Properties

Label 6864.2.a.bk
Level $6864$
Weight $2$
Character orbit 6864.a
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
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: \(0\)
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 429)
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.

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

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 2.00000 0 1.00000 0
1.2 0 1.00000 0 0.732051 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.bk 2
4.b odd 2 1 429.2.a.d 2
12.b even 2 1 1287.2.a.f 2
44.c even 2 1 4719.2.a.n 2
52.b odd 2 1 5577.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.d 2 4.b odd 2 1
1287.2.a.f 2 12.b even 2 1
4719.2.a.n 2 44.c even 2 1
5577.2.a.h 2 52.b odd 2 1
6864.2.a.bk 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{2} + 2 T_{5} - 2 \)
\( T_{7} - 2 \)
\( T_{17}^{2} - 10 T_{17} + 22 \)
\( T_{19}^{2} - 8 T_{19} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -2 + 2 T + T^{2} \)
$7$ \( ( -2 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 22 - 10 T + T^{2} \)
$19$ \( 4 - 8 T + T^{2} \)
$23$ \( ( -2 + T )^{2} \)
$29$ \( -26 - 2 T + T^{2} \)
$31$ \( -18 - 6 T + T^{2} \)
$37$ \( ( 2 + T )^{2} \)
$41$ \( -8 + 4 T + T^{2} \)
$43$ \( -26 - 2 T + T^{2} \)
$47$ \( -8 + 4 T + T^{2} \)
$53$ \( ( -2 + T )^{2} \)
$59$ \( 88 - 20 T + T^{2} \)
$61$ \( -44 + 4 T + T^{2} \)
$67$ \( 22 - 14 T + T^{2} \)
$71$ \( -48 + T^{2} \)
$73$ \( -32 + 8 T + T^{2} \)
$79$ \( -50 + 10 T + T^{2} \)
$83$ \( -192 + T^{2} \)
$89$ \( -26 + 14 T + T^{2} \)
$97$ \( -44 - 4 T + T^{2} \)
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