Properties

Label 9216.2.a.bf
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $4$
CM discriminant -3
Inner twists $4$

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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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2304)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 \beta_{1} - \beta_{2} ) q^{7} +O(q^{10})\) \( q + ( -2 \beta_{1} - \beta_{2} ) q^{7} + ( \beta_{1} + 2 \beta_{2} ) q^{13} -\beta_{3} q^{19} -5 q^{25} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{31} + ( 5 \beta_{1} + 2 \beta_{2} ) q^{37} + 3 \beta_{3} q^{43} + ( 7 + 4 \beta_{3} ) q^{49} + ( 7 \beta_{1} - 2 \beta_{2} ) q^{61} -16 q^{67} -4 \beta_{3} q^{73} + ( 2 \beta_{1} + 5 \beta_{2} ) q^{79} + ( -16 - 5 \beta_{3} ) q^{91} -4 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 20q^{25} + 28q^{49} - 64q^{67} - 64q^{91} + O(q^{100}) \)

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
1.93185
−0.517638
0.517638
−1.93185
0 0 0 0 0 −5.27792 0 0 0
1.2 0 0 0 0 0 −0.378937 0 0 0
1.3 0 0 0 0 0 0.378937 0 0 0
1.4 0 0 0 0 0 5.27792 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
8.d odd 2 1 inner
24.f 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.bf 4
3.b odd 2 1 CM 9216.2.a.bf 4
4.b odd 2 1 9216.2.a.bg 4
8.b even 2 1 9216.2.a.bg 4
8.d odd 2 1 inner 9216.2.a.bf 4
12.b even 2 1 9216.2.a.bg 4
24.f even 2 1 inner 9216.2.a.bf 4
24.h odd 2 1 9216.2.a.bg 4
32.g even 8 2 2304.2.k.g 8
32.g even 8 2 2304.2.k.j yes 8
32.h odd 8 2 2304.2.k.g 8
32.h odd 8 2 2304.2.k.j yes 8
96.o even 8 2 2304.2.k.g 8
96.o even 8 2 2304.2.k.j yes 8
96.p odd 8 2 2304.2.k.g 8
96.p odd 8 2 2304.2.k.j yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2304.2.k.g 8 32.g even 8 2
2304.2.k.g 8 32.h odd 8 2
2304.2.k.g 8 96.o even 8 2
2304.2.k.g 8 96.p odd 8 2
2304.2.k.j yes 8 32.g even 8 2
2304.2.k.j yes 8 32.h odd 8 2
2304.2.k.j yes 8 96.o even 8 2
2304.2.k.j yes 8 96.p odd 8 2
9216.2.a.bf 4 1.a even 1 1 trivial
9216.2.a.bf 4 3.b odd 2 1 CM
9216.2.a.bf 4 8.d odd 2 1 inner
9216.2.a.bf 4 24.f even 2 1 inner
9216.2.a.bg 4 4.b odd 2 1
9216.2.a.bg 4 8.b even 2 1
9216.2.a.bg 4 12.b even 2 1
9216.2.a.bg 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9216))\):

\( T_{5} \)
\( T_{7}^{4} - 28 T_{7}^{2} + 4 \)
\( T_{11} \)
\( T_{13}^{4} - 52 T_{13}^{2} + 484 \)
\( T_{17} \)
\( T_{19}^{2} - 12 \)
\( T_{67} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 4 - 28 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 484 - 52 T^{2} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( ( -12 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( 2116 - 124 T^{2} + T^{4} \)
$37$ \( 676 - 148 T^{2} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( -108 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( 5476 - 244 T^{2} + T^{4} \)
$67$ \( ( 16 + T )^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -192 + T^{2} )^{2} \)
$79$ \( 20164 - 316 T^{2} + T^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( -192 + T^{2} )^{2} \)
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