Properties

Label 9216.2.a.u
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 512)
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.

\(f(q)\) \(=\) \( q + \beta q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} + 4 q^{7} - 2 \beta q^{11} - 3 \beta q^{13} + 2 \beta q^{19} - 4 q^{23} - 3 q^{25} + 3 \beta q^{29} - 8 q^{31} + 4 \beta q^{35} + \beta q^{37} - 8 q^{41} - 2 \beta q^{43} - 8 q^{47} + 9 q^{49} + \beta q^{53} - 4 q^{55} + 6 \beta q^{59} + 3 \beta q^{61} - 6 q^{65} - 2 \beta q^{67} - 12 q^{71} - 2 q^{73} - 8 \beta q^{77} + 10 \beta q^{83} - 14 q^{89} - 12 \beta q^{91} + 4 q^{95} - 16 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{7} - 8 q^{23} - 6 q^{25} - 16 q^{31} - 16 q^{41} - 16 q^{47} + 18 q^{49} - 8 q^{55} - 12 q^{65} - 24 q^{71} - 4 q^{73} - 28 q^{89} + 8 q^{95} - 32 q^{97}+O(q^{100}) \) 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 −1.41421 0 4.00000 0 0 0
1.2 0 0 0 1.41421 0 4.00000 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.b 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.u 2
3.b odd 2 1 1024.2.a.f 2
4.b odd 2 1 9216.2.a.b 2
8.b even 2 1 inner 9216.2.a.u 2
8.d odd 2 1 9216.2.a.b 2
12.b even 2 1 1024.2.a.a 2
24.f even 2 1 1024.2.a.a 2
24.h odd 2 1 1024.2.a.f 2
32.g even 8 2 4608.2.k.f 2
32.g even 8 2 4608.2.k.s 2
32.h odd 8 2 4608.2.k.j 2
32.h odd 8 2 4608.2.k.o 2
48.i odd 4 2 1024.2.b.a 2
48.k even 4 2 1024.2.b.f 2
96.o even 8 2 512.2.e.b yes 2
96.o even 8 2 512.2.e.g yes 2
96.p odd 8 2 512.2.e.a 2
96.p odd 8 2 512.2.e.h yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.a 2 96.p odd 8 2
512.2.e.b yes 2 96.o even 8 2
512.2.e.g yes 2 96.o even 8 2
512.2.e.h yes 2 96.p odd 8 2
1024.2.a.a 2 12.b even 2 1
1024.2.a.a 2 24.f even 2 1
1024.2.a.f 2 3.b odd 2 1
1024.2.a.f 2 24.h odd 2 1
1024.2.b.a 2 48.i odd 4 2
1024.2.b.f 2 48.k even 4 2
4608.2.k.f 2 32.g even 8 2
4608.2.k.j 2 32.h odd 8 2
4608.2.k.o 2 32.h odd 8 2
4608.2.k.s 2 32.g even 8 2
9216.2.a.b 2 4.b odd 2 1
9216.2.a.b 2 8.d odd 2 1
9216.2.a.u 2 1.a even 1 1 trivial
9216.2.a.u 2 8.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 8 \) Copy content Toggle raw display
\( T_{13}^{2} - 18 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{19}^{2} - 8 \) Copy content Toggle raw display
\( T_{67}^{2} - 8 \) 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
$5$ \( T^{2} - 2 \) Copy content Toggle raw display
$7$ \( (T - 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 8 \) Copy content Toggle raw display
$13$ \( T^{2} - 18 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 8 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 18 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 2 \) Copy content Toggle raw display
$41$ \( (T + 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 8 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 2 \) Copy content Toggle raw display
$59$ \( T^{2} - 72 \) Copy content Toggle raw display
$61$ \( T^{2} - 18 \) Copy content Toggle raw display
$67$ \( T^{2} - 8 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 200 \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( (T + 16)^{2} \) Copy content Toggle raw display
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