Properties

Label 9216.2.a.bo
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 48)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} ) q^{5} + ( 1 + \beta_{2} ) q^{7} +O(q^{10})\) \( q + ( 1 + \beta_{3} ) q^{5} + ( 1 + \beta_{2} ) q^{7} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( -2 + \beta_{2} + \beta_{3} ) q^{13} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + 2 \beta_{1} q^{23} + ( 1 + 2 \beta_{1} + 2 \beta_{3} ) q^{25} + ( 3 - 2 \beta_{1} - \beta_{3} ) q^{29} + ( 3 - \beta_{2} ) q^{31} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{35} + ( -4 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{37} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{41} + ( \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{43} -2 \beta_{1} q^{47} + ( 1 - 4 \beta_{1} + 2 \beta_{2} ) q^{49} + ( 5 + 2 \beta_{1} + \beta_{3} ) q^{53} + ( 4 + 4 \beta_{1} ) q^{55} -4 \beta_{1} q^{59} + ( -4 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{61} + ( 2 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{65} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( 2 + 2 \beta_{2} ) q^{71} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( 3 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{79} + ( 5 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{83} + ( -6 + 2 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 2 - 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{91} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{95} + ( -6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{5} + 4q^{7} - 8q^{13} + 4q^{25} + 12q^{29} + 12q^{31} - 16q^{37} + 4q^{49} + 20q^{53} + 16q^{55} - 16q^{61} + 8q^{65} + 16q^{67} + 8q^{71} - 8q^{73} + 24q^{77} + 12q^{79} - 24q^{85} + 8q^{89} + 16q^{91} + 24q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 6 x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 2 \nu + 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 2 \nu^{2} + 4 \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(3 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 3\)

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
−1.27133
0.334904
−1.74912
2.68554
0 0 0 −1.79793 0 −0.158942 0 0 0
1.2 0 0 0 −0.473626 0 4.55765 0 0 0
1.3 0 0 0 2.47363 0 −2.55765 0 0 0
1.4 0 0 0 3.79793 0 2.15894 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 9216.2.a.bo 4
3.b odd 2 1 3072.2.a.i 4
4.b odd 2 1 9216.2.a.bn 4
8.b even 2 1 9216.2.a.y 4
8.d odd 2 1 9216.2.a.x 4
12.b even 2 1 3072.2.a.o 4
24.f even 2 1 3072.2.a.n 4
24.h odd 2 1 3072.2.a.t 4
32.g even 8 2 144.2.k.b 8
32.g even 8 2 1152.2.k.c 8
32.h odd 8 2 576.2.k.b 8
32.h odd 8 2 1152.2.k.f 8
48.i odd 4 2 3072.2.d.f 8
48.k even 4 2 3072.2.d.i 8
96.o even 8 2 192.2.j.a 8
96.o even 8 2 384.2.j.a 8
96.p odd 8 2 48.2.j.a 8
96.p odd 8 2 384.2.j.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 96.p odd 8 2
144.2.k.b 8 32.g even 8 2
192.2.j.a 8 96.o even 8 2
384.2.j.a 8 96.o even 8 2
384.2.j.b 8 96.p odd 8 2
576.2.k.b 8 32.h odd 8 2
1152.2.k.c 8 32.g even 8 2
1152.2.k.f 8 32.h odd 8 2
3072.2.a.i 4 3.b odd 2 1
3072.2.a.n 4 24.f even 2 1
3072.2.a.o 4 12.b even 2 1
3072.2.a.t 4 24.h odd 2 1
3072.2.d.f 8 48.i odd 4 2
3072.2.d.i 8 48.k even 4 2
9216.2.a.x 4 8.d odd 2 1
9216.2.a.y 4 8.b even 2 1
9216.2.a.bn 4 4.b odd 2 1
9216.2.a.bo 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9216))\):

\( T_{5}^{4} - 4 T_{5}^{3} - 4 T_{5}^{2} + 16 T_{5} + 8 \)
\( T_{7}^{4} - 4 T_{7}^{3} - 8 T_{7}^{2} + 24 T_{7} + 4 \)
\( T_{11}^{4} - 24 T_{11}^{2} + 32 T_{11} + 32 \)
\( T_{13}^{4} + 8 T_{13}^{3} + 4 T_{13}^{2} - 48 T_{13} + 4 \)
\( T_{17}^{4} - 32 T_{17}^{2} + 64 T_{17} + 16 \)
\( T_{19}^{4} - 32 T_{19}^{2} - 64 T_{19} + 16 \)
\( T_{67}^{4} - 16 T_{67}^{3} + 256 T_{67} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 8 + 16 T - 4 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 4 + 24 T - 8 T^{2} - 4 T^{3} + T^{4} \)
$11$ \( 32 + 32 T - 24 T^{2} + T^{4} \)
$13$ \( 4 - 48 T + 4 T^{2} + 8 T^{3} + T^{4} \)
$17$ \( 16 + 64 T - 32 T^{2} + T^{4} \)
$19$ \( 16 - 64 T - 32 T^{2} + T^{4} \)
$23$ \( ( -8 + T^{2} )^{2} \)
$29$ \( -248 + 80 T + 28 T^{2} - 12 T^{3} + T^{4} \)
$31$ \( -28 - 24 T + 40 T^{2} - 12 T^{3} + T^{4} \)
$37$ \( -1052 - 224 T + 52 T^{2} + 16 T^{3} + T^{4} \)
$41$ \( -112 - 192 T - 64 T^{2} + T^{4} \)
$43$ \( -112 - 256 T - 96 T^{2} + T^{4} \)
$47$ \( ( -8 + T^{2} )^{2} \)
$53$ \( 136 - 272 T + 124 T^{2} - 20 T^{3} + T^{4} \)
$59$ \( ( -32 + T^{2} )^{2} \)
$61$ \( -1052 - 224 T + 52 T^{2} + 16 T^{3} + T^{4} \)
$67$ \( 256 + 256 T - 16 T^{3} + T^{4} \)
$71$ \( 64 + 192 T - 32 T^{2} - 8 T^{3} + T^{4} \)
$73$ \( 64 + 64 T - 96 T^{2} + 8 T^{3} + T^{4} \)
$79$ \( -10108 + 2888 T - 168 T^{2} - 12 T^{3} + T^{4} \)
$83$ \( 32 + 160 T - 216 T^{2} + T^{4} \)
$89$ \( -1904 + 1632 T - 200 T^{2} - 8 T^{3} + T^{4} \)
$97$ \( 512 + 768 T - 224 T^{2} + T^{4} \)
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