Properties

Label 161.2.k.b
Level $161$
Weight $2$
Character orbit 161.k
Analytic conductor $1.286$
Analytic rank $0$
Dimension $120$
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] = [161,2,Mod(20,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.20");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 161.k (of order \(22\), degree \(10\), 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: \(1.28559147254\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(12\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 22 q^{2} - 30 q^{4} - 11 q^{7} - 10 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 22 q^{2} - 30 q^{4} - 11 q^{7} - 10 q^{8} + 6 q^{9} - 22 q^{11} - 11 q^{14} - 22 q^{15} + 38 q^{16} + 2 q^{18} + 22 q^{21} - 36 q^{23} - 20 q^{25} + 33 q^{28} - 38 q^{29} - 66 q^{30} - 126 q^{32} - 13 q^{35} + 120 q^{36} + 22 q^{37} - 14 q^{39} - 121 q^{42} + 22 q^{43} + 110 q^{44} - 66 q^{46} - 69 q^{49} + 40 q^{50} + 22 q^{51} - 22 q^{53} - 132 q^{56} + 22 q^{57} - 66 q^{58} - 22 q^{60} - 66 q^{63} + 62 q^{64} + 132 q^{65} - 22 q^{67} - 54 q^{70} + 26 q^{71} + 280 q^{72} + 154 q^{74} - 4 q^{77} + 162 q^{78} + 66 q^{79} - 78 q^{81} - 77 q^{84} + 110 q^{85} + 22 q^{86} - 66 q^{88} + 236 q^{92} - 108 q^{93} - 80 q^{95} - 56 q^{98} + 132 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −0.347190 2.41476i −0.922369 0.421232i −3.79153 + 1.11329i −1.48641 + 1.71541i −0.696936 + 2.37354i −1.44308 + 2.21755i 1.97783 + 4.33085i −1.29125 1.49019i 4.65837 + 2.99375i
20.2 −0.347190 2.41476i 0.922369 + 0.421232i −3.79153 + 1.11329i 1.48641 1.71541i 0.696936 2.37354i 2.61663 0.391471i 1.97783 + 4.33085i −1.29125 1.49019i −4.65837 2.99375i
20.3 −0.198303 1.37923i −2.35718 1.07649i 0.0560394 0.0164547i 2.67181 3.08344i −1.01729 + 3.46456i −1.66841 + 2.05339i −1.19150 2.60901i 2.43290 + 2.80772i −4.78259 3.07359i
20.4 −0.198303 1.37923i 2.35718 + 1.07649i 0.0560394 0.0164547i −2.67181 + 3.08344i 1.01729 3.46456i 2.56091 0.664634i −1.19150 2.60901i 2.43290 + 2.80772i 4.78259 + 3.07359i
20.5 −0.0343259 0.238742i −2.02155 0.923209i 1.86317 0.547075i −0.355896 + 0.410726i −0.151017 + 0.514318i 0.655104 2.56336i −0.394959 0.864839i 1.26975 + 1.46537i 0.110274 + 0.0708688i
20.6 −0.0343259 0.238742i 2.02155 + 0.923209i 1.86317 0.547075i 0.355896 0.410726i 0.151017 0.514318i −2.60386 0.468957i −0.394959 0.864839i 1.26975 + 1.46537i −0.110274 0.0708688i
20.7 0.187562 + 1.30452i −1.36235 0.622165i 0.252392 0.0741091i 1.67205 1.92964i 0.556102 1.89391i 0.444888 2.60808i 1.23900 + 2.71302i −0.495672 0.572036i 2.83087 + 1.81929i
20.8 0.187562 + 1.30452i 1.36235 + 0.622165i 0.252392 0.0741091i −1.67205 + 1.92964i −0.556102 + 1.89391i −2.55720 0.678751i 1.23900 + 2.71302i −0.495672 0.572036i −2.83087 1.81929i
20.9 0.259711 + 1.80633i −0.739551 0.337742i −1.27639 + 0.374782i −2.45337 + 2.83134i 0.418003 1.42359i 2.48097 + 0.919137i 0.507712 + 1.11173i −1.53172 1.76769i −5.75151 3.69627i
