Properties

Label 9216.2.a.z
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + ( \beta_{2} - \beta_{3} ) q^{5} + ( -2 - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( \beta_{2} - \beta_{3} ) q^{5} + ( -2 - \beta_{3} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{11} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{13} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{17} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{19} + 4 q^{23} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{25} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{29} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{31} + ( 4 - \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{35} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{37} + ( -3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{41} + ( 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{43} + ( 4 - 4 \beta_{2} ) q^{47} + ( 1 - 2 \beta_{2} + 4 \beta_{3} ) q^{49} + ( -2 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{53} + ( -4 + 2 \beta_{1} ) q^{55} + ( 4 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{59} + ( -6 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{61} + ( 2 + \beta_{1} - 6 \beta_{2} + 5 \beta_{3} ) q^{65} + ( 4 + 4 \beta_{2} + 4 \beta_{3} ) q^{67} + ( 4 + 4 \beta_{2} - 2 \beta_{3} ) q^{71} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{73} + ( -4 - 2 \beta_{1} - 2 \beta_{3} ) q^{77} + ( -6 - 4 \beta_{2} - 3 \beta_{3} ) q^{79} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{83} -2 \beta_{3} q^{85} + ( 2 + 4 \beta_{1} + 4 \beta_{2} ) q^{89} + ( 4 - 5 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{91} + ( 8 - 4 \beta_{1} + 2 \beta_{3} ) q^{95} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{7} + O(q^{10}) \) \( 4q - 8q^{7} - 8q^{13} + 16q^{23} + 4q^{25} - 8q^{31} + 16q^{35} - 8q^{37} + 16q^{47} + 4q^{49} - 16q^{55} + 16q^{59} - 24q^{61} + 8q^{65} + 16q^{67} + 16q^{71} - 8q^{73} - 16q^{77} - 24q^{79} + 8q^{89} + 16q^{91} + 32q^{95} - 16q^{97} + O(q^{100}) \)

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
0.765367
1.84776
−0.765367
−1.84776
0 0 0 −4.02734 0 −4.61313 0 0 0
1.2 0 0 0 0.331821 0 −3.08239 0 0 0
1.3 0 0 0 1.19891 0 0.613126 0 0 0
1.4 0 0 0 2.49661 0 −0.917608 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.z 4
3.b odd 2 1 3072.2.a.p 4
4.b odd 2 1 9216.2.a.bl 4
8.b even 2 1 9216.2.a.ba 4
8.d odd 2 1 9216.2.a.bm 4
12.b even 2 1 3072.2.a.m 4
24.f even 2 1 3072.2.a.s 4
24.h odd 2 1 3072.2.a.j 4
32.g even 8 2 4608.2.k.bc 8
32.g even 8 2 4608.2.k.bj 8
32.h odd 8 2 4608.2.k.be 8
32.h odd 8 2 4608.2.k.bh 8
48.i odd 4 2 3072.2.d.j 8
48.k even 4 2 3072.2.d.e 8
96.o even 8 2 1536.2.j.e 8
96.o even 8 2 1536.2.j.j yes 8
96.p odd 8 2 1536.2.j.f yes 8
96.p odd 8 2 1536.2.j.i yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.e 8 96.o even 8 2
1536.2.j.f yes 8 96.p odd 8 2
1536.2.j.i yes 8 96.p odd 8 2
1536.2.j.j yes 8 96.o even 8 2
3072.2.a.j 4 24.h odd 2 1
3072.2.a.m 4 12.b even 2 1
3072.2.a.p 4 3.b odd 2 1
3072.2.a.s 4 24.f even 2 1
3072.2.d.e 8 48.k even 4 2
3072.2.d.j 8 48.i odd 4 2
4608.2.k.bc 8 32.g even 8 2
4608.2.k.be 8 32.h odd 8 2
4608.2.k.bh 8 32.h odd 8 2
4608.2.k.bj 8 32.g even 8 2
9216.2.a.z 4 1.a even 1 1 trivial
9216.2.a.ba 4 8.b even 2 1
9216.2.a.bl 4 4.b odd 2 1
9216.2.a.bm 4 8.d odd 2 1

Hecke kernels

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

\( T_{5}^{4} - 12 T_{5}^{2} + 16 T_{5} - 4 \)
\( T_{7}^{4} + 8 T_{7}^{3} + 16 T_{7}^{2} - 8 \)
\( T_{11}^{4} - 16 T_{11}^{2} + 32 \)
\( T_{13}^{4} + 8 T_{13}^{3} - 12 T_{13}^{2} - 176 T_{13} - 188 \)
\( T_{17}^{4} - 32 T_{17}^{2} + 64 T_{17} - 32 \)
\( T_{19}^{4} - 32 T_{19}^{2} - 64 T_{19} - 32 \)
\( T_{67}^{4} - 16 T_{67}^{3} - 96 T_{67}^{2} + 2304 T_{67} - 7936 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( -4 + 16 T - 12 T^{2} + T^{4} \)
$7$ \( -8 + 16 T^{2} + 8 T^{3} + T^{4} \)
$11$ \( 32 - 16 T^{2} + T^{4} \)
$13$ \( -188 - 176 T - 12 T^{2} + 8 T^{3} + T^{4} \)
$17$ \( -32 + 64 T - 32 T^{2} + T^{4} \)
$19$ \( -32 - 64 T - 32 T^{2} + T^{4} \)
$23$ \( ( -4 + T )^{4} \)
$29$ \( -68 + 112 T - 44 T^{2} + T^{4} \)
$31$ \( 248 - 128 T - 16 T^{2} + 8 T^{3} + T^{4} \)
$37$ \( 388 - 112 T - 44 T^{2} + 8 T^{3} + T^{4} \)
$41$ \( 992 - 64 T - 96 T^{2} + T^{4} \)
$43$ \( 992 + 64 T - 96 T^{2} + T^{4} \)
$47$ \( ( -16 - 8 T + T^{2} )^{2} \)
$53$ \( 188 + 272 T - 108 T^{2} + T^{4} \)
$59$ \( -4352 + 1280 T - 32 T^{2} - 16 T^{3} + T^{4} \)
$61$ \( 4 + 176 T + 148 T^{2} + 24 T^{3} + T^{4} \)
$67$ \( -7936 + 2304 T - 96 T^{2} - 16 T^{3} + T^{4} \)
$71$ \( -2176 + 768 T - 16 T^{3} + T^{4} \)
$73$ \( 1552 - 32 T - 120 T^{2} + 8 T^{3} + T^{4} \)
$79$ \( -7688 - 1344 T + 80 T^{2} + 24 T^{3} + T^{4} \)
$83$ \( 544 + 128 T - 80 T^{2} + T^{4} \)
$89$ \( 272 - 288 T - 168 T^{2} - 8 T^{3} + T^{4} \)
$97$ \( -256 - 256 T - 32 T^{2} + 16 T^{3} + T^{4} \)
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