Properties

Label 6864.2.a.bn
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: 3.3.404.1
Defining polynomial: \(x^{3} - x^{2} - 5 x - 1\)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 - \beta_{2} ) q^{5} + ( 2 + \beta_{1} - \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 - \beta_{2} ) q^{5} + ( 2 + \beta_{1} - \beta_{2} ) q^{7} + q^{9} - q^{11} + q^{13} + ( 1 + \beta_{2} ) q^{15} + ( -1 + 3 \beta_{1} ) q^{17} + ( 2 - \beta_{1} + \beta_{2} ) q^{19} + ( -2 - \beta_{1} + \beta_{2} ) q^{21} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{23} + ( 1 + \beta_{1} + \beta_{2} ) q^{25} - q^{27} + ( -1 + \beta_{1} ) q^{29} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{31} + q^{33} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{35} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{37} - q^{39} + ( -3 \beta_{1} - \beta_{2} ) q^{41} + ( -3 + 3 \beta_{1} ) q^{43} + ( -1 - \beta_{2} ) q^{45} + ( 4 - 2 \beta_{2} ) q^{47} + ( 3 + 2 \beta_{1} - 4 \beta_{2} ) q^{49} + ( 1 - 3 \beta_{1} ) q^{51} + ( -4 - 2 \beta_{1} ) q^{53} + ( 1 + \beta_{2} ) q^{55} + ( -2 + \beta_{1} - \beta_{2} ) q^{57} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 2 + 2 \beta_{1} - 4 \beta_{2} ) q^{61} + ( 2 + \beta_{1} - \beta_{2} ) q^{63} + ( -1 - \beta_{2} ) q^{65} + ( 9 - 4 \beta_{1} - \beta_{2} ) q^{67} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{69} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 2 - \beta_{1} - \beta_{2} ) q^{73} + ( -1 - \beta_{1} - \beta_{2} ) q^{75} + ( -2 - \beta_{1} + \beta_{2} ) q^{77} + ( 7 - \beta_{1} + 2 \beta_{2} ) q^{79} + q^{81} + ( -6 + 4 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -2 - 9 \beta_{1} + \beta_{2} ) q^{85} + ( 1 - \beta_{1} ) q^{87} + ( -1 + 4 \beta_{1} - 3 \beta_{2} ) q^{89} + ( 2 + \beta_{1} - \beta_{2} ) q^{91} + ( -5 + 2 \beta_{1} - \beta_{2} ) q^{93} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{95} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 4q^{5} + 6q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 4q^{5} + 6q^{7} + 3q^{9} - 3q^{11} + 3q^{13} + 4q^{15} + 6q^{19} - 6q^{21} + 5q^{25} - 3q^{27} - 2q^{29} + 14q^{31} + 3q^{33} + 2q^{35} - 8q^{37} - 3q^{39} - 4q^{41} - 6q^{43} - 4q^{45} + 10q^{47} + 7q^{49} - 14q^{53} + 4q^{55} - 6q^{57} - 4q^{59} + 4q^{61} + 6q^{63} - 4q^{65} + 22q^{67} + 6q^{71} + 4q^{73} - 5q^{75} - 6q^{77} + 22q^{79} + 3q^{81} - 16q^{83} - 14q^{85} + 2q^{87} - 2q^{89} + 6q^{91} - 14q^{93} - 18q^{95} - 24q^{97} - 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)

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
2.86620
−1.65544
−0.210756
0 −1.00000 0 −3.34889 0 2.51730 0 1.00000 0
1.2 0 −1.00000 0 −2.39593 0 −1.05137 0 1.00000 0
1.3 0 −1.00000 0 1.74483 0 4.53407 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.bn 3
4.b odd 2 1 3432.2.a.n 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.n 3 4.b odd 2 1
6864.2.a.bn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{3} + 4 T_{5}^{2} - 2 T_{5} - 14 \)
\( T_{7}^{3} - 6 T_{7}^{2} + 4 T_{7} + 12 \)
\( T_{17}^{3} - 48 T_{17} - 74 \)
\( T_{19}^{3} - 6 T_{19}^{2} + 4 T_{19} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -14 - 2 T + 4 T^{2} + T^{3} \)
$7$ \( 12 + 4 T - 6 T^{2} + T^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( ( -1 + T )^{3} \)
$17$ \( -74 - 48 T + T^{3} \)
$19$ \( 4 + 4 T - 6 T^{2} + T^{3} \)
$23$ \( 108 - 72 T + T^{3} \)
$29$ \( -6 - 4 T + 2 T^{2} + T^{3} \)
$31$ \( -42 + 46 T - 14 T^{2} + T^{3} \)
$37$ \( -288 - 48 T + 8 T^{2} + T^{3} \)
$41$ \( 132 - 64 T + 4 T^{2} + T^{3} \)
$43$ \( -162 - 36 T + 6 T^{2} + T^{3} \)
$47$ \( 8 + 4 T - 10 T^{2} + T^{3} \)
$53$ \( 24 + 44 T + 14 T^{2} + T^{3} \)
$59$ \( 144 - 72 T + 4 T^{2} + T^{3} \)
$61$ \( -144 - 96 T - 4 T^{2} + T^{3} \)
$67$ \( 862 + 50 T - 22 T^{2} + T^{3} \)
$71$ \( 24 - 20 T - 6 T^{2} + T^{3} \)
$73$ \( 36 - 12 T - 4 T^{2} + T^{3} \)
$79$ \( -174 + 136 T - 22 T^{2} + T^{3} \)
$83$ \( -16 + 8 T + 16 T^{2} + T^{3} \)
$89$ \( 258 - 94 T + 2 T^{2} + T^{3} \)
$97$ \( 288 + 160 T + 24 T^{2} + T^{3} \)
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