# Properties

 Label 9408.2.a.dn Level $9408$ Weight $2$ Character orbit 9408.a Self dual yes Analytic conductor $75.123$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9408 = 2^{6} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9408.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.1232582216$$ 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 4704) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{5} + q^{9} +O(q^{10})$$ $$q - q^{3} + \beta q^{5} + q^{9} -2 \beta q^{11} -3 \beta q^{13} -\beta q^{15} -3 \beta q^{17} + 8 q^{19} -2 \beta q^{23} -3 q^{25} - q^{27} -4 q^{31} + 2 \beta q^{33} + 4 q^{37} + 3 \beta q^{39} -5 \beta q^{41} -4 \beta q^{43} + \beta q^{45} + 4 q^{47} + 3 \beta q^{51} + 6 q^{53} -4 q^{55} -8 q^{57} + 8 q^{59} -3 \beta q^{61} -6 q^{65} + 2 \beta q^{69} + 10 \beta q^{71} + \beta q^{73} + 3 q^{75} -8 \beta q^{79} + q^{81} + 12 q^{83} -6 q^{85} + \beta q^{89} + 4 q^{93} + 8 \beta q^{95} + 9 \beta q^{97} -2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{9} + 16q^{19} - 6q^{25} - 2q^{27} - 8q^{31} + 8q^{37} + 8q^{47} + 12q^{53} - 8q^{55} - 16q^{57} + 16q^{59} - 12q^{65} + 6q^{75} + 2q^{81} + 24q^{83} - 12q^{85} + 8q^{93} + O(q^{100})$$

## 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 −1.00000 0 −1.41421 0 0 0 1.00000 0
1.2 0 −1.00000 0 1.41421 0 0 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9408.2.a.dn 2
4.b odd 2 1 9408.2.a.dy 2
7.b odd 2 1 9408.2.a.dy 2
8.b even 2 1 4704.2.a.bo yes 2
8.d odd 2 1 4704.2.a.bl 2
28.d even 2 1 inner 9408.2.a.dn 2
56.e even 2 1 4704.2.a.bo yes 2
56.h odd 2 1 4704.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4704.2.a.bl 2 8.d odd 2 1
4704.2.a.bl 2 56.h odd 2 1
4704.2.a.bo yes 2 8.b even 2 1
4704.2.a.bo yes 2 56.e even 2 1
9408.2.a.dn 2 1.a even 1 1 trivial
9408.2.a.dn 2 28.d even 2 1 inner
9408.2.a.dy 2 4.b odd 2 1
9408.2.a.dy 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9408))$$:

 $$T_{5}^{2} - 2$$ $$T_{11}^{2} - 8$$ $$T_{13}^{2} - 18$$ $$T_{17}^{2} - 18$$ $$T_{19} - 8$$ $$T_{31} + 4$$