Properties

Label 6840.2.a.bh
Level $6840$
Weight $2$
Character orbit 6840.a
Self dual yes
Analytic conductor $54.618$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.6176749826\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1772.1
Defining polynomial: \(x^{3} - x^{2} - 12 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
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.

\(f(q)\) \(=\) \( q - q^{5} + \beta_{2} q^{7} +O(q^{10})\) \( q - q^{5} + \beta_{2} q^{7} + ( -2 - \beta_{2} ) q^{11} + ( 1 - \beta_{1} ) q^{13} -4 q^{17} + q^{19} + ( 1 - \beta_{1} + \beta_{2} ) q^{23} + q^{25} + \beta_{2} q^{29} -2 q^{31} -\beta_{2} q^{35} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{37} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{41} + ( 1 + \beta_{1} ) q^{43} + ( -1 + \beta_{1} - \beta_{2} ) q^{47} + ( 2 - \beta_{1} - \beta_{2} ) q^{49} + ( 3 - \beta_{1} + \beta_{2} ) q^{53} + ( 2 + \beta_{2} ) q^{55} + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{59} + ( 5 + \beta_{1} + \beta_{2} ) q^{61} + ( -1 + \beta_{1} ) q^{65} + 4 q^{67} + ( 2 + 2 \beta_{1} ) q^{71} + 6 q^{73} + ( -9 + \beta_{1} - \beta_{2} ) q^{77} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{79} + ( 2 + 2 \beta_{2} ) q^{83} + 4 q^{85} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{89} + ( -3 - \beta_{1} + 5 \beta_{2} ) q^{91} - q^{95} + ( 11 + \beta_{1} - 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{5} + O(q^{10}) \) \( 3q - 3q^{5} - 6q^{11} + 4q^{13} - 12q^{17} + 3q^{19} + 4q^{23} + 3q^{25} - 6q^{31} - 4q^{37} - 2q^{41} + 2q^{43} - 4q^{47} + 7q^{49} + 10q^{53} + 6q^{55} - 2q^{59} + 14q^{61} - 4q^{65} + 12q^{67} + 4q^{71} + 18q^{73} - 28q^{77} - 2q^{79} + 6q^{83} + 12q^{85} - 10q^{89} - 8q^{91} - 3q^{95} + 32q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 12 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} + 3 \nu - 10 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{2} + \beta_{1} + 17\)\()/2\)

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
0.654334
3.67370
−3.32803
0 0 0 −1.00000 0 −4.11309 0 0 0
1.2 0 0 0 −1.00000 0 0.911179 0 0 0
1.3 0 0 0 −1.00000 0 3.20191 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(19\) \(-1\)

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 6840.2.a.bh 3
3.b odd 2 1 2280.2.a.u 3
12.b even 2 1 4560.2.a.bs 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.u 3 3.b odd 2 1
4560.2.a.bs 3 12.b even 2 1
6840.2.a.bh 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6840))\):

\( T_{7}^{3} - 14 T_{7} + 12 \)
\( T_{11}^{3} + 6 T_{11}^{2} - 2 T_{11} - 32 \)
\( T_{13}^{3} - 4 T_{13}^{2} - 38 T_{13} + 164 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( 12 - 14 T + T^{3} \)
$11$ \( -32 - 2 T + 6 T^{2} + T^{3} \)
$13$ \( 164 - 38 T - 4 T^{2} + T^{3} \)
$17$ \( ( 4 + T )^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( 32 - 44 T - 4 T^{2} + T^{3} \)
$29$ \( 12 - 14 T + T^{3} \)
$31$ \( ( 2 + T )^{3} \)
$37$ \( 180 - 78 T + 4 T^{2} + T^{3} \)
$41$ \( -344 - 82 T + 2 T^{2} + T^{3} \)
$43$ \( -80 - 42 T - 2 T^{2} + T^{3} \)
$47$ \( -32 - 44 T + 4 T^{2} + T^{3} \)
$53$ \( 96 - 16 T - 10 T^{2} + T^{3} \)
$59$ \( -688 - 144 T + 2 T^{2} + T^{3} \)
$61$ \( 144 - 14 T^{2} + T^{3} \)
$67$ \( ( -4 + T )^{3} \)
$71$ \( -640 - 168 T - 4 T^{2} + T^{3} \)
$73$ \( ( -6 + T )^{3} \)
$79$ \( -1024 - 192 T + 2 T^{2} + T^{3} \)
$83$ \( 200 - 44 T - 6 T^{2} + T^{3} \)
$89$ \( 48 - 50 T + 10 T^{2} + T^{3} \)
$97$ \( -36 + 258 T - 32 T^{2} + T^{3} \)
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