Properties

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

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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: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1536)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( - \beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + ( - \beta_{2} - \beta_1) q^{7} + (\beta_{3} - 2) q^{11} - \beta_1 q^{13} + (\beta_{3} + 2) q^{17} + (\beta_{3} - 2) q^{19} - 4 \beta_1 q^{23} + 5 q^{25} + (\beta_{2} - 2 \beta_1) q^{29} + ( - \beta_{2} - 5 \beta_1) q^{31} + ( - \beta_{3} - 10) q^{35} + (2 \beta_{2} - 3 \beta_1) q^{37} + ( - \beta_{3} - 2) q^{41} + (\beta_{3} + 6) q^{43} + (2 \beta_{3} + 5) q^{49} + ( - \beta_{2} + 6 \beta_1) q^{53} + ( - 2 \beta_{2} + 10 \beta_1) q^{55} + 2 \beta_{3} q^{59} + (2 \beta_{2} + 3 \beta_1) q^{61} - \beta_{3} q^{65} + 12 q^{67} + ( - 2 \beta_{2} + 2 \beta_1) q^{71} + ( - 2 \beta_{3} - 6) q^{73} - 8 \beta_1 q^{77} + (\beta_{2} - 3 \beta_1) q^{79} + ( - \beta_{3} + 2) q^{83} + (2 \beta_{2} + 10 \beta_1) q^{85} + 10 q^{89} + (\beta_{3} + 2) q^{91} + ( - 2 \beta_{2} + 10 \beta_1) q^{95} + (2 \beta_{3} - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{11} + 8 q^{17} - 8 q^{19} + 20 q^{25} - 40 q^{35} - 8 q^{41} + 24 q^{43} + 20 q^{49} + 48 q^{67} - 24 q^{73} + 8 q^{83} + 40 q^{89} + 8 q^{91} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 4\beta_1 \) 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
−0.874032
−2.28825
2.28825
0.874032
0 0 0 −3.16228 0 1.74806 0 0 0
1.2 0 0 0 −3.16228 0 4.57649 0 0 0
1.3 0 0 0 3.16228 0 −4.57649 0 0 0
1.4 0 0 0 3.16228 0 −1.74806 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 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.bd 4
3.b odd 2 1 3072.2.a.q 4
4.b odd 2 1 9216.2.a.bj 4
8.b even 2 1 9216.2.a.bj 4
8.d odd 2 1 inner 9216.2.a.bd 4
12.b even 2 1 3072.2.a.k 4
24.f even 2 1 3072.2.a.q 4
24.h odd 2 1 3072.2.a.k 4
32.g even 8 2 4608.2.k.bf 8
32.g even 8 2 4608.2.k.bg 8
32.h odd 8 2 4608.2.k.bf 8
32.h odd 8 2 4608.2.k.bg 8
48.i odd 4 2 3072.2.d.g 8
48.k even 4 2 3072.2.d.g 8
96.o even 8 2 1536.2.j.g 8
96.o even 8 2 1536.2.j.h yes 8
96.p odd 8 2 1536.2.j.g 8
96.p odd 8 2 1536.2.j.h yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.g 8 96.o even 8 2
1536.2.j.g 8 96.p odd 8 2
1536.2.j.h yes 8 96.o even 8 2
1536.2.j.h yes 8 96.p odd 8 2
3072.2.a.k 4 12.b even 2 1
3072.2.a.k 4 24.h odd 2 1
3072.2.a.q 4 3.b odd 2 1
3072.2.a.q 4 24.f even 2 1
3072.2.d.g 8 48.i odd 4 2
3072.2.d.g 8 48.k even 4 2
4608.2.k.bf 8 32.g even 8 2
4608.2.k.bf 8 32.h odd 8 2
4608.2.k.bg 8 32.g even 8 2
4608.2.k.bg 8 32.h odd 8 2
9216.2.a.bd 4 1.a even 1 1 trivial
9216.2.a.bd 4 8.d odd 2 1 inner
9216.2.a.bj 4 4.b odd 2 1
9216.2.a.bj 4 8.b even 2 1

Hecke kernels

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

\( T_{5}^{2} - 10 \) Copy content Toggle raw display
\( T_{7}^{4} - 24T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 16 \) Copy content Toggle raw display
\( T_{13}^{2} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} - 16 \) Copy content Toggle raw display
\( T_{19}^{2} + 4T_{19} - 16 \) Copy content Toggle raw display
\( T_{67} - 12 \) 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
$5$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 24T^{2} + 64 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 36T^{2} + 4 \) Copy content Toggle raw display
$31$ \( T^{4} - 120T^{2} + 1600 \) Copy content Toggle raw display
$37$ \( T^{4} - 116T^{2} + 484 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 164T^{2} + 3844 \) Copy content Toggle raw display
$59$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 116T^{2} + 484 \) Copy content Toggle raw display
$67$ \( (T - 12)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 96T^{2} + 1024 \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T - 44)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 56T^{2} + 64 \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$89$ \( (T - 10)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 64)^{2} \) Copy content Toggle raw display
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