Properties

Label 9216.2.a.bb
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 256)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + ( -\beta_{2} - \beta_{3} ) q^{7} +O(q^{10})\) \( q + \beta_{3} q^{5} + ( -\beta_{2} - \beta_{3} ) q^{7} + ( -3 - \beta_{1} ) q^{11} + ( -2 \beta_{2} + \beta_{3} ) q^{13} + 2 \beta_{1} q^{17} + ( 3 + \beta_{1} ) q^{19} + ( 3 \beta_{2} - \beta_{3} ) q^{23} + q^{25} -\beta_{3} q^{29} + 4 \beta_{2} q^{31} + ( -6 - 2 \beta_{1} ) q^{35} + ( 2 \beta_{2} + \beta_{3} ) q^{37} -4 \beta_{1} q^{41} + ( 3 - 3 \beta_{1} ) q^{43} -4 \beta_{3} q^{47} + ( 1 + 4 \beta_{1} ) q^{49} + ( 6 \beta_{2} + \beta_{3} ) q^{53} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{55} + ( -9 + \beta_{1} ) q^{59} + ( -2 \beta_{2} - \beta_{3} ) q^{61} + ( 6 - 4 \beta_{1} ) q^{65} + ( 1 + 3 \beta_{1} ) q^{67} + ( -3 \beta_{2} + \beta_{3} ) q^{71} + ( -6 + 2 \beta_{1} ) q^{73} + ( 6 \beta_{2} + 4 \beta_{3} ) q^{77} + ( 4 \beta_{2} - 4 \beta_{3} ) q^{79} + ( -3 + 3 \beta_{1} ) q^{83} + 6 \beta_{2} q^{85} + ( 6 - 2 \beta_{1} ) q^{89} + ( -2 + 2 \beta_{1} ) q^{91} + ( 3 \beta_{2} + 3 \beta_{3} ) q^{95} + 2 \beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 12q^{11} + 12q^{19} + 4q^{25} - 24q^{35} + 12q^{43} + 4q^{49} - 36q^{59} + 24q^{65} + 4q^{67} - 24q^{73} - 12q^{83} + 24q^{89} - 8q^{91} + O(q^{100}) \)

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
−0.517638
−1.93185
1.93185
0.517638
0 0 0 −2.44949 0 1.03528 0 0 0
1.2 0 0 0 −2.44949 0 3.86370 0 0 0
1.3 0 0 0 2.44949 0 −3.86370 0 0 0
1.4 0 0 0 2.44949 0 −1.03528 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.bb 4
3.b odd 2 1 1024.2.a.j 4
4.b odd 2 1 9216.2.a.bk 4
8.b even 2 1 9216.2.a.bk 4
8.d odd 2 1 inner 9216.2.a.bb 4
12.b even 2 1 1024.2.a.g 4
24.f even 2 1 1024.2.a.j 4
24.h odd 2 1 1024.2.a.g 4
32.g even 8 2 2304.2.k.f 8
32.g even 8 2 2304.2.k.k 8
32.h odd 8 2 2304.2.k.f 8
32.h odd 8 2 2304.2.k.k 8
48.i odd 4 2 1024.2.b.h 8
48.k even 4 2 1024.2.b.h 8
96.o even 8 2 256.2.e.a 8
96.o even 8 2 256.2.e.b yes 8
96.p odd 8 2 256.2.e.a 8
96.p odd 8 2 256.2.e.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
256.2.e.a 8 96.o even 8 2
256.2.e.a 8 96.p odd 8 2
256.2.e.b yes 8 96.o even 8 2
256.2.e.b yes 8 96.p odd 8 2
1024.2.a.g 4 12.b even 2 1
1024.2.a.g 4 24.h odd 2 1
1024.2.a.j 4 3.b odd 2 1
1024.2.a.j 4 24.f even 2 1
1024.2.b.h 8 48.i odd 4 2
1024.2.b.h 8 48.k even 4 2
2304.2.k.f 8 32.g even 8 2
2304.2.k.f 8 32.h odd 8 2
2304.2.k.k 8 32.g even 8 2
2304.2.k.k 8 32.h odd 8 2
9216.2.a.bb 4 1.a even 1 1 trivial
9216.2.a.bb 4 8.d odd 2 1 inner
9216.2.a.bk 4 4.b odd 2 1
9216.2.a.bk 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9216))\):

\( T_{5}^{2} - 6 \)
\( T_{7}^{4} - 16 T_{7}^{2} + 16 \)
\( T_{11}^{2} + 6 T_{11} + 6 \)
\( T_{13}^{4} - 28 T_{13}^{2} + 4 \)
\( T_{17}^{2} - 12 \)
\( T_{19}^{2} - 6 T_{19} + 6 \)
\( T_{67}^{2} - 2 T_{67} - 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -6 + T^{2} )^{2} \)
$7$ \( 16 - 16 T^{2} + T^{4} \)
$11$ \( ( 6 + 6 T + T^{2} )^{2} \)
$13$ \( 4 - 28 T^{2} + T^{4} \)
$17$ \( ( -12 + T^{2} )^{2} \)
$19$ \( ( 6 - 6 T + T^{2} )^{2} \)
$23$ \( 144 - 48 T^{2} + T^{4} \)
$29$ \( ( -6 + T^{2} )^{2} \)
$31$ \( ( -32 + T^{2} )^{2} \)
$37$ \( 4 - 28 T^{2} + T^{4} \)
$41$ \( ( -48 + T^{2} )^{2} \)
$43$ \( ( -18 - 6 T + T^{2} )^{2} \)
$47$ \( ( -96 + T^{2} )^{2} \)
$53$ \( 4356 - 156 T^{2} + T^{4} \)
$59$ \( ( 78 + 18 T + T^{2} )^{2} \)
$61$ \( 4 - 28 T^{2} + T^{4} \)
$67$ \( ( -26 - 2 T + T^{2} )^{2} \)
$71$ \( 144 - 48 T^{2} + T^{4} \)
$73$ \( ( 24 + 12 T + T^{2} )^{2} \)
$79$ \( 4096 - 256 T^{2} + T^{4} \)
$83$ \( ( -18 + 6 T + T^{2} )^{2} \)
$89$ \( ( 24 - 12 T + T^{2} )^{2} \)
$97$ \( ( -12 + T^{2} )^{2} \)
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