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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Newspace parameters

Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.8093159474\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1620.1
Defining polynomial: \(x^{3} - 12 x - 14\)
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

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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 8\)

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.

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

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

Hecke characteristic polynomials

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