Properties

Label 6240.2.a.bh
Level $6240$
Weight $2$
Character orbit 6240.a
Self dual yes
Analytic conductor $49.827$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.8266508613\)
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: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + \beta q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{5} + \beta q^{7} + q^{9} + \beta q^{11} + q^{13} + q^{15} + ( - 2 \beta + 2) q^{17} + (\beta - 4) q^{19} - \beta q^{21} + 4 q^{23} + q^{25} - q^{27} + ( - 2 \beta - 2) q^{29} + ( - \beta - 4) q^{31} - \beta q^{33} - \beta q^{35} - 2 q^{37} - q^{39} - 2 q^{41} + ( - 2 \beta - 4) q^{43} - q^{45} + ( - \beta - 4) q^{47} + q^{49} + (2 \beta - 2) q^{51} - 2 q^{53} - \beta q^{55} + ( - \beta + 4) q^{57} - 3 \beta q^{59} + 6 q^{61} + \beta q^{63} - q^{65} + \beta q^{67} - 4 q^{69} + ( - \beta + 8) q^{71} + (2 \beta + 10) q^{73} - q^{75} + 8 q^{77} + ( - 2 \beta + 8) q^{79} + q^{81} + (\beta - 4) q^{83} + (2 \beta - 2) q^{85} + (2 \beta + 2) q^{87} + (4 \beta + 6) q^{89} + \beta q^{91} + (\beta + 4) q^{93} + ( - \beta + 4) q^{95} + ( - 2 \beta + 2) q^{97} + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 8 q^{19} + 8 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 8 q^{31} - 4 q^{37} - 2 q^{39} - 4 q^{41} - 8 q^{43} - 2 q^{45} - 8 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} + 8 q^{57} + 12 q^{61} - 2 q^{65} - 8 q^{69} + 16 q^{71} + 20 q^{73} - 2 q^{75} + 16 q^{77} + 16 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{85} + 4 q^{87} + 12 q^{89} + 8 q^{93} + 8 q^{95} + 4 q^{97}+O(q^{100}) \) 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 −1.00000 0 −1.00000 0 −2.82843 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 2.82843 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(13\) \(-1\)

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 6240.2.a.bh 2
4.b odd 2 1 6240.2.a.br yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6240.2.a.bh 2 1.a even 1 1 trivial
6240.2.a.br yes 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6240))\):

\( T_{7}^{2} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 8 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} - 28 \) Copy content Toggle raw display
\( T_{19}^{2} + 8T_{19} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 8 \) Copy content Toggle raw display
$11$ \( T^{2} - 8 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 72 \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 8 \) Copy content Toggle raw display
$71$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$73$ \( T^{2} - 20T + 68 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 32 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T - 92 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
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