# Properties

 Label 9408.2.a.cg Level $9408$ Weight $2$ Character orbit 9408.a Self dual yes Analytic conductor $75.123$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more about

## 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + q^{9} + O(q^{10})$$ $$q + q^{3} - q^{5} + q^{9} - q^{11} - q^{15} - 8q^{17} + 4q^{19} + 4q^{23} - 4q^{25} + q^{27} + 5q^{29} + 7q^{31} - q^{33} - 8q^{37} + 4q^{41} - 10q^{43} - q^{45} + 6q^{47} - 8q^{51} + q^{53} + q^{55} + 4q^{57} - 9q^{59} + 2q^{61} - 2q^{67} + 4q^{69} + 6q^{71} + 2q^{73} - 4q^{75} - 9q^{79} + q^{81} + 3q^{83} + 8q^{85} + 5q^{87} - 6q^{89} + 7q^{93} - 4q^{95} - q^{97} - q^{99} + O(q^{100})$$

## 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
 0
0 1.00000 0 −1.00000 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9408.2.a.cg 1
4.b odd 2 1 9408.2.a.o 1
7.b odd 2 1 9408.2.a.bb 1
7.c even 3 2 1344.2.q.h 2
8.b even 2 1 4704.2.a.l 1
8.d odd 2 1 4704.2.a.bc 1
28.d even 2 1 9408.2.a.cp 1
28.g odd 6 2 1344.2.q.r 2
56.e even 2 1 4704.2.a.f 1
56.h odd 2 1 4704.2.a.w 1
56.k odd 6 2 672.2.q.b 2
56.p even 6 2 672.2.q.g yes 2
168.s odd 6 2 2016.2.s.j 2
168.v even 6 2 2016.2.s.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.b 2 56.k odd 6 2
672.2.q.g yes 2 56.p even 6 2
1344.2.q.h 2 7.c even 3 2
1344.2.q.r 2 28.g odd 6 2
2016.2.s.i 2 168.v even 6 2
2016.2.s.j 2 168.s odd 6 2
4704.2.a.f 1 56.e even 2 1
4704.2.a.l 1 8.b even 2 1
4704.2.a.w 1 56.h odd 2 1
4704.2.a.bc 1 8.d odd 2 1
9408.2.a.o 1 4.b odd 2 1
9408.2.a.bb 1 7.b odd 2 1
9408.2.a.cg 1 1.a even 1 1 trivial
9408.2.a.cp 1 28.d even 2 1

## Hecke kernels

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

 $$T_{5} + 1$$ $$T_{11} + 1$$ $$T_{13}$$ $$T_{17} + 8$$ $$T_{19} - 4$$ $$T_{31} - 7$$