Properties

Label 8280.2.a.bh
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: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - q^{5} -\beta_{1} q^{7} +O(q^{10})\) \( q - q^{5} -\beta_{1} q^{7} + ( 1 - \beta_{2} ) q^{13} + ( -2 + \beta_{1} ) q^{17} - q^{23} + q^{25} + ( -2 - \beta_{1} ) q^{29} + ( 1 + \beta_{1} + \beta_{2} ) q^{31} + \beta_{1} q^{35} + ( 4 + \beta_{1} + 2 \beta_{2} ) q^{37} + ( -1 - \beta_{1} + \beta_{2} ) q^{41} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{43} + 2 \beta_{1} q^{47} + ( 2 + \beta_{1} + \beta_{2} ) q^{49} + ( -5 - \beta_{1} + \beta_{2} ) q^{53} + ( -1 + \beta_{1} - \beta_{2} ) q^{59} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{61} + ( -1 + \beta_{2} ) q^{65} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{67} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{71} + ( 2 - 2 \beta_{1} ) q^{73} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{79} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{83} + ( 2 - \beta_{1} ) q^{85} + ( -8 - 2 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -2 - 2 \beta_{2} ) q^{91} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + q^{7} + O(q^{10}) \) \( 3 q - 3 q^{5} + q^{7} + 4 q^{13} - 7 q^{17} - 3 q^{23} + 3 q^{25} - 5 q^{29} + q^{31} - q^{35} + 9 q^{37} - 3 q^{41} - 2 q^{47} + 4 q^{49} - 15 q^{53} - 3 q^{59} - 6 q^{61} - 4 q^{65} - 3 q^{67} - 5 q^{71} + 8 q^{73} + 8 q^{79} - 7 q^{83} + 7 q^{85} - 20 q^{89} - 4 q^{91} + 6 q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{2}\)\(=\)\( -2 \nu^{2} + 2 \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{2} + 2 \beta_{1} + 11\)\()/4\)

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
2.11491
−1.86081
−0.254102
0 0 0 −1.00000 0 −3.58774 0 0 0
1.2 0 0 0 −1.00000 0 1.39821 0 0 0
1.3 0 0 0 −1.00000 0 3.18953 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.bh 3
3.b odd 2 1 2760.2.a.t 3
12.b even 2 1 5520.2.a.ca 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.t 3 3.b odd 2 1
5520.2.a.ca 3 12.b even 2 1
8280.2.a.bh 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

\( T_{7}^{3} - T_{7}^{2} - 12 T_{7} + 16 \)
\( T_{11} \)
\( T_{13}^{3} - 4 T_{13}^{2} - 20 T_{13} + 16 \)
\( T_{17}^{3} + 7 T_{17}^{2} + 4 T_{17} - 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( 16 - 12 T - T^{2} + T^{3} \)
$11$ \( T^{3} \)
$13$ \( 16 - 20 T - 4 T^{2} + T^{3} \)
$17$ \( -28 + 4 T + 7 T^{2} + T^{3} \)
$19$ \( T^{3} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( -4 - 4 T + 5 T^{2} + T^{3} \)
$31$ \( 64 - 32 T - T^{2} + T^{3} \)
$37$ \( 676 - 76 T - 9 T^{2} + T^{3} \)
$41$ \( -148 - 40 T + 3 T^{2} + T^{3} \)
$43$ \( -64 - 64 T + T^{3} \)
$47$ \( -128 - 48 T + 2 T^{2} + T^{3} \)
$53$ \( -196 + 32 T + 15 T^{2} + T^{3} \)
$59$ \( 64 - 40 T + 3 T^{2} + T^{3} \)
$61$ \( -184 - 52 T + 6 T^{2} + T^{3} \)
$67$ \( -496 - 100 T + 3 T^{2} + T^{3} \)
$71$ \( 112 - 116 T + 5 T^{2} + T^{3} \)
$73$ \( 208 - 28 T - 8 T^{2} + T^{3} \)
$79$ \( 448 - 64 T - 8 T^{2} + T^{3} \)
$83$ \( -592 - 108 T + 7 T^{2} + T^{3} \)
$89$ \( -992 + 4 T + 20 T^{2} + T^{3} \)
$97$ \( 184 - 52 T - 6 T^{2} + T^{3} \)
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