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 $1$

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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: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - q^{5} + ( - \beta_{2} + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + ( - \beta_{2} + 2) q^{7} + ( - \beta_1 - 1) q^{11} + (\beta_1 + 2) q^{13} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{17} + ( - \beta_{2} - 2 \beta_1 + 2) q^{19} + q^{23} + q^{25} + ( - \beta_{2} + 5 \beta_1) q^{29} + (2 \beta_{2} - \beta_1 + 4) q^{31} + (\beta_{2} - 2) q^{35} + ( - 2 \beta_{2} + 6 \beta_1) q^{37} + ( - \beta_{2} - 4 \beta_1 + 3) q^{41} + ( - 4 \beta_1 + 4) q^{43} + (\beta_{2} - \beta_1) q^{47} + ( - 6 \beta_{2} + \beta_1) q^{49} + (4 \beta_{2} - 2) q^{53} + (\beta_1 + 1) q^{55} + (4 \beta_{2} - 6 \beta_1 - 2) q^{59} + (4 \beta_{2} - 3 \beta_1 - 3) q^{61} + ( - \beta_1 - 2) q^{65} + ( - 6 \beta_{2} + 2 \beta_1 - 4) q^{67} + ( - 5 \beta_{2} - 5) q^{71} + ( - \beta_{2} - 3 \beta_1 + 2) q^{73} + (\beta_{2} - \beta_1 - 1) q^{77} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{79} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{83} + (2 \beta_{2} - 3 \beta_1 - 1) q^{85} + (4 \beta_{2} - 4 \beta_1 + 8) q^{89} + ( - 2 \beta_{2} + \beta_1 + 3) q^{91} + (\beta_{2} + 2 \beta_1 - 2) q^{95} + ( - 2 \beta_{2} - \beta_1 + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + 7 q^{7} - 3 q^{11} + 6 q^{13} + 5 q^{17} + 7 q^{19} + 3 q^{23} + 3 q^{25} + q^{29} + 10 q^{31} - 7 q^{35} + 2 q^{37} + 10 q^{41} + 12 q^{43} - q^{47} + 6 q^{49} - 10 q^{53} + 3 q^{55} - 10 q^{59} - 13 q^{61} - 6 q^{65} - 6 q^{67} - 10 q^{71} + 7 q^{73} - 4 q^{77} + 14 q^{79} - 16 q^{83} - 5 q^{85} + 20 q^{89} + 11 q^{91} - 7 q^{95} + 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

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