Properties

Label 6864.2.a.bv
Level $6864$
Weight $2$
Character orbit 6864.a
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6864,2,Mod(1,6864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6864, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("6864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.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: \(54.8093159474\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1620.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 12x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1716)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} - \beta_{2} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_1 q^{5} - \beta_{2} q^{7} + q^{9} + q^{11} - q^{13} + \beta_1 q^{15} + (\beta_{2} + \beta_1 + 2) q^{17} + \beta_{2} q^{19} - \beta_{2} q^{21} - \beta_{2} q^{23} + (\beta_{2} + 2 \beta_1 + 3) q^{25} + q^{27} + ( - \beta_{2} + \beta_1) q^{29} + (\beta_1 - 2) q^{31} + q^{33} + (2 \beta_{2} + 2) q^{35} + 4 q^{37} - q^{39} + ( - \beta_{2} + 2) q^{41} + ( - \beta_{2} + \beta_1) q^{43} + \beta_1 q^{45} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{47} + ( - 2 \beta_1 + 1) q^{49} + (\beta_{2} + \beta_1 + 2) q^{51} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{53} + \beta_1 q^{55} + \beta_{2} q^{57} + ( - 2 \beta_1 - 4) q^{59} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{61} - \beta_{2} q^{63} - \beta_1 q^{65} + ( - 2 \beta_{2} + \beta_1 - 2) q^{67} - \beta_{2} q^{69} + (4 \beta_{2} + 2) q^{71} + (\beta_{2} - 2 \beta_1 + 8) q^{73} + (\beta_{2} + 2 \beta_1 + 3) q^{75} - \beta_{2} q^{77} + ( - \beta_{2} - \beta_1 - 8) q^{79} + q^{81} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{83} + ( - \beta_{2} + 4 \beta_1 + 6) q^{85} + ( - \beta_{2} + \beta_1) q^{87} + ( - 3 \beta_1 + 2) q^{89} + \beta_{2} q^{91} + (\beta_1 - 2) q^{93} + ( - 2 \beta_{2} - 2) q^{95} + 8 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 3 q^{9} + 3 q^{11} - 3 q^{13} + 6 q^{17} + 9 q^{25} + 3 q^{27} - 6 q^{31} + 3 q^{33} + 6 q^{35} + 12 q^{37} - 3 q^{39} + 6 q^{41} - 6 q^{47} + 3 q^{49} + 6 q^{51} + 6 q^{53} - 12 q^{59} + 12 q^{61} - 6 q^{67} + 6 q^{71} + 24 q^{73} + 9 q^{75} - 24 q^{79} + 3 q^{81} + 12 q^{83} + 18 q^{85} + 6 q^{89} - 6 q^{93} - 6 q^{95} + 24 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 12x - 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 8 \) 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
−2.55247
−1.39091
3.94338
0 1.00000 0 −2.55247 0 −3.62008 0 1.00000 0
1.2 0 1.00000 0 −1.39091 0 3.28357 0 1.00000 0
1.3 0 1.00000 0 3.94338 0 0.336509 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6864.2.a.bv 3
4.b odd 2 1 1716.2.a.f 3
12.b even 2 1 5148.2.a.n 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.a.f 3 4.b odd 2 1
5148.2.a.n 3 12.b even 2 1
6864.2.a.bv 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{3} - 12T_{5} - 14 \) Copy content Toggle raw display
\( T_{7}^{3} - 12T_{7} + 4 \) Copy content Toggle raw display
\( T_{17}^{3} - 6T_{17}^{2} - 6T_{17} + 46 \) Copy content Toggle raw display
\( T_{19}^{3} - 12T_{19} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 12T - 14 \) Copy content Toggle raw display
$7$ \( T^{3} - 12T + 4 \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 6 T^{2} - 6 T + 46 \) Copy content Toggle raw display
$19$ \( T^{3} - 12T - 4 \) Copy content Toggle raw display
$23$ \( T^{3} - 12T + 4 \) Copy content Toggle raw display
$29$ \( T^{3} - 30T + 50 \) Copy content Toggle raw display
$31$ \( T^{3} + 6T^{2} - 30 \) Copy content Toggle raw display
$37$ \( (T - 4)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 6T^{2} + 20 \) Copy content Toggle raw display
$43$ \( T^{3} - 30T + 50 \) Copy content Toggle raw display
$47$ \( T^{3} + 6 T^{2} - 60 T - 280 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} - 60 T - 8 \) Copy content Toggle raw display
$59$ \( T^{3} + 12T^{2} - 16 \) Copy content Toggle raw display
$61$ \( T^{3} - 12 T^{2} - 24 T + 80 \) Copy content Toggle raw display
$67$ \( T^{3} + 6 T^{2} - 60 T + 98 \) Copy content Toggle raw display
$71$ \( T^{3} - 6 T^{2} - 180 T + 120 \) Copy content Toggle raw display
$73$ \( T^{3} - 24 T^{2} + 120 T + 28 \) Copy content Toggle raw display
$79$ \( T^{3} + 24 T^{2} + 174 T + 350 \) Copy content Toggle raw display
$83$ \( T^{3} - 12 T^{2} - 72 T + 816 \) Copy content Toggle raw display
$89$ \( T^{3} - 6 T^{2} - 96 T + 586 \) Copy content Toggle raw display
$97$ \( (T - 8)^{3} \) Copy content Toggle raw display
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