6.30053i −36.1353 −27.4247 + 47.5010i 34.2982 + 59.4062i
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.a ✓ 44
7.c even 3 1 inner 161.4.e.a ✓ 44
7.c even 3 1 1127.4.a.l 22
7.d odd 6 1 1127.4.a.k 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.4.e.a ✓ 44 1.a even 1 1 trivial
161.4.e.a ✓ 44 7.c even 3 1 inner
1127.4.a.k 22 7.d odd 6 1
1127.4.a.l 22 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{44} + 132 T_{2}^{42} - 14 T_{2}^{41} + 10053 T_{2}^{40} - 1739 T_{2}^{39} + 519681 T_{2}^{38} + \cdots + 28\!\cdots\!64 \) acting on \(S_{4}^{\mathrm{new}}(161, [\chi])\). Copy content Toggle raw display