# Properties

 Label 9216.2.a.v Level $9216$ Weight $2$ Character orbit 9216.a Self dual yes Analytic conductor $73.590$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$0$$ 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 4608) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} + 4 q^{7} +O(q^{10})$$ $$q + \beta q^{5} + 4 q^{7} + 4 \beta q^{11} -3 \beta q^{13} + 6 q^{17} + 4 \beta q^{19} -8 q^{23} -3 q^{25} -3 \beta q^{29} + 4 q^{31} + 4 \beta q^{35} -\beta q^{37} + 2 q^{41} -4 \beta q^{43} + 8 q^{47} + 9 q^{49} + 7 \beta q^{53} + 8 q^{55} -3 \beta q^{61} -6 q^{65} + 8 \beta q^{67} + 10 q^{73} + 16 \beta q^{77} + 12 q^{79} + 4 \beta q^{83} + 6 \beta q^{85} -16 q^{89} -12 \beta q^{91} + 8 q^{95} + 8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{7} + O(q^{10})$$ $$2q + 8q^{7} + 12q^{17} - 16q^{23} - 6q^{25} + 8q^{31} + 4q^{41} + 16q^{47} + 18q^{49} + 16q^{55} - 12q^{65} + 20q^{73} + 24q^{79} - 32q^{89} + 16q^{95} + 16q^{97} + O(q^{100})$$

## 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 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.v 2
3.b odd 2 1 9216.2.a.t 2
4.b odd 2 1 9216.2.a.c 2
8.b even 2 1 inner 9216.2.a.v 2
8.d odd 2 1 9216.2.a.c 2
12.b even 2 1 9216.2.a.a 2
24.f even 2 1 9216.2.a.a 2
24.h odd 2 1 9216.2.a.t 2
32.g even 8 2 4608.2.k.l yes 2
32.g even 8 2 4608.2.k.m yes 2
32.h odd 8 2 4608.2.k.e yes 2
32.h odd 8 2 4608.2.k.t yes 2
96.o even 8 2 4608.2.k.d 2
96.o even 8 2 4608.2.k.u yes 2
96.p odd 8 2 4608.2.k.k yes 2
96.p odd 8 2 4608.2.k.n yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.k.d 2 96.o even 8 2
4608.2.k.e yes 2 32.h odd 8 2
4608.2.k.k yes 2 96.p odd 8 2
4608.2.k.l yes 2 32.g even 8 2
4608.2.k.m yes 2 32.g even 8 2
4608.2.k.n yes 2 96.p odd 8 2
4608.2.k.t yes 2 32.h odd 8 2
4608.2.k.u yes 2 96.o even 8 2
9216.2.a.a 2 12.b even 2 1
9216.2.a.a 2 24.f even 2 1
9216.2.a.c 2 4.b odd 2 1
9216.2.a.c 2 8.d odd 2 1
9216.2.a.t 2 3.b odd 2 1
9216.2.a.t 2 24.h odd 2 1
9216.2.a.v 2 1.a even 1 1 trivial
9216.2.a.v 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9216))$$:

 $$T_{5}^{2} - 2$$ $$T_{7} - 4$$ $$T_{11}^{2} - 32$$ $$T_{13}^{2} - 18$$ $$T_{17} - 6$$ $$T_{19}^{2} - 32$$ $$T_{67}^{2} - 128$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-2 + T^{2}$$
$7$ $$( -4 + T )^{2}$$
$11$ $$-32 + T^{2}$$
$13$ $$-18 + T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$-32 + T^{2}$$
$23$ $$( 8 + T )^{2}$$
$29$ $$-18 + T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$-2 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$-32 + T^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$-98 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$-18 + T^{2}$$
$67$ $$-128 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$( -12 + T )^{2}$$
$83$ $$-32 + T^{2}$$
$89$ $$( 16 + T )^{2}$$
$97$ $$( -8 + T )^{2}$$