Properties

Label 8280.2.a.bn
Level $8280$
Weight $2$
Character orbit 8280.a
Self dual yes
Analytic conductor $66.116$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1161328736\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1436.1
Defining polynomial: \(x^{3} - 11 x - 12\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2760)
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.

\(f(q)\) \(=\) \( q + q^{5} + ( -1 + \beta_{2} ) q^{7} +O(q^{10})\) \( q + q^{5} + ( -1 + \beta_{2} ) q^{7} + ( -1 - \beta_{1} - \beta_{2} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} ) q^{13} + ( -2 + \beta_{1} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} ) q^{19} - q^{23} + q^{25} + \beta_{1} q^{29} + ( 1 - \beta_{2} ) q^{31} + ( -1 + \beta_{2} ) q^{35} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{37} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -3 - \beta_{1} + 3 \beta_{2} ) q^{47} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{49} + ( -2 + 3 \beta_{1} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} ) q^{55} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{59} + ( 1 + \beta_{1} + \beta_{2} ) q^{61} + ( -1 + \beta_{1} + \beta_{2} ) q^{65} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{67} + ( -6 + \beta_{1} ) q^{71} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{73} + ( -5 + 3 \beta_{1} + \beta_{2} ) q^{77} -8 q^{79} + ( -8 - \beta_{1} ) q^{83} + ( -2 + \beta_{1} ) q^{85} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 7 - 3 \beta_{1} - 3 \beta_{2} ) q^{91} + ( 1 - \beta_{1} - \beta_{2} ) q^{95} + ( 4 + 4 \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 2 q^{7} + O(q^{10}) \) \( 3 q + 3 q^{5} - 2 q^{7} - 4 q^{11} - 2 q^{13} - 6 q^{17} + 2 q^{19} - 3 q^{23} + 3 q^{25} + 2 q^{31} - 2 q^{35} + 8 q^{37} - 2 q^{41} + 10 q^{43} - 6 q^{47} + 5 q^{49} - 6 q^{53} - 4 q^{55} - 8 q^{59} + 4 q^{61} - 2 q^{65} + 8 q^{67} - 18 q^{71} + 8 q^{73} - 14 q^{77} - 24 q^{79} - 24 q^{83} - 6 q^{85} - 4 q^{89} + 18 q^{91} + 2 q^{95} + 12 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 11 x - 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 7\)

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
−1.28282
3.76644
−2.48361
0 0 0 1.00000 0 −3.78872 0 0 0
1.2 0 0 0 1.00000 0 −1.34683 0 0 0
1.3 0 0 0 1.00000 0 3.13555 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

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 8280.2.a.bn 3
3.b odd 2 1 2760.2.a.s 3
12.b even 2 1 5520.2.a.bw 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.s 3 3.b odd 2 1
5520.2.a.bw 3 12.b even 2 1
8280.2.a.bn 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

\( T_{7}^{3} + 2 T_{7}^{2} - 11 T_{7} - 16 \)
\( T_{11}^{3} + 4 T_{11}^{2} - 10 T_{11} - 36 \)
\( T_{13}^{3} + 2 T_{13}^{2} - 14 T_{13} + 8 \)
\( T_{17}^{3} + 6 T_{17}^{2} + T_{17} - 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( -16 - 11 T + 2 T^{2} + T^{3} \)
$11$ \( -36 - 10 T + 4 T^{2} + T^{3} \)
$13$ \( 8 - 14 T + 2 T^{2} + T^{3} \)
$17$ \( -26 + T + 6 T^{2} + T^{3} \)
$19$ \( -8 - 14 T - 2 T^{2} + T^{3} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( -12 - 11 T + T^{3} \)
$31$ \( 16 - 11 T - 2 T^{2} + T^{3} \)
$37$ \( 214 - 51 T - 8 T^{2} + T^{3} \)
$41$ \( -82 - 99 T + 2 T^{2} + T^{3} \)
$43$ \( 24 - 28 T - 10 T^{2} + T^{3} \)
$47$ \( -936 - 134 T + 6 T^{2} + T^{3} \)
$53$ \( -514 - 87 T + 6 T^{2} + T^{3} \)
$59$ \( 72 - 55 T + 8 T^{2} + T^{3} \)
$61$ \( 36 - 10 T - 4 T^{2} + T^{3} \)
$67$ \( 214 - 51 T - 8 T^{2} + T^{3} \)
$71$ \( 138 + 97 T + 18 T^{2} + T^{3} \)
$73$ \( 484 - 66 T - 8 T^{2} + T^{3} \)
$79$ \( ( 8 + T )^{3} \)
$83$ \( 436 + 181 T + 24 T^{2} + T^{3} \)
$89$ \( -576 - 120 T + 4 T^{2} + T^{3} \)
$97$ \( -128 - 128 T - 12 T^{2} + T^{3} \)
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