Properties

Label 8280.2.a.bo
Level $8280$
Weight $2$
Character orbit 8280.a
Self dual yes
Analytic conductor $66.116$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \(x^{3} - 6 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
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.

\(f(q)\) \(=\) \( q + q^{5} + ( 1 + \beta_{1} ) q^{7} +O(q^{10})\) \( q + q^{5} + ( 1 + \beta_{1} ) q^{7} + ( -1 + \beta_{1} - \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} + ( 1 - \beta_{1} + \beta_{2} ) q^{17} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{19} - q^{23} + q^{25} + ( -1 - \beta_{2} ) q^{29} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{31} + ( 1 + \beta_{1} ) q^{35} + 4 q^{37} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 6 + 2 \beta_{2} ) q^{43} + ( -5 - \beta_{2} ) q^{47} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{49} + ( -2 - 4 \beta_{2} ) q^{53} + ( -1 + \beta_{1} - \beta_{2} ) q^{55} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 9 - \beta_{1} + \beta_{2} ) q^{61} + ( \beta_{1} - \beta_{2} ) q^{65} + ( 4 + 4 \beta_{1} - 4 \beta_{2} ) q^{67} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -5 - 4 \beta_{1} + \beta_{2} ) q^{73} + ( 4 + \beta_{2} ) q^{77} + ( 4 + 4 \beta_{1} + 2 \beta_{2} ) q^{83} + ( 1 - \beta_{1} + \beta_{2} ) q^{85} + ( 10 + 2 \beta_{2} ) q^{89} + ( 5 + \beta_{1} + \beta_{2} ) q^{91} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{95} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} + 3 q^{7} + O(q^{10}) \) \( 3 q + 3 q^{5} + 3 q^{7} - 3 q^{11} + 3 q^{17} + 3 q^{19} - 3 q^{23} + 3 q^{25} - 3 q^{29} + 6 q^{31} + 3 q^{35} + 12 q^{37} + 6 q^{41} + 18 q^{43} - 15 q^{47} - 6 q^{49} - 6 q^{53} - 3 q^{55} - 6 q^{59} + 27 q^{61} + 12 q^{67} - 6 q^{71} - 15 q^{73} + 12 q^{77} + 12 q^{83} + 3 q^{85} + 30 q^{89} + 15 q^{91} + 3 q^{95} + 9 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 6 x - 3\):

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

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
−2.14510
−0.523976
2.66908
0 0 0 1.00000 0 −1.14510 0 0 0
1.2 0 0 0 1.00000 0 0.476024 0 0 0
1.3 0 0 0 1.00000 0 3.66908 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.bo 3
3.b odd 2 1 920.2.a.g 3
12.b even 2 1 1840.2.a.t 3
15.d odd 2 1 4600.2.a.y 3
15.e even 4 2 4600.2.e.r 6
24.f even 2 1 7360.2.a.ca 3
24.h odd 2 1 7360.2.a.cb 3
60.h even 2 1 9200.2.a.cd 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.g 3 3.b odd 2 1
1840.2.a.t 3 12.b even 2 1
4600.2.a.y 3 15.d odd 2 1
4600.2.e.r 6 15.e even 4 2
7360.2.a.ca 3 24.f even 2 1
7360.2.a.cb 3 24.h odd 2 1
8280.2.a.bo 3 1.a even 1 1 trivial
9200.2.a.cd 3 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

\( T_{7}^{3} - 3 T_{7}^{2} - 3 T_{7} + 2 \)
\( T_{11}^{3} + 3 T_{11}^{2} - 15 T_{11} + 12 \)
\( T_{13}^{3} - 18 T_{13} + 29 \)
\( T_{17}^{3} - 3 T_{17}^{2} - 15 T_{17} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( 2 - 3 T - 3 T^{2} + T^{3} \)
$11$ \( 12 - 15 T + 3 T^{2} + T^{3} \)
$13$ \( 29 - 18 T + T^{3} \)
$17$ \( -12 - 15 T - 3 T^{2} + T^{3} \)
$19$ \( -48 - 33 T - 3 T^{2} + T^{3} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( -12 - 6 T + 3 T^{2} + T^{3} \)
$31$ \( 249 - 42 T - 6 T^{2} + T^{3} \)
$37$ \( ( -4 + T )^{3} \)
$41$ \( 69 - 60 T - 6 T^{2} + T^{3} \)
$43$ \( 32 + 72 T - 18 T^{2} + T^{3} \)
$47$ \( 76 + 66 T + 15 T^{2} + T^{3} \)
$53$ \( -536 - 132 T + 6 T^{2} + T^{3} \)
$59$ \( -32 - 36 T + 6 T^{2} + T^{3} \)
$61$ \( -596 + 225 T - 27 T^{2} + T^{3} \)
$67$ \( 2944 - 240 T - 12 T^{2} + T^{3} \)
$71$ \( -203 - 60 T + 6 T^{2} + T^{3} \)
$73$ \( -588 - 42 T + 15 T^{2} + T^{3} \)
$79$ \( T^{3} \)
$83$ \( 64 - 60 T - 12 T^{2} + T^{3} \)
$89$ \( -608 + 264 T - 30 T^{2} + T^{3} \)
$97$ \( 138 - 69 T - 9 T^{2} + T^{3} \)
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