# Properties

 Label 9408.2.a.cz 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

 $$f(q)$$ $$=$$ $$q + q^{3} + 2q^{5} + q^{9} + O(q^{10})$$ $$q + q^{3} + 2q^{5} + q^{9} + 2q^{13} + 2q^{15} - 2q^{17} - 4q^{19} - q^{25} + q^{27} - 6q^{29} - 6q^{37} + 2q^{39} + 6q^{41} + 8q^{43} + 2q^{45} + 8q^{47} - 2q^{51} - 6q^{53} - 4q^{57} + 12q^{59} + 10q^{61} + 4q^{65} + 16q^{67} + 8q^{71} + 6q^{73} - q^{75} - 8q^{79} + q^{81} + 12q^{83} - 4q^{85} - 6q^{87} + 14q^{89} - 8q^{95} + 6q^{97} + O(q^{100})$$

## 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 2.00000 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

## 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.cz 1
4.b odd 2 1 9408.2.a.bd 1
7.b odd 2 1 1344.2.a.d 1
8.b even 2 1 4704.2.a.c 1
8.d odd 2 1 4704.2.a.v 1
21.c even 2 1 4032.2.a.bi 1
28.d even 2 1 1344.2.a.l 1
56.e even 2 1 672.2.a.d 1
56.h odd 2 1 672.2.a.h yes 1
84.h odd 2 1 4032.2.a.bd 1
112.j even 4 2 5376.2.c.u 2
112.l odd 4 2 5376.2.c.o 2
168.e odd 2 1 2016.2.a.a 1
168.i even 2 1 2016.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.d 1 56.e even 2 1
672.2.a.h yes 1 56.h odd 2 1
1344.2.a.d 1 7.b odd 2 1
1344.2.a.l 1 28.d even 2 1
2016.2.a.a 1 168.e odd 2 1
2016.2.a.b 1 168.i even 2 1
4032.2.a.bd 1 84.h odd 2 1
4032.2.a.bi 1 21.c even 2 1
4704.2.a.c 1 8.b even 2 1
4704.2.a.v 1 8.d odd 2 1
5376.2.c.o 2 112.l odd 4 2
5376.2.c.u 2 112.j even 4 2
9408.2.a.bd 1 4.b odd 2 1
9408.2.a.cz 1 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T$$
$5$ $$1 - 2 T + 5 T^{2}$$
$7$ 1
$11$ $$1 + 11 T^{2}$$
$13$ $$1 - 2 T + 13 T^{2}$$
$17$ $$1 + 2 T + 17 T^{2}$$
$19$ $$1 + 4 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 + 6 T + 29 T^{2}$$
$31$ $$1 + 31 T^{2}$$
$37$ $$1 + 6 T + 37 T^{2}$$
$41$ $$1 - 6 T + 41 T^{2}$$
$43$ $$1 - 8 T + 43 T^{2}$$
$47$ $$1 - 8 T + 47 T^{2}$$
$53$ $$1 + 6 T + 53 T^{2}$$
$59$ $$1 - 12 T + 59 T^{2}$$
$61$ $$1 - 10 T + 61 T^{2}$$
$67$ $$1 - 16 T + 67 T^{2}$$
$71$ $$1 - 8 T + 71 T^{2}$$
$73$ $$1 - 6 T + 73 T^{2}$$
$79$ $$1 + 8 T + 79 T^{2}$$
$83$ $$1 - 12 T + 83 T^{2}$$
$89$ $$1 - 14 T + 89 T^{2}$$
$97$ $$1 - 6 T + 97 T^{2}$$