Properties

Label 8280.2.a.bb
Level $8280$
Weight $2$
Character orbit 8280.a
Self dual yes
Analytic conductor $66.116$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q - q^{5} + 2 \beta q^{7} +O(q^{10})\) \( q - q^{5} + 2 \beta q^{7} + 4 q^{11} + ( 2 - \beta ) q^{13} + ( -2 + 2 \beta ) q^{17} + 4 q^{19} - q^{23} + q^{25} + ( 6 + \beta ) q^{29} + ( 4 - \beta ) q^{31} -2 \beta q^{35} + ( -2 - 2 \beta ) q^{37} + ( 2 - \beta ) q^{41} + ( -4 - 2 \beta ) q^{43} + ( -4 + 3 \beta ) q^{47} + ( 9 + 4 \beta ) q^{49} + ( -6 + 4 \beta ) q^{53} -4 q^{55} + ( -4 + 4 \beta ) q^{59} + ( 6 + 2 \beta ) q^{61} + ( -2 + \beta ) q^{65} + ( -4 + 4 \beta ) q^{67} + ( 4 - 3 \beta ) q^{71} + ( 14 + \beta ) q^{73} + 8 \beta q^{77} -4 \beta q^{79} -12 q^{83} + ( 2 - 2 \beta ) q^{85} -10 q^{89} + ( -8 + 2 \beta ) q^{91} -4 q^{95} + ( -6 - 4 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{5} + 2 q^{7} + 8 q^{11} + 3 q^{13} - 2 q^{17} + 8 q^{19} - 2 q^{23} + 2 q^{25} + 13 q^{29} + 7 q^{31} - 2 q^{35} - 6 q^{37} + 3 q^{41} - 10 q^{43} - 5 q^{47} + 22 q^{49} - 8 q^{53} - 8 q^{55} - 4 q^{59} + 14 q^{61} - 3 q^{65} - 4 q^{67} + 5 q^{71} + 29 q^{73} + 8 q^{77} - 4 q^{79} - 24 q^{83} + 2 q^{85} - 20 q^{89} - 14 q^{91} - 8 q^{95} - 16 q^{97} + O(q^{100}) \)

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
−1.56155
2.56155
0 0 0 −1.00000 0 −3.12311 0 0 0
1.2 0 0 0 −1.00000 0 5.12311 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.bb 2
3.b odd 2 1 920.2.a.f 2
12.b even 2 1 1840.2.a.k 2
15.d odd 2 1 4600.2.a.r 2
15.e even 4 2 4600.2.e.m 4
24.f even 2 1 7360.2.a.bm 2
24.h odd 2 1 7360.2.a.bj 2
60.h even 2 1 9200.2.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.f 2 3.b odd 2 1
1840.2.a.k 2 12.b even 2 1
4600.2.a.r 2 15.d odd 2 1
4600.2.e.m 4 15.e even 4 2
7360.2.a.bj 2 24.h odd 2 1
7360.2.a.bm 2 24.f even 2 1
8280.2.a.bb 2 1.a even 1 1 trivial
9200.2.a.bx 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

\( T_{7}^{2} - 2 T_{7} - 16 \)
\( T_{11} - 4 \)
\( T_{13}^{2} - 3 T_{13} - 2 \)
\( T_{17}^{2} + 2 T_{17} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -16 - 2 T + T^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( -2 - 3 T + T^{2} \)
$17$ \( -16 + 2 T + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( 38 - 13 T + T^{2} \)
$31$ \( 8 - 7 T + T^{2} \)
$37$ \( -8 + 6 T + T^{2} \)
$41$ \( -2 - 3 T + T^{2} \)
$43$ \( 8 + 10 T + T^{2} \)
$47$ \( -32 + 5 T + T^{2} \)
$53$ \( -52 + 8 T + T^{2} \)
$59$ \( -64 + 4 T + T^{2} \)
$61$ \( 32 - 14 T + T^{2} \)
$67$ \( -64 + 4 T + T^{2} \)
$71$ \( -32 - 5 T + T^{2} \)
$73$ \( 206 - 29 T + T^{2} \)
$79$ \( -64 + 4 T + T^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( -4 + 16 T + T^{2} \)
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