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Properties

Label 187.2.j.a
Level $187$
Weight $2$
Character orbit 187.j
Analytic conductor $1.493$
Analytic rank $0$
Dimension $64$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [187,2,Mod(16,187)] 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("187.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(187, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([4, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 187 = 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 187.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.49320251780\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 8 q^{2} - 20 q^{4} - 4 q^{8} + 30 q^{9} - 4 q^{13} - 34 q^{15} - 16 q^{16} - 14 q^{17} + 10 q^{18} - 72 q^{21} + 36 q^{25} + 22 q^{26} - 8 q^{30} + 84 q^{32} - 44 q^{33} - 44 q^{34} + 18 q^{35} - 32 q^{36}+ \cdots + 72 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −0.781136 − 2.40409i −0.850854 + 1.17110i −3.55144 + 2.58027i 2.42532 + 0.788033i 3.48006 + 1.13074i 0.460130 + 0.633314i 4.88728 + 3.55082i 0.279529 + 0.860301i − 6.44624i
16.2 −0.781136 − 2.40409i 0.850854 − 1.17110i −3.55144 + 2.58027i −2.42532 − 0.788033i −3.48006 − 1.13074i −0.460130 − 0.633314i 4.88728 + 3.55082i 0.279529 + 0.860301i 6.44624i
16.3 −0.609729 − 1.87655i −1.09362 + 1.50523i −1.53164 + 1.11281i −2.63951 − 0.857628i 3.49146 + 1.13444i 1.01174 + 1.39254i −0.170457 − 0.123844i −0.142682 − 0.439131i 5.47610i
16.4 −0.609729 − 1.87655i 1.09362 − 1.50523i −1.53164 + 1.11281i 2.63951 + 0.857628i −3.49146 − 1.13444i −1.01174 − 1.39254i −0.170457 − 0.123844i −0.142682 − 0.439131i − 5.47610i
16.5 −0.299814 − 0.922734i −0.345028 + 0.474890i 0.856485 − 0.622273i −2.20747 − 0.717252i 0.541642 + 0.175990i −2.36486 − 3.25494i −2.40083 − 1.74430i 0.820575 + 2.52547i 2.25195i
16.6 −0.299814 − 0.922734i 0.345028 − 0.474890i 0.856485 − 0.622273i 2.20747 + 0.717252i −0.541642 − 0.175990i 2.36486 + 3.25494i −2.40083 − 1.74430i 0.820575 + 2.52547i − 2.25195i
16.7 −0.150351 − 0.462733i −1.01595 + 1.39834i 1.42652 − 1.03643i 0.411135 + 0.133586i 0.799806 + 0.259873i 1.36610 + 1.88027i −1.48131 − 1.07624i 0.00385940 + 0.0118780i − 0.210330i
16.8 −0.150351 − 0.462733i 1.01595 − 1.39834i 1.42652 − 1.03643i −0.411135 − 0.133586i −0.799806 − 0.259873i −1.36610 − 1.88027i −1.48131 − 1.07624i 0.00385940 + 0.0118780i 0.210330i
16.9 0.0862831 + 0.265552i −1.41541 + 1.94814i 1.55496 − 1.12975i 3.81388 + 1.23921i −0.639458 − 0.207772i −2.14946 − 2.95848i 0.885957 + 0.643686i −0.864821 − 2.66164i 1.11971i
16.10 0.0862831 + 0.265552i 1.41541 − 1.94814i 1.55496 − 1.12975i −3.81388 − 1.23921i 0.639458 + 0.207772i 2.14946 + 2.95848i 0.885957 + 0.643686i −0.864821 − 2.66164i − 1.11971i
16.11 0.418751 + 1.28878i −0.465953 + 0.641330i 0.132428 − 0.0962145i 0.416272 + 0.135255i −1.02165 − 0.331955i 0.474970 + 0.653740i 2.37206 + 1.72340i 0.732860 + 2.25551i 0.593122i
16.12 0.418751 + 1.28878i 0.465953 − 0.641330i 0.132428 − 0.0962145i −0.416272 − 0.135255i 1.02165 + 0.331955i −0.474970 − 0.653740i 2.37206 + 1.72340i 0.732860 + 2.25551i − 0.593122i
16.13 0.636250 + 1.95818i −1.78295 + 2.45402i −1.81160 + 1.31621i −0.308808 − 0.100338i −5.93981 − 1.92996i 0.911451 + 1.25450i −0.398550 − 0.289564i −1.91626 − 5.89763i − 0.668541i
16.14 0.636250 + 1.95818i 1.78295 − 2.45402i −1.81160 + 1.31621i 0.308808 + 0.100338i 5.93981 + 1.92996i −0.911451 − 1.25450i −0.398550 − 0.289564i −1.91626 − 5.89763i 0.668541i
16.15 0.817781 + 2.51687i −0.0601072 + 0.0827304i −4.04784 + 2.94093i 3.67505 + 1.19410i −0.257376 − 0.0836266i −1.36656 − 1.88091i −6.43022 − 4.67183i 0.923820 + 2.84322i 10.2261i
16.16 0.817781 + 2.51687i 0.0601072 − 0.0827304i −4.04784 + 2.94093i −3.67505 − 1.19410i 0.257376 + 0.0836266i 1.36656 + 1.88091i −6.43022 − 4.67183i 0.923820 + 2.84322i − 10.2261i
135.1 −2.20158 + 1.59954i −2.67663 + 0.869690i 1.67040 − 5.14095i 0.525344 − 0.723074i 4.50172 − 6.19609i 1.60348 + 0.521001i 2.86380 + 8.81387i 3.98094 − 2.89232i 2.43222i
135.2 −2.20158 + 1.59954i 2.67663 − 0.869690i 1.67040 − 5.14095i −0.525344 + 0.723074i −4.50172 + 6.19609i −1.60348 − 0.521001i 2.86380 + 8.81387i 3.98094 − 2.89232i − 2.43222i
135.3 −1.68340 + 1.22306i −0.00997681 + 0.00324166i 0.719917 − 2.21568i 0.572549 − 0.788046i 0.0128302 − 0.0176592i −3.51500 − 1.14209i 0.211997 + 0.652459i −2.42696 + 1.76329i 2.02686i
135.4 −1.68340 + 1.22306i 0.00997681 − 0.00324166i 0.719917 − 2.21568i −0.572549 + 0.788046i −0.0128302 + 0.0176592i 3.51500 + 1.14209i 0.211997 + 0.652459i −2.42696 + 1.76329i − 2.02686i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
17.b even 2 1 inner
187.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 187.2.j.a ✓ 64
11.c even 5 1 inner 187.2.j.a ✓ 64
17.b even 2 1 inner 187.2.j.a ✓ 64
187.j even 10 1 inner 187.2.j.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.2.j.a ✓ 64 1.a even 1 1 trivial
187.2.j.a ✓ 64 11.c even 5 1 inner
187.2.j.a ✓ 64 17.b even 2 1 inner
187.2.j.a ✓ 64 187.j even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(187, [\chi])\).