Properties

 Label 9408.2.a.x 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

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{9} + O(q^{10})$$ $$q - q^{3} + q^{9} + 2 q^{11} + q^{13} + 2 q^{17} + 5 q^{19} + 6 q^{23} - 5 q^{25} - q^{27} + 8 q^{29} + 3 q^{31} - 2 q^{33} + 9 q^{37} - q^{39} - 2 q^{41} - q^{43} - 8 q^{47} - 2 q^{51} - 6 q^{53} - 5 q^{57} + 6 q^{59} - 2 q^{61} + 5 q^{67} - 6 q^{69} + 4 q^{71} + 11 q^{73} + 5 q^{75} - 5 q^{79} + q^{81} - 8 q^{87} - 12 q^{89} - 3 q^{93} - 18 q^{97} + 2 q^{99} + O(q^{100})$$

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 0 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.x 1
4.b odd 2 1 9408.2.a.ci 1
7.b odd 2 1 9408.2.a.cn 1
7.d odd 6 2 1344.2.q.e 2
8.b even 2 1 4704.2.a.y 1
8.d odd 2 1 4704.2.a.j 1
28.d even 2 1 9408.2.a.t 1
28.f even 6 2 1344.2.q.q 2
56.e even 2 1 4704.2.a.ba 1
56.h odd 2 1 4704.2.a.g 1
56.j odd 6 2 672.2.q.h yes 2
56.m even 6 2 672.2.q.d 2
168.ba even 6 2 2016.2.s.e 2
168.be odd 6 2 2016.2.s.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.d 2 56.m even 6 2
672.2.q.h yes 2 56.j odd 6 2
1344.2.q.e 2 7.d odd 6 2
1344.2.q.q 2 28.f even 6 2
2016.2.s.e 2 168.ba even 6 2
2016.2.s.h 2 168.be odd 6 2
4704.2.a.g 1 56.h odd 2 1
4704.2.a.j 1 8.d odd 2 1
4704.2.a.y 1 8.b even 2 1
4704.2.a.ba 1 56.e even 2 1
9408.2.a.t 1 28.d even 2 1
9408.2.a.x 1 1.a even 1 1 trivial
9408.2.a.ci 1 4.b odd 2 1
9408.2.a.cn 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9408))$$:

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-2 + T$$
$13$ $$-1 + T$$
$17$ $$-2 + T$$
$19$ $$-5 + T$$
$23$ $$-6 + T$$
$29$ $$-8 + T$$
$31$ $$-3 + T$$
$37$ $$-9 + T$$
$41$ $$2 + T$$
$43$ $$1 + T$$
$47$ $$8 + T$$
$53$ $$6 + T$$
$59$ $$-6 + T$$
$61$ $$2 + T$$
$67$ $$-5 + T$$
$71$ $$-4 + T$$
$73$ $$-11 + T$$
$79$ $$5 + T$$
$83$ $$T$$
$89$ $$12 + T$$
$97$ $$18 + T$$