Properties

Label 6864.2.a.bt
Level $6864$
Weight $2$
Character orbit 6864.a
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Defining polynomial: \(x^{3} - x^{2} - 5 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{5} + ( 2 - \beta_{1} - \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{5} + ( 2 - \beta_{1} - \beta_{2} ) q^{7} + q^{9} + q^{11} + q^{13} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{15} + ( 1 - \beta_{1} ) q^{17} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{19} + ( -2 + \beta_{1} + \beta_{2} ) q^{21} + ( -4 + \beta_{1} - \beta_{2} ) q^{23} + ( 9 + \beta_{1} + \beta_{2} ) q^{25} - q^{27} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{29} + ( -7 + 2 \beta_{1} + \beta_{2} ) q^{31} - q^{33} -6 \beta_{2} q^{35} + ( 2 - 2 \beta_{2} ) q^{37} - q^{39} + ( 4 - \beta_{1} - \beta_{2} ) q^{41} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{43} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{45} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 7 + 2 \beta_{1} - 4 \beta_{2} ) q^{49} + ( -1 + \beta_{1} ) q^{51} + ( 4 - 4 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{55} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{57} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{59} -4 \beta_{1} q^{61} + ( 2 - \beta_{1} - \beta_{2} ) q^{63} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{65} + ( 5 - 4 \beta_{1} - \beta_{2} ) q^{67} + ( 4 - \beta_{1} + \beta_{2} ) q^{69} -2 \beta_{2} q^{71} + ( 4 + \beta_{1} + 3 \beta_{2} ) q^{73} + ( -9 - \beta_{1} - \beta_{2} ) q^{75} + ( 2 - \beta_{1} - \beta_{2} ) q^{77} + ( 3 + 7 \beta_{1} - 4 \beta_{2} ) q^{79} + q^{81} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -4 + \beta_{1} - 3 \beta_{2} ) q^{85} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{87} + ( 1 + 2 \beta_{1} + 5 \beta_{2} ) q^{89} + ( 2 - \beta_{1} - \beta_{2} ) q^{91} + ( 7 - 2 \beta_{1} - \beta_{2} ) q^{93} + ( -12 + 12 \beta_{1} + 2 \beta_{2} ) q^{95} + ( -6 - 4 \beta_{1} + 2 \beta_{2} ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 4q^{5} + 4q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 4q^{5} + 4q^{7} + 3q^{9} + 3q^{11} + 3q^{13} - 4q^{15} + 2q^{17} + 8q^{19} - 4q^{21} - 12q^{23} + 29q^{25} - 3q^{27} + 4q^{29} - 18q^{31} - 3q^{33} - 6q^{35} + 4q^{37} - 3q^{39} + 10q^{41} + 16q^{43} + 4q^{45} - 2q^{47} + 19q^{49} - 2q^{51} + 6q^{53} + 4q^{55} - 8q^{57} + 8q^{59} - 4q^{61} + 4q^{63} + 4q^{65} + 10q^{67} + 12q^{69} - 2q^{71} + 16q^{73} - 29q^{75} + 4q^{77} + 12q^{79} + 3q^{81} + 8q^{83} - 14q^{85} - 4q^{87} + 10q^{89} + 4q^{91} + 18q^{93} - 22q^{95} - 20q^{97} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)

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
−1.65544
−0.210756
2.86620
0 −1.00000 0 −3.70682 0 2.25951 0 1.00000 0
1.2 0 −1.00000 0 3.32331 0 4.95558 0 1.00000 0
1.3 0 −1.00000 0 4.38350 0 −3.21509 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.bt 3
4.b odd 2 1 1716.2.a.h 3
12.b even 2 1 5148.2.a.l 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.a.h 3 4.b odd 2 1
5148.2.a.l 3 12.b even 2 1
6864.2.a.bt 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{3} - 4 T_{5}^{2} - 14 T_{5} + 54 \)
\( T_{7}^{3} - 4 T_{7}^{2} - 12 T_{7} + 36 \)
\( T_{17}^{3} - 2 T_{17}^{2} - 4 T_{17} + 6 \)
\( T_{19}^{3} - 8 T_{19}^{2} - 36 T_{19} + 292 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 54 - 14 T - 4 T^{2} + T^{3} \)
$7$ \( 36 - 12 T - 4 T^{2} + T^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( ( -1 + T )^{3} \)
$17$ \( 6 - 4 T - 2 T^{2} + T^{3} \)
$19$ \( 292 - 36 T - 8 T^{2} + T^{3} \)
$23$ \( 36 + 40 T + 12 T^{2} + T^{3} \)
$29$ \( 66 - 20 T - 4 T^{2} + T^{3} \)
$31$ \( -98 + 70 T + 18 T^{2} + T^{3} \)
$37$ \( -16 - 24 T - 4 T^{2} + T^{3} \)
$41$ \( 36 + 16 T - 10 T^{2} + T^{3} \)
$43$ \( 18 + 60 T - 16 T^{2} + T^{3} \)
$47$ \( -168 - 68 T + 2 T^{2} + T^{3} \)
$53$ \( 984 - 140 T - 6 T^{2} + T^{3} \)
$59$ \( 432 - 56 T - 8 T^{2} + T^{3} \)
$61$ \( 64 - 80 T + 4 T^{2} + T^{3} \)
$67$ \( 774 - 78 T - 10 T^{2} + T^{3} \)
$71$ \( -72 - 28 T + 2 T^{2} + T^{3} \)
$73$ \( 404 - 16 T^{2} + T^{3} \)
$79$ \( 2422 - 200 T - 12 T^{2} + T^{3} \)
$83$ \( 432 - 56 T - 8 T^{2} + T^{3} \)
$89$ \( 1134 - 218 T - 10 T^{2} + T^{3} \)
$97$ \( -464 + 56 T + 20 T^{2} + T^{3} \)
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