Properties

Label 6240.2.a.bp
Level $6240$
Weight $2$
Character orbit 6240.a
Self dual yes
Analytic conductor $49.827$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). 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 \( q + q^{3} - q^{5} - \beta q^{7} + q^{9} + (\beta - 4) q^{11} - q^{13} - q^{15} + ( - \beta - 2) q^{17} - 2 \beta q^{19} - \beta q^{21} + (\beta + 4) q^{23} + q^{25} + q^{27} + (2 \beta + 2) q^{29} + (\beta - 4) q^{33} + \beta q^{35} + ( - \beta - 6) q^{37} - q^{39} + ( - 3 \beta + 2) q^{41} + (2 \beta + 4) q^{43} - q^{45} + 4 q^{47} + (\beta + 1) q^{49} + ( - \beta - 2) q^{51} + ( - \beta + 2) q^{53} + ( - \beta + 4) q^{55} - 2 \beta q^{57} - 4 q^{59} + ( - \beta + 6) q^{61} - \beta q^{63} + q^{65} + ( - 4 \beta + 4) q^{67} + (\beta + 4) q^{69} + (\beta + 8) q^{71} + 10 q^{73} + q^{75} + (3 \beta - 8) q^{77} - \beta q^{79} + q^{81} + (4 \beta - 4) q^{83} + (\beta + 2) q^{85} + (2 \beta + 2) q^{87} + ( - \beta - 6) q^{89} + \beta q^{91} + 2 \beta q^{95} + (\beta + 2) q^{97} + (\beta - 4) q^{99} +O(q^{100}) \) 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 \( 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} - 7 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} - 2 q^{19} - q^{21} + 9 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} - 7 q^{33} + q^{35} - 13 q^{37} - 2 q^{39} + q^{41} + 10 q^{43} - 2 q^{45} + 8 q^{47} + 3 q^{49} - 5 q^{51} + 3 q^{53} + 7 q^{55} - 2 q^{57} - 8 q^{59} + 11 q^{61} - q^{63} + 2 q^{65} + 4 q^{67} + 9 q^{69} + 17 q^{71} + 20 q^{73} + 2 q^{75} - 13 q^{77} - q^{79} + 2 q^{81} - 4 q^{83} + 5 q^{85} + 6 q^{87} - 13 q^{89} + q^{91} + 2 q^{95} + 5 q^{97} - 7 q^{99}+O(q^{100}) \) 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
3.37228
−2.37228
0 1.00000 0 −1.00000 0 −3.37228 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 2.37228 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.bp yes 2
4.b odd 2 1 6240.2.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6240.2.a.bj 2 4.b odd 2 1
6240.2.a.bp yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6240))\):

\( T_{7}^{2} + T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 7T_{11} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 5T_{17} - 2 \) Copy content Toggle raw display
\( T_{19}^{2} + 2T_{19} - 32 \) 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
$7$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$11$ \( T^{2} + 7T + 4 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 5T - 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$23$ \( T^{2} - 9T + 12 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 13T + 34 \) Copy content Toggle raw display
$41$ \( T^{2} - T - 74 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T - 8 \) Copy content Toggle raw display
$47$ \( (T - 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 11T + 22 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 128 \) Copy content Toggle raw display
$71$ \( T^{2} - 17T + 64 \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 128 \) Copy content Toggle raw display
$89$ \( T^{2} + 13T + 34 \) Copy content Toggle raw display
$97$ \( T^{2} - 5T - 2 \) Copy content Toggle raw display
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