Properties

Label 6864.2.a.bq
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: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3432)
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.

\(f(q)\) \(=\) \( q - q^{3} -\beta_{2} q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} -\beta_{2} q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + q^{9} + q^{11} - q^{13} + \beta_{2} q^{15} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{17} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} ) q^{21} + ( 1 + \beta_{1} + 3 \beta_{2} ) q^{23} + ( -2 - \beta_{1} - \beta_{2} ) q^{25} - q^{27} + ( -3 + 3 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{31} - q^{33} + ( 2 - 2 \beta_{2} ) q^{35} + ( 2 - 2 \beta_{1} - 4 \beta_{2} ) q^{37} + q^{39} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{41} + ( 5 - \beta_{1} - 2 \beta_{2} ) q^{43} -\beta_{2} q^{45} + ( 2 - 4 \beta_{2} ) q^{47} + ( -3 - 2 \beta_{2} ) q^{49} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{51} + ( -2 + 2 \beta_{2} ) q^{53} -\beta_{2} q^{55} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{57} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{59} -4 \beta_{2} q^{61} + ( 1 - \beta_{1} - \beta_{2} ) q^{63} + \beta_{2} q^{65} + ( 6 - 4 \beta_{1} - \beta_{2} ) q^{67} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{69} + ( 4 - 6 \beta_{1} - 4 \beta_{2} ) q^{71} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{73} + ( 2 + \beta_{1} + \beta_{2} ) q^{75} + ( 1 - \beta_{1} - \beta_{2} ) q^{77} + ( 1 + 3 \beta_{1} ) q^{79} + q^{81} + ( 2 - 2 \beta_{1} - 4 \beta_{2} ) q^{83} + ( -7 + 3 \beta_{1} + 3 \beta_{2} ) q^{85} + ( 3 - 3 \beta_{1} - 2 \beta_{2} ) q^{87} + ( -2 + 8 \beta_{1} - \beta_{2} ) q^{89} + ( -1 + \beta_{1} + \beta_{2} ) q^{91} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{93} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{95} + ( 6 - 2 \beta_{1} ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 2q^{7} + 3q^{9} + 3q^{11} - 3q^{13} - 4q^{17} + 6q^{19} - 2q^{21} + 4q^{23} - 7q^{25} - 3q^{27} - 6q^{29} + 10q^{31} - 3q^{33} + 6q^{35} + 4q^{37} + 3q^{39} + 14q^{43} + 6q^{47} - 9q^{49} + 4q^{51} - 6q^{53} - 6q^{57} - 8q^{59} + 2q^{63} + 14q^{67} - 4q^{69} + 6q^{71} + 7q^{75} + 2q^{77} + 6q^{79} + 3q^{81} + 4q^{83} - 18q^{85} + 6q^{87} + 2q^{89} - 2q^{91} - 10q^{93} + 14q^{95} + 16q^{97} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

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

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
−1.48119
2.17009
0.311108
0 −1.00000 0 −1.67513 0 0.806063 0 1.00000 0
1.2 0 −1.00000 0 −0.539189 0 −1.70928 0 1.00000 0
1.3 0 −1.00000 0 2.21432 0 2.90321 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.bq 3
4.b odd 2 1 3432.2.a.p 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.p 3 4.b odd 2 1
6864.2.a.bq 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{3} - 4 T_{5} - 2 \)
\( T_{7}^{3} - 2 T_{7}^{2} - 4 T_{7} + 4 \)
\( T_{17}^{3} + 4 T_{17}^{2} - 18 T_{17} - 46 \)
\( T_{19}^{3} - 6 T_{19}^{2} - 28 T_{19} + 148 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -2 - 4 T + T^{3} \)
$7$ \( 4 - 4 T - 2 T^{2} + T^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( ( 1 + T )^{3} \)
$17$ \( -46 - 18 T + 4 T^{2} + T^{3} \)
$19$ \( 148 - 28 T - 6 T^{2} + T^{3} \)
$23$ \( 116 - 28 T - 4 T^{2} + T^{3} \)
$29$ \( -122 - 22 T + 6 T^{2} + T^{3} \)
$31$ \( -2 + 12 T - 10 T^{2} + T^{3} \)
$37$ \( -80 - 56 T - 4 T^{2} + T^{3} \)
$41$ \( -76 - 40 T + T^{3} \)
$43$ \( -50 + 50 T - 14 T^{2} + T^{3} \)
$47$ \( -8 - 52 T - 6 T^{2} + T^{3} \)
$53$ \( -8 - 4 T + 6 T^{2} + T^{3} \)
$59$ \( -272 - 32 T + 8 T^{2} + T^{3} \)
$61$ \( -128 - 64 T + T^{3} \)
$67$ \( 230 + 16 T - 14 T^{2} + T^{3} \)
$71$ \( 760 - 124 T - 6 T^{2} + T^{3} \)
$73$ \( 52 - 28 T + T^{3} \)
$79$ \( 50 - 18 T - 6 T^{2} + T^{3} \)
$83$ \( -80 - 56 T - 4 T^{2} + T^{3} \)
$89$ \( 622 - 232 T - 2 T^{2} + T^{3} \)
$97$ \( -80 + 72 T - 16 T^{2} + T^{3} \)
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