Properties

Label 9216.2.a.f
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} +O(q^{10})\) \( q + \beta q^{5} -4 q^{11} + \beta q^{13} + 4 q^{19} + 4 \beta q^{23} -3 q^{25} -5 \beta q^{29} -4 \beta q^{31} -3 \beta q^{37} + 12 q^{43} -8 \beta q^{47} -7 q^{49} + \beta q^{53} -4 \beta q^{55} + 4 q^{59} -9 \beta q^{61} + 2 q^{65} + 4 q^{67} + 4 \beta q^{71} -10 q^{73} + 12 \beta q^{79} -12 q^{83} -6 q^{89} + 4 \beta q^{95} + 8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 8q^{11} + 8q^{19} - 6q^{25} + 24q^{43} - 14q^{49} + 8q^{59} + 4q^{65} + 8q^{67} - 20q^{73} - 24q^{83} - 12q^{89} + 16q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 0 0 −1.41421 0 0 0 0 0
1.2 0 0 0 1.41421 0 0 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.f 2
3.b odd 2 1 3072.2.a.c 2
4.b odd 2 1 9216.2.a.r 2
8.b even 2 1 9216.2.a.r 2
8.d odd 2 1 inner 9216.2.a.f 2
12.b even 2 1 3072.2.a.e 2
24.f even 2 1 3072.2.a.c 2
24.h odd 2 1 3072.2.a.e 2
32.g even 8 2 4608.2.k.z 4
32.g even 8 2 4608.2.k.ba 4
32.h odd 8 2 4608.2.k.z 4
32.h odd 8 2 4608.2.k.ba 4
48.i odd 4 2 3072.2.d.b 4
48.k even 4 2 3072.2.d.b 4
96.o even 8 2 1536.2.j.a 4
96.o even 8 2 1536.2.j.d yes 4
96.p odd 8 2 1536.2.j.a 4
96.p odd 8 2 1536.2.j.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.a 4 96.o even 8 2
1536.2.j.a 4 96.p odd 8 2
1536.2.j.d yes 4 96.o even 8 2
1536.2.j.d yes 4 96.p odd 8 2
3072.2.a.c 2 3.b odd 2 1
3072.2.a.c 2 24.f even 2 1
3072.2.a.e 2 12.b even 2 1
3072.2.a.e 2 24.h odd 2 1
3072.2.d.b 4 48.i odd 4 2
3072.2.d.b 4 48.k even 4 2
4608.2.k.z 4 32.g even 8 2
4608.2.k.z 4 32.h odd 8 2
4608.2.k.ba 4 32.g even 8 2
4608.2.k.ba 4 32.h odd 8 2
9216.2.a.f 2 1.a even 1 1 trivial
9216.2.a.f 2 8.d odd 2 1 inner
9216.2.a.r 2 4.b odd 2 1
9216.2.a.r 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9216))\):

\( T_{5}^{2} - 2 \)
\( T_{7} \)
\( T_{11} + 4 \)
\( T_{13}^{2} - 2 \)
\( T_{17} \)
\( T_{19} - 4 \)
\( T_{67} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -2 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( -2 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( -32 + T^{2} \)
$29$ \( -50 + T^{2} \)
$31$ \( -32 + T^{2} \)
$37$ \( -18 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -12 + T )^{2} \)
$47$ \( -128 + T^{2} \)
$53$ \( -2 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( -162 + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( -32 + T^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( -288 + T^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( ( -8 + T )^{2} \)
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