Properties

Label 8280.2.a.bu
Level $8280$
Weight $2$
Character orbit 8280.a
Self dual yes
Analytic conductor $66.116$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 16x^{4} + 26x^{3} + 52x^{2} - 48x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - \beta_{2} q^{7} + ( - \beta_{3} - 1) q^{11} + \beta_{4} q^{13} + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{17} + ( - \beta_{3} + 1) q^{19} + q^{23} + q^{25} + (\beta_{5} + \beta_{3} - \beta_{2} + \beta_1 + 4) q^{29} + ( - \beta_{2} - \beta_1 - 1) q^{31} - \beta_{2} q^{35} + (\beta_{5} + \beta_{2} - \beta_1 + 1) q^{37} + ( - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 + 1) q^{41} + ( - \beta_{4} - \beta_{3} + 1) q^{43} + ( - \beta_{5} - \beta_{3} + \beta_1 - 2) q^{47} + ( - \beta_{5} + \beta_{2} + 1) q^{49} + (\beta_{3} - \beta_{2} + \beta_1 + 2) q^{53} + ( - \beta_{3} - 1) q^{55} + ( - \beta_{4} + \beta_{2}) q^{59} + (\beta_{5} + \beta_{4} + \beta_1 - 1) q^{61} + \beta_{4} q^{65} + ( - \beta_{5} + \beta_{2} + \beta_1 - 1) q^{67} + (2 \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{71} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{73} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{77} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_1 - 4) q^{79} + (2 \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 + 5) q^{83} + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{85} - 2 \beta_{5} q^{89} + ( - 2 \beta_{4} + \beta_{3} - 1) q^{91} + ( - \beta_{3} + 1) q^{95} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{5} - 2 q^{7} - 4 q^{11} + 4 q^{17} + 8 q^{19} + 6 q^{23} + 6 q^{25} + 18 q^{29} - 8 q^{31} - 2 q^{35} + 6 q^{37} + 6 q^{41} + 8 q^{43} - 8 q^{47} + 10 q^{49} + 8 q^{53} - 4 q^{55} + 2 q^{59} - 8 q^{61} - 2 q^{67} + 2 q^{71} + 8 q^{73} + 16 q^{77} - 28 q^{79} + 24 q^{83} + 4 q^{85} + 4 q^{89} - 8 q^{91} + 8 q^{95} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 16x^{4} + 26x^{3} + 52x^{2} - 48x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{5} + \nu^{4} + 47\nu^{3} - 22\nu^{2} - 245\nu + 66 ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{5} - 2\nu^{4} - 67\nu^{3} + 17\nu^{2} + 247\nu + 3 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -13\nu^{5} + 20\nu^{4} + 211\nu^{3} - 251\nu^{2} - 688\nu + 348 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\nu^{5} - 20\nu^{4} - 211\nu^{3} + 251\nu^{2} + 742\nu - 375 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -20\nu^{5} + 37\nu^{4} + 335\nu^{3} - 436\nu^{2} - 1235\nu + 471 ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{4} - 2\beta_{2} - \beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + 11\beta_{4} + 9\beta_{3} + 3\beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 15\beta_{5} + 26\beta_{4} - 16\beta_{2} - 13\beta _1 + 133 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 20\beta_{5} + 127\beta_{4} + 89\beta_{3} + 14\beta_{2} + 48\beta _1 + 149 ) / 2 \) 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
0.485614
0.287750
3.58888
−3.32949
−1.88788
2.85512
0 0 0 1.00000 0 −4.41777 0 0 0
1.2 0 0 0 1.00000 0 −2.73629 0 0 0
1.3 0 0 0 1.00000 0 −2.26189 0 0 0
1.4 0 0 0 1.00000 0 1.49704 0 0 0
1.5 0 0 0 1.00000 0 2.71231 0 0 0
1.6 0 0 0 1.00000 0 3.20659 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.bu yes 6
3.b odd 2 1 8280.2.a.bt 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8280.2.a.bt 6 3.b odd 2 1
8280.2.a.bu yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

\( T_{7}^{6} + 2T_{7}^{5} - 24T_{7}^{4} - 30T_{7}^{3} + 171T_{7}^{2} + 112T_{7} - 356 \) Copy content Toggle raw display
\( T_{11}^{6} + 4T_{11}^{5} - 38T_{11}^{4} - 100T_{11}^{3} + 364T_{11}^{2} + 160T_{11} + 16 \) Copy content Toggle raw display
\( T_{13}^{6} - 42T_{13}^{4} - 60T_{13}^{3} + 60T_{13}^{2} + 48T_{13} - 32 \) Copy content Toggle raw display
\( T_{17}^{6} - 4T_{17}^{5} - 72T_{17}^{4} + 276T_{17}^{3} + 1239T_{17}^{2} - 3848T_{17} - 3380 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} - 24 T^{4} - 30 T^{3} + \cdots - 356 \) Copy content Toggle raw display
$11$ \( T^{6} + 4 T^{5} - 38 T^{4} - 100 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{6} - 42 T^{4} - 60 T^{3} + 60 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} - 72 T^{4} + \cdots - 3380 \) Copy content Toggle raw display
$19$ \( T^{6} - 8 T^{5} - 18 T^{4} + \cdots + 1280 \) Copy content Toggle raw display
$23$ \( (T - 1)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} - 18 T^{5} - 10 T^{4} + \cdots + 49348 \) Copy content Toggle raw display
$31$ \( T^{6} + 8 T^{5} - 34 T^{4} + \cdots + 7376 \) Copy content Toggle raw display
$37$ \( T^{6} - 6 T^{5} - 156 T^{4} + \cdots + 8728 \) Copy content Toggle raw display
$41$ \( T^{6} - 6 T^{5} - 126 T^{4} + \cdots - 4832 \) Copy content Toggle raw display
$43$ \( T^{6} - 8 T^{5} - 44 T^{4} + \cdots + 1408 \) Copy content Toggle raw display
$47$ \( T^{6} + 8 T^{5} - 122 T^{4} + \cdots + 22528 \) Copy content Toggle raw display
$53$ \( T^{6} - 8 T^{5} - 88 T^{4} + \cdots + 8732 \) Copy content Toggle raw display
$59$ \( T^{6} - 2 T^{5} - 58 T^{4} - 22 T^{3} + \cdots - 396 \) Copy content Toggle raw display
$61$ \( T^{6} + 8 T^{5} - 142 T^{4} + \cdots + 39888 \) Copy content Toggle raw display
$67$ \( T^{6} + 2 T^{5} - 132 T^{4} + \cdots - 200 \) Copy content Toggle raw display
$71$ \( T^{6} - 2 T^{5} - 174 T^{4} + \cdots + 34344 \) Copy content Toggle raw display
$73$ \( T^{6} - 8 T^{5} - 238 T^{4} + \cdots - 305680 \) Copy content Toggle raw display
$79$ \( T^{6} + 28 T^{5} + 152 T^{4} + \cdots - 47104 \) Copy content Toggle raw display
$83$ \( T^{6} - 24 T^{5} - 116 T^{4} + \cdots + 1080216 \) Copy content Toggle raw display
$89$ \( T^{6} - 4 T^{5} - 272 T^{4} + \cdots - 367616 \) Copy content Toggle raw display
$97$ \( T^{6} - 24 T^{5} - 88 T^{4} + \cdots + 25792 \) Copy content Toggle raw display
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