Properties

Label 9216.2.a.e
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: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 768)
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.

\(f(q)\) \(=\) \( q + 2 \beta q^{5} + 3 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{5} + 3 \beta q^{7} - 4 q^{11} - 3 \beta q^{13} - 6 q^{17} + 2 q^{19} - 2 \beta q^{23} + 3 q^{25} - 4 \beta q^{29} - 3 \beta q^{31} + 12 q^{35} - 3 \beta q^{37} + 10 q^{41} - 6 q^{43} + 2 \beta q^{47} + 11 q^{49} - 4 \beta q^{53} - 8 \beta q^{55} + 3 \beta q^{61} - 12 q^{65} + 4 q^{67} - 2 \beta q^{71} + 16 q^{73} - 12 \beta q^{77} - 3 \beta q^{79} - 16 q^{83} - 12 \beta q^{85} - 14 q^{89} - 18 q^{91} + 4 \beta q^{95} - 4 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{11} - 12 q^{17} + 4 q^{19} + 6 q^{25} + 24 q^{35} + 20 q^{41} - 12 q^{43} + 22 q^{49} - 24 q^{65} + 8 q^{67} + 32 q^{73} - 32 q^{83} - 28 q^{89} - 36 q^{91} - 8 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
0 0 0 −2.82843 0 −4.24264 0 0 0
1.2 0 0 0 2.82843 0 4.24264 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.e 2
3.b odd 2 1 3072.2.a.d 2
4.b odd 2 1 9216.2.a.q 2
8.b even 2 1 9216.2.a.q 2
8.d odd 2 1 inner 9216.2.a.e 2
12.b even 2 1 3072.2.a.f 2
24.f even 2 1 3072.2.a.d 2
24.h odd 2 1 3072.2.a.f 2
32.g even 8 2 2304.2.k.a 4
32.g even 8 2 2304.2.k.d 4
32.h odd 8 2 2304.2.k.a 4
32.h odd 8 2 2304.2.k.d 4
48.i odd 4 2 3072.2.d.d 4
48.k even 4 2 3072.2.d.d 4
96.o even 8 2 768.2.j.a 4
96.o even 8 2 768.2.j.d yes 4
96.p odd 8 2 768.2.j.a 4
96.p odd 8 2 768.2.j.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.a 4 96.o even 8 2
768.2.j.a 4 96.p odd 8 2
768.2.j.d yes 4 96.o even 8 2
768.2.j.d yes 4 96.p odd 8 2
2304.2.k.a 4 32.g even 8 2
2304.2.k.a 4 32.h odd 8 2
2304.2.k.d 4 32.g even 8 2
2304.2.k.d 4 32.h odd 8 2
3072.2.a.d 2 3.b odd 2 1
3072.2.a.d 2 24.f even 2 1
3072.2.a.f 2 12.b even 2 1
3072.2.a.f 2 24.h odd 2 1
3072.2.d.d 4 48.i odd 4 2
3072.2.d.d 4 48.k even 4 2
9216.2.a.e 2 1.a even 1 1 trivial
9216.2.a.e 2 8.d odd 2 1 inner
9216.2.a.q 2 4.b odd 2 1
9216.2.a.q 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9216))\):

\( T_{5}^{2} - 8 \) Copy content Toggle raw display
\( T_{7}^{2} - 18 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 18 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display
\( T_{19} - 2 \) Copy content Toggle raw display
\( T_{67} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 8 \) Copy content Toggle raw display
$7$ \( T^{2} - 18 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 18 \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 32 \) Copy content Toggle raw display
$31$ \( T^{2} - 18 \) Copy content Toggle raw display
$37$ \( T^{2} - 18 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( (T + 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8 \) Copy content Toggle raw display
$53$ \( T^{2} - 32 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 18 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 8 \) Copy content Toggle raw display
$73$ \( (T - 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 18 \) Copy content Toggle raw display
$83$ \( (T + 16)^{2} \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( (T + 4)^{2} \) Copy content Toggle raw display
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