# Properties

 Label 9408.2.a.bp Level $9408$ Weight $2$ Character orbit 9408.a Self dual yes Analytic conductor $75.123$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - q^{3} + 4q^{5} + q^{9} + O(q^{10})$$ $$q - q^{3} + 4q^{5} + q^{9} + 6q^{11} - 5q^{13} - 4q^{15} + 2q^{17} + q^{19} + 6q^{23} + 11q^{25} - q^{27} + 3q^{31} - 6q^{33} - 3q^{37} + 5q^{39} - 6q^{41} + 5q^{43} + 4q^{45} + 4q^{47} - 2q^{51} + 6q^{53} + 24q^{55} - q^{57} - 6q^{59} + 2q^{61} - 20q^{65} + 7q^{67} - 6q^{69} - 16q^{71} - 3q^{73} - 11q^{75} - 11q^{79} + q^{81} + 12q^{83} + 8q^{85} + 4q^{89} - 3q^{93} + 4q^{95} - 6q^{97} + 6q^{99} + O(q^{100})$$

## 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 4.00000 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.bp 1
4.b odd 2 1 9408.2.a.dd 1
7.b odd 2 1 9408.2.a.bs 1
7.c even 3 2 1344.2.q.l 2
8.b even 2 1 4704.2.a.r 1
8.d odd 2 1 4704.2.a.a 1
28.d even 2 1 9408.2.a.a 1
28.g odd 6 2 1344.2.q.a 2
56.e even 2 1 4704.2.a.bh 1
56.h odd 2 1 4704.2.a.p 1
56.k odd 6 2 672.2.q.j yes 2
56.p even 6 2 672.2.q.e 2
168.s odd 6 2 2016.2.s.b 2
168.v even 6 2 2016.2.s.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.e 2 56.p even 6 2
672.2.q.j yes 2 56.k odd 6 2
1344.2.q.a 2 28.g odd 6 2
1344.2.q.l 2 7.c even 3 2
2016.2.s.a 2 168.v even 6 2
2016.2.s.b 2 168.s odd 6 2
4704.2.a.a 1 8.d odd 2 1
4704.2.a.p 1 56.h odd 2 1
4704.2.a.r 1 8.b even 2 1
4704.2.a.bh 1 56.e even 2 1
9408.2.a.a 1 28.d even 2 1
9408.2.a.bp 1 1.a even 1 1 trivial
9408.2.a.bs 1 7.b odd 2 1
9408.2.a.dd 1 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9408))$$:

 $$T_{5} - 4$$ $$T_{11} - 6$$ $$T_{13} + 5$$ $$T_{17} - 2$$ $$T_{19} - 1$$ $$T_{31} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T$$
$5$ $$1 - 4 T + 5 T^{2}$$
$7$ 1
$11$ $$1 - 6 T + 11 T^{2}$$
$13$ $$1 + 5 T + 13 T^{2}$$
$17$ $$1 - 2 T + 17 T^{2}$$
$19$ $$1 - T + 19 T^{2}$$
$23$ $$1 - 6 T + 23 T^{2}$$
$29$ $$1 + 29 T^{2}$$
$31$ $$1 - 3 T + 31 T^{2}$$
$37$ $$1 + 3 T + 37 T^{2}$$
$41$ $$1 + 6 T + 41 T^{2}$$
$43$ $$1 - 5 T + 43 T^{2}$$
$47$ $$1 - 4 T + 47 T^{2}$$
$53$ $$1 - 6 T + 53 T^{2}$$
$59$ $$1 + 6 T + 59 T^{2}$$
$61$ $$1 - 2 T + 61 T^{2}$$
$67$ $$1 - 7 T + 67 T^{2}$$
$71$ $$1 + 16 T + 71 T^{2}$$
$73$ $$1 + 3 T + 73 T^{2}$$
$79$ $$1 + 11 T + 79 T^{2}$$
$83$ $$1 - 12 T + 83 T^{2}$$
$89$ $$1 - 4 T + 89 T^{2}$$
$97$ $$1 + 6 T + 97 T^{2}$$