Properties

Label 6864.2.a.bf
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: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 858)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{41}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−2.70156
3.70156
0 −1.00000 0 2.00000 0 0 0 1.00000 0
1.2 0 −1.00000 0 2.00000 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.bf 2
4.b odd 2 1 858.2.a.q 2
12.b even 2 1 2574.2.a.z 2
44.c even 2 1 9438.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
858.2.a.q 2 4.b odd 2 1
2574.2.a.z 2 12.b even 2 1
6864.2.a.bf 2 1.a even 1 1 trivial
9438.2.a.bs 2 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5} - 2 \)
\( T_{7} \)
\( T_{17}^{2} - 2 T_{17} - 40 \)
\( T_{19}^{2} + 2 T_{19} - 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -2 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -40 - 2 T + T^{2} \)
$19$ \( -40 + 2 T + T^{2} \)
$23$ \( -32 - 6 T + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( -40 - 2 T + T^{2} \)
$37$ \( -40 - 2 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( -32 - 6 T + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( -164 + T^{2} \)
$67$ \( T^{2} \)
$71$ \( -160 - 4 T + T^{2} \)
$73$ \( 8 - 14 T + T^{2} \)
$79$ \( -160 + 4 T + T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( ( -10 + T )^{2} \)
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