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Properties

Label 322.3.f.a
Level $322$
Weight $3$
Character orbit 322.f
Analytic conductor $8.774$
Analytic rank $0$
Dimension $64$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [322,3,Mod(137,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.137");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 322.f (of order \(6\), degree \(2\), 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: \(8.77386451240\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 64 q^{4} - 16 q^{6} - 112 q^{9} - 128 q^{16} + 48 q^{18} - 2 q^{23} + 16 q^{24} + 132 q^{25} - 80 q^{26} - 120 q^{27} + 120 q^{29} + 12 q^{31} + 96 q^{35} + 448 q^{36} - 104 q^{39} - 248 q^{41} + 60 q^{46}+ \cdots - 32 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 −0.707107 − 1.22474i −2.51318 + 4.35296i −1.00000 + 1.73205i −0.546184 + 0.315339i 7.10835 −6.22568 − 3.20014i 2.82843 −8.13216 − 14.0853i 0.772420 + 0.445957i
137.2 −0.707107 − 1.22474i −2.51318 + 4.35296i −1.00000 + 1.73205i 0.546184 − 0.315339i 7.10835 6.22568 + 3.20014i 2.82843 −8.13216 − 14.0853i −0.772420 − 0.445957i
137.3 −0.707107 − 1.22474i −1.54948 + 2.68377i −1.00000 + 1.73205i −7.12361 + 4.11282i 4.38258 −5.34633 + 4.51849i 2.82843 −0.301758 − 0.522661i 10.0743 + 5.81640i
137.4 −0.707107 − 1.22474i −1.54948 + 2.68377i −1.00000 + 1.73205i 7.12361 − 4.11282i 4.38258 5.34633 − 4.51849i 2.82843 −0.301758 − 0.522661i −10.0743 − 5.81640i
137.5 −0.707107 − 1.22474i −1.32093 + 2.28791i −1.00000 + 1.73205i −1.64114 + 0.947513i 3.73614 1.24703 − 6.88803i 2.82843 1.01031 + 1.74991i 2.32092 + 1.33999i
137.6 −0.707107 − 1.22474i −1.32093 + 2.28791i −1.00000 + 1.73205i 1.64114 − 0.947513i 3.73614 −1.24703 + 6.88803i 2.82843 1.01031 + 1.74991i −2.32092 − 1.33999i
137.7 −0.707107 − 1.22474i 0.0188989 − 0.0327339i −1.00000 + 1.73205i −2.42489 + 1.40001i −0.0534542 6.92977 + 0.989096i 2.82843 4.49929 + 7.79299i 3.42931 + 1.97992i
137.8 −0.707107 − 1.22474i 0.0188989 − 0.0327339i −1.00000 + 1.73205i 2.42489 − 1.40001i −0.0534542 −6.92977 − 0.989096i 2.82843 4.49929 + 7.79299i −3.42931 − 1.97992i
137.9 −0.707107 − 1.22474i 0.304956 − 0.528200i −1.00000 + 1.73205i −7.28595 + 4.20655i −0.862547 1.93587 − 6.72699i 2.82843 4.31400 + 7.47207i 10.3039 + 5.94895i
137.10 −0.707107 − 1.22474i 0.304956 − 0.528200i −1.00000 + 1.73205i 7.28595 − 4.20655i −0.862547 −1.93587 + 6.72699i 2.82843 4.31400 + 7.47207i −10.3039 − 5.94895i
137.11 −0.707107 − 1.22474i 1.29385 − 2.24101i −1.00000 + 1.73205i −1.29217 + 0.746033i −3.65956 6.89147 + 1.22785i 2.82843 1.15191 + 1.99516i 1.82740 + 1.05505i
137.12 −0.707107 − 1.22474i 1.29385 − 2.24101i −1.00000 + 1.73205i 1.29217 − 0.746033i −3.65956 −6.89147 − 1.22785i 2.82843 1.15191 + 1.99516i −1.82740 − 1.05505i
137.13 −0.707107 − 1.22474i 1.66831 − 2.88959i −1.00000 + 1.73205i −5.05968 + 2.92121i −4.71868 −3.03583 + 6.30743i 2.82843 −1.06649 − 1.84722i 7.15547 + 4.13121i
137.14 −0.707107 − 1.22474i 1.66831 − 2.88959i −1.00000 + 1.73205i 5.05968 − 2.92121i −4.71868 3.03583 − 6.30743i 2.82843 −1.06649 − 1.84722i −7.15547 − 4.13121i
137.15 −0.707107 − 1.22474i 2.80468 − 4.85785i −1.00000 + 1.73205i −5.52581 + 3.19033i −7.93283 −2.28263 − 6.61737i 2.82843 −11.2325 − 19.4552i 7.81468 + 4.51181i
137.16 −0.707107 − 1.22474i 2.80468 − 4.85785i −1.00000 + 1.73205i 5.52581 − 3.19033i −7.93283 2.28263 + 6.61737i 2.82843 −11.2325 − 19.4552i −7.81468 − 4.51181i
137.17 0.707107 + 1.22474i −2.94029 + 5.09274i −1.00000 + 1.73205i −2.72860 + 1.57536i −8.31641 −4.41595 + 5.43133i −2.82843 −12.7907 − 22.1541i −3.85882 − 2.22789i
137.18 0.707107 + 1.22474i −2.94029 + 5.09274i −1.00000 + 1.73205i 2.72860 − 1.57536i −8.31641 4.41595 − 5.43133i −2.82843 −12.7907 − 22.1541i 3.85882 + 2.22789i
137.19 0.707107 + 1.22474i −1.53650 + 2.66130i −1.00000 + 1.73205i −6.58899 + 3.80416i −4.34588 2.27828 + 6.61887i −2.82843 −0.221676 − 0.383954i −9.31824 − 5.37989i
137.20 0.707107 + 1.22474i −1.53650 + 2.66130i −1.00000 + 1.73205i 6.58899 − 3.80416i −4.34588 −2.27828 − 6.61887i −2.82843 −0.221676 − 0.383954i 9.31824 + 5.37989i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
23.b odd 2 1 inner
161.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.3.f.a ✓ 64
7.c even 3 1 inner 322.3.f.a ✓ 64
23.b odd 2 1 inner 322.3.f.a ✓ 64
161.f odd 6 1 inner 322.3.f.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.3.f.a ✓ 64 1.a even 1 1 trivial
322.3.f.a ✓ 64 7.c even 3 1 inner
322.3.f.a ✓ 64 23.b odd 2 1 inner
322.3.f.a ✓ 64 161.f odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(322, [\chi])\).