# Properties

 Label 9216.2.a.g Level $9216$ Weight $2$ Character orbit 9216.a Self dual yes Analytic conductor $73.590$ Analytic rank $1$ Dimension $2$ CM discriminant -4 Inner twists $4$

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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: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 512) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 3 \beta q^{5} +O(q^{10})$$ $$q + 3 \beta q^{5} -\beta q^{13} -8 q^{17} + 13 q^{25} -7 \beta q^{29} -5 \beta q^{37} -8 q^{41} -7 q^{49} -5 \beta q^{53} + \beta q^{61} -6 q^{65} + 6 q^{73} -24 \beta q^{85} + 10 q^{89} + 8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 16q^{17} + 26q^{25} - 16q^{41} - 14q^{49} - 12q^{65} + 12q^{73} + 20q^{89} + 16q^{97} + O(q^{100})$$

## 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 −4.24264 0 0 0 0 0
1.2 0 0 0 4.24264 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
8.b even 2 1 inner
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.g 2
3.b odd 2 1 1024.2.a.d 2
4.b odd 2 1 CM 9216.2.a.g 2
8.b even 2 1 inner 9216.2.a.g 2
8.d odd 2 1 inner 9216.2.a.g 2
12.b even 2 1 1024.2.a.d 2
24.f even 2 1 1024.2.a.d 2
24.h odd 2 1 1024.2.a.d 2
32.g even 8 2 4608.2.k.b 2
32.g even 8 2 4608.2.k.w 2
32.h odd 8 2 4608.2.k.b 2
32.h odd 8 2 4608.2.k.w 2
48.i odd 4 2 1024.2.b.d 2
48.k even 4 2 1024.2.b.d 2
96.o even 8 2 512.2.e.c 2
96.o even 8 2 512.2.e.f yes 2
96.p odd 8 2 512.2.e.c 2
96.p odd 8 2 512.2.e.f yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.c 2 96.o even 8 2
512.2.e.c 2 96.p odd 8 2
512.2.e.f yes 2 96.o even 8 2
512.2.e.f yes 2 96.p odd 8 2
1024.2.a.d 2 3.b odd 2 1
1024.2.a.d 2 12.b even 2 1
1024.2.a.d 2 24.f even 2 1
1024.2.a.d 2 24.h odd 2 1
1024.2.b.d 2 48.i odd 4 2
1024.2.b.d 2 48.k even 4 2
4608.2.k.b 2 32.g even 8 2
4608.2.k.b 2 32.h odd 8 2
4608.2.k.w 2 32.g even 8 2
4608.2.k.w 2 32.h odd 8 2
9216.2.a.g 2 1.a even 1 1 trivial
9216.2.a.g 2 4.b odd 2 1 CM
9216.2.a.g 2 8.b even 2 1 inner
9216.2.a.g 2 8.d odd 2 1 inner

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-18 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$-2 + T^{2}$$
$17$ $$( 8 + T )^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$-98 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$-50 + T^{2}$$
$41$ $$( 8 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$-50 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$-2 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$( -8 + T )^{2}$$
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