Properties

Label 8280.2.a.bq
Level $8280$
Weight $2$
Character orbit 8280.a
Self dual yes
Analytic conductor $66.116$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.54764.1
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - \beta_1 q^{7} + \beta_{3} q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{13} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{17} + (\beta_{2} + \beta_1 - 1) q^{19} - q^{23} + q^{25} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{29} + (\beta_{3} + \beta_{2} - 1) q^{31} - \beta_1 q^{35} + (2 \beta_{2} + \beta_1) q^{37} + ( - \beta_{3} + \beta_1 - 2) q^{41} + ( - 2 \beta_{2} - 6) q^{43} + (\beta_{2} + \beta_1 - 1) q^{47} + ( - 3 \beta_{3} - \beta_{2} + 2) q^{49} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{53} + \beta_{3} q^{55} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{59} + (\beta_{3} + 2 \beta_{2}) q^{61} + ( - \beta_{2} + \beta_1 - 1) q^{65} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 - 7) q^{67} + ( - \beta_{3} + \beta_1) q^{71} + ( - \beta_{2} + \beta_1 - 5) q^{73} + (\beta_{2} + 3 \beta_1 - 1) q^{77} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 3) q^{79} + ( - 2 \beta_{3} + \beta_{2} + 7) q^{83} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{85} + (\beta_{3} + \beta_{2} + \beta_1 - 3) q^{89} + (2 \beta_{3} + \beta_{2} + \beta_1 - 9) q^{91} + (\beta_{2} + \beta_1 - 1) q^{95} + ( - 2 \beta_{3} - 2 \beta_{2} - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} - 2 q^{13} - 2 q^{17} - 6 q^{19} - 4 q^{23} + 4 q^{25} - 4 q^{29} - 6 q^{31} - 4 q^{37} - 8 q^{41} - 20 q^{43} - 6 q^{47} + 10 q^{49} + 4 q^{53} + 4 q^{59} - 4 q^{61} - 2 q^{65} - 26 q^{67} - 18 q^{73} - 6 q^{77} - 18 q^{79} + 26 q^{83} - 2 q^{85} - 14 q^{89} - 38 q^{91} - 6 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 11\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} + 9\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 8\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3\beta_{2} + \beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{3} + 11\beta_{2} - 7\beta _1 + 15 ) / 2 \) 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
−2.67673
−0.339102
3.36007
0.655762
0 0 0 1.00000 0 −4.13277 0 0 0
1.2 0 0 0 1.00000 0 −0.845563 0 0 0
1.3 0 0 0 1.00000 0 0.512641 0 0 0
1.4 0 0 0 1.00000 0 4.46569 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.bq 4
3.b odd 2 1 2760.2.a.v 4
12.b even 2 1 5520.2.a.cb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.v 4 3.b odd 2 1
5520.2.a.cb 4 12.b even 2 1
8280.2.a.bq 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

\( T_{7}^{4} - 19T_{7}^{2} - 6T_{7} + 8 \) Copy content Toggle raw display
\( T_{11}^{4} - 22T_{11}^{2} - 12T_{11} + 80 \) Copy content Toggle raw display
\( T_{13}^{4} + 2T_{13}^{3} - 38T_{13}^{2} - 56T_{13} + 296 \) Copy content Toggle raw display
\( T_{17}^{4} + 2T_{17}^{3} - 71T_{17}^{2} - 58T_{17} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 19 T^{2} - 6 T + 8 \) Copy content Toggle raw display
$11$ \( T^{4} - 22 T^{2} - 12 T + 80 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} - 38 T^{2} - 56 T + 296 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} - 71 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} - 26 T^{2} - 112 T + 256 \) Copy content Toggle raw display
$23$ \( (T + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} - 75 T^{2} - 400 T - 164 \) Copy content Toggle raw display
$31$ \( T^{4} + 6 T^{3} - 11 T^{2} - 104 T - 128 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} - 95 T^{2} + \cdots + 2104 \) Copy content Toggle raw display
$41$ \( T^{4} + 8 T^{3} - 11 T^{2} - 204 T - 380 \) Copy content Toggle raw display
$43$ \( T^{4} + 20 T^{3} + 68 T^{2} + \cdots - 2048 \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} - 26 T^{2} - 112 T + 256 \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} - 147 T^{2} + \cdots + 1468 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} - 75 T^{2} - 84 T + 320 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} - 62 T^{2} - 364 T - 400 \) Copy content Toggle raw display
$67$ \( T^{4} + 26 T^{3} + 141 T^{2} + \cdots - 6976 \) Copy content Toggle raw display
$71$ \( T^{4} - 35 T^{2} - 96 T - 64 \) Copy content Toggle raw display
$73$ \( T^{4} + 18 T^{3} + 82 T^{2} + \cdots - 152 \) Copy content Toggle raw display
$79$ \( T^{4} + 18 T^{3} - 56 T^{2} + \cdots + 512 \) Copy content Toggle raw display
$83$ \( T^{4} - 26 T^{3} + 109 T^{2} + \cdots - 9176 \) Copy content Toggle raw display
$89$ \( T^{4} + 14 T^{3} + 24 T^{2} - 48 T - 64 \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} - 44 T^{2} - 128 T + 64 \) Copy content Toggle raw display
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