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Properties

Label 165.2.j.a
Level $165$
Weight $2$
Character orbit 165.j
Analytic conductor $1.318$
Analytic rank $0$
Dimension $24$
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] = [165,2,Mod(43,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 165.j (of order \(4\), 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: \(1.31753163335\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{5} - 8 q^{11} + 16 q^{12} - 8 q^{15} - 8 q^{16} - 40 q^{20} + 32 q^{22} - 32 q^{23} - 8 q^{25} - 16 q^{31} + 16 q^{33} - 24 q^{36} - 8 q^{37} - 56 q^{38} + 8 q^{42} + 32 q^{48} + 24 q^{53} - 40 q^{55} + 160 q^{56} - 24 q^{58} + 32 q^{60} - 16 q^{66} - 32 q^{67} - 8 q^{70} - 32 q^{71} + 24 q^{77} - 24 q^{78} - 120 q^{80} - 24 q^{81} - 72 q^{82} + 80 q^{86} + 104 q^{88} - 16 q^{91} + 80 q^{92} - 48 q^{93} + 88 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.91746 − 1.91746i −0.707107 − 0.707107i 5.35328i 2.23563 − 0.0440169i 2.71169i 2.40424 + 2.40424i 6.42978 − 6.42978i 1.00000i −4.37113 − 4.20233i
43.2 −1.45644 − 1.45644i 0.707107 + 0.707107i 2.24244i −1.57013 + 1.59207i − 2.05972i 1.85474 + 1.85474i 0.353101 − 0.353101i 1.00000i 4.60556 − 0.0319594i
43.3 −1.24819 − 1.24819i −0.707107 − 0.707107i 1.11598i −0.445397 + 2.19126i 1.76521i −2.93706 − 2.93706i −1.10343 + 1.10343i 1.00000i 3.29106 − 2.17918i
43.4 −0.857692 − 0.857692i 0.707107 + 0.707107i − 0.528729i 2.20289 + 0.383740i − 1.21296i 1.82630 + 1.82630i −2.16887 + 2.16887i 1.00000i −1.56027 − 2.21854i
43.5 −0.824369 − 0.824369i −0.707107 − 0.707107i − 0.640833i 0.623976 − 2.14724i 1.16583i 0.423225 + 0.423225i −2.17702 + 2.17702i 1.00000i −2.28451 + 1.25573i
43.6 −0.478466 − 0.478466i 0.707107 + 0.707107i − 1.54214i −1.04698 − 1.97581i − 0.676654i −3.26171 − 3.26171i −1.69480 + 1.69480i 1.00000i −0.444416 + 1.44630i
43.7 0.478466 + 0.478466i 0.707107 + 0.707107i − 1.54214i −1.04698 − 1.97581i 0.676654i 3.26171 + 3.26171i 1.69480 − 1.69480i 1.00000i 0.444416 − 1.44630i
43.8 0.824369 + 0.824369i −0.707107 − 0.707107i − 0.640833i 0.623976 − 2.14724i − 1.16583i −0.423225 − 0.423225i 2.17702 − 2.17702i 1.00000i 2.28451 − 1.25573i
43.9 0.857692 + 0.857692i 0.707107 + 0.707107i − 0.528729i 2.20289 + 0.383740i 1.21296i −1.82630 − 1.82630i 2.16887 − 2.16887i 1.00000i 1.56027 + 2.21854i
43.10 1.24819 + 1.24819i −0.707107 − 0.707107i 1.11598i −0.445397 + 2.19126i − 1.76521i 2.93706 + 2.93706i 1.10343 − 1.10343i 1.00000i −3.29106 + 2.17918i
43.11 1.45644 + 1.45644i 0.707107 + 0.707107i 2.24244i −1.57013 + 1.59207i 2.05972i −1.85474 − 1.85474i −0.353101 + 0.353101i 1.00000i −4.60556 + 0.0319594i
43.12 1.91746 + 1.91746i −0.707107 − 0.707107i 5.35328i 2.23563 − 0.0440169i − 2.71169i −2.40424 − 2.40424i −6.42978 + 6.42978i 1.00000i 4.37113 + 4.20233i
142.1 −1.91746 + 1.91746i −0.707107 + 0.707107i − 5.35328i 2.23563 + 0.0440169i − 2.71169i 2.40424 − 2.40424i 6.42978 + 6.42978i − 1.00000i −4.37113 + 4.20233i
142.2 −1.45644 + 1.45644i 0.707107 − 0.707107i − 2.24244i −1.57013 − 1.59207i 2.05972i 1.85474 − 1.85474i 0.353101 + 0.353101i − 1.00000i 4.60556 + 0.0319594i
142.3 −1.24819 + 1.24819i −0.707107 + 0.707107i − 1.11598i −0.445397 − 2.19126i − 1.76521i −2.93706 + 2.93706i −1.10343 − 1.10343i − 1.00000i 3.29106 + 2.17918i
142.4 −0.857692 + 0.857692i 0.707107 − 0.707107i 0.528729i 2.20289 − 0.383740i 1.21296i 1.82630 − 1.82630i −2.16887 − 2.16887i − 1.00000i −1.56027 + 2.21854i
142.5 −0.824369 + 0.824369i −0.707107 + 0.707107i 0.640833i 0.623976 + 2.14724i − 1.16583i 0.423225 − 0.423225i −2.17702 − 2.17702i − 1.00000i −2.28451 − 1.25573i
142.6 −0.478466 + 0.478466i 0.707107 − 0.707107i 1.54214i −1.04698 + 1.97581i 0.676654i −3.26171 + 3.26171i −1.69480 − 1.69480i − 1.00000i −0.444416 − 1.44630i
142.7 0.478466 − 0.478466i 0.707107 − 0.707107i 1.54214i −1.04698 + 1.97581i − 0.676654i 3.26171 − 3.26171i 1.69480 + 1.69480i − 1.00000i 0.444416 + 1.44630i
142.8 0.824369 − 0.824369i −0.707107 + 0.707107i 0.640833i 0.623976 + 2.14724i 1.16583i −0.423225 + 0.423225i 2.17702 + 2.17702i − 1.00000i 2.28451 + 1.25573i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.b odd 2 1 inner
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.j.a ✓ 24
3.b odd 2 1 495.2.k.c 24
5.b even 2 1 825.2.j.c 24
5.c odd 4 1 inner 165.2.j.a ✓ 24
5.c odd 4 1 825.2.j.c 24
11.b odd 2 1 inner 165.2.j.a ✓ 24
15.e even 4 1 495.2.k.c 24
33.d even 2 1 495.2.k.c 24
55.d odd 2 1 825.2.j.c 24
55.e even 4 1 inner 165.2.j.a ✓ 24
55.e even 4 1 825.2.j.c 24
165.l odd 4 1 495.2.k.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.j.a ✓ 24 1.a even 1 1 trivial
165.2.j.a ✓ 24 5.c odd 4 1 inner
165.2.j.a ✓ 24 11.b odd 2 1 inner
165.2.j.a ✓ 24 55.e even 4 1 inner
495.2.k.c 24 3.b odd 2 1
495.2.k.c 24 15.e even 4 1
495.2.k.c 24 33.d even 2 1
495.2.k.c 24 165.l odd 4 1
825.2.j.c 24 5.b even 2 1
825.2.j.c 24 5.c odd 4 1
825.2.j.c 24 55.d odd 2 1
825.2.j.c 24 55.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(165, [\chi])\).