Properties

Label 6240.2.a.bo
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: \(\Q(\sqrt{33}) \)
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.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + ( - \beta - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + ( - \beta - 1) q^{7} + q^{9} + (\beta + 1) q^{11} - q^{13} - q^{15} + (\beta - 3) q^{17} - 2 q^{19} + ( - \beta - 1) q^{21} + ( - \beta + 1) q^{23} + q^{25} + q^{27} + ( - 2 \beta + 4) q^{29} + 2 \beta q^{31} + (\beta + 1) q^{33} + (\beta + 1) q^{35} + (\beta + 1) q^{37} - q^{39} + ( - \beta + 3) q^{41} + (2 \beta - 2) q^{43} - q^{45} + (2 \beta - 4) q^{47} + (3 \beta + 2) q^{49} + (\beta - 3) q^{51} + (\beta - 11) q^{53} + ( - \beta - 1) q^{55} - 2 q^{57} + ( - 2 \beta + 4) q^{59} + ( - 3 \beta + 1) q^{61} + ( - \beta - 1) q^{63} + q^{65} + ( - 2 \beta + 8) q^{67} + ( - \beta + 1) q^{69} + (\beta + 1) q^{71} - 6 q^{73} + q^{75} + ( - 3 \beta - 9) q^{77} + (\beta - 13) q^{79} + q^{81} + ( - 2 \beta - 8) q^{83} + ( - \beta + 3) q^{85} + ( - 2 \beta + 4) q^{87} + ( - 3 \beta - 3) q^{89} + (\beta + 1) q^{91} + 2 \beta q^{93} + 2 q^{95} + (3 \beta - 1) q^{97} + (\beta + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 2 q^{9} + 3 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} - 4 q^{19} - 3 q^{21} + q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} + 2 q^{31} + 3 q^{33} + 3 q^{35} + 3 q^{37} - 2 q^{39} + 5 q^{41} - 2 q^{43} - 2 q^{45} - 6 q^{47} + 7 q^{49} - 5 q^{51} - 21 q^{53} - 3 q^{55} - 4 q^{57} + 6 q^{59} - q^{61} - 3 q^{63} + 2 q^{65} + 14 q^{67} + q^{69} + 3 q^{71} - 12 q^{73} + 2 q^{75} - 21 q^{77} - 25 q^{79} + 2 q^{81} - 18 q^{83} + 5 q^{85} + 6 q^{87} - 9 q^{89} + 3 q^{91} + 2 q^{93} + 4 q^{95} + q^{97} + 3 q^{99}+O(q^{100}) \) 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
3.37228
−2.37228
0 1.00000 0 −1.00000 0 −4.37228 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 1.37228 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.bo yes 2
4.b odd 2 1 6240.2.a.bk 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6240.2.a.bk 2 4.b odd 2 1
6240.2.a.bo 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6240))\):

\( T_{7}^{2} + 3T_{7} - 6 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 6 \) Copy content Toggle raw display
\( T_{17}^{2} + 5T_{17} - 2 \) 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
$7$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 5T - 2 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$37$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$41$ \( T^{2} - 5T - 2 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T - 24 \) Copy content Toggle raw display
$53$ \( T^{2} + 21T + 102 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$61$ \( T^{2} + T - 74 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 25T + 148 \) Copy content Toggle raw display
$83$ \( T^{2} + 18T + 48 \) Copy content Toggle raw display
$89$ \( T^{2} + 9T - 54 \) Copy content Toggle raw display
$97$ \( T^{2} - T - 74 \) Copy content Toggle raw display
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