↑

Properties

Label 416.2.z.a
Level $416$
Weight $2$
Character orbit 416.z
Analytic conductor $3.322$
Analytic rank $0$
Dimension $24$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(81,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.z (of order \(6\), degree \(2\), not 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: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 104)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{7} + 6 q^{9} - 4 q^{15} + 14 q^{23} - 12 q^{25} + 8 q^{31} - 14 q^{33} + 34 q^{39} - 4 q^{41} + 8 q^{47} + 6 q^{49} - 8 q^{55} - 52 q^{57} - 32 q^{63} + 30 q^{65} - 30 q^{71} - 12 q^{73} + 48 q^{79} + 8 q^{81} + 26 q^{87} - 22 q^{89} - 60 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 0 −2.64382 + 1.52641i 0 − 0.497079i 0 0.845740 − 1.46486i 0 3.15986 − 5.47304i 0
81.2 0 −2.00439 + 1.15724i 0 4.18204i 0 −0.818571 + 1.41781i 0 1.17840 − 2.04104i 0
81.3 0 −1.47584 + 0.852079i 0 − 2.59989i 0 0.300588 − 0.520633i 0 −0.0479214 + 0.0830022i 0
81.4 0 −1.39066 + 0.802895i 0 0.556100i 0 2.30251 − 3.98807i 0 −0.210719 + 0.364976i 0
81.5 0 −0.609172 + 0.351705i 0 − 2.24007i 0 −0.471952 + 0.817445i 0 −1.25261 + 2.16958i 0
81.6 0 −0.509400 + 0.294102i 0 − 1.78237i 0 −1.65832 + 2.87229i 0 −1.32701 + 2.29844i 0
81.7 0 0.509400 − 0.294102i 0 1.78237i 0 −1.65832 + 2.87229i 0 −1.32701 + 2.29844i 0
81.8 0 0.609172 − 0.351705i 0 2.24007i 0 −0.471952 + 0.817445i 0 −1.25261 + 2.16958i 0
81.9 0 1.39066 − 0.802895i 0 − 0.556100i 0 2.30251 − 3.98807i 0 −0.210719 + 0.364976i 0
81.10 0 1.47584 − 0.852079i 0 2.59989i 0 0.300588 − 0.520633i 0 −0.0479214 + 0.0830022i 0
81.11 0 2.00439 − 1.15724i 0 − 4.18204i 0 −0.818571 + 1.41781i 0 1.17840 − 2.04104i 0
81.12 0 2.64382 − 1.52641i 0 0.497079i 0 0.845740 − 1.46486i 0 3.15986 − 5.47304i 0
113.1 0 −2.64382 − 1.52641i 0 0.497079i 0 0.845740 + 1.46486i 0 3.15986 + 5.47304i 0
113.2 0 −2.00439 − 1.15724i 0 − 4.18204i 0 −0.818571 − 1.41781i 0 1.17840 + 2.04104i 0
113.3 0 −1.47584 − 0.852079i 0 2.59989i 0 0.300588 + 0.520633i 0 −0.0479214 − 0.0830022i 0
113.4 0 −1.39066 − 0.802895i 0 − 0.556100i 0 2.30251 + 3.98807i 0 −0.210719 − 0.364976i 0
113.5 0 −0.609172 − 0.351705i 0 2.24007i 0 −0.471952 − 0.817445i 0 −1.25261 − 2.16958i 0
113.6 0 −0.509400 − 0.294102i 0 1.78237i 0 −1.65832 − 2.87229i 0 −1.32701 − 2.29844i 0
113.7 0 0.509400 + 0.294102i 0 − 1.78237i 0 −1.65832 − 2.87229i 0 −1.32701 − 2.29844i 0
113.8 0 0.609172 + 0.351705i 0 − 2.24007i 0 −0.471952 − 0.817445i 0 −1.25261 − 2.16958i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
13.c even 3 1 inner
104.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.z.a 24
4.b odd 2 1 104.2.r.a ✓ 24
8.b even 2 1 inner 416.2.z.a 24
8.d odd 2 1 104.2.r.a ✓ 24
12.b even 2 1 936.2.be.a 24
13.c even 3 1 inner 416.2.z.a 24
24.f even 2 1 936.2.be.a 24
52.j odd 6 1 104.2.r.a ✓ 24
104.n odd 6 1 104.2.r.a ✓ 24
104.r even 6 1 inner 416.2.z.a 24
156.p even 6 1 936.2.be.a 24
312.bn even 6 1 936.2.be.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.r.a ✓ 24 4.b odd 2 1
104.2.r.a ✓ 24 8.d odd 2 1
104.2.r.a ✓ 24 52.j odd 6 1
104.2.r.a ✓ 24 104.n odd 6 1
416.2.z.a 24 1.a even 1 1 trivial
416.2.z.a 24 8.b even 2 1 inner
416.2.z.a 24 13.c even 3 1 inner
416.2.z.a 24 104.r even 6 1 inner
936.2.be.a 24 12.b even 2 1
936.2.be.a 24 24.f even 2 1
936.2.be.a 24 156.p even 6 1
936.2.be.a 24 312.bn even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(416, [\chi])\).