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Properties

Label 161.4.c.c
Level $161$
Weight $4$
Character orbit 161.c
Analytic conductor $9.499$
Analytic rank $0$
Dimension $36$
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] = [161,4,Mod(160,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.160");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 161.c (of order \(2\), degree \(1\), 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: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{2} + 184 q^{4} - 228 q^{8} - 436 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{2} + 184 q^{4} - 228 q^{8} - 436 q^{9} + 168 q^{16} - 404 q^{18} + 204 q^{23} - 140 q^{25} + 256 q^{29} + 1008 q^{32} - 2040 q^{35} - 4412 q^{36} + 1296 q^{39} - 888 q^{46} + 1916 q^{49} + 4080 q^{50} + 6020 q^{58} + 3988 q^{64} - 1712 q^{70} - 5040 q^{71} - 8636 q^{72} - 2864 q^{77} + 8820 q^{78} + 9380 q^{81} - 5032 q^{85} - 3672 q^{92} - 5352 q^{93} - 3840 q^{95} + 6884 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
160.1 −5.29875 − 6.28001i 20.0768 −10.4216 33.2762i 11.5274 − 14.4955i −63.9920 −12.4385 55.2216
160.2 −5.29875 − 6.28001i 20.0768 10.4216 33.2762i −11.5274 + 14.4955i −63.9920 −12.4385 −55.2216
160.3 −5.29875 6.28001i 20.0768 −10.4216 − 33.2762i 11.5274 + 14.4955i −63.9920 −12.4385 55.2216
160.4 −5.29875 6.28001i 20.0768 10.4216 − 33.2762i −11.5274 − 14.4955i −63.9920 −12.4385 −55.2216
160.5 −4.42537 − 3.10698i 11.5839 −13.0444 13.7495i −12.3048 + 13.8417i −15.8600 17.3467 57.7264
160.6 −4.42537 − 3.10698i 11.5839 13.0444 13.7495i 12.3048 − 13.8417i −15.8600 17.3467 −57.7264
160.7 −4.42537 3.10698i 11.5839 −13.0444 − 13.7495i −12.3048 − 13.8417i −15.8600 17.3467 57.7264
160.8 −4.42537 3.10698i 11.5839 13.0444 − 13.7495i 12.3048 + 13.8417i −15.8600 17.3467 −57.7264
160.9 −3.36931 − 9.90574i 3.35226 −8.06501 33.3755i 16.4867 + 8.43733i 15.6597 −71.1238 27.1735
160.10 −3.36931 − 9.90574i 3.35226 8.06501 33.3755i −16.4867 − 8.43733i 15.6597 −71.1238 −27.1735
160.11 −3.36931 9.90574i 3.35226 −8.06501 − 33.3755i 16.4867 − 8.43733i 15.6597 −71.1238 27.1735
160.12 −3.36931 9.90574i 3.35226 8.06501 − 33.3755i −16.4867 + 8.43733i 15.6597 −71.1238 −27.1735
160.13 −2.89408 − 3.10338i 0.375694 −8.39365 8.98142i 13.9217 + 12.2141i 22.0653 17.3690 24.2919
160.14 −2.89408 − 3.10338i 0.375694 8.39365 8.98142i −13.9217 − 12.2141i 22.0653 17.3690 −24.2919
160.15 −2.89408 3.10338i 0.375694 −8.39365 − 8.98142i 13.9217 − 12.2141i 22.0653 17.3690 24.2919
160.16 −2.89408 3.10338i 0.375694 8.39365 − 8.98142i −13.9217 + 12.2141i 22.0653 17.3690 −24.2919
160.17 0.765757 − 0.826637i −7.41362 −5.17868 − 0.633003i 15.9670 + 9.38369i −11.8031 26.3167 −3.96561
160.18 0.765757 − 0.826637i −7.41362 5.17868 − 0.633003i −15.9670 − 9.38369i −11.8031 26.3167 3.96561
160.19 0.765757 0.826637i −7.41362 −5.17868 0.633003i 15.9670 − 9.38369i −11.8031 26.3167 −3.96561
160.20 0.765757 0.826637i −7.41362 5.17868 0.633003i −15.9670 + 9.38369i −11.8031 26.3167 3.96561
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 160.36
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.4.c.c ✓ 36
7.b odd 2 1 inner 161.4.c.c ✓ 36
23.b odd 2 1 inner 161.4.c.c ✓ 36
161.c even 2 1 inner 161.4.c.c ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.4.c.c ✓ 36 1.a even 1 1 trivial
161.4.c.c ✓ 36 7.b odd 2 1 inner
161.4.c.c ✓ 36 23.b odd 2 1 inner
161.4.c.c ✓ 36 161.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} + 2T_{2}^{8} - 57T_{2}^{7} - 87T_{2}^{6} + 1108T_{2}^{5} + 1005T_{2}^{4} - 8880T_{2}^{3} - 1572T_{2}^{2} + 25504T_{2} - 15232 \) acting on \(S_{4}^{\mathrm{new}}(161, [\chi])\). Copy content Toggle raw display