Properties

Label 9216.2.a.d
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $0$
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: \(0\)
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 16)
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.

\(f(q)\) \(=\) \( q + \beta q^{5} -2 q^{7} +O(q^{10})\) \( q + \beta q^{5} -2 q^{7} + \beta q^{11} -\beta q^{13} -2 q^{17} + 3 \beta q^{19} + 6 q^{23} -3 q^{25} + 3 \beta q^{29} -8 q^{31} -2 \beta q^{35} + 3 \beta q^{37} -5 \beta q^{43} + 8 q^{47} -3 q^{49} -5 \beta q^{53} + 2 q^{55} + 3 \beta q^{59} + 9 \beta q^{61} -2 q^{65} + 5 \beta q^{67} + 10 q^{71} + 4 q^{73} -2 \beta q^{77} + \beta q^{83} -2 \beta q^{85} -4 q^{89} + 2 \beta q^{91} + 6 q^{95} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{7} + O(q^{10}) \) \( 2q - 4q^{7} - 4q^{17} + 12q^{23} - 6q^{25} - 16q^{31} + 16q^{47} - 6q^{49} + 4q^{55} - 4q^{65} + 20q^{71} + 8q^{73} - 8q^{89} + 12q^{95} - 4q^{97} + O(q^{100}) \)

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 −2.00000 0 0 0
1.2 0 0 0 1.41421 0 −2.00000 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.d 2
3.b odd 2 1 1024.2.a.b 2
4.b odd 2 1 9216.2.a.s 2
8.b even 2 1 inner 9216.2.a.d 2
8.d odd 2 1 9216.2.a.s 2
12.b even 2 1 1024.2.a.e 2
24.f even 2 1 1024.2.a.e 2
24.h odd 2 1 1024.2.a.b 2
32.g even 8 2 144.2.k.a 2
32.g even 8 2 1152.2.k.b 2
32.h odd 8 2 576.2.k.a 2
32.h odd 8 2 1152.2.k.a 2
48.i odd 4 2 1024.2.b.e 2
48.k even 4 2 1024.2.b.b 2
96.o even 8 2 64.2.e.a 2
96.o even 8 2 128.2.e.a 2
96.p odd 8 2 16.2.e.a 2
96.p odd 8 2 128.2.e.b 2
480.bq odd 8 2 1600.2.q.b 2
480.br even 8 2 400.2.q.b 2
480.bs even 8 2 1600.2.l.a 2
480.bu odd 8 2 400.2.l.c 2
480.ca odd 8 2 1600.2.q.a 2
480.cb even 8 2 400.2.q.a 2
672.bo even 8 2 784.2.m.b 2
672.ce odd 24 4 784.2.x.f 4
672.cl even 24 4 784.2.x.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 96.p odd 8 2
64.2.e.a 2 96.o even 8 2
128.2.e.a 2 96.o even 8 2
128.2.e.b 2 96.p odd 8 2
144.2.k.a 2 32.g even 8 2
400.2.l.c 2 480.bu odd 8 2
400.2.q.a 2 480.cb even 8 2
400.2.q.b 2 480.br even 8 2
576.2.k.a 2 32.h odd 8 2
784.2.m.b 2 672.bo even 8 2
784.2.x.c 4 672.cl even 24 4
784.2.x.f 4 672.ce odd 24 4
1024.2.a.b 2 3.b odd 2 1
1024.2.a.b 2 24.h odd 2 1
1024.2.a.e 2 12.b even 2 1
1024.2.a.e 2 24.f even 2 1
1024.2.b.b 2 48.k even 4 2
1024.2.b.e 2 48.i odd 4 2
1152.2.k.a 2 32.h odd 8 2
1152.2.k.b 2 32.g even 8 2
1600.2.l.a 2 480.bs even 8 2
1600.2.q.a 2 480.ca odd 8 2
1600.2.q.b 2 480.bq odd 8 2
9216.2.a.d 2 1.a even 1 1 trivial
9216.2.a.d 2 8.b even 2 1 inner
9216.2.a.s 2 4.b odd 2 1
9216.2.a.s 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9216))\):

\( T_{5}^{2} - 2 \)
\( T_{7} + 2 \)
\( T_{11}^{2} - 2 \)
\( T_{13}^{2} - 2 \)
\( T_{17} + 2 \)
\( T_{19}^{2} - 18 \)
\( T_{67}^{2} - 50 \)

Hecke characteristic polynomials

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