Properties

Label 6864.2.a.br
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: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) 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.

\(f(q)\) \(=\) \( q - q^{3} + ( - \beta_1 + 1) q^{5} + (\beta_{2} - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + ( - \beta_1 + 1) q^{5} + (\beta_{2} - 1) q^{7} + q^{9} - q^{11} - q^{13} + (\beta_1 - 1) q^{15} + ( - \beta_{2} - \beta_1) q^{17} + ( - \beta_{2} - 3) q^{19} + ( - \beta_{2} + 1) q^{21} + (\beta_{2} - 4 \beta_1 - 1) q^{23} + (\beta_{2} - 2 \beta_1) q^{25} - q^{27} + (3 \beta_{2} - \beta_1 + 6) q^{29} + (\beta_1 - 3) q^{31} + q^{33} - 2 q^{35} + (2 \beta_{2} - 2) q^{37} + q^{39} + ( - \beta_{2} + 4 \beta_1 + 3) q^{41} + ( - \beta_{2} + \beta_1) q^{43} + ( - \beta_1 + 1) q^{45} - 2 \beta_1 q^{47} + ( - 4 \beta_{2} + 2 \beta_1 - 1) q^{49} + (\beta_{2} + \beta_1) q^{51} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{53} + (\beta_1 - 1) q^{55} + (\beta_{2} + 3) q^{57} + (2 \beta_1 + 2) q^{59} + (4 \beta_{2} - 2 \beta_1 + 2) q^{61} + (\beta_{2} - 1) q^{63} + (\beta_1 - 1) q^{65} + ( - 2 \beta_{2} - 3 \beta_1 + 3) q^{67} + ( - \beta_{2} + 4 \beta_1 + 1) q^{69} + ( - 2 \beta_{2} - 4 \beta_1) q^{71} + (3 \beta_{2} + 2 \beta_1 - 1) q^{73} + ( - \beta_{2} + 2 \beta_1) q^{75} + ( - \beta_{2} + 1) q^{77} + (3 \beta_{2} + 3 \beta_1 + 2) q^{79} + q^{81} + (2 \beta_{2} - 2 \beta_1) q^{83} + (\beta_{2} + 5) q^{85} + ( - 3 \beta_{2} + \beta_1 - 6) q^{87} + (2 \beta_{2} - 5 \beta_1 + 5) q^{89} + ( - \beta_{2} + 1) q^{91} + ( - \beta_1 + 3) q^{93} + (4 \beta_1 - 2) q^{95} + ( - 2 \beta_{2} - 2) q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 2 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 2 q^{5} - 4 q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} - 2 q^{15} - 8 q^{19} + 4 q^{21} - 8 q^{23} - 3 q^{25} - 3 q^{27} + 14 q^{29} - 8 q^{31} + 3 q^{33} - 6 q^{35} - 8 q^{37} + 3 q^{39} + 14 q^{41} + 2 q^{43} + 2 q^{45} - 2 q^{47} + 3 q^{49} + 6 q^{53} - 2 q^{55} + 8 q^{57} + 8 q^{59} - 4 q^{63} - 2 q^{65} + 8 q^{67} + 8 q^{69} - 2 q^{71} - 4 q^{73} + 3 q^{75} + 4 q^{77} + 6 q^{79} + 3 q^{81} - 4 q^{83} + 14 q^{85} - 14 q^{87} + 8 q^{89} + 4 q^{91} + 8 q^{93} - 2 q^{95} - 4 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
2.51414
0.571993
−2.08613
0 −1.00000 0 −1.51414 0 1.32088 0 1.00000 0
1.2 0 −1.00000 0 0.428007 0 −4.67282 0 1.00000 0
1.3 0 −1.00000 0 3.08613 0 −0.648061 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.br 3
4.b odd 2 1 1716.2.a.g 3
12.b even 2 1 5148.2.a.m 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.a.g 3 4.b odd 2 1
5148.2.a.m 3 12.b even 2 1
6864.2.a.br 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{3} - 2T_{5}^{2} - 4T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{3} + 4T_{7}^{2} - 4T_{7} - 4 \) Copy content Toggle raw display
\( T_{17}^{3} - 18T_{17} + 26 \) Copy content Toggle raw display
\( T_{19}^{3} + 8T_{19}^{2} + 12T_{19} - 12 \) 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
$5$ \( T^{3} - 2 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$7$ \( T^{3} + 4 T^{2} - 4 T - 4 \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 18T + 26 \) Copy content Toggle raw display
$19$ \( T^{3} + 8 T^{2} + 12 T - 12 \) Copy content Toggle raw display
$23$ \( T^{3} + 8 T^{2} - 60 T - 468 \) Copy content Toggle raw display
$29$ \( T^{3} - 14 T^{2} - 14 T + 534 \) Copy content Toggle raw display
$31$ \( T^{3} + 8 T^{2} + 16 T + 6 \) Copy content Toggle raw display
$37$ \( T^{3} + 8 T^{2} - 16 T - 32 \) Copy content Toggle raw display
$41$ \( T^{3} - 14 T^{2} - 16 T + 548 \) Copy content Toggle raw display
$43$ \( T^{3} - 2 T^{2} - 10 T + 2 \) Copy content Toggle raw display
$47$ \( T^{3} + 2 T^{2} - 20 T - 24 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} - 60 T + 344 \) Copy content Toggle raw display
$59$ \( T^{3} - 8T^{2} + 48 \) Copy content Toggle raw display
$61$ \( T^{3} - 144T + 656 \) Copy content Toggle raw display
$67$ \( T^{3} - 8 T^{2} - 84 T + 678 \) Copy content Toggle raw display
$71$ \( T^{3} + 2 T^{2} - 148 T + 568 \) Copy content Toggle raw display
$73$ \( T^{3} + 4 T^{2} - 120 T - 492 \) Copy content Toggle raw display
$79$ \( T^{3} - 6 T^{2} - 150 T - 386 \) Copy content Toggle raw display
$83$ \( T^{3} + 4 T^{2} - 40 T - 16 \) Copy content Toggle raw display
$89$ \( T^{3} - 8 T^{2} - 116 T - 246 \) Copy content Toggle raw display
$97$ \( T^{3} + 4 T^{2} - 32 T - 96 \) Copy content Toggle raw display
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