Properties

Label 6240.2.a.bg
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: \(\Q(\sqrt{57}) \)
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). 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 \( q - q^{3} - q^{5} - \beta q^{7} + q^{9} + (\beta - 2) q^{11} + q^{13} + q^{15} + ( - \beta - 2) q^{17} + 2 q^{19} + \beta q^{21} + (\beta - 4) q^{23} + q^{25} - q^{27} + 4 q^{31} + ( - \beta + 2) q^{33} + \beta q^{35} + (\beta + 4) q^{37} - q^{39} + (\beta + 4) q^{41} - 2 \beta q^{43} - q^{45} + 4 q^{47} + (\beta + 7) q^{49} + (\beta + 2) q^{51} + (\beta + 4) q^{53} + ( - \beta + 2) q^{55} - 2 q^{57} - 4 q^{59} + (3 \beta - 4) q^{61} - \beta q^{63} - q^{65} - 8 q^{67} + ( - \beta + 4) q^{69} + (\beta - 6) q^{71} + ( - 2 \beta - 4) q^{73} - q^{75} + (\beta - 14) q^{77} + ( - \beta - 10) q^{79} + q^{81} - 12 q^{83} + (\beta + 2) q^{85} + ( - \beta - 4) q^{89} - \beta q^{91} - 4 q^{93} - 2 q^{95} + (\beta + 10) q^{97} + (\beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} - 3 q^{11} + 2 q^{13} + 2 q^{15} - 5 q^{17} + 4 q^{19} + q^{21} - 7 q^{23} + 2 q^{25} - 2 q^{27} + 8 q^{31} + 3 q^{33} + q^{35} + 9 q^{37} - 2 q^{39} + 9 q^{41} - 2 q^{43} - 2 q^{45} + 8 q^{47} + 15 q^{49} + 5 q^{51} + 9 q^{53} + 3 q^{55} - 4 q^{57} - 8 q^{59} - 5 q^{61} - q^{63} - 2 q^{65} - 16 q^{67} + 7 q^{69} - 11 q^{71} - 10 q^{73} - 2 q^{75} - 27 q^{77} - 21 q^{79} + 2 q^{81} - 24 q^{83} + 5 q^{85} - 9 q^{89} - q^{91} - 8 q^{93} - 4 q^{95} + 21 q^{97} - 3 q^{99}+O(q^{100}) \) 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
4.27492
−3.27492
0 −1.00000 0 −1.00000 0 −4.27492 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 3.27492 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.bg 2
4.b odd 2 1 6240.2.a.bs yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6240.2.a.bg 2 1.a even 1 1 trivial
6240.2.a.bs 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6240))\):

\( T_{7}^{2} + T_{7} - 14 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 12 \) Copy content Toggle raw display
\( T_{17}^{2} + 5T_{17} - 8 \) Copy content Toggle raw display
\( T_{19} - 2 \) 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
$7$ \( T^{2} + T - 14 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T - 12 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 5T - 8 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 7T - 2 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 9T + 6 \) Copy content Toggle raw display
$41$ \( T^{2} - 9T + 6 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 56 \) Copy content Toggle raw display
$47$ \( (T - 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 6 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 5T - 122 \) Copy content Toggle raw display
$67$ \( (T + 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 11T + 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 10T - 32 \) Copy content Toggle raw display
$79$ \( T^{2} + 21T + 96 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 9T + 6 \) Copy content Toggle raw display
$97$ \( T^{2} - 21T + 96 \) Copy content Toggle raw display
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