Properties

Label 6864.2.a.bs
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: 3.3.148.1
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 429)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 + \beta_{1} ) q^{5} + ( 1 - \beta_{1} + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 + \beta_{1} ) q^{5} + ( 1 - \beta_{1} + \beta_{2} ) q^{7} + q^{9} + q^{11} - q^{13} + ( -1 - \beta_{1} ) q^{15} -\beta_{2} q^{17} + ( -1 + \beta_{1} - \beta_{2} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} ) q^{21} + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{23} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{25} - q^{27} + ( 2 - 4 \beta_{1} - 3 \beta_{2} ) q^{29} + ( -5 + \beta_{1} ) q^{31} - q^{33} -2 q^{35} -4 \beta_{2} q^{37} + q^{39} + ( -1 - \beta_{1} + \beta_{2} ) q^{41} + ( 4 - 6 \beta_{1} - 3 \beta_{2} ) q^{43} + ( 1 + \beta_{1} ) q^{45} + ( -6 + 4 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 1 - 4 \beta_{1} + 2 \beta_{2} ) q^{49} + \beta_{2} q^{51} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 1 + \beta_{1} ) q^{55} + ( 1 - \beta_{1} + \beta_{2} ) q^{57} + ( -10 + 2 \beta_{1} ) q^{59} + ( -2 - 2 \beta_{1} + 4 \beta_{2} ) q^{61} + ( 1 - \beta_{1} + \beta_{2} ) q^{63} + ( -1 - \beta_{1} ) q^{65} + ( -3 + 7 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{69} + ( -2 - 4 \beta_{1} - 8 \beta_{2} ) q^{71} + ( -7 - 3 \beta_{1} - 5 \beta_{2} ) q^{73} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{75} + ( 1 - \beta_{1} + \beta_{2} ) q^{77} + ( 6 + 4 \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( -6 + 2 \beta_{1} ) q^{83} + ( 1 - \beta_{1} - \beta_{2} ) q^{85} + ( -2 + 4 \beta_{1} + 3 \beta_{2} ) q^{87} + ( 1 - \beta_{1} ) q^{89} + ( -1 + \beta_{1} - \beta_{2} ) q^{91} + ( 5 - \beta_{1} ) q^{93} + 2 q^{95} + ( -4 + 4 \beta_{2} ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 4q^{5} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 4q^{5} + 2q^{7} + 3q^{9} + 3q^{11} - 3q^{13} - 4q^{15} - 2q^{19} - 2q^{21} - 4q^{23} - 3q^{25} - 3q^{27} + 2q^{29} - 14q^{31} - 3q^{33} - 6q^{35} + 3q^{39} - 4q^{41} + 6q^{43} + 4q^{45} - 14q^{47} - q^{49} - 2q^{53} + 4q^{55} + 2q^{57} - 28q^{59} - 8q^{61} + 2q^{63} - 4q^{65} - 2q^{67} + 4q^{69} - 10q^{71} - 24q^{73} + 3q^{75} + 2q^{77} + 22q^{79} + 3q^{81} - 16q^{83} + 2q^{85} - 2q^{87} + 2q^{89} - 2q^{91} + 14q^{93} + 6q^{95} - 12q^{97} + 3q^{99} + O(q^{100}) \)

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
0.311108
2.17009
0 −1.00000 0 −0.481194 0 4.15633 0 1.00000 0
1.2 0 −1.00000 0 1.31111 0 −1.52543 0 1.00000 0
1.3 0 −1.00000 0 3.17009 0 −0.630898 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.bs 3
4.b odd 2 1 429.2.a.g 3
12.b even 2 1 1287.2.a.h 3
44.c even 2 1 4719.2.a.q 3
52.b odd 2 1 5577.2.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.g 3 4.b odd 2 1
1287.2.a.h 3 12.b even 2 1
4719.2.a.q 3 44.c even 2 1
5577.2.a.j 3 52.b odd 2 1
6864.2.a.bs 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 2 + 2 T - 4 T^{2} + T^{3} \)
$7$ \( -4 - 8 T - 2 T^{2} + T^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( ( 1 + T )^{3} \)
$17$ \( -2 - 4 T + T^{3} \)
$19$ \( 4 - 8 T + 2 T^{2} + T^{3} \)
$23$ \( -68 - 40 T + 4 T^{2} + T^{3} \)
$29$ \( 178 - 64 T - 2 T^{2} + T^{3} \)
$31$ \( 86 + 62 T + 14 T^{2} + T^{3} \)
$37$ \( -128 - 64 T + T^{3} \)
$41$ \( -20 - 4 T + 4 T^{2} + T^{3} \)
$43$ \( 734 - 108 T - 6 T^{2} + T^{3} \)
$47$ \( -296 + 12 T + 14 T^{2} + T^{3} \)
$53$ \( 232 - 84 T + 2 T^{2} + T^{3} \)
$59$ \( 688 + 248 T + 28 T^{2} + T^{3} \)
$61$ \( -368 - 72 T + 8 T^{2} + T^{3} \)
$67$ \( -698 - 150 T + 2 T^{2} + T^{3} \)
$71$ \( -2056 - 212 T + 10 T^{2} + T^{3} \)
$73$ \( -556 + 92 T + 24 T^{2} + T^{3} \)
$79$ \( -134 + 112 T - 22 T^{2} + T^{3} \)
$83$ \( 80 + 72 T + 16 T^{2} + T^{3} \)
$89$ \( 2 - 2 T - 2 T^{2} + T^{3} \)
$97$ \( -64 - 16 T + 12 T^{2} + T^{3} \)
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