Properties

Label 6840.2.a.x
Level $6840$
Weight $2$
Character orbit 6840.a
Self dual yes
Analytic conductor $54.618$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + (\beta - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + (\beta - 2) q^{7} - \beta q^{11} - 3 \beta q^{13} + 2 \beta q^{17} - q^{19} + ( - 2 \beta + 2) q^{23} + q^{25} + (3 \beta + 2) q^{29} + (2 \beta - 6) q^{31} + (\beta - 2) q^{35} + 5 \beta q^{37} + (3 \beta + 2) q^{41} + ( - 3 \beta - 6) q^{43} + (2 \beta - 2) q^{47} + ( - 4 \beta - 1) q^{49} + ( - 6 \beta + 4) q^{53} - \beta q^{55} - 2 \beta q^{59} - 8 q^{61} - 3 \beta q^{65} + 8 \beta q^{67} + (2 \beta + 8) q^{71} + (4 \beta + 2) q^{73} + (2 \beta - 2) q^{77} + ( - 4 \beta - 8) q^{79} + 6 q^{83} + 2 \beta q^{85} + ( - 11 \beta - 2) q^{89} + (6 \beta - 6) q^{91} - q^{95} - 7 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 4 q^{7} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 4 q^{29} - 12 q^{31} - 4 q^{35} + 4 q^{41} - 12 q^{43} - 4 q^{47} - 2 q^{49} + 8 q^{53} - 16 q^{61} + 16 q^{71} + 4 q^{73} - 4 q^{77} - 16 q^{79} + 12 q^{83} - 4 q^{89} - 12 q^{91} - 2 q^{95}+O(q^{100}) \) 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
−1.41421
1.41421
0 0 0 1.00000 0 −3.41421 0 0 0
1.2 0 0 0 1.00000 0 −0.585786 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.x 2
3.b odd 2 1 2280.2.a.o 2
12.b even 2 1 4560.2.a.bg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.o 2 3.b odd 2 1
4560.2.a.bg 2 12.b even 2 1
6840.2.a.x 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6840))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 2 \) Copy content Toggle raw display
$13$ \( T^{2} - 18 \) Copy content Toggle raw display
$17$ \( T^{2} - 8 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$31$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$37$ \( T^{2} - 50 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$43$ \( T^{2} + 12T + 18 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
$59$ \( T^{2} - 8 \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 128 \) Copy content Toggle raw display
$71$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$79$ \( T^{2} + 16T + 32 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 4T - 238 \) Copy content Toggle raw display
$97$ \( T^{2} - 98 \) Copy content Toggle raw display
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