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Properties

Label 6724.2.a.m
Level $6724$
Weight $2$
Character orbit 6724.a
Self dual yes
Analytic conductor $53.691$
Analytic rank $1$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6724,2,Mod(1,6724)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6724, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6724.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6724 = 2^{2} \cdot 41^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6724.a (trivial)

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: yes
Analytic conductor: \(53.6914103191\)
Analytic rank: \(1\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 164)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{5} + 8 q^{9} - 52 q^{21} + 4 q^{23} + 16 q^{25} - 12 q^{31} - 44 q^{33} - 4 q^{37} + 8 q^{39} - 20 q^{43} - 16 q^{49} - 72 q^{51} + 64 q^{57} - 56 q^{59} - 72 q^{61} - 84 q^{73} + 12 q^{77} - 12 q^{81} - 56 q^{83} + 24 q^{87} - 36 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −3.16282 0 0.0632719 0 2.78154 0 7.00341 0
1.2 0 −2.60707 0 −0.310959 0 0.526804 0 3.79682 0
1.3 0 −2.43921 0 0.403665 0 1.41851 0 2.94974 0
1.4 0 −2.40745 0 −4.12436 0 2.00315 0 2.79580 0
1.5 0 −1.92504 0 −2.18745 0 −1.10117 0 0.705788 0
1.6 0 −1.72139 0 −0.751274 0 4.71038 0 −0.0368254 0
1.7 0 −1.61482 0 2.79466 0 1.11944 0 −0.392355 0
1.8 0 −0.939743 0 1.72070 0 −0.584637 0 −2.11688 0
1.9 0 −0.754701 0 2.05504 0 2.36774 0 −2.43043 0
1.10 0 −0.737603 0 0.124173 0 −3.88271 0 −2.45594 0
1.11 0 −0.419436 0 −4.04339 0 3.59094 0 −2.82407 0
1.12 0 −0.0703578 0 −3.74408 0 −1.84093 0 −2.99505 0
1.13 0 0.0703578 0 −3.74408 0 1.84093 0 −2.99505 0
1.14 0 0.419436 0 −4.04339 0 −3.59094 0 −2.82407 0
1.15 0 0.737603 0 0.124173 0 3.88271 0 −2.45594 0
1.16 0 0.754701 0 2.05504 0 −2.36774 0 −2.43043 0
1.17 0 0.939743 0 1.72070 0 0.584637 0 −2.11688 0
1.18 0 1.61482 0 2.79466 0 −1.11944 0 −0.392355 0
1.19 0 1.72139 0 −0.751274 0 −4.71038 0 −0.0368254 0
1.20 0 1.92504 0 −2.18745 0 1.10117 0 0.705788 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(41\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6724.2.a.m 24
41.b even 2 1 inner 6724.2.a.m 24
41.h odd 40 2 164.2.m.a ✓ 24
164.o even 40 2 656.2.bs.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.m.a ✓ 24 41.h odd 40 2
656.2.bs.c 24 164.o even 40 2
6724.2.a.m 24 1.a even 1 1 trivial
6724.2.a.m 24 41.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 40 T_{3}^{22} + 677 T_{3}^{20} - 6352 T_{3}^{18} + 36334 T_{3}^{16} - 131248 T_{3}^{14} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6724))\). Copy content Toggle raw display