Properties

Label 8280.2.a.z
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: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). 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 \( q - q^{5} + \beta q^{7} - 4 q^{11} + (2 \beta - 2) q^{13} + ( - \beta - 2) q^{17} + 4 q^{19} - q^{23} + q^{25} + (\beta + 2) q^{29} + \beta q^{31} - \beta q^{35} + ( - \beta - 2) q^{37} + (3 \beta - 2) q^{41} - 4 q^{43} + 2 \beta q^{47} + (\beta + 1) q^{49} + ( - \beta + 2) q^{53} + 4 q^{55} + (\beta - 12) q^{59} - 2 q^{61} + ( - 2 \beta + 2) q^{65} + (\beta + 12) q^{67} - \beta q^{71} + ( - 2 \beta - 6) q^{73} - 4 \beta q^{77} - 4 \beta q^{79} + (5 \beta - 4) q^{83} + (\beta + 2) q^{85} + (2 \beta - 2) q^{89} + 16 q^{91} - 4 q^{95} + (4 \beta + 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + q^{7} - 8 q^{11} - 2 q^{13} - 5 q^{17} + 8 q^{19} - 2 q^{23} + 2 q^{25} + 5 q^{29} + q^{31} - q^{35} - 5 q^{37} - q^{41} - 8 q^{43} + 2 q^{47} + 3 q^{49} + 3 q^{53} + 8 q^{55} - 23 q^{59} - 4 q^{61} + 2 q^{65} + 25 q^{67} - q^{71} - 14 q^{73} - 4 q^{77} - 4 q^{79} - 3 q^{83} + 5 q^{85} - 2 q^{89} + 32 q^{91} - 8 q^{95} + 8 q^{97}+O(q^{100}) \) 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
−2.37228
3.37228
0 0 0 −1.00000 0 −2.37228 0 0 0
1.2 0 0 0 −1.00000 0 3.37228 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.z 2
3.b odd 2 1 2760.2.a.o 2
12.b even 2 1 5520.2.a.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.o 2 3.b odd 2 1
5520.2.a.bq 2 12.b even 2 1
8280.2.a.z 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

\( T_{7}^{2} - T_{7} - 8 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 32 \) Copy content Toggle raw display
\( T_{17}^{2} + 5T_{17} - 2 \) 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
$7$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$17$ \( T^{2} + 5T - 2 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 5T - 2 \) Copy content Toggle raw display
$31$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 5T - 2 \) Copy content Toggle raw display
$41$ \( T^{2} + T - 74 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$59$ \( T^{2} + 23T + 124 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 25T + 148 \) Copy content Toggle raw display
$71$ \( T^{2} + T - 8 \) Copy content Toggle raw display
$73$ \( T^{2} + 14T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 128 \) Copy content Toggle raw display
$83$ \( T^{2} + 3T - 204 \) Copy content Toggle raw display
$89$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 116 \) Copy content Toggle raw display
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