Properties

Label 6840.2.a.bk
Level $6840$
Weight $2$
Character orbit 6840.a
Self dual yes
Analytic conductor $54.618$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{5} + ( - \beta_{2} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{2} - 1) q^{11} + (\beta_1 - 2) q^{13} - 2 \beta_1 q^{17} + q^{19} + (\beta_{2} + \beta_1 + 3) q^{23} + q^{25} + (\beta_{2} + 2 \beta_1 - 3) q^{29} + ( - 2 \beta_1 - 2) q^{31} + ( - \beta_{2} - 1) q^{35} + ( - 3 \beta_1 - 2) q^{37} + ( - 2 \beta_{2} + \beta_1 - 2) q^{41} + 3 \beta_1 q^{43} + ( - \beta_{2} - \beta_1 + 5) q^{47} + ( - \beta_{2} + \beta_1 + 6) q^{49} + (\beta_{2} + \beta_1 - 7) q^{53} + (\beta_{2} - 1) q^{55} + (\beta_{2} - 3 \beta_1 - 7) q^{59} + (\beta_{2} + 3 \beta_1 + 1) q^{61} + (\beta_1 - 2) q^{65} + (2 \beta_{2} + 2 \beta_1 + 2) q^{67} + ( - 2 \beta_{2} - 4 \beta_1 + 2) q^{71} + (2 \beta_{2} - 2 \beta_1 + 4) q^{73} + (3 \beta_{2} - \beta_1 - 11) q^{77} + (\beta_{2} - \beta_1 - 3) q^{79} + (2 \beta_{2} + 2 \beta_1) q^{83} - 2 \beta_1 q^{85} + ( - 3 \beta_1 - 8) q^{89} + (\beta_{2} - 3 \beta_1 + 3) q^{91} + q^{95} + ( - 3 \beta_1 - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} - 2 q^{7} - 4 q^{11} - 6 q^{13} + 3 q^{19} + 8 q^{23} + 3 q^{25} - 10 q^{29} - 6 q^{31} - 2 q^{35} - 6 q^{37} - 4 q^{41} + 16 q^{47} + 19 q^{49} - 22 q^{53} - 4 q^{55} - 22 q^{59} + 2 q^{61} - 6 q^{65} + 4 q^{67} + 8 q^{71} + 10 q^{73} - 36 q^{77} - 10 q^{79} - 2 q^{83} - 24 q^{89} + 8 q^{91} + 3 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 3\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 3\beta _1 + 9 ) / 2 \) Copy content Toggle raw display

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
3.12489
−0.363328
−1.76156
0 0 0 1.00000 0 −3.60975 0 0 0
1.2 0 0 0 1.00000 0 −2.77801 0 0 0
1.3 0 0 0 1.00000 0 4.38776 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.bk 3
3.b odd 2 1 2280.2.a.r 3
12.b even 2 1 4560.2.a.bt 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.r 3 3.b odd 2 1
4560.2.a.bt 3 12.b even 2 1
6840.2.a.bk 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6840))\):

\( T_{7}^{3} + 2T_{7}^{2} - 18T_{7} - 44 \) Copy content Toggle raw display
\( T_{11}^{3} + 4T_{11}^{2} - 14T_{11} + 8 \) Copy content Toggle raw display
\( T_{13}^{3} + 6T_{13}^{2} + 2T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 18 T - 44 \) Copy content Toggle raw display
$11$ \( T^{3} + 4 T^{2} - 14 T + 8 \) Copy content Toggle raw display
$13$ \( T^{3} + 6 T^{2} + 2 T - 4 \) Copy content Toggle raw display
$17$ \( T^{3} - 40T - 64 \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 8 T^{2} - 4 T + 16 \) Copy content Toggle raw display
$29$ \( T^{3} + 10 T^{2} - 18 T - 268 \) Copy content Toggle raw display
$31$ \( T^{3} + 6 T^{2} - 28 T - 136 \) Copy content Toggle raw display
$37$ \( T^{3} + 6 T^{2} - 78 T - 388 \) Copy content Toggle raw display
$41$ \( T^{3} + 4 T^{2} - 90 T - 400 \) Copy content Toggle raw display
$43$ \( T^{3} - 90T + 216 \) Copy content Toggle raw display
$47$ \( T^{3} - 16 T^{2} + 60 T + 16 \) Copy content Toggle raw display
$53$ \( T^{3} + 22 T^{2} + 136 T + 176 \) Copy content Toggle raw display
$59$ \( T^{3} + 22 T^{2} + 40 T - 976 \) Copy content Toggle raw display
$61$ \( T^{3} - 2 T^{2} - 96 T - 160 \) Copy content Toggle raw display
$67$ \( T^{3} - 4 T^{2} - 96 T - 128 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} - 184 T + 1600 \) Copy content Toggle raw display
$73$ \( T^{3} - 10 T^{2} - 100 T + 488 \) Copy content Toggle raw display
$79$ \( T^{3} + 10T^{2} - 64 \) Copy content Toggle raw display
$83$ \( T^{3} + 2 T^{2} - 100 T - 328 \) Copy content Toggle raw display
$89$ \( T^{3} + 24 T^{2} + 102 T - 424 \) Copy content Toggle raw display
$97$ \( T^{3} + 6 T^{2} - 78 T - 388 \) Copy content Toggle raw display
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