20.10 0.259711 + 1.80633i 0.739551 + 0.337742i −1.27639 + 0.374782i 2.45337 2.83134i −0.418003 + 1.42359i −0.194554 + 2.63859i 0.507712 + 1.11173i −1.53172 1.76769i 5.75151 + 3.69627i
20.11 0.365139 + 2.53960i −2.08739 0.953277i −4.39725 + 1.29115i −0.106291 + 0.122667i 1.65876 5.64920i −2.64064 0.164315i −2.75294 6.02810i 1.48386 + 1.71246i −0.350335 0.225147i
20.12 0.365139 + 2.53960i 2.08739 + 0.953277i −4.39725 + 1.29115i 0.106291 0.122667i −1.65876 + 5.64920i 0.947497 2.47027i −2.75294 6.02810i 1.48386 + 1.71246i 0.350335 + 0.225147i
34.1 −1.68017 + 1.07978i −0.664315 0.0955140i 0.826210 1.80915i −3.17412 + 0.932004i 1.21929 0.556832i 2.63780 + 0.204993i −0.00316123 0.0219868i −2.44629 0.718295i 4.32668 4.99326i
34.2 −1.68017 + 1.07978i 0.664315 + 0.0955140i 0.826210 1.80915i 3.17412 0.932004i −1.21929 + 0.556832i 0.172491 2.64012i −0.00316123 0.0219868i −2.44629 0.718295i −4.32668 + 4.99326i
34.3 −1.53131 + 0.984115i −3.07833 0.442597i 0.545606 1.19471i −0.648243 + 0.190341i 5.14945 2.35168i −2.63540 + 0.233843i −0.177865 1.23708i 6.40174 + 1.87972i 0.805344 0.929417i
34.4 −1.53131 + 0.984115i 3.07833 + 0.442597i 0.545606 1.19471i 0.648243 0.190341i −5.14945 + 2.35168i −0.606519 + 2.57529i −0.177865 1.23708i 6.40174 + 1.87972i −0.805344 + 0.929417i
34.5 −0.494143 + 0.317566i −1.18392 0.170222i −0.687501 + 1.50542i −0.804140 + 0.236117i 0.639083 0.291860i −1.03103 2.43659i −0.305534 2.12504i −1.50578 0.442138i 0.322377 0.372043i
34.6 −0.494143 + 0.317566i 1.18392 + 0.170222i −0.687501 + 1.50542i 0.804140 0.236117i −0.639083 + 0.291860i 2.26506 + 1.36730i −0.305534 2.12504i −1.50578 0.442138i −0.322377 + 0.372043i
34.7 0.430824 0.276874i −2.93167 0.421510i −0.721880 + 1.58070i 3.09475 0.908700i −1.37974 + 0.630106i 2.40804 1.09607i 0.272415 + 1.89469i 5.53854 + 1.62626i 1.08170 1.24835i
34.8 0.430824 0.276874i 2.93167 + 0.421510i −0.721880 + 1.58070i −3.09475 + 0.908700i 1.37974 0.630106i 1.42761 2.22754i 0.272415 + 1.89469i 5.53854 + 1.62626i −1.08170 + 1.24835i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.d odd 22 1 inner
161.k even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.2.k.b 120
7.b odd 2 1 inner 161.2.k.b 120
23.d odd 22 1 inner 161.2.k.b 120
161.k even 22 1 inner 161.2.k.b 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.k.b 120 1.a even 1 1 trivial
161.2.k.b 120 7.b odd 2 1 inner
161.2.k.b 120 23.d odd 22 1 inner
161.2.k.b 120 161.k even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + 11 T_{2}^{59} + 74 T_{2}^{58} + 372 T_{2}^{57} + 1533 T_{2}^{56} + 5465 T_{2}^{55} + 17463 T_{2}^{54} + 51343 T_{2}^{53} + 142150 T_{2}^{52} + 374850 T_{2}^{51} + 948807 T_{2}^{50} + 2306112 T_{2}^{49} + \cdots + 1849 \) acting on \(S_{2}^{\mathrm{new}}(161, [\chi])\). Copy content Toggle raw display