# Properties

 Label 9216.2.a.b Level $9216$ Weight $2$ Character orbit 9216.a Self dual yes Analytic conductor $73.590$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## 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 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} -4 q^{7} +O(q^{10})$$ $$q + \beta q^{5} -4 q^{7} + 2 \beta q^{11} -3 \beta q^{13} -2 \beta q^{19} + 4 q^{23} -3 q^{25} + 3 \beta q^{29} + 8 q^{31} -4 \beta q^{35} + \beta q^{37} -8 q^{41} + 2 \beta q^{43} + 8 q^{47} + 9 q^{49} + \beta q^{53} + 4 q^{55} -6 \beta q^{59} + 3 \beta q^{61} -6 q^{65} + 2 \beta q^{67} + 12 q^{71} -2 q^{73} -8 \beta q^{77} -10 \beta q^{83} -14 q^{89} + 12 \beta q^{91} -4 q^{95} -16 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{7} + O(q^{10})$$ $$2q - 8q^{7} + 8q^{23} - 6q^{25} + 16q^{31} - 16q^{41} + 16q^{47} + 18q^{49} + 8q^{55} - 12q^{65} + 24q^{71} - 4q^{73} - 28q^{89} - 8q^{95} - 32q^{97} + O(q^{100})$$

## 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 −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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b 2
3.b odd 2 1 1024.2.a.a 2
4.b odd 2 1 9216.2.a.u 2
8.b even 2 1 inner 9216.2.a.b 2
8.d odd 2 1 9216.2.a.u 2
12.b even 2 1 1024.2.a.f 2
24.f even 2 1 1024.2.a.f 2
24.h odd 2 1 1024.2.a.a 2
32.g even 8 2 4608.2.k.j 2
32.g even 8 2 4608.2.k.o 2
32.h odd 8 2 4608.2.k.f 2
32.h odd 8 2 4608.2.k.s 2
48.i odd 4 2 1024.2.b.f 2
48.k even 4 2 1024.2.b.a 2
96.o even 8 2 512.2.e.a 2
96.o even 8 2 512.2.e.h yes 2
96.p odd 8 2 512.2.e.b yes 2
96.p odd 8 2 512.2.e.g yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.a 2 96.o even 8 2
512.2.e.b yes 2 96.p odd 8 2
512.2.e.g yes 2 96.p odd 8 2
512.2.e.h yes 2 96.o even 8 2
1024.2.a.a 2 3.b odd 2 1
1024.2.a.a 2 24.h odd 2 1
1024.2.a.f 2 12.b even 2 1
1024.2.a.f 2 24.f even 2 1
1024.2.b.a 2 48.k even 4 2
1024.2.b.f 2 48.i odd 4 2
4608.2.k.f 2 32.h odd 8 2
4608.2.k.j 2 32.g even 8 2
4608.2.k.o 2 32.g even 8 2
4608.2.k.s 2 32.h odd 8 2
9216.2.a.b 2 1.a even 1 1 trivial
9216.2.a.b 2 8.b even 2 1 inner
9216.2.a.u 2 4.b odd 2 1
9216.2.a.u 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9216))$$:

 $$T_{5}^{2} - 2$$ $$T_{7} + 4$$ $$T_{11}^{2} - 8$$ $$T_{13}^{2} - 18$$ $$T_{17}$$ $$T_{19}^{2} - 8$$ $$T_{67}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-2 + T^{2}$$
$7$ $$( 4 + T )^{2}$$
$11$ $$-8 + T^{2}$$
$13$ $$-18 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$-18 + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$-2 + T^{2}$$
$41$ $$( 8 + T )^{2}$$
$43$ $$-8 + T^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$-2 + T^{2}$$
$59$ $$-72 + T^{2}$$
$61$ $$-18 + T^{2}$$
$67$ $$-8 + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$-200 + T^{2}$$
$89$ $$( 14 + T )^{2}$$
$97$ $$( 16 + T )^{2}$$