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Properties

Label 342.3.ba.a
Level $342$
Weight $3$
Character orbit 342.ba
Analytic conductor $9.319$
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] = [342,3,Mod(17,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 10]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 342.ba (of order \(18\), degree \(6\), 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: \(9.31882504112\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 12 q^{7} + 18 q^{13} - 78 q^{19} - 120 q^{25} + 108 q^{28} + 60 q^{31} - 60 q^{34} - 96 q^{37} + 102 q^{43} + 192 q^{46} + 210 q^{49} + 72 q^{52} - 396 q^{55} + 120 q^{58} - 720 q^{61} + 144 q^{64} - 54 q^{67} + 96 q^{70} - 390 q^{73} - 24 q^{76} - 624 q^{79} - 168 q^{82} + 1596 q^{85} + 120 q^{88} - 120 q^{91} - 408 q^{94} + 888 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −1.39273 − 0.245576i 0 1.87939 + 0.684040i −2.28459 − 6.27686i 0 −1.01042 − 1.75010i −2.44949 − 1.41421i 0 1.64037 + 9.30300i
17.2 −1.39273 − 0.245576i 0 1.87939 + 0.684040i 1.29578 + 3.56012i 0 6.59082 + 11.4156i −2.44949 − 1.41421i 0 −0.930387 − 5.27649i
17.3 −1.39273 − 0.245576i 0 1.87939 + 0.684040i 1.47250 + 4.04567i 0 −3.24038 − 5.61249i −2.44949 − 1.41421i 0 −1.05728 − 5.99613i
17.4 1.39273 + 0.245576i 0 1.87939 + 0.684040i −1.47250 − 4.04567i 0 −3.24038 − 5.61249i 2.44949 + 1.41421i 0 −1.05728 − 5.99613i
17.5 1.39273 + 0.245576i 0 1.87939 + 0.684040i −1.29578 − 3.56012i 0 6.59082 + 11.4156i 2.44949 + 1.41421i 0 −0.930387 − 5.27649i
17.6 1.39273 + 0.245576i 0 1.87939 + 0.684040i 2.28459 + 6.27686i 0 −1.01042 − 1.75010i 2.44949 + 1.41421i 0 1.64037 + 9.30300i
35.1 −0.909039 + 1.08335i 0 −0.347296 − 1.96962i −2.49017 − 0.439085i 0 2.19511 + 3.80205i 2.44949 + 1.41421i 0 2.73935 − 2.29858i
35.2 −0.909039 + 1.08335i 0 −0.347296 − 1.96962i −2.32637 − 0.410202i 0 −2.60864 − 4.51830i 2.44949 + 1.41421i 0 2.55915 − 2.14739i
35.3 −0.909039 + 1.08335i 0 −0.347296 − 1.96962i 6.20927 + 1.09486i 0 1.19072 + 2.06239i 2.44949 + 1.41421i 0 −6.83059 + 5.73155i
35.4 0.909039 − 1.08335i 0 −0.347296 − 1.96962i −6.20927 − 1.09486i 0 1.19072 + 2.06239i −2.44949 − 1.41421i 0 −6.83059 + 5.73155i
35.5 0.909039 − 1.08335i 0 −0.347296 − 1.96962i 2.32637 + 0.410202i 0 −2.60864 − 4.51830i −2.44949 − 1.41421i 0 2.55915 − 2.14739i
35.6 0.909039 − 1.08335i 0 −0.347296 − 1.96962i 2.49017 + 0.439085i 0 2.19511 + 3.80205i −2.44949 − 1.41421i 0 2.73935 − 2.29858i
161.1 −1.39273 + 0.245576i 0 1.87939 − 0.684040i −2.28459 + 6.27686i 0 −1.01042 + 1.75010i −2.44949 + 1.41421i 0 1.64037 − 9.30300i
161.2 −1.39273 + 0.245576i 0 1.87939 − 0.684040i 1.29578 − 3.56012i 0 6.59082 − 11.4156i −2.44949 + 1.41421i 0 −0.930387 + 5.27649i
161.3 −1.39273 + 0.245576i 0 1.87939 − 0.684040i 1.47250 − 4.04567i 0 −3.24038 + 5.61249i −2.44949 + 1.41421i 0 −1.05728 + 5.99613i
161.4 1.39273 − 0.245576i 0 1.87939 − 0.684040i −1.47250 + 4.04567i 0 −3.24038 + 5.61249i 2.44949 − 1.41421i 0 −1.05728 + 5.99613i
161.5 1.39273 − 0.245576i 0 1.87939 − 0.684040i −1.29578 + 3.56012i 0 6.59082 − 11.4156i 2.44949 − 1.41421i 0 −0.930387 + 5.27649i
161.6 1.39273 − 0.245576i 0 1.87939 − 0.684040i 2.28459 − 6.27686i 0 −1.01042 + 1.75010i 2.44949 − 1.41421i 0 1.64037 − 9.30300i
215.1 −0.909039 − 1.08335i 0 −0.347296 + 1.96962i −2.49017 + 0.439085i 0 2.19511 − 3.80205i 2.44949 − 1.41421i 0 2.73935 + 2.29858i
215.2 −0.909039 − 1.08335i 0 −0.347296 + 1.96962i −2.32637 + 0.410202i 0 −2.60864 + 4.51830i 2.44949 − 1.41421i 0 2.55915 + 2.14739i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.e even 9 1 inner
57.l odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.3.ba.a ✓ 36
3.b odd 2 1 inner 342.3.ba.a ✓ 36
19.e even 9 1 inner 342.3.ba.a ✓ 36
57.l odd 18 1 inner 342.3.ba.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.3.ba.a ✓ 36 1.a even 1 1 trivial
342.3.ba.a ✓ 36 3.b odd 2 1 inner
342.3.ba.a ✓ 36 19.e even 9 1 inner
342.3.ba.a ✓ 36 57.l odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{36} + 60 T_{5}^{34} + 5676 T_{5}^{32} + 106886 T_{5}^{30} + 293004 T_{5}^{28} + \cdots + 80\!\cdots\!64 \) acting on \(S_{3}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display