Properties

Label 6840.2.a.bf
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: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) 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.

\(f(q)\) \(=\) \( q - q^{5} + (\beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + (\beta_1 - 1) q^{7} + (\beta_{2} + 1) q^{11} + (2 \beta_{2} - \beta_1 + 1) q^{13} + (\beta_{2} + \beta_1 + 2) q^{17} - q^{19} + (\beta_{2} + \beta_1 + 4) q^{23} + q^{25} + (\beta_{2} - 1) q^{29} + (3 \beta_{2} - \beta_1) q^{31} + ( - \beta_1 + 1) q^{35} + (\beta_1 - 3) q^{37} + ( - \beta_{2} + 2 \beta_1 - 1) q^{41} + ( - 2 \beta_{2} - \beta_1 - 5) q^{43} + ( - \beta_{2} - \beta_1 + 4) q^{47} + ( - \beta_{2} + \beta_1 + 3) q^{49} + ( - \beta_{2} - \beta_1 - 2) q^{53} + ( - \beta_{2} - 1) q^{55} + (2 \beta_{2} + 6) q^{59} + ( - 3 \beta_{2} + 3 \beta_1 - 4) q^{61} + ( - 2 \beta_{2} + \beta_1 - 1) q^{65} + 4 \beta_{2} q^{67} + ( - 4 \beta_{2} + 2 \beta_1 + 2) q^{71} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{73} + ( - 3 \beta_{2} + \beta_1) q^{77} + ( - \beta_{2} - 3 \beta_1 + 2) q^{83} + ( - \beta_{2} - \beta_1 - 2) q^{85} + (5 \beta_{2} - 2 \beta_1 + 1) q^{89} + ( - 5 \beta_{2} - \beta_1 - 8) q^{91} + q^{95} + ( - 4 \beta_{2} + \beta_1 - 3) q^{97}+O(q^{100}) \) 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 \( 3 q - 3 q^{5} - 2 q^{7} + 2 q^{11} + 6 q^{17} - 3 q^{19} + 12 q^{23} + 3 q^{25} - 4 q^{29} - 4 q^{31} + 2 q^{35} - 8 q^{37} - 14 q^{43} + 12 q^{47} + 11 q^{49} - 6 q^{53} - 2 q^{55} + 16 q^{59} - 6 q^{61} - 4 q^{67} + 12 q^{71} - 2 q^{73} + 4 q^{77} + 4 q^{83} - 6 q^{85} - 4 q^{89} - 20 q^{91} + 3 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 6 ) / 2 \) 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
0.470683
−1.81361
2.34292
0 0 0 −1.00000 0 −3.30777 0 0 0
1.2 0 0 0 −1.00000 0 −2.52444 0 0 0
1.3 0 0 0 −1.00000 0 3.83221 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.bf 3
3.b odd 2 1 2280.2.a.s 3
12.b even 2 1 4560.2.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.s 3 3.b odd 2 1
4560.2.a.bv 3 12.b even 2 1
6840.2.a.bf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6840))\):

\( T_{7}^{3} + 2T_{7}^{2} - 14T_{7} - 32 \) Copy content Toggle raw display
\( T_{11}^{3} - 2T_{11}^{2} - 6T_{11} + 8 \) Copy content Toggle raw display
\( T_{13}^{3} - 34T_{13} - 76 \) 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
$7$ \( T^{3} + 2 T^{2} - 14 T - 32 \) Copy content Toggle raw display
$11$ \( T^{3} - 2 T^{2} - 6 T + 8 \) Copy content Toggle raw display
$13$ \( T^{3} - 34T - 76 \) Copy content Toggle raw display
$17$ \( T^{3} - 6 T^{2} - 16 T + 64 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 12 T^{2} + 20 T + 64 \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} - 2 T - 4 \) Copy content Toggle raw display
$31$ \( T^{3} + 4 T^{2} - 60 T - 256 \) Copy content Toggle raw display
$37$ \( T^{3} + 8 T^{2} + 6 T - 44 \) Copy content Toggle raw display
$41$ \( T^{3} - 58T - 124 \) Copy content Toggle raw display
$43$ \( T^{3} + 14 T^{2} + 10 T - 296 \) Copy content Toggle raw display
$47$ \( T^{3} - 12 T^{2} + 20 T + 32 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 16 T - 64 \) Copy content Toggle raw display
$59$ \( T^{3} - 16 T^{2} + 56 T + 32 \) Copy content Toggle raw display
$61$ \( T^{3} + 6 T^{2} - 144 T - 176 \) Copy content Toggle raw display
$67$ \( T^{3} + 4 T^{2} - 112 T + 64 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} - 88 T + 1088 \) Copy content Toggle raw display
$73$ \( T^{3} + 2 T^{2} - 68 T - 8 \) Copy content Toggle raw display
$79$ \( T^{3} \) Copy content Toggle raw display
$83$ \( T^{3} - 4 T^{2} - 156 T + 688 \) Copy content Toggle raw display
$89$ \( T^{3} + 4 T^{2} - 186 T - 1228 \) Copy content Toggle raw display
$97$ \( T^{3} + 4 T^{2} - 106 T + 124 \) Copy content Toggle raw display
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