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Properties

Label 161.3.f.a
Level $161$
Weight $3$
Character orbit 161.f
Analytic conductor $4.387$
Analytic rank $0$
Dimension $60$
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] = [161,3,Mod(114,161)]
 
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, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.114");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 161.f (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: \(4.38693225620\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 4 q^{2} - 2 q^{3} - 64 q^{4} + 24 q^{6} + 4 q^{8} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 4 q^{2} - 2 q^{3} - 64 q^{4} + 24 q^{6} + 4 q^{8} - 64 q^{9} + 26 q^{12} - 8 q^{13} - 120 q^{16} - 82 q^{18} + 16 q^{23} - 84 q^{24} + 106 q^{25} + 80 q^{26} + 88 q^{27} - 220 q^{29} - 8 q^{31} + 18 q^{32} - 226 q^{35} + 376 q^{36} + 84 q^{39} + 108 q^{41} - 82 q^{46} - 48 q^{47} + 92 q^{48} + 300 q^{49} - 332 q^{50} + 208 q^{52} - 134 q^{54} - 276 q^{55} + 58 q^{58} + 242 q^{59} + 464 q^{62} + 188 q^{64} + 700 q^{69} + 290 q^{70} + 48 q^{71} - 396 q^{72} + 94 q^{73} + 368 q^{75} - 308 q^{77} - 324 q^{78} + 70 q^{81} - 92 q^{82} - 676 q^{85} - 446 q^{87} - 220 q^{92} - 314 q^{93} + 120 q^{94} + 180 q^{95} + 370 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
114.1 −1.81241 + 3.13919i 1.76847 + 3.06309i −4.56967 − 7.91489i −5.25325 − 3.03296i −12.8208 −0.111938 − 6.99910i 18.6292 −1.75500 + 3.03976i 19.0421 − 10.9939i
114.2 −1.81241 + 3.13919i 1.76847 + 3.06309i −4.56967 − 7.91489i 5.25325 + 3.03296i −12.8208 0.111938 + 6.99910i 18.6292 −1.75500 + 3.03976i −19.0421 + 10.9939i
114.3 −1.72321 + 2.98469i −2.51755 − 4.36052i −3.93890 − 6.82238i −7.28801 − 4.20774i 17.3530 6.96685 + 0.680489i 13.3645 −8.17607 + 14.1614i 25.1175 − 14.5016i
114.4 −1.72321 + 2.98469i −2.51755 − 4.36052i −3.93890 − 6.82238i 7.28801 + 4.20774i 17.3530 −6.96685 − 0.680489i 13.3645 −8.17607 + 14.1614i −25.1175 + 14.5016i
114.5 −1.67274 + 2.89726i −0.275252 − 0.476751i −3.59609 − 6.22861i −2.86755 − 1.65558i 1.84170 −5.58955 + 4.21390i 10.6793 4.34847 − 7.53177i 9.59332 − 5.53870i
114.6 −1.67274 + 2.89726i −0.275252 − 0.476751i −3.59609 − 6.22861i 2.86755 + 1.65558i 1.84170 5.58955 − 4.21390i 10.6793 4.34847 − 7.53177i −9.59332 + 5.53870i
114.7 −1.10982 + 1.92227i −1.04515 − 1.81026i −0.463404 − 0.802639i −1.02606 − 0.592395i 4.63973 0.226114 + 6.99635i −6.82139 2.31531 − 4.01024i 2.27748 − 1.31490i
114.8 −1.10982 + 1.92227i −1.04515 − 1.81026i −0.463404 − 0.802639i 1.02606 + 0.592395i 4.63973 −0.226114 − 6.99635i −6.82139 2.31531 − 4.01024i −2.27748 + 1.31490i
114.9 −1.05847 + 1.83333i 2.49580 + 4.32286i −0.240734 − 0.416963i −3.37289 − 1.94734i −10.5670 6.98998 + 0.374495i −7.44855 −7.95808 + 13.7838i 7.14024 − 4.12242i
114.10 −1.05847 + 1.83333i 2.49580 + 4.32286i −0.240734 − 0.416963i 3.37289 + 1.94734i −10.5670 −6.98998 − 0.374495i −7.44855 −7.95808 + 13.7838i −7.14024 + 4.12242i
114.11 −0.759261 + 1.31508i 0.739823 + 1.28141i 0.847047 + 1.46713i −5.37678 − 3.10428i −2.24687 −6.72748 − 1.93415i −8.64660 3.40533 − 5.89820i 8.16475 − 4.71392i
114.12 −0.759261 + 1.31508i 0.739823 + 1.28141i 0.847047 + 1.46713i 5.37678 + 3.10428i −2.24687 6.72748 + 1.93415i −8.64660 3.40533 − 5.89820i −8.16475 + 4.71392i
114.13 −0.373993 + 0.647775i −2.42362 − 4.19783i 1.72026 + 2.97958i −1.52351 − 0.879598i 3.62566 −0.628153 − 6.97176i −5.56540 −7.24783 + 12.5536i 1.13956 − 0.657926i
114.14 −0.373993 + 0.647775i −2.42362 − 4.19783i 1.72026 + 2.97958i 1.52351 + 0.879598i 3.62566 0.628153 + 6.97176i −5.56540 −7.24783 + 12.5536i −1.13956 + 0.657926i
114.15 −0.244824 + 0.424048i −0.388520 − 0.672936i 1.88012 + 3.25647i −7.94195 − 4.58529i 0.380476 6.75757 − 1.82626i −3.79979 4.19810 − 7.27133i 3.88876 − 2.24518i
114.16 −0.244824 + 0.424048i −0.388520 − 0.672936i 1.88012 + 3.25647i 7.94195 + 4.58529i 0.380476 −6.75757 + 1.82626i −3.79979 4.19810 − 7.27133i −3.88876 + 2.24518i
114.17 0.184766 − 0.320025i 1.77215 + 3.06946i 1.93172 + 3.34584i −2.76342 − 1.59546i 1.30974 −1.93384 + 6.72758i 2.90580 −1.78105 + 3.08487i −1.02117 + 0.589574i
114.18 0.184766 − 0.320025i 1.77215 + 3.06946i 1.93172 + 3.34584i 2.76342 + 1.59546i 1.30974 1.93384 − 6.72758i 2.90580 −1.78105 + 3.08487i 1.02117 − 0.589574i
114.19 0.682469 − 1.18207i −2.02313 − 3.50416i 1.06847 + 1.85065i −5.84666 − 3.37557i −5.52289 −6.03601 + 3.54494i 8.37655 −3.68610 + 6.38451i −7.98033 + 4.60744i
114.20 0.682469 − 1.18207i −2.02313 − 3.50416i 1.06847 + 1.85065i 5.84666 + 3.37557i −5.52289 6.03601 − 3.54494i 8.37655 −3.68610 + 6.38451i 7.98033 − 4.60744i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 114.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
23.b odd 2 1 inner
161.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.3.f.a ✓ 60
7.c even 3 1 inner 161.3.f.a ✓ 60
23.b odd 2 1 inner 161.3.f.a ✓ 60
161.f odd 6 1 inner 161.3.f.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.3.f.a ✓ 60 1.a even 1 1 trivial
161.3.f.a ✓ 60 7.c even 3 1 inner
161.3.f.a ✓ 60 23.b odd 2 1 inner
161.3.f.a ✓ 60 161.f odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(161, [\chi])\).