Properties

Label 6912.2.a.bu
Level $6912$
Weight $2$
Character orbit 6912.a
Self dual yes
Analytic conductor $55.193$
Analytic rank $1$
Dimension $2$
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] = [6912,2,Mod(1,6912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6912.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6912 = 2^{8} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6912.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: \(55.1925978771\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 216)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - q^{7} + 3 q^{11} - \beta q^{13} + \beta q^{17} + \beta q^{19} - \beta q^{23} - 4 q^{25} - 6 q^{29} - 7 q^{31} - q^{35} + \beta q^{37} + \beta q^{41} - 2 \beta q^{43} - 6 q^{49} - 9 q^{53} + 3 q^{55} - 4 q^{59} - \beta q^{65} + 3 \beta q^{71} - 3 q^{73} - 3 q^{77} + 4 q^{79} + 7 q^{83} + \beta q^{85} - 2 \beta q^{89} + \beta q^{91} + \beta q^{95} + 7 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{7} + 6 q^{11} - 8 q^{25} - 12 q^{29} - 14 q^{31} - 2 q^{35} - 12 q^{49} - 18 q^{53} + 6 q^{55} - 8 q^{59} - 6 q^{73} - 6 q^{77} + 8 q^{79} + 14 q^{83} + 14 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.64575
−2.64575
0 0 0 1.00000 0 −1.00000 0 0 0
1.2 0 0 0 1.00000 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6912.2.a.bu 2
3.b odd 2 1 6912.2.a.bc 2
4.b odd 2 1 6912.2.a.bv 2
8.b even 2 1 6912.2.a.bc 2
8.d odd 2 1 6912.2.a.bd 2
12.b even 2 1 6912.2.a.bd 2
16.e even 4 2 216.2.d.b 4
16.f odd 4 2 864.2.d.a 4
24.f even 2 1 6912.2.a.bv 2
24.h odd 2 1 inner 6912.2.a.bu 2
48.i odd 4 2 216.2.d.b 4
48.k even 4 2 864.2.d.a 4
144.u even 12 4 2592.2.r.p 8
144.v odd 12 4 2592.2.r.p 8
144.w odd 12 4 648.2.n.n 8
144.x even 12 4 648.2.n.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.b 4 16.e even 4 2
216.2.d.b 4 48.i odd 4 2
648.2.n.n 8 144.w odd 12 4
648.2.n.n 8 144.x even 12 4
864.2.d.a 4 16.f odd 4 2
864.2.d.a 4 48.k even 4 2
2592.2.r.p 8 144.u even 12 4
2592.2.r.p 8 144.v odd 12 4
6912.2.a.bc 2 3.b odd 2 1
6912.2.a.bc 2 8.b even 2 1
6912.2.a.bd 2 8.d odd 2 1
6912.2.a.bd 2 12.b even 2 1
6912.2.a.bu 2 1.a even 1 1 trivial
6912.2.a.bu 2 24.h odd 2 1 inner
6912.2.a.bv 2 4.b odd 2 1
6912.2.a.bv 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6912))\):

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{13}^{2} - 28 \) Copy content Toggle raw display
\( T_{17}^{2} - 28 \) Copy content Toggle raw display
\( T_{19}^{2} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 28 \) Copy content Toggle raw display
$17$ \( T^{2} - 28 \) Copy content Toggle raw display
$19$ \( T^{2} - 28 \) Copy content Toggle raw display
$23$ \( T^{2} - 28 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 28 \) Copy content Toggle raw display
$41$ \( T^{2} - 28 \) Copy content Toggle raw display
$43$ \( T^{2} - 112 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 252 \) Copy content Toggle raw display
$73$ \( (T + 3)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( (T - 7)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 112 \) Copy content Toggle raw display
$97$ \( (T - 7)^{2} \) Copy content Toggle raw display
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