Properties

Label 6864.2.a.bu
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: 3.3.564.1
Defining polynomial: \(x^{3} - x^{2} - 5 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( -\beta_{1} + \beta_{2} ) q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -\beta_{1} + \beta_{2} ) q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} - q^{11} - q^{13} + ( -\beta_{1} + \beta_{2} ) q^{15} + ( -1 + \beta_{1} ) q^{17} + ( -3 + \beta_{2} ) q^{19} + ( -1 - \beta_{2} ) q^{21} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{23} + ( 2 - 3 \beta_{2} ) q^{25} + q^{27} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{29} + ( -6 + \beta_{1} - \beta_{2} ) q^{31} - q^{33} + ( -4 + 2 \beta_{2} ) q^{35} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{37} - q^{39} + ( -1 + 4 \beta_{1} + \beta_{2} ) q^{41} + ( 5 - \beta_{1} - 2 \beta_{2} ) q^{43} + ( -\beta_{1} + \beta_{2} ) q^{45} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{47} + ( -1 + 2 \beta_{1} ) q^{49} + ( -1 + \beta_{1} ) q^{51} + ( -4 + 2 \beta_{1} ) q^{53} + ( \beta_{1} - \beta_{2} ) q^{55} + ( -3 + \beta_{2} ) q^{57} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 2 + 2 \beta_{1} ) q^{61} + ( -1 - \beta_{2} ) q^{63} + ( \beta_{1} - \beta_{2} ) q^{65} + ( -8 - 3 \beta_{1} + \beta_{2} ) q^{67} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{69} -2 \beta_{2} q^{71} + ( 3 - 4 \beta_{1} - 3 \beta_{2} ) q^{73} + ( 2 - 3 \beta_{2} ) q^{75} + ( 1 + \beta_{2} ) q^{77} + ( -7 + \beta_{1} ) q^{79} + q^{81} + ( -8 - 2 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{85} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{87} + ( -8 + \beta_{1} - 3 \beta_{2} ) q^{89} + ( 1 + \beta_{2} ) q^{91} + ( -6 + \beta_{1} - \beta_{2} ) q^{93} + ( 4 + 4 \beta_{1} - 6 \beta_{2} ) q^{95} + ( -2 - 4 \beta_{1} + 2 \beta_{2} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} - 2q^{5} - 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} - 2q^{5} - 2q^{7} + 3q^{9} - 3q^{11} - 3q^{13} - 2q^{15} - 2q^{17} - 10q^{19} - 2q^{21} + 12q^{23} + 9q^{25} + 3q^{27} + 12q^{29} - 16q^{31} - 3q^{33} - 14q^{35} - 3q^{39} + 16q^{43} - 2q^{45} + 2q^{47} - q^{49} - 2q^{51} - 10q^{53} + 2q^{55} - 10q^{57} - 16q^{59} + 8q^{61} - 2q^{63} + 2q^{65} - 28q^{67} + 12q^{69} + 2q^{71} + 8q^{73} + 9q^{75} + 2q^{77} - 20q^{79} + 3q^{81} - 24q^{83} - 6q^{85} + 12q^{87} - 20q^{89} + 2q^{91} - 16q^{93} + 22q^{95} - 12q^{97} - 3q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)

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
0.571993
2.51414
−2.08613
0 1.00000 0 −4.24482 0 2.67282 0 1.00000 0
1.2 0 1.00000 0 −0.193252 0 −3.32088 0 1.00000 0
1.3 0 1.00000 0 2.43807 0 −1.35194 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.bu 3
4.b odd 2 1 429.2.a.e 3
12.b even 2 1 1287.2.a.j 3
44.c even 2 1 4719.2.a.u 3
52.b odd 2 1 5577.2.a.l 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.e 3 4.b odd 2 1
1287.2.a.j 3 12.b even 2 1
4719.2.a.u 3 44.c even 2 1
5577.2.a.l 3 52.b odd 2 1
6864.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{3} + 2 T_{5}^{2} - 10 T_{5} - 2 \)
\( T_{7}^{3} + 2 T_{7}^{2} - 8 T_{7} - 12 \)
\( T_{17}^{3} + 2 T_{17}^{2} - 4 T_{17} - 2 \)
\( T_{19}^{3} + 10 T_{19}^{2} + 24 T_{19} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( -2 - 10 T + 2 T^{2} + T^{3} \)
$7$ \( -12 - 8 T + 2 T^{2} + T^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( ( 1 + T )^{3} \)
$17$ \( -2 - 4 T + 2 T^{2} + T^{3} \)
$19$ \( 12 + 24 T + 10 T^{2} + T^{3} \)
$23$ \( 68 + 24 T - 12 T^{2} + T^{3} \)
$29$ \( 162 + 12 T - 12 T^{2} + T^{3} \)
$31$ \( 86 + 74 T + 16 T^{2} + T^{3} \)
$37$ \( 288 - 96 T + T^{3} \)
$41$ \( -244 - 108 T + T^{3} \)
$43$ \( 162 + 36 T - 16 T^{2} + T^{3} \)
$47$ \( 72 - 44 T - 2 T^{2} + T^{3} \)
$53$ \( -24 + 12 T + 10 T^{2} + T^{3} \)
$59$ \( -48 + 40 T + 16 T^{2} + T^{3} \)
$61$ \( 48 - 8 T^{2} + T^{3} \)
$67$ \( 246 + 214 T + 28 T^{2} + T^{3} \)
$71$ \( -24 - 36 T - 2 T^{2} + T^{3} \)
$73$ \( 1692 - 188 T - 8 T^{2} + T^{3} \)
$79$ \( 262 + 128 T + 20 T^{2} + T^{3} \)
$83$ \( 144 + 120 T + 24 T^{2} + T^{3} \)
$89$ \( -498 + 54 T + 20 T^{2} + T^{3} \)
$97$ \( -608 - 48 T + 12 T^{2} + T^{3} \)
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