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Properties

Label 392.2.p.h
Level $392$
Weight $2$
Character orbit 392.p
Analytic conductor $3.130$
Analytic rank $0$
Dimension $24$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [392,2,Mod(165,392)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("392.165"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,2,0,-2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.13013575923\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} - 2 q^{4} - 4 q^{8} + 12 q^{9} - 32 q^{15} - 2 q^{16} - 22 q^{18} - 48 q^{22} - 32 q^{23} + 12 q^{25} + 8 q^{30} + 42 q^{32} + 84 q^{36} - 32 q^{39} - 8 q^{44} + 56 q^{46} + 44 q^{50} - 16 q^{58}+ \cdots + 64 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1 −1.27930 − 0.602812i −0.456937 − 0.263813i 1.27324 + 1.54236i −1.30721 + 0.754717i 0.425532 + 0.612943i 0 −0.699104 − 2.74067i −1.36081 − 2.35698i 2.12727 − 0.177512i
165.2 −1.27930 − 0.602812i 0.456937 + 0.263813i 1.27324 + 1.54236i 1.30721 − 0.754717i −0.425532 − 0.612943i 0 −0.699104 − 2.74067i −1.36081 − 2.35698i −2.12727 + 0.177512i
165.3 −1.03225 + 0.966672i −2.48442 − 1.43438i 0.131089 − 1.99570i 2.76990 − 1.59920i 3.95113 − 0.920979i 0 1.79387 + 2.18678i 2.61491 + 4.52915i −1.31333 + 4.32837i
165.4 −1.03225 + 0.966672i 2.48442 + 1.43438i 0.131089 − 1.99570i −2.76990 + 1.59920i −3.95113 + 0.920979i 0 1.79387 + 2.18678i 2.61491 + 4.52915i 1.31333 − 4.32837i
165.5 0.117602 − 1.40932i −0.456937 − 0.263813i −1.97234 − 0.331475i −1.30721 + 0.754717i −0.425532 + 0.612943i 0 −0.699104 + 2.74067i −1.36081 − 2.35698i 0.909904 + 1.93102i
165.6 0.117602 − 1.40932i 0.456937 + 0.263813i −1.97234 − 0.331475i 1.30721 − 0.754717i 0.425532 − 0.612943i 0 −0.699104 + 2.74067i −1.36081 − 2.35698i −0.909904 − 1.93102i
165.7 0.280509 + 1.38611i −1.61829 − 0.934317i −1.84263 + 0.777636i −2.02949 + 1.17173i 0.841128 − 2.50521i 0 −1.59477 − 2.33596i 0.245898 + 0.425908i −2.19344 − 2.48443i
165.8 0.280509 + 1.38611i 1.61829 + 0.934317i −1.84263 + 0.777636i 2.02949 − 1.17173i −0.841128 + 2.50521i 0 −1.59477 − 2.33596i 0.245898 + 0.425908i 2.19344 + 2.48443i
165.9 1.06016 + 0.935986i −1.61829 − 0.934317i 0.247862 + 1.98458i −2.02949 + 1.17173i −0.841128 − 2.50521i 0 −1.59477 + 2.33596i 0.245898 + 0.425908i −3.24830 − 0.657361i
165.10 1.06016 + 0.935986i 1.61829 + 0.934317i 0.247862 + 1.98458i 2.02949 − 1.17173i 0.841128 + 2.50521i 0 −1.59477 + 2.33596i 0.245898 + 0.425908i 3.24830 + 0.657361i
165.11 1.35329 − 0.410620i −2.48442 − 1.43438i 1.66278 − 1.11138i 2.76990 − 1.59920i −3.95113 − 0.920979i 0 1.79387 − 2.18678i 2.61491 + 4.52915i 3.09181 − 3.30156i
165.12 1.35329 − 0.410620i 2.48442 + 1.43438i 1.66278 − 1.11138i −2.76990 + 1.59920i 3.95113 + 0.920979i 0 1.79387 − 2.18678i 2.61491 + 4.52915i −3.09181 + 3.30156i
373.1 −1.27930 + 0.602812i −0.456937 + 0.263813i 1.27324 − 1.54236i −1.30721 − 0.754717i 0.425532 − 0.612943i 0 −0.699104 + 2.74067i −1.36081 + 2.35698i 2.12727 + 0.177512i
