# Properties

 Label 9216.2.a.f 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 1536) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} +O(q^{10})$$ $$q + \beta q^{5} -4 q^{11} + \beta q^{13} + 4 q^{19} + 4 \beta q^{23} -3 q^{25} -5 \beta q^{29} -4 \beta q^{31} -3 \beta q^{37} + 12 q^{43} -8 \beta q^{47} -7 q^{49} + \beta q^{53} -4 \beta q^{55} + 4 q^{59} -9 \beta q^{61} + 2 q^{65} + 4 q^{67} + 4 \beta q^{71} -10 q^{73} + 12 \beta q^{79} -12 q^{83} -6 q^{89} + 4 \beta q^{95} + 8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 8q^{11} + 8q^{19} - 6q^{25} + 24q^{43} - 14q^{49} + 8q^{59} + 4q^{65} + 8q^{67} - 20q^{73} - 24q^{83} - 12q^{89} + 16q^{97} + O(q^{100})$$

## 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 0 0 0 0
1.2 0 0 0 1.41421 0 0 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.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.f 2
3.b odd 2 1 3072.2.a.c 2
4.b odd 2 1 9216.2.a.r 2
8.b even 2 1 9216.2.a.r 2
8.d odd 2 1 inner 9216.2.a.f 2
12.b even 2 1 3072.2.a.e 2
24.f even 2 1 3072.2.a.c 2
24.h odd 2 1 3072.2.a.e 2
32.g even 8 2 4608.2.k.z 4
32.g even 8 2 4608.2.k.ba 4
32.h odd 8 2 4608.2.k.z 4
32.h odd 8 2 4608.2.k.ba 4
48.i odd 4 2 3072.2.d.b 4
48.k even 4 2 3072.2.d.b 4
96.o even 8 2 1536.2.j.a 4
96.o even 8 2 1536.2.j.d yes 4
96.p odd 8 2 1536.2.j.a 4
96.p odd 8 2 1536.2.j.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.a 4 96.o even 8 2
1536.2.j.a 4 96.p odd 8 2
1536.2.j.d yes 4 96.o even 8 2
1536.2.j.d yes 4 96.p odd 8 2
3072.2.a.c 2 3.b odd 2 1
3072.2.a.c 2 24.f even 2 1
3072.2.a.e 2 12.b even 2 1
3072.2.a.e 2 24.h odd 2 1
3072.2.d.b 4 48.i odd 4 2
3072.2.d.b 4 48.k even 4 2
4608.2.k.z 4 32.g even 8 2
4608.2.k.z 4 32.h odd 8 2
4608.2.k.ba 4 32.g even 8 2
4608.2.k.ba 4 32.h odd 8 2
9216.2.a.f 2 1.a even 1 1 trivial
9216.2.a.f 2 8.d odd 2 1 inner
9216.2.a.r 2 4.b odd 2 1
9216.2.a.r 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9216))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-2 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$-2 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$-32 + T^{2}$$
$29$ $$-50 + T^{2}$$
$31$ $$-32 + T^{2}$$
$37$ $$-18 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -12 + T )^{2}$$
$47$ $$-128 + T^{2}$$
$53$ $$-2 + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$-162 + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$-32 + T^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$-288 + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$( -8 + T )^{2}$$