Properties

Label 6864.2.a.bo
Level $6864$
Weight $2$
Character orbit 6864.a
Self dual yes
Analytic conductor $54.809$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \(x^{3} - x^{2} - 8 x + 10\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 8 x + 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 6\)

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
2.59774
1.31955
−2.91729
0 −1.00000 0 −2.59774 0 −2.74823 0 1.00000 0
1.2 0 −1.00000 0 −1.31955 0 2.25879 0 1.00000 0
1.3 0 −1.00000 0 2.91729 0 −4.51056 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.bo 3
4.b odd 2 1 3432.2.a.o 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.o 3 4.b odd 2 1
6864.2.a.bo 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{3} + T_{5}^{2} - 8 T_{5} - 10 \)
\( T_{7}^{3} + 5 T_{7}^{2} - 4 T_{7} - 28 \)
\( T_{17}^{3} + 6 T_{17}^{2} + 2 T_{17} - 16 \)
\( T_{19} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -10 - 8 T + T^{2} + T^{3} \)
$7$ \( -28 - 4 T + 5 T^{2} + T^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( ( -1 + T )^{3} \)
$17$ \( -16 + 2 T + 6 T^{2} + T^{3} \)
$19$ \( ( 2 + T )^{3} \)
$23$ \( 76 - 16 T - 5 T^{2} + T^{3} \)
$29$ \( -2 - 20 T - 7 T^{2} + T^{3} \)
$31$ \( -16 + 2 T + 6 T^{2} + T^{3} \)
$37$ \( ( -6 + T )^{3} \)
$41$ \( -32 - 24 T - T^{2} + T^{3} \)
$43$ \( -350 - 20 T + 11 T^{2} + T^{3} \)
$47$ \( -32 - 40 T + T^{3} \)
$53$ \( ( -2 + T )^{3} \)
$59$ \( -808 - 76 T + 11 T^{2} + T^{3} \)
$61$ \( 4 - 8 T - 7 T^{2} + T^{3} \)
$67$ \( -1622 - 168 T + 11 T^{2} + T^{3} \)
$71$ \( 608 - 64 T - 10 T^{2} + T^{3} \)
$73$ \( 608 - 200 T - 5 T^{2} + T^{3} \)
$79$ \( 28 + 38 T + 12 T^{2} + T^{3} \)
$83$ \( -64 - 48 T - 2 T^{2} + T^{3} \)
$89$ \( -16 - 54 T - 2 T^{2} + T^{3} \)
$97$ \( 904 - 196 T - 2 T^{2} + T^{3} \)
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