# Properties

 Label 9216.2.a.e 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: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 768) 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{5} + 3 \beta q^{7}+O(q^{10})$$ q + 2*b * q^5 + 3*b * q^7 $$q + 2 \beta q^{5} + 3 \beta q^{7} - 4 q^{11} - 3 \beta q^{13} - 6 q^{17} + 2 q^{19} - 2 \beta q^{23} + 3 q^{25} - 4 \beta q^{29} - 3 \beta q^{31} + 12 q^{35} - 3 \beta q^{37} + 10 q^{41} - 6 q^{43} + 2 \beta q^{47} + 11 q^{49} - 4 \beta q^{53} - 8 \beta q^{55} + 3 \beta q^{61} - 12 q^{65} + 4 q^{67} - 2 \beta q^{71} + 16 q^{73} - 12 \beta q^{77} - 3 \beta q^{79} - 16 q^{83} - 12 \beta q^{85} - 14 q^{89} - 18 q^{91} + 4 \beta q^{95} - 4 q^{97} +O(q^{100})$$ q + 2*b * q^5 + 3*b * q^7 - 4 * q^11 - 3*b * q^13 - 6 * q^17 + 2 * q^19 - 2*b * q^23 + 3 * q^25 - 4*b * q^29 - 3*b * q^31 + 12 * q^35 - 3*b * q^37 + 10 * q^41 - 6 * q^43 + 2*b * q^47 + 11 * q^49 - 4*b * q^53 - 8*b * q^55 + 3*b * q^61 - 12 * q^65 + 4 * q^67 - 2*b * q^71 + 16 * q^73 - 12*b * q^77 - 3*b * q^79 - 16 * q^83 - 12*b * q^85 - 14 * q^89 - 18 * q^91 + 4*b * q^95 - 4 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 8 q^{11} - 12 q^{17} + 4 q^{19} + 6 q^{25} + 24 q^{35} + 20 q^{41} - 12 q^{43} + 22 q^{49} - 24 q^{65} + 8 q^{67} + 32 q^{73} - 32 q^{83} - 28 q^{89} - 36 q^{91} - 8 q^{97}+O(q^{100})$$ 2 * q - 8 * q^11 - 12 * q^17 + 4 * q^19 + 6 * q^25 + 24 * q^35 + 20 * q^41 - 12 * q^43 + 22 * q^49 - 24 * q^65 + 8 * q^67 + 32 * q^73 - 32 * q^83 - 28 * q^89 - 36 * q^91 - 8 * q^97

## 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 −2.82843 0 −4.24264 0 0 0
1.2 0 0 0 2.82843 0 4.24264 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.e 2
3.b odd 2 1 3072.2.a.d 2
4.b odd 2 1 9216.2.a.q 2
8.b even 2 1 9216.2.a.q 2
8.d odd 2 1 inner 9216.2.a.e 2
12.b even 2 1 3072.2.a.f 2
24.f even 2 1 3072.2.a.d 2
24.h odd 2 1 3072.2.a.f 2
32.g even 8 2 2304.2.k.a 4
32.g even 8 2 2304.2.k.d 4
32.h odd 8 2 2304.2.k.a 4
32.h odd 8 2 2304.2.k.d 4
48.i odd 4 2 3072.2.d.d 4
48.k even 4 2 3072.2.d.d 4
96.o even 8 2 768.2.j.a 4
96.o even 8 2 768.2.j.d yes 4
96.p odd 8 2 768.2.j.a 4
96.p odd 8 2 768.2.j.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.a 4 96.o even 8 2
768.2.j.a 4 96.p odd 8 2
768.2.j.d yes 4 96.o even 8 2
768.2.j.d yes 4 96.p odd 8 2
2304.2.k.a 4 32.g even 8 2
2304.2.k.a 4 32.h odd 8 2
2304.2.k.d 4 32.g even 8 2
2304.2.k.d 4 32.h odd 8 2
3072.2.a.d 2 3.b odd 2 1
3072.2.a.d 2 24.f even 2 1
3072.2.a.f 2 12.b even 2 1
3072.2.a.f 2 24.h odd 2 1
3072.2.d.d 4 48.i odd 4 2
3072.2.d.d 4 48.k even 4 2
9216.2.a.e 2 1.a even 1 1 trivial
9216.2.a.e 2 8.d odd 2 1 inner
9216.2.a.q 2 4.b odd 2 1
9216.2.a.q 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9216))$$:

 $$T_{5}^{2} - 8$$ T5^2 - 8 $$T_{7}^{2} - 18$$ T7^2 - 18 $$T_{11} + 4$$ T11 + 4 $$T_{13}^{2} - 18$$ T13^2 - 18 $$T_{17} + 6$$ T17 + 6 $$T_{19} - 2$$ T19 - 2 $$T_{67} - 4$$ T67 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 8$$
$7$ $$T^{2} - 18$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} - 18$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T - 2)^{2}$$
$23$ $$T^{2} - 8$$
$29$ $$T^{2} - 32$$
$31$ $$T^{2} - 18$$
$37$ $$T^{2} - 18$$
$41$ $$(T - 10)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} - 8$$
$53$ $$T^{2} - 32$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 18$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} - 8$$
$73$ $$(T - 16)^{2}$$
$79$ $$T^{2} - 18$$
$83$ $$(T + 16)^{2}$$
$89$ $$(T + 14)^{2}$$
$97$ $$(T + 4)^{2}$$