Properties

Label 8280.2.a.bl
Level $8280$
Weight $2$
Character orbit 8280.a
Self dual yes
Analytic conductor $66.116$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
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.

\(f(q)\) \(=\) \( q - q^{5} + ( 2 - \beta_{2} ) q^{7} +O(q^{10})\) \( q - q^{5} + ( 2 - \beta_{2} ) q^{7} + ( -1 - \beta_{1} ) q^{11} + ( 2 + \beta_{1} ) q^{13} + ( 1 + 3 \beta_{1} - 2 \beta_{2} ) q^{17} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{19} + q^{23} + q^{25} + ( 5 \beta_{1} - \beta_{2} ) q^{29} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{31} + ( -2 + \beta_{2} ) q^{35} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{37} + ( 3 - 4 \beta_{1} - \beta_{2} ) q^{41} + ( 4 - 4 \beta_{1} ) q^{43} + ( -\beta_{1} + \beta_{2} ) q^{47} + ( \beta_{1} - 6 \beta_{2} ) q^{49} + ( -2 + 4 \beta_{2} ) q^{53} + ( 1 + \beta_{1} ) q^{55} + ( -2 - 6 \beta_{1} + 4 \beta_{2} ) q^{59} + ( -3 - 3 \beta_{1} + 4 \beta_{2} ) q^{61} + ( -2 - \beta_{1} ) q^{65} + ( -4 + 2 \beta_{1} - 6 \beta_{2} ) q^{67} + ( -5 - 5 \beta_{2} ) q^{71} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{73} + ( -1 - \beta_{1} + \beta_{2} ) q^{77} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{79} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -1 - 3 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 8 - 4 \beta_{1} + 4 \beta_{2} ) q^{89} + ( 3 + \beta_{1} - 2 \beta_{2} ) q^{91} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{95} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 7 q^{7} + O(q^{10}) \) \( 3 q - 3 q^{5} + 7 q^{7} - 3 q^{11} + 6 q^{13} + 5 q^{17} + 7 q^{19} + 3 q^{23} + 3 q^{25} + q^{29} + 10 q^{31} - 7 q^{35} + 2 q^{37} + 10 q^{41} + 12 q^{43} - q^{47} + 6 q^{49} - 10 q^{53} + 3 q^{55} - 10 q^{59} - 13 q^{61} - 6 q^{65} - 6 q^{67} - 10 q^{71} + 7 q^{73} - 4 q^{77} + 14 q^{79} - 16 q^{83} - 5 q^{85} + 20 q^{89} + 11 q^{91} - 7 q^{95} + 5 q^{97} + O(q^{100}) \)

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

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

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 0.527166 0 0 0
1.2 0 0 0 −1.00000 0 1.53740 0 0 0
1.3 0 0 0 −1.00000 0 4.93543 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.bl 3
3.b odd 2 1 920.2.a.i 3
12.b even 2 1 1840.2.a.q 3
15.d odd 2 1 4600.2.a.v 3
15.e even 4 2 4600.2.e.q 6
24.f even 2 1 7360.2.a.cf 3
24.h odd 2 1 7360.2.a.bw 3
60.h even 2 1 9200.2.a.ci 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.i 3 3.b odd 2 1
1840.2.a.q 3 12.b even 2 1
4600.2.a.v 3 15.d odd 2 1
4600.2.e.q 6 15.e even 4 2
7360.2.a.bw 3 24.h odd 2 1
7360.2.a.cf 3 24.f even 2 1
8280.2.a.bl 3 1.a even 1 1 trivial
9200.2.a.ci 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

\( T_{7}^{3} - 7 T_{7}^{2} + 11 T_{7} - 4 \)
\( T_{11}^{3} + 3 T_{11}^{2} - T_{11} - 2 \)
\( T_{13}^{3} - 6 T_{13}^{2} + 8 T_{13} - 1 \)
\( T_{17}^{3} - 5 T_{17}^{2} - 31 T_{17} + 148 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( -4 + 11 T - 7 T^{2} + T^{3} \)
$11$ \( -2 - T + 3 T^{2} + T^{3} \)
$13$ \( -1 + 8 T - 6 T^{2} + T^{3} \)
$17$ \( 148 - 31 T - 5 T^{2} + T^{3} \)
$19$ \( 106 - 11 T - 7 T^{2} + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( 148 - 90 T - T^{2} + T^{3} \)
$31$ \( 53 + 14 T - 10 T^{2} + T^{3} \)
$37$ \( 512 - 128 T - 2 T^{2} + T^{3} \)
$41$ \( 481 - 48 T - 10 T^{2} + T^{3} \)
$43$ \( 256 - 16 T - 12 T^{2} + T^{3} \)
$47$ \( -4 - 6 T + T^{2} + T^{3} \)
$53$ \( -8 - 52 T + 10 T^{2} + T^{3} \)
$59$ \( -1184 - 124 T + 10 T^{2} + T^{3} \)
$61$ \( -214 - 29 T + 13 T^{2} + T^{3} \)
$67$ \( -1184 - 160 T + 6 T^{2} + T^{3} \)
$71$ \( -875 - 100 T + 10 T^{2} + T^{3} \)
$73$ \( 236 - 34 T - 7 T^{2} + T^{3} \)
$79$ \( 224 + 16 T - 14 T^{2} + T^{3} \)
$83$ \( 32 + 60 T + 16 T^{2} + T^{3} \)
$89$ \( 256 + 32 T - 20 T^{2} + T^{3} \)
$97$ \( 56 - 23 T - 5 T^{2} + T^{3} \)
show more
show less