Properties

Label 8280.2.a.bf
Level $8280$
Weight $2$
Character orbit 8280.a
Self dual yes
Analytic conductor $66.116$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} - \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - \beta q^{7} + 2 q^{11} + ( - \beta - 1) q^{13} + (\beta - 4) q^{17} - 2 q^{19} + q^{23} + q^{25} + (2 \beta + 3) q^{29} + (2 \beta - 5) q^{31} - \beta q^{35} + (\beta - 2) q^{37} + ( - 2 \beta + 1) q^{41} + (4 \beta - 4) q^{43} + (3 \beta + 1) q^{47} + (\beta - 3) q^{49} + ( - \beta - 6) q^{53} + 2 q^{55} + ( - \beta + 4) q^{59} + (2 \beta - 6) q^{61} + ( - \beta - 1) q^{65} + (\beta - 10) q^{67} + 5 q^{71} + (\beta + 5) q^{73} - 2 \beta q^{77} + (2 \beta - 6) q^{79} - 3 \beta q^{83} + (\beta - 4) q^{85} - 8 q^{89} + (2 \beta + 4) q^{91} - 2 q^{95} + 2 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - q^{7} + 4 q^{11} - 3 q^{13} - 7 q^{17} - 4 q^{19} + 2 q^{23} + 2 q^{25} + 8 q^{29} - 8 q^{31} - q^{35} - 3 q^{37} - 4 q^{43} + 5 q^{47} - 5 q^{49} - 13 q^{53} + 4 q^{55} + 7 q^{59} - 10 q^{61} - 3 q^{65} - 19 q^{67} + 10 q^{71} + 11 q^{73} - 2 q^{77} - 10 q^{79} - 3 q^{83} - 7 q^{85} - 16 q^{89} + 10 q^{91} - 4 q^{95} + 2 q^{97}+O(q^{100}) \) 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
2.56155
−1.56155
0 0 0 1.00000 0 −2.56155 0 0 0
1.2 0 0 0 1.00000 0 1.56155 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.bf 2
3.b odd 2 1 920.2.a.e 2
12.b even 2 1 1840.2.a.o 2
15.d odd 2 1 4600.2.a.t 2
15.e even 4 2 4600.2.e.n 4
24.f even 2 1 7360.2.a.bl 2
24.h odd 2 1 7360.2.a.bp 2
60.h even 2 1 9200.2.a.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.e 2 3.b odd 2 1
1840.2.a.o 2 12.b even 2 1
4600.2.a.t 2 15.d odd 2 1
4600.2.e.n 4 15.e even 4 2
7360.2.a.bl 2 24.f even 2 1
7360.2.a.bp 2 24.h odd 2 1
8280.2.a.bf 2 1.a even 1 1 trivial
9200.2.a.bq 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$17$ \( T^{2} + 7T + 8 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 8T - 1 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T - 1 \) Copy content Toggle raw display
$37$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$41$ \( T^{2} - 17 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$47$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$53$ \( T^{2} + 13T + 38 \) Copy content Toggle raw display
$59$ \( T^{2} - 7T + 8 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$67$ \( T^{2} + 19T + 86 \) Copy content Toggle raw display
$71$ \( (T - 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 11T + 26 \) Copy content Toggle raw display
$79$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$83$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$89$ \( (T + 8)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
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