Properties

Label 9216.2.a.br
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.5901305028\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.3288334336.1
Defining polynomial: \(x^{8} - 12 x^{6} - 8 x^{5} + 24 x^{4} + 8 x^{3} - 16 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 4608)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{4} q^{5} -\beta_{7} q^{7} +O(q^{10})\) \( q -\beta_{4} q^{5} -\beta_{7} q^{7} + \beta_{2} q^{11} + ( -2 + \beta_{5} ) q^{13} + ( \beta_{4} + \beta_{6} ) q^{17} + ( \beta_{1} + 2 \beta_{7} ) q^{19} + ( -\beta_{2} + \beta_{3} ) q^{23} + ( 1 - 2 \beta_{5} ) q^{25} + ( \beta_{4} - 2 \beta_{6} ) q^{29} + ( -2 \beta_{1} - \beta_{7} ) q^{31} + ( -\beta_{2} - 2 \beta_{3} ) q^{35} + ( -2 - \beta_{5} ) q^{37} + ( 3 \beta_{4} - \beta_{6} ) q^{41} + ( -\beta_{1} + 2 \beta_{7} ) q^{43} + ( -3 \beta_{2} - \beta_{3} ) q^{47} + ( 1 + 2 \beta_{5} ) q^{49} + ( -\beta_{4} + 2 \beta_{6} ) q^{53} + ( 2 \beta_{1} + 2 \beta_{7} ) q^{55} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{59} + ( -6 + \beta_{5} ) q^{61} + ( 3 \beta_{4} - \beta_{6} ) q^{65} -2 \beta_{1} q^{67} -4 \beta_{3} q^{71} + ( 2 - 6 \beta_{5} ) q^{73} + ( 2 \beta_{4} - 2 \beta_{6} ) q^{77} + \beta_{7} q^{79} + ( \beta_{2} + 4 \beta_{3} ) q^{83} + ( -8 - 2 \beta_{5} ) q^{85} + 4 \beta_{4} q^{89} + ( \beta_{1} + 2 \beta_{7} ) q^{91} + ( 5 \beta_{2} + 3 \beta_{3} ) q^{95} + ( -4 + 8 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 16q^{13} + 8q^{25} - 16q^{37} + 8q^{49} - 48q^{61} + 16q^{73} - 64q^{85} - 32q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 12 x^{6} - 8 x^{5} + 24 x^{4} + 8 x^{3} - 16 x^{2} + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -2 \nu^{6} + 2 \nu^{5} + 18 \nu^{4} + 4 \nu^{3} - 18 \nu^{2} - 8 \nu + 4 \)
\(\beta_{2}\)\(=\)\( -2 \nu^{7} + 22 \nu^{5} + 20 \nu^{4} - 32 \nu^{3} - 30 \nu^{2} + 12 \nu + 8 \)
\(\beta_{3}\)\(=\)\( 2 \nu^{7} + 2 \nu^{6} - 22 \nu^{5} - 40 \nu^{4} + 12 \nu^{3} + 42 \nu^{2} - 8 \)
\(\beta_{4}\)\(=\)\( -2 \nu^{7} - \nu^{6} + 24 \nu^{5} + 27 \nu^{4} - 38 \nu^{3} - 32 \nu^{2} + 16 \nu + 6 \)
\(\beta_{5}\)\(=\)\( 2 \nu^{7} + 2 \nu^{6} - 24 \nu^{5} - 37 \nu^{4} + 28 \nu^{3} + 38 \nu^{2} - 12 \nu - 6 \)
\(\beta_{6}\)\(=\)\( -4 \nu^{7} - \nu^{6} + 44 \nu^{5} + 47 \nu^{4} - 50 \nu^{3} - 42 \nu^{2} + 16 \nu + 4 \)
\(\beta_{7}\)\(=\)\( 4 \nu^{7} + 3 \nu^{6} - 46 \nu^{5} - 66 \nu^{4} + 48 \nu^{3} + 68 \nu^{2} - 14 \nu - 14 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{2} + \beta_{1} + 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(6 \beta_{7} + 4 \beta_{6} - 16 \beta_{5} - 16 \beta_{4} + 7 \beta_{3} + 11 \beta_{2} + 6 \beta_{1} + 12\)\()/4\)
\(\nu^{4}\)\(=\)\(10 \beta_{7} + 5 \beta_{6} - 13 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} + 8 \beta_{2} + 6 \beta_{1} + 24\)
\(\nu^{5}\)\(=\)\((\)\(50 \beta_{7} + 29 \beta_{6} - 94 \beta_{5} - 83 \beta_{4} + 32 \beta_{3} + 63 \beta_{2} + 40 \beta_{1} + 110\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(218 \beta_{7} + 114 \beta_{6} - 322 \beta_{5} - 248 \beta_{4} + 73 \beta_{3} + 207 \beta_{2} + 144 \beta_{1} + 504\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(672 \beta_{7} + 372 \beta_{6} - 1142 \beta_{5} - 956 \beta_{4} + 339 \beta_{3} + 752 \beta_{2} + 497 \beta_{1} + 1512\)\()/2\)

