Properties

Label 6864.2.a.bg
Level $6864$
Weight $2$
Character orbit 6864.a
Self dual yes
Analytic conductor $54.809$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−2.44949
2.44949
0 1.00000 0 −4.44949 0 2.00000 0 1.00000 0
1.2 0 1.00000 0 0.449490 0 2.00000 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.bg 2
4.b odd 2 1 3432.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.j 2 4.b odd 2 1
6864.2.a.bg 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{2} + 4 T_{5} - 2 \)
\( T_{7} - 2 \)
\( T_{17}^{2} + 8 T_{17} + 10 \)
\( T_{19}^{2} + 4 T_{19} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -2 + 4 T + T^{2} \)
$7$ \( ( -2 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( 10 + 8 T + T^{2} \)
$19$ \( -20 + 4 T + T^{2} \)
$23$ \( -20 + 4 T + T^{2} \)
$29$ \( 10 - 8 T + T^{2} \)
$31$ \( 10 + 8 T + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( -8 + 8 T + T^{2} \)
$43$ \( -2 + 4 T + T^{2} \)
$47$ \( -24 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( -24 + T^{2} \)
$61$ \( -92 + 4 T + T^{2} \)
$67$ \( 10 - 8 T + T^{2} \)
$71$ \( -96 + T^{2} \)
$73$ \( -8 - 8 T + T^{2} \)
$79$ \( -50 - 4 T + T^{2} \)
$83$ \( -96 + T^{2} \)
$89$ \( -2 + 4 T + T^{2} \)
$97$ \( ( 10 + T )^{2} \)
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