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: \(x^{6} - 2 x^{5} - 16 x^{4} + 26 x^{3} + 52 x^{2} - 48 x + 9\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{5} + \nu^{4} + 47 \nu^{3} - 22 \nu^{2} - 245 \nu + 66 \)\()/27\)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{5} - 2 \nu^{4} - 67 \nu^{3} + 17 \nu^{2} + 247 \nu + 3 \)\()/27\)
\(\beta_{3}\)\(=\)\((\)\( -13 \nu^{5} + 20 \nu^{4} + 211 \nu^{3} - 251 \nu^{2} - 688 \nu + 348 \)\()/27\)
\(\beta_{4}\)\(=\)\((\)\( 13 \nu^{5} - 20 \nu^{4} - 211 \nu^{3} + 251 \nu^{2} + 742 \nu - 375 \)\()/27\)
\(\beta_{5}\)\(=\)\((\)\( -20 \nu^{5} + 37 \nu^{4} + 335 \nu^{3} - 436 \nu^{2} - 1235 \nu + 471 \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 2 \beta_{4} - 2 \beta_{2} - \beta_{1} + 13\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} + 11 \beta_{4} + 9 \beta_{3} + 3 \beta_{1} + 12\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(15 \beta_{5} + 26 \beta_{4} - 16 \beta_{2} - 13 \beta_{1} + 133\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(20 \beta_{5} + 127 \beta_{4} + 89 \beta_{3} + 14 \beta_{2} + 48 \beta_{1} + 149\)\()/2\)

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8280))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( ( -1 + T )^{6} \)
$7$ \( -356 + 112 T + 171 T^{2} - 30 T^{3} - 24 T^{4} + 2 T^{5} + T^{6} \)
$11$ \( 16 + 160 T + 364 T^{2} - 100 T^{3} - 38 T^{4} + 4 T^{5} + T^{6} \)
$13$ \( -32 + 48 T + 60 T^{2} - 60 T^{3} - 42 T^{4} + T^{6} \)
$17$ \( -3380 - 3848 T + 1239 T^{2} + 276 T^{3} - 72 T^{4} - 4 T^{5} + T^{6} \)
$19$ \( 1280 - 1152 T - 28 T^{2} + 204 T^{3} - 18 T^{4} - 8 T^{5} + T^{6} \)
$23$ \( ( -1 + T )^{6} \)
$29$ \( 49348 + 4928 T - 10331 T^{2} + 1838 T^{3} - 10 T^{4} - 18 T^{5} + T^{6} \)
$31$ \( 7376 + 3632 T - 383 T^{2} - 408 T^{3} - 34 T^{4} + 8 T^{5} + T^{6} \)
$37$ \( 8728 - 33228 T + 4611 T^{2} + 1038 T^{3} - 156 T^{4} - 6 T^{5} + T^{6} \)
$41$ \( -4832 - 2712 T + 1773 T^{2} + 594 T^{3} - 126 T^{4} - 6 T^{5} + T^{6} \)
$43$ \( 1408 - 2112 T + 464 T^{2} + 304 T^{3} - 44 T^{4} - 8 T^{5} + T^{6} \)
$47$ \( 22528 + 15488 T + 76 T^{2} - 1124 T^{3} - 122 T^{4} + 8 T^{5} + T^{6} \)
$53$ \( 8732 - 12524 T + 1399 T^{2} + 708 T^{3} - 88 T^{4} - 8 T^{5} + T^{6} \)
$59$ \( -396 + 660 T + 637 T^{2} - 22 T^{3} - 58 T^{4} - 2 T^{5} + T^{6} \)
$61$ \( 39888 + 15024 T - 2756 T^{2} - 1628 T^{3} - 142 T^{4} + 8 T^{5} + T^{6} \)
$67$ \( -200 + 1852 T + 1755 T^{2} + 126 T^{3} - 132 T^{4} + 2 T^{5} + T^{6} \)
$71$ \( 34344 + 32076 T + 7717 T^{2} - 230 T^{3} - 174 T^{4} - 2 T^{5} + T^{6} \)
$73$ \( -305680 - 62144 T + 15460 T^{2} + 1548 T^{3} - 238 T^{4} - 8 T^{5} + T^{6} \)
$79$ \( -47104 - 57344 T - 19328 T^{2} - 1728 T^{3} + 152 T^{4} + 28 T^{5} + T^{6} \)
$83$ \( 1080216 - 98604 T - 43209 T^{2} + 6780 T^{3} - 116 T^{4} - 24 T^{5} + T^{6} \)
$89$ \( -367616 + 12800 T + 21568 T^{2} + 352 T^{3} - 272 T^{4} - 4 T^{5} + T^{6} \)
$97$ \( 25792 - 35584 T - 21248 T^{2} + 4736 T^{3} - 88 T^{4} - 24 T^{5} + T^{6} \)
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