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
−2.08509
0.528036
−2.13875
−1.05636
0.724535
−0.357857
3.49930
0.886177
0 0 0 −2.97127 0 −2.27411 0 0 0
1.2 0 0 0 −2.97127 0 2.27411 0 0 0
1.3 0 0 0 −1.78089 0 −3.29066 0 0 0
1.4 0 0 0 −1.78089 0 3.29066 0 0 0
1.5 0 0 0 1.78089 0 −3.29066 0 0 0
1.6 0 0 0 1.78089 0 3.29066 0 0 0
1.7 0 0 0 2.97127 0 −2.27411 0 0 0
1.8 0 0 0 2.97127 0 2.27411 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9216.2.a.br 8
3.b odd 2 1 inner 9216.2.a.br 8
4.b odd 2 1 inner 9216.2.a.br 8
8.b even 2 1 9216.2.a.bs 8
8.d odd 2 1 9216.2.a.bs 8
12.b even 2 1 inner 9216.2.a.br 8
24.f even 2 1 9216.2.a.bs 8
24.h odd 2 1 9216.2.a.bs 8
32.g even 8 2 4608.2.k.bk 16
32.g even 8 2 4608.2.k.bl yes 16
32.h odd 8 2 4608.2.k.bk 16
32.h odd 8 2 4608.2.k.bl yes 16
96.o even 8 2 4608.2.k.bk 16
96.o even 8 2 4608.2.k.bl yes 16
96.p odd 8 2 4608.2.k.bk 16
96.p odd 8 2 4608.2.k.bl yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.k.bk 16 32.g even 8 2
4608.2.k.bk 16 32.h odd 8 2
4608.2.k.bk 16 96.o even 8 2
4608.2.k.bk 16 96.p odd 8 2
4608.2.k.bl yes 16 32.g even 8 2
4608.2.k.bl yes 16 32.h odd 8 2
4608.2.k.bl yes 16 96.o even 8 2
4608.2.k.bl yes 16 96.p odd 8 2
9216.2.a.br 8 1.a even 1 1 trivial
9216.2.a.br 8 3.b odd 2 1 inner
9216.2.a.br 8 4.b odd 2 1 inner
9216.2.a.br 8 12.b even 2 1 inner
9216.2.a.bs 8 8.b even 2 1
9216.2.a.bs 8 8.d odd 2 1
9216.2.a.bs 8 24.f even 2 1
9216.2.a.bs 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9216))\):

\( T_{5}^{4} - 12 T_{5}^{2} + 28 \)
\( T_{7}^{4} - 16 T_{7}^{2} + 56 \)
\( T_{11}^{4} - 16 T_{11}^{2} + 32 \)
\( T_{13}^{2} + 4 T_{13} + 2 \)
\( T_{17}^{4} - 40 T_{17}^{2} + 112 \)
\( T_{19}^{4} - 64 T_{19}^{2} + 224 \)
\( T_{67}^{4} - 128 T_{67}^{2} + 3584 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 28 - 12 T^{2} + T^{4} )^{2} \)
$7$ \( ( 56 - 16 T^{2} + T^{4} )^{2} \)
$11$ \( ( 32 - 16 T^{2} + T^{4} )^{2} \)
$13$ \( ( 2 + 4 T + T^{2} )^{4} \)
$17$ \( ( 112 - 40 T^{2} + T^{4} )^{2} \)
$19$ \( ( 224 - 64 T^{2} + T^{4} )^{2} \)
$23$ \( ( 128 - 32 T^{2} + T^{4} )^{2} \)
$29$ \( ( 1372 - 76 T^{2} + T^{4} )^{2} \)
$31$ \( ( 2744 - 112 T^{2} + T^{4} )^{2} \)
$37$ \( ( 2 + 4 T + T^{2} )^{4} \)
$41$ \( ( 112 - 104 T^{2} + T^{4} )^{2} \)
$43$ \( ( 224 - 128 T^{2} + T^{4} )^{2} \)
$47$ \( ( 128 - 160 T^{2} + T^{4} )^{2} \)
$53$ \( ( 1372 - 76 T^{2} + T^{4} )^{2} \)
$59$ \( ( 2048 - 128 T^{2} + T^{4} )^{2} \)
$61$ \( ( 34 + 12 T + T^{2} )^{4} \)
$67$ \( ( 3584 - 128 T^{2} + T^{4} )^{2} \)
$71$ \( ( 8192 - 256 T^{2} + T^{4} )^{2} \)
$73$ \( ( -68 - 4 T + T^{2} )^{4} \)
$79$ \( ( 56 - 16 T^{2} + T^{4} )^{2} \)
$83$ \( ( 16928 - 272 T^{2} + T^{4} )^{2} \)
$89$ \( ( 7168 - 192 T^{2} + T^{4} )^{2} \)
$97$ \( ( -112 + 8 T + T^{2} )^{4} \)
show more
show less