Properties

Label 6864.2.a.bp
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.

\(f(q)\) \(=\) \( q - q^{3} -\beta_{2} q^{5} + ( 1 - \beta_{1} + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} -\beta_{2} q^{5} + ( 1 - \beta_{1} + \beta_{2} ) q^{7} + q^{9} - q^{11} + q^{13} + \beta_{2} q^{15} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{17} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} ) q^{21} + ( -3 + \beta_{1} - \beta_{2} ) q^{23} + ( -2 - \beta_{1} - \beta_{2} ) q^{25} - q^{27} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{29} + ( 2 - \beta_{2} ) q^{31} + q^{33} + ( -4 + 2 \beta_{1} ) q^{35} + ( -4 - 2 \beta_{2} ) q^{37} - q^{39} + ( 5 - 3 \beta_{1} - \beta_{2} ) q^{41} + ( -1 + \beta_{1} - 4 \beta_{2} ) q^{43} -\beta_{2} q^{45} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 1 - 4 \beta_{1} + 2 \beta_{2} ) q^{49} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{51} + ( -2 + 4 \beta_{1} + 6 \beta_{2} ) q^{53} + \beta_{2} q^{55} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{57} + ( -4 + 4 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( 1 - \beta_{1} + \beta_{2} ) q^{63} -\beta_{2} q^{65} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{67} + ( 3 - \beta_{1} + \beta_{2} ) q^{69} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -3 + 5 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 2 + \beta_{1} + \beta_{2} ) q^{75} + ( -1 + \beta_{1} - \beta_{2} ) q^{77} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{79} + q^{81} + ( 2 - 2 \beta_{1} ) q^{83} + ( -7 + 3 \beta_{1} - \beta_{2} ) q^{85} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{87} + ( -6 + 8 \beta_{1} + 3 \beta_{2} ) q^{89} + ( 1 - \beta_{1} + \beta_{2} ) q^{91} + ( -2 + \beta_{2} ) q^{93} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{95} + ( -4 - 2 \beta_{2} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 2q^{7} + 3q^{9} - 3q^{11} + 3q^{13} + 8q^{17} - 6q^{19} - 2q^{21} - 8q^{23} - 7q^{25} - 3q^{27} + 2q^{29} + 6q^{31} + 3q^{33} - 10q^{35} - 12q^{37} - 3q^{39} + 12q^{41} - 2q^{43} - 2q^{47} - q^{49} - 8q^{51} - 2q^{53} + 6q^{57} - 8q^{59} - 4q^{61} + 2q^{63} - 2q^{67} + 8q^{69} - 2q^{71} - 4q^{73} + 7q^{75} - 2q^{77} - 14q^{79} + 3q^{81} + 4q^{83} - 18q^{85} - 2q^{87} - 10q^{89} + 2q^{91} - 6q^{93} - 2q^{95} - 12q^{97} - 3q^{99} + O(q^{100}) \)

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
2.17009
0.311108
0 −1.00000 0 −1.67513 0 4.15633 0 1.00000 0
1.2 0 −1.00000 0 −0.539189 0 −0.630898 0 1.00000 0
1.3 0 −1.00000 0 2.21432 0 −1.52543 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.bp 3
4.b odd 2 1 429.2.a.f 3
12.b even 2 1 1287.2.a.i 3
44.c even 2 1 4719.2.a.t 3
52.b odd 2 1 5577.2.a.k 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.f 3 4.b odd 2 1
1287.2.a.i 3 12.b even 2 1
4719.2.a.t 3 44.c even 2 1
5577.2.a.k 3 52.b odd 2 1
6864.2.a.bp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{3} - 4 T_{5} - 2 \)
\( T_{7}^{3} - 2 T_{7}^{2} - 8 T_{7} - 4 \)
\( T_{17}^{3} - 8 T_{17}^{2} - 2 T_{17} + 26 \)
\( T_{19}^{3} + 6 T_{19}^{2} - 16 T_{19} - 100 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -2 - 4 T + T^{3} \)
$7$ \( -4 - 8 T - 2 T^{2} + T^{3} \)
$11$ \( ( 1 + T )^{3} \)
$13$ \( ( -1 + T )^{3} \)
$17$ \( 26 - 2 T - 8 T^{2} + T^{3} \)
$19$ \( -100 - 16 T + 6 T^{2} + T^{3} \)
$23$ \( 4 + 12 T + 8 T^{2} + T^{3} \)
$29$ \( -10 - 14 T - 2 T^{2} + T^{3} \)
$31$ \( -2 + 8 T - 6 T^{2} + T^{3} \)
$37$ \( -16 + 32 T + 12 T^{2} + T^{3} \)
$41$ \( 100 + 20 T - 12 T^{2} + T^{3} \)
$43$ \( -74 - 74 T + 2 T^{2} + T^{3} \)
$47$ \( -104 - 36 T + 2 T^{2} + T^{3} \)
$53$ \( 296 - 148 T + 2 T^{2} + T^{3} \)
$59$ \( 80 - 64 T + 8 T^{2} + T^{3} \)
$61$ \( -32 - 16 T + 4 T^{2} + T^{3} \)
$67$ \( -74 - 36 T + 2 T^{2} + T^{3} \)
$71$ \( -184 - 52 T + 2 T^{2} + T^{3} \)
$73$ \( -412 - 84 T + 4 T^{2} + T^{3} \)
$79$ \( 10 + 42 T + 14 T^{2} + T^{3} \)
$83$ \( 16 - 8 T - 4 T^{2} + T^{3} \)
$89$ \( -1690 - 168 T + 10 T^{2} + T^{3} \)
$97$ \( -16 + 32 T + 12 T^{2} + T^{3} \)
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