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: \( x^{4} - 6x^{2} - 4x + 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{5} + (\beta_{2} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{5} + (\beta_{2} + 1) q^{7} + (\beta_{3} + \beta_{2} - \beta_1) q^{11} + (\beta_{3} + \beta_{2} - 2) q^{13} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{17} + (\beta_{3} - \beta_{2} - \beta_1) q^{19} + 2 \beta_1 q^{23} + (2 \beta_{3} + 2 \beta_1 + 1) q^{25} + ( - \beta_{3} - 2 \beta_1 + 3) q^{29} + ( - \beta_{2} + 3) q^{31} + (\beta_{3} + \beta_{2} + 3 \beta_1) q^{35} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{37} + ( - \beta_{3} + \beta_{2} - 3 \beta_1) q^{41} + (3 \beta_{3} + \beta_{2} + \beta_1) q^{43} - 2 \beta_1 q^{47} + (2 \beta_{2} - 4 \beta_1 + 1) q^{49} + (\beta_{3} + 2 \beta_1 + 5) q^{53} + (4 \beta_1 + 4) q^{55} - 4 \beta_1 q^{59} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 4) q^{61} + ( - \beta_{3} + \beta_{2} + 5 \beta_1 + 2) q^{65} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{67} + (2 \beta_{2} + 2) q^{71} + (2 \beta_{2} + 4 \beta_1 - 2) q^{73} + (2 \beta_{2} - 2 \beta_1 + 6) q^{77} + ( - 4 \beta_{3} - \beta_{2} - 4 \beta_1 + 3) q^{79} + (3 \beta_{3} - \beta_{2} + 5 \beta_1) q^{83} + (2 \beta_{2} + 2 \beta_1 - 6) q^{85} + (2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 2) q^{89} + (\beta_{3} - \beta_{2} - \beta_1 + 4) q^{91} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{95} + ( - 2 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} + 4 q^{7} - 8 q^{13} + 4 q^{25} + 12 q^{29} + 12 q^{31} - 16 q^{37} + 4 q^{49} + 20 q^{53} + 16 q^{55} - 16 q^{61} + 8 q^{65} + 16 q^{67} + 8 q^{71} - 8 q^{73} + 24 q^{77} + 12 q^{79} - 24 q^{85} + 8 q^{89} + 16 q^{91} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 2\nu^{2} + 4\nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + 2\beta_{2} + 4\beta _1 + 3 \) 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
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9216))\):

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