Properties

Label 6864.2.a.bi
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1716)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 + \beta ) q^{5} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 + \beta ) q^{5} + q^{9} - q^{11} - q^{13} + ( -1 + \beta ) q^{15} + ( -1 - 3 \beta ) q^{17} + ( 2 + 2 \beta ) q^{19} -2 q^{23} + ( -1 - 2 \beta ) q^{25} + q^{27} + ( 3 - 3 \beta ) q^{29} + ( 1 - 3 \beta ) q^{31} - q^{33} + 6 \beta q^{37} - q^{39} + ( -8 + 2 \beta ) q^{41} + ( 1 + \beta ) q^{43} + ( -1 + \beta ) q^{45} -4 \beta q^{47} -7 q^{49} + ( -1 - 3 \beta ) q^{51} + ( 2 - 4 \beta ) q^{53} + ( 1 - \beta ) q^{55} + ( 2 + 2 \beta ) q^{57} + ( -6 + 2 \beta ) q^{59} + ( -4 - 2 \beta ) q^{61} + ( 1 - \beta ) q^{65} + ( 9 + \beta ) q^{67} -2 q^{69} + ( 2 - 2 \beta ) q^{71} + ( -6 - 2 \beta ) q^{73} + ( -1 - 2 \beta ) q^{75} + ( 11 - \beta ) q^{79} + q^{81} + 4 \beta q^{83} + ( -8 + 2 \beta ) q^{85} + ( 3 - 3 \beta ) q^{87} + ( -11 + 3 \beta ) q^{89} + ( 1 - 3 \beta ) q^{93} + 4 q^{95} + ( -4 - 6 \beta ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{5} + 2q^{9} - 2q^{11} - 2q^{13} - 2q^{15} - 2q^{17} + 4q^{19} - 4q^{23} - 2q^{25} + 2q^{27} + 6q^{29} + 2q^{31} - 2q^{33} - 2q^{39} - 16q^{41} + 2q^{43} - 2q^{45} - 14q^{49} - 2q^{51} + 4q^{53} + 2q^{55} + 4q^{57} - 12q^{59} - 8q^{61} + 2q^{65} + 18q^{67} - 4q^{69} + 4q^{71} - 12q^{73} - 2q^{75} + 22q^{79} + 2q^{81} - 16q^{85} + 6q^{87} - 22q^{89} + 2q^{93} + 8q^{95} - 8q^{97} - 2q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
0 1.00000 0 −2.73205 0 0 0 1.00000 0
1.2 0 1.00000 0 0.732051 0 0 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.bi 2
4.b odd 2 1 1716.2.a.d 2
12.b even 2 1 5148.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.a.d 2 4.b odd 2 1
5148.2.a.i 2 12.b even 2 1
6864.2.a.bi 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -2 + 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -26 + 2 T + T^{2} \)
$19$ \( -8 - 4 T + T^{2} \)
$23$ \( ( 2 + T )^{2} \)
$29$ \( -18 - 6 T + T^{2} \)
$31$ \( -26 - 2 T + T^{2} \)
$37$ \( -108 + T^{2} \)
$41$ \( 52 + 16 T + T^{2} \)
$43$ \( -2 - 2 T + T^{2} \)
$47$ \( -48 + T^{2} \)
$53$ \( -44 - 4 T + T^{2} \)
$59$ \( 24 + 12 T + T^{2} \)
$61$ \( 4 + 8 T + T^{2} \)
$67$ \( 78 - 18 T + T^{2} \)
$71$ \( -8 - 4 T + T^{2} \)
$73$ \( 24 + 12 T + T^{2} \)
$79$ \( 118 - 22 T + T^{2} \)
$83$ \( -48 + T^{2} \)
$89$ \( 94 + 22 T + T^{2} \)
$97$ \( -92 + 8 T + T^{2} \)
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