Properties

Label 8280.2.a.bi
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: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - q^{5} + ( - \beta_{2} - 2 \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + ( - \beta_{2} - 2 \beta_1 + 1) q^{7} + (\beta_{2} - \beta_1 - 2) q^{11} + ( - \beta_{2} + \beta_1) q^{13} + (\beta_1 + 1) q^{17} + (\beta_{2} + \beta_1) q^{19} + q^{23} + q^{25} + (\beta_1 - 7) q^{29} + (3 \beta_{2} + 2 \beta_1 + 5) q^{31} + (\beta_{2} + 2 \beta_1 - 1) q^{35} + ( - \beta_{2} - 2 \beta_1 - 1) q^{37} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{41} + (2 \beta_{2} - 2) q^{43} + (\beta_{2} + \beta_1 - 4) q^{47} + ( - 3 \beta_{2} + 2 \beta_1 + 8) q^{49} + ( - 2 \beta_{2} + \beta_1 - 1) q^{53} + ( - \beta_{2} + \beta_1 + 2) q^{55} + ( - 4 \beta_{2} - 3 \beta_1 - 5) q^{59} + (3 \beta_{2} - \beta_1 - 6) q^{61} + (\beta_{2} - \beta_1) q^{65} + (5 \beta_{2} + 2 \beta_1 + 3) q^{67} + ( - 2 \beta_{2} - 5 \beta_1 + 3) q^{71} + (3 \beta_{2} + 5 \beta_1 - 4) q^{73} + (7 \beta_{2} + 9 \beta_1 + 2) q^{77} + 4 \beta_1 q^{79} + ( - \beta_1 - 7) q^{83} + ( - \beta_1 - 1) q^{85} + (4 \beta_{2} + 2 \beta_1 + 4) q^{89} + ( - 5 \beta_{2} - 5 \beta_1 - 4) q^{91} + ( - \beta_{2} - \beta_1) q^{95} + (4 \beta_{2} + 6 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + 2 q^{7} - 8 q^{11} + 2 q^{13} + 4 q^{17} + 3 q^{23} + 3 q^{25} - 20 q^{29} + 14 q^{31} - 2 q^{35} - 4 q^{37} + 8 q^{41} - 8 q^{43} - 12 q^{47} + 29 q^{49} + 8 q^{55} - 14 q^{59} - 22 q^{61} - 2 q^{65} + 6 q^{67} + 6 q^{71} - 10 q^{73} + 8 q^{77} + 4 q^{79} - 22 q^{83} - 4 q^{85} + 10 q^{89} - 12 q^{91} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) 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
3.12489
−1.76156
−0.363328
0 0 0 −1.00000 0 −4.76491 0 0 0
1.2 0 0 0 −1.00000 0 1.89692 0 0 0
1.3 0 0 0 −1.00000 0 4.86799 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.bi 3
3.b odd 2 1 2760.2.a.u 3
12.b even 2 1 5520.2.a.bz 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.u 3 3.b odd 2 1
5520.2.a.bz 3 12.b even 2 1
8280.2.a.bi 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

\( T_{7}^{3} - 2T_{7}^{2} - 23T_{7} + 44 \) Copy content Toggle raw display
\( T_{11}^{3} + 8T_{11}^{2} + 2T_{11} - 64 \) Copy content Toggle raw display
\( T_{13}^{3} - 2T_{13}^{2} - 18T_{13} + 44 \) Copy content Toggle raw display
\( T_{17}^{3} - 4T_{17}^{2} - T_{17} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} - 23 T + 44 \) Copy content Toggle raw display
$11$ \( T^{3} + 8 T^{2} + 2 T - 64 \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 18 T + 44 \) Copy content Toggle raw display
$17$ \( T^{3} - 4T^{2} - T + 2 \) Copy content Toggle raw display
$19$ \( T^{3} - 10T + 8 \) Copy content Toggle raw display
$23$ \( (T - 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 20 T^{2} + 127 T + 250 \) Copy content Toggle raw display
$31$ \( T^{3} - 14 T^{2} - 7 T + 472 \) Copy content Toggle raw display
$37$ \( T^{3} + 4 T^{2} - 19 T - 2 \) Copy content Toggle raw display
$41$ \( T^{3} - 8 T^{2} - 97 T + 670 \) Copy content Toggle raw display
$43$ \( T^{3} + 8 T^{2} - 12 T - 80 \) Copy content Toggle raw display
$47$ \( T^{3} + 12 T^{2} + 38 T + 32 \) Copy content Toggle raw display
$53$ \( T^{3} - 49T + 122 \) Copy content Toggle raw display
$59$ \( T^{3} + 14 T^{2} - 69 T - 1100 \) Copy content Toggle raw display
$61$ \( T^{3} + 22 T^{2} + 66 T - 580 \) Copy content Toggle raw display
$67$ \( T^{3} - 6 T^{2} - 175 T + 1156 \) Copy content Toggle raw display
$71$ \( T^{3} - 6 T^{2} - 133 T + 848 \) Copy content Toggle raw display
$73$ \( T^{3} + 10 T^{2} - 130 T - 764 \) Copy content Toggle raw display
$79$ \( T^{3} - 4 T^{2} - 96 T - 128 \) Copy content Toggle raw display
$83$ \( T^{3} + 22 T^{2} + 155 T + 352 \) Copy content Toggle raw display
$89$ \( T^{3} - 10 T^{2} - 88 T + 848 \) Copy content Toggle raw display
$97$ \( T^{3} - 2 T^{2} - 248 T - 16 \) Copy content Toggle raw display
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