# Properties

 Label 9216.2.a.u 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: $$\Z[a_1, \ldots, a_{5}]$$ 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 + b * q^5 + 4 * q^7 $$q + \beta q^{5} + 4 q^{7} - 2 \beta q^{11} - 3 \beta q^{13} + 2 \beta q^{19} - 4 q^{23} - 3 q^{25} + 3 \beta q^{29} - 8 q^{31} + 4 \beta q^{35} + \beta q^{37} - 8 q^{41} - 2 \beta q^{43} - 8 q^{47} + 9 q^{49} + \beta q^{53} - 4 q^{55} + 6 \beta q^{59} + 3 \beta q^{61} - 6 q^{65} - 2 \beta q^{67} - 12 q^{71} - 2 q^{73} - 8 \beta q^{77} + 10 \beta q^{83} - 14 q^{89} - 12 \beta q^{91} + 4 q^{95} - 16 q^{97} +O(q^{100})$$ q + b * q^5 + 4 * q^7 - 2*b * q^11 - 3*b * q^13 + 2*b * q^19 - 4 * q^23 - 3 * q^25 + 3*b * q^29 - 8 * q^31 + 4*b * q^35 + b * q^37 - 8 * q^41 - 2*b * q^43 - 8 * q^47 + 9 * q^49 + b * q^53 - 4 * q^55 + 6*b * q^59 + 3*b * q^61 - 6 * q^65 - 2*b * q^67 - 12 * q^71 - 2 * q^73 - 8*b * q^77 + 10*b * q^83 - 14 * q^89 - 12*b * q^91 + 4 * q^95 - 16 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{7}+O(q^{10})$$ 2 * q + 8 * q^7 $$2 q + 8 q^{7} - 8 q^{23} - 6 q^{25} - 16 q^{31} - 16 q^{41} - 16 q^{47} + 18 q^{49} - 8 q^{55} - 12 q^{65} - 24 q^{71} - 4 q^{73} - 28 q^{89} + 8 q^{95} - 32 q^{97}+O(q^{100})$$ 2 * q + 8 * q^7 - 8 * q^23 - 6 * q^25 - 16 * q^31 - 16 * q^41 - 16 * q^47 + 18 * q^49 - 8 * q^55 - 12 * q^65 - 24 * q^71 - 4 * q^73 - 28 * q^89 + 8 * q^95 - 32 * 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 −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.u 2
3.b odd 2 1 1024.2.a.f 2
4.b odd 2 1 9216.2.a.b 2
8.b even 2 1 inner 9216.2.a.u 2
8.d odd 2 1 9216.2.a.b 2
12.b even 2 1 1024.2.a.a 2
24.f even 2 1 1024.2.a.a 2
24.h odd 2 1 1024.2.a.f 2
32.g even 8 2 4608.2.k.f 2
32.g even 8 2 4608.2.k.s 2
32.h odd 8 2 4608.2.k.j 2
32.h odd 8 2 4608.2.k.o 2
48.i odd 4 2 1024.2.b.a 2
48.k even 4 2 1024.2.b.f 2
96.o even 8 2 512.2.e.b yes 2
96.o even 8 2 512.2.e.g yes 2
96.p odd 8 2 512.2.e.a 2
96.p odd 8 2 512.2.e.h yes 2

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

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

## Hecke characteristic polynomials

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