Properties

Label 342.2.x.b
Level $342$
Weight $2$
Character orbit 342.x
Analytic conductor $2.731$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [342,2,Mod(29,342)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("342.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([3, 17])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{3} - 9 q^{5} - 9 q^{6} + 9 q^{7} + 24 q^{8} - 9 q^{10} - 6 q^{12} + 15 q^{13} - 6 q^{14} + 12 q^{15} - 27 q^{17} - 9 q^{18} - 12 q^{19} - 9 q^{20} - 15 q^{21} + 18 q^{22} - 3 q^{24} - 9 q^{25}+ \cdots + 15 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 0.939693 + 0.342020i −1.73163 + 0.0383566i 0.766044 + 0.642788i −1.94422 + 0.342818i −1.64031 − 0.556208i −2.01725 − 3.49397i 0.500000 + 0.866025i 2.99706 − 0.132839i −1.94422 − 0.342818i
29.2 0.939693 + 0.342020i −1.51766 + 0.834696i 0.766044 + 0.642788i 2.58978 − 0.456648i −1.71161 + 0.265288i 0.132058 + 0.228731i 0.500000 + 0.866025i 1.60657 − 2.53356i 2.58978 + 0.456648i
29.3 0.939693 + 0.342020i −1.48223 − 0.896107i 0.766044 + 0.642788i −1.60519 + 0.283038i −1.08635 − 1.34902i 1.45402 + 2.51844i 0.500000 + 0.866025i 1.39399 + 2.65646i −1.60519 − 0.283038i
29.4 0.939693 + 0.342020i −0.671718 + 1.59649i 0.766044 + 0.642788i −3.58027 + 0.631298i −1.17724 + 1.27047i 1.05892 + 1.83410i 0.500000 + 0.866025i −2.09759 − 2.14479i −3.58027 − 0.631298i
29.5 0.939693 + 0.342020i −0.0815983 − 1.73013i 0.766044 + 0.642788i −0.114049 + 0.0201098i 0.515061 − 1.65370i −2.28181 − 3.95221i 0.500000 + 0.866025i −2.98668 + 0.282351i −0.114049 − 0.0201098i
29.6 0.939693 + 0.342020i 0.441173 + 1.67492i 0.766044 + 0.642788i 2.80119 − 0.493925i −0.158290 + 1.72480i 0.343965 + 0.595764i 0.500000 + 0.866025i −2.61073 + 1.47786i 2.80119 + 0.493925i
29.7 0.939693 + 0.342020i 0.809890 + 1.53104i 0.766044 + 0.642788i −1.35684 + 0.239247i 0.237401 + 1.71570i −0.687596 − 1.19095i 0.500000 + 0.866025i −1.68816 + 2.47994i −1.35684 − 0.239247i
29.8 0.939693 + 0.342020i 1.35438 − 1.07966i 0.766044 + 0.642788i −1.10948 + 0.195632i 1.64196 − 0.551323i 1.61830 + 2.80298i 0.500000 + 0.866025i 0.668674 − 2.92453i −1.10948 − 0.195632i
41.1 −0.173648 + 0.984808i −1.62939 + 0.587428i −0.939693 − 0.342020i 1.91646 − 2.28395i −0.295563 − 1.70665i −1.85811 + 3.21834i 0.500000 − 0.866025i 2.30986 − 1.91431i 1.91646 + 2.28395i
41.2 −0.173648 + 0.984808i −1.55230 − 0.768348i −0.939693 − 0.342020i −0.0329165 + 0.0392284i 1.02623 − 1.39530i −0.586286 + 1.01548i 0.500000 − 0.866025i 1.81928 + 2.38542i −0.0329165 − 0.0392284i
41.3 −0.173648 + 0.984808i −1.23042 − 1.21904i −0.939693 − 0.342020i −1.78525 + 2.12758i 1.41418 − 1.00004i 2.27701 − 3.94391i 0.500000 − 0.866025i 0.0278586 + 2.99987i −1.78525 − 2.12758i
41.4 −0.173648 + 0.984808i −0.809257 + 1.53137i −0.939693 − 0.342020i 1.61926 − 1.92976i −1.36758 − 1.06288i 1.64390 − 2.84732i 0.500000 − 0.866025i −1.69021 − 2.47855i 1.61926 + 1.92976i
41.5 −0.173648 + 0.984808i −0.105335 + 1.72884i −0.939693 − 0.342020i −1.71500 + 2.04386i −1.68429 − 0.403945i 0.0718646 − 0.124473i 0.500000 − 0.866025i −2.97781 − 0.364216i −1.71500 − 2.04386i
41.6 −0.173648 + 0.984808i 1.27863 − 1.16838i −0.939693 − 0.342020i 0.249090 − 0.296853i 0.928597 + 1.46209i 1.37093 − 2.37452i 0.500000 − 0.866025i 0.269780 − 2.98785i 0.249090 + 0.296853i
41.7 −0.173648 + 0.984808i 1.66799 + 0.466718i −0.939693 − 0.342020i −1.31219 + 1.56380i −0.749270 + 1.56160i −2.37786 + 4.11857i 0.500000 − 0.866025i 2.56435 + 1.55696i −1.31219 − 1.56380i
41.8 −0.173648 + 0.984808i 1.72739 + 0.126984i −0.939693 − 0.342020i 0.0814829 − 0.0971075i −0.425013 + 1.67910i 1.30583 − 2.26177i 0.500000 − 0.866025i 2.96775 + 0.438701i 0.0814829 + 0.0971075i
59.1 0.939693 − 0.342020i −1.73163 − 0.0383566i 0.766044 − 0.642788i −1.94422 − 0.342818i −1.64031 + 0.556208i −2.01725 + 3.49397i 0.500000 − 0.866025i 2.99706 + 0.132839i −1.94422 + 0.342818i
59.2 0.939693 − 0.342020i −1.51766 − 0.834696i 0.766044 − 0.642788i 2.58978 + 0.456648i −1.71161 − 0.265288i 0.132058 − 0.228731i 0.500000 − 0.866025i 1.60657 + 2.53356i 2.58978 − 0.456648i
59.3 0.939693 − 0.342020i −1.48223 + 0.896107i 0.766044 − 0.642788i −1.60519 − 0.283038i −1.08635 + 1.34902i 1.45402 − 2.51844i 0.500000 − 0.866025i 1.39399 − 2.65646i −1.60519 + 0.283038i
59.4 0.939693 − 0.342020i −0.671718 − 1.59649i 0.766044 − 0.642788i −3.58027 − 0.631298i −1.17724 − 1.27047i 1.05892 − 1.83410i 0.500000 − 0.866025i −2.09759 + 2.14479i −3.58027 + 0.631298i
See all 48 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 29.8
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.x even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.x.b ✓ 48
9.d odd 6 1 342.2.bf.b yes 48
19.f odd 18 1 342.2.bf.b yes 48
171.x even 18 1 inner 342.2.x.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.x.b ✓ 48 1.a even 1 1 trivial
342.2.x.b ✓ 48 171.x even 18 1 inner
342.2.bf.b yes 48 9.d odd 6 1
342.2.bf.b yes 48 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} + 9 T_{5}^{47} + 45 T_{5}^{46} + 171 T_{5}^{45} + 504 T_{5}^{44} + 1323 T_{5}^{43} + \cdots + 5184 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display