373.2 −1.27930 + 0.602812i 0.456937 − 0.263813i 1.27324 − 1.54236i 1.30721 + 0.754717i −0.425532 + 0.612943i 0 −0.699104 + 2.74067i −1.36081 + 2.35698i −2.12727 − 0.177512i
373.3 −1.03225 − 0.966672i −2.48442 + 1.43438i 0.131089 + 1.99570i 2.76990 + 1.59920i 3.95113 + 0.920979i 0 1.79387 − 2.18678i 2.61491 − 4.52915i −1.31333 − 4.32837i
373.4 −1.03225 − 0.966672i 2.48442 − 1.43438i 0.131089 + 1.99570i −2.76990 − 1.59920i −3.95113 − 0.920979i 0 1.79387 − 2.18678i 2.61491 − 4.52915i 1.31333 + 4.32837i
373.5 0.117602 + 1.40932i −0.456937 + 0.263813i −1.97234 + 0.331475i −1.30721 − 0.754717i −0.425532 − 0.612943i 0 −0.699104 − 2.74067i −1.36081 + 2.35698i 0.909904 − 1.93102i
373.6 0.117602 + 1.40932i 0.456937 − 0.263813i −1.97234 + 0.331475i 1.30721 + 0.754717i 0.425532 + 0.612943i 0 −0.699104 − 2.74067i −1.36081 + 2.35698i −0.909904 + 1.93102i
373.7 0.280509 − 1.38611i −1.61829 + 0.934317i −1.84263 − 0.777636i −2.02949 − 1.17173i 0.841128 + 2.50521i 0 −1.59477 + 2.33596i 0.245898 − 0.425908i −2.19344 + 2.48443i
373.8 0.280509 − 1.38611i 1.61829 − 0.934317i −1.84263 − 0.777636i 2.02949 + 1.17173i −0.841128 − 2.50521i 0 −1.59477 + 2.33596i 0.245898 − 0.425908i 2.19344 − 2.48443i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
56.h odd 2 1 inner
56.j odd 6 1 inner
56.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.2.p.h 24
4.b odd 2 1 1568.2.t.h 24
7.b odd 2 1 inner 392.2.p.h 24
7.c even 3 1 392.2.b.g ✓ 12
7.c even 3 1 inner 392.2.p.h 24
7.d odd 6 1 392.2.b.g ✓ 12
7.d odd 6 1 inner 392.2.p.h 24
8.b even 2 1 inner 392.2.p.h 24
8.d odd 2 1 1568.2.t.h 24
28.d even 2 1 1568.2.t.h 24
28.f even 6 1 1568.2.b.g 12
28.f even 6 1 1568.2.t.h 24
28.g odd 6 1 1568.2.b.g 12
28.g odd 6 1 1568.2.t.h 24
56.e even 2 1 1568.2.t.h 24
56.h odd 2 1 inner 392.2.p.h 24
56.j odd 6 1 392.2.b.g ✓ 12
56.j odd 6 1 inner 392.2.p.h 24
56.k odd 6 1 1568.2.b.g 12
56.k odd 6 1 1568.2.t.h 24
56.m even 6 1 1568.2.b.g 12
56.m even 6 1 1568.2.t.h 24
56.p even 6 1 392.2.b.g ✓ 12
56.p even 6 1 inner 392.2.p.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.b.g ✓ 12 7.c even 3 1
392.2.b.g ✓ 12 7.d odd 6 1
392.2.b.g ✓ 12 56.j odd 6 1
392.2.b.g ✓ 12 56.p even 6 1
392.2.p.h 24 1.a even 1 1 trivial
392.2.p.h 24 7.b odd 2 1 inner
392.2.p.h 24 7.c even 3 1 inner
392.2.p.h 24 7.d odd 6 1 inner
392.2.p.h 24 8.b even 2 1 inner
392.2.p.h 24 56.h odd 2 1 inner
392.2.p.h 24 56.j odd 6 1 inner
392.2.p.h 24 56.p even 6 1 inner
1568.2.b.g 12 28.f even 6 1
1568.2.b.g 12 28.g odd 6 1
1568.2.b.g 12 56.k odd 6 1
1568.2.b.g 12 56.m even 6 1
1568.2.t.h 24 4.b odd 2 1
1568.2.t.h 24 8.d odd 2 1
1568.2.t.h 24 28.d even 2 1
1568.2.t.h 24 28.f even 6 1
1568.2.t.h 24 28.g odd 6 1
1568.2.t.h 24 56.e even 2 1
1568.2.t.h 24 56.k odd 6 1
1568.2.t.h 24 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{12} - 12T_{3}^{10} + 112T_{3}^{8} - 368T_{3}^{6} + 928T_{3}^{4} - 256T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{12} + 86T_{17}^{10} + 5336T_{17}^{8} + 155256T_{17}^{6} + 3301728T_{17}^{4} + 22561120T_{17}^{2} + 119946304 \) Copy content Toggle raw display