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Properties

Label 164.3.p.a
Level $164$
Weight $3$
Character orbit 164.p
Analytic conductor $4.469$
Analytic rank $0$
Dimension $112$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [164,3,Mod(13,164)] 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("164.13"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(164, base_ring=CyclotomicField(40)) chi = DirichletCharacter(H, H._module([0, 31])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 164.p (of order \(40\), degree \(16\), 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: \(4.46867633551\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(7\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 8 q^{3} - 24 q^{9} - 36 q^{13} - 12 q^{15} + 56 q^{17} + 48 q^{19} - 52 q^{21} - 52 q^{27} + 52 q^{29} + 180 q^{31} + 304 q^{33} + 184 q^{35} + 68 q^{37} + 64 q^{39} - 4 q^{41} + 80 q^{43} - 400 q^{45}+ \cdots + 888 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0 −1.82897 + 4.41553i 0 4.72709 + 0.748697i 0 6.86243 + 5.86107i 0 −9.78777 − 9.78777i 0
13.2 0 −1.54809 + 3.73742i 0 −3.32602 − 0.526789i 0 −6.11952 − 5.22656i 0 −5.20775 − 5.20775i 0
13.3 0 −0.155806 + 0.376149i 0 −3.98726 − 0.631520i 0 3.14094 + 2.68261i 0 6.24675 + 6.24675i 0
13.4 0 0.116511 − 0.281281i 0 6.74584 + 1.06844i 0 −3.28559 − 2.80616i 0 6.29842 + 6.29842i 0
13.5 0 1.01026 − 2.43897i 0 −7.34954 − 1.16405i 0 −0.253886 − 0.216839i 0 1.43598 + 1.43598i 0
13.6 0 1.56061 − 3.76765i 0 4.24816 + 0.672843i 0 9.38440 + 8.01503i 0 −5.39573 − 5.39573i 0
13.7 0 2.00947 − 4.85128i 0 −0.400423 − 0.0634208i 0 −9.72877 − 8.30915i 0 −13.1330 − 13.1330i 0
17.1 0 −1.74685 − 4.21727i 0 −4.27795 − 2.17973i 0 −1.85942 − 1.13945i 0 −8.36992 + 8.36992i 0
17.2 0 −1.50568 − 3.63502i 0 7.12193 + 3.62880i 0 5.35696 + 3.28275i 0 −4.58236 + 4.58236i 0
17.3 0 −0.378915 − 0.914782i 0 0.598740 + 0.305073i 0 −8.88688 − 5.44589i 0 5.67071 − 5.67071i 0
17.4 0 0.131950 + 0.318555i 0 −7.59062 − 3.86762i 0 9.54444 + 5.84884i 0 6.27989 − 6.27989i 0
17.5 0 0.716369 + 1.72947i 0 1.44524 + 0.736388i 0 2.96256 + 1.81546i 0 3.88609 − 3.88609i 0
17.6 0 2.03675 + 4.91716i 0 7.12792 + 3.63186i 0 0.954557 + 0.584953i 0 −13.6661 + 13.6661i 0
17.7 0 2.05153 + 4.95284i 0 −8.35605 − 4.25762i 0 −8.07223 − 4.94667i 0 −13.9579 + 13.9579i 0
29.1 0 −1.74685 + 4.21727i 0 −4.27795 + 2.17973i 0 −1.85942 + 1.13945i 0 −8.36992 − 8.36992i 0
29.2 0 −1.50568 + 3.63502i 0 7.12193 − 3.62880i 0 5.35696 − 3.28275i 0 −4.58236 − 4.58236i 0
29.3 0 −0.378915 + 0.914782i 0 0.598740 − 0.305073i 0 −8.88688 + 5.44589i 0 5.67071 + 5.67071i 0
29.4 0 0.131950 − 0.318555i 0 −7.59062 + 3.86762i 0 9.54444 − 5.84884i 0 6.27989 + 6.27989i 0
29.5 0 0.716369 − 1.72947i 0 1.44524 − 0.736388i 0 2.96256 − 1.81546i 0 3.88609 + 3.88609i 0
29.6 0 2.03675 − 4.91716i 0 7.12792 − 3.63186i 0 0.954557 − 0.584953i 0 −13.6661 − 13.6661i 0
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.h odd 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.3.p.a ✓ 112
41.h odd 40 1 inner 164.3.p.a ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.3.p.a ✓ 112 1.a even 1 1 trivial
164.3.p.a ✓ 112 41.h odd 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(164, [\chi])\).