Properties

Label 6240.2.a.bw
Level $6240$
Weight $2$
Character orbit 6240.a
Self dual yes
Analytic conductor $49.827$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + ( - \beta_{2} - 2) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{5} + ( - \beta_{2} - 2) q^{7} + q^{9} - \beta_1 q^{11} + q^{13} + q^{15} + (\beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{2} - \beta_1 - 2) q^{19} + (\beta_{2} + 2) q^{21} + ( - 3 \beta_{2} - 2) q^{23} + q^{25} - q^{27} + ( - 3 \beta_{2} + \beta_1) q^{29} - 4 q^{31} + \beta_1 q^{33} + (\beta_{2} + 2) q^{35} + ( - \beta_1 + 2) q^{37} - q^{39} + ( - \beta_1 - 2) q^{41} - 2 \beta_1 q^{43} - q^{45} + 4 \beta_{2} q^{47} + (4 \beta_{2} - \beta_1 + 1) q^{49} + ( - \beta_{2} + 2 \beta_1) q^{51} + (3 \beta_1 + 2) q^{53} + \beta_1 q^{55} + (\beta_{2} + \beta_1 + 2) q^{57} + ( - 2 \beta_{2} + 2 \beta_1) q^{59} + ( - 2 \beta_{2} - \beta_1 - 2) q^{61} + ( - \beta_{2} - 2) q^{63} - q^{65} + (3 \beta_{2} + 2) q^{69} + ( - 2 \beta_{2} + \beta_1 - 4) q^{71} + ( - 5 \beta_{2} + \beta_1) q^{73} - q^{75} + ( - 2 \beta_{2} + \beta_1) q^{77} + (2 \beta_{2} - \beta_1 - 8) q^{79} + q^{81} + (4 \beta_{2} + 8) q^{83} + ( - \beta_{2} + 2 \beta_1) q^{85} + (3 \beta_{2} - \beta_1) q^{87} + (2 \beta_{2} - \beta_1 - 2) q^{89} + ( - \beta_{2} - 2) q^{91} + 4 q^{93} + (\beta_{2} + \beta_1 + 2) q^{95} + (\beta_{2} + 8) q^{97} - \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9} + q^{11} + 3 q^{13} + 3 q^{15} + q^{17} - 4 q^{19} + 5 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 12 q^{31} - q^{33} + 5 q^{35} + 7 q^{37} - 3 q^{39} - 5 q^{41} + 2 q^{43} - 3 q^{45} - 4 q^{47} - q^{51} + 3 q^{53} - q^{55} + 4 q^{57} - 3 q^{61} - 5 q^{63} - 3 q^{65} + 3 q^{69} - 11 q^{71} + 4 q^{73} - 3 q^{75} + q^{77} - 25 q^{79} + 3 q^{81} + 20 q^{83} - q^{85} - 2 q^{87} - 7 q^{89} - 5 q^{91} + 12 q^{93} + 4 q^{95} + 23 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 6 ) / 2 \) 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
−1.86081
2.11491
−0.254102
0 −1.00000 0 −1.00000 0 −4.32340 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −1.35793 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 0.681331 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.bw 3
4.b odd 2 1 6240.2.a.cb yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6240.2.a.bw 3 1.a even 1 1 trivial
6240.2.a.cb yes 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6240))\):

\( T_{7}^{3} + 5T_{7}^{2} + 2T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{3} - T_{11}^{2} - 12T_{11} + 16 \) Copy content Toggle raw display
\( T_{17}^{3} - T_{17}^{2} - 50T_{17} + 148 \) Copy content Toggle raw display
\( T_{19}^{3} + 4T_{19}^{2} - 16T_{19} - 56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 5 T^{2} + 2 T - 4 \) Copy content Toggle raw display
$11$ \( T^{3} - T^{2} - 12 T + 16 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - T^{2} - 50 T + 148 \) Copy content Toggle raw display
$19$ \( T^{3} + 4 T^{2} - 16 T - 56 \) Copy content Toggle raw display
$23$ \( T^{3} + 3 T^{2} - 54 T - 4 \) Copy content Toggle raw display
$29$ \( T^{3} - 2 T^{2} - 60 T + 224 \) Copy content Toggle raw display
$31$ \( (T + 4)^{3} \) Copy content Toggle raw display
$37$ \( T^{3} - 7 T^{2} + 4 T + 28 \) Copy content Toggle raw display
$41$ \( T^{3} + 5 T^{2} - 4 T - 4 \) Copy content Toggle raw display
$43$ \( T^{3} - 2 T^{2} - 48 T + 128 \) Copy content Toggle raw display
$47$ \( T^{3} + 4 T^{2} - 96 T - 256 \) Copy content Toggle raw display
$53$ \( T^{3} - 3 T^{2} - 108 T - 212 \) Copy content Toggle raw display
$59$ \( T^{3} - 64T - 64 \) Copy content Toggle raw display
$61$ \( T^{3} + 3 T^{2} - 40 T - 148 \) Copy content Toggle raw display
$67$ \( T^{3} \) Copy content Toggle raw display
$71$ \( T^{3} + 11 T^{2} + 8 T - 16 \) Copy content Toggle raw display
$73$ \( T^{3} - 4 T^{2} - 152 T + 904 \) Copy content Toggle raw display
$79$ \( T^{3} + 25 T^{2} + 176 T + 256 \) Copy content Toggle raw display
$83$ \( T^{3} - 20 T^{2} + 32 T + 256 \) Copy content Toggle raw display
$89$ \( T^{3} + 7 T^{2} - 16 T - 116 \) Copy content Toggle raw display
$97$ \( T^{3} - 23 T^{2} + 170 T - 404 \) Copy content Toggle raw display
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