Properties

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

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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1176) 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 $$q - q^{3} + q^{9} + 4 q^{13} - 4 q^{17} + 4 q^{19} - 4 q^{23} - 5 q^{25} - q^{27} - 2 q^{29} + 8 q^{31} + 6 q^{37} - 4 q^{39} - 12 q^{41} + 4 q^{43} - 8 q^{47} + 4 q^{51} - 6 q^{53} - 4 q^{57} - 12 q^{59} + 4 q^{61} - 4 q^{67} + 4 q^{69} + 12 q^{71} - 8 q^{73} + 5 q^{75} + 16 q^{79} + q^{81} + 4 q^{83} + 2 q^{87} - 4 q^{89} - 8 q^{93} - 16 q^{97}+O(q^{100})$$ q - q^3 + q^9 + 4 * q^13 - 4 * q^17 + 4 * q^19 - 4 * q^23 - 5 * q^25 - q^27 - 2 * q^29 + 8 * q^31 + 6 * q^37 - 4 * q^39 - 12 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^51 - 6 * q^53 - 4 * q^57 - 12 * q^59 + 4 * q^61 - 4 * q^67 + 4 * q^69 + 12 * q^71 - 8 * q^73 + 5 * q^75 + 16 * q^79 + q^81 + 4 * q^83 + 2 * q^87 - 4 * q^89 - 8 * q^93 - 16 * q^97

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.v 1
4.b odd 2 1 9408.2.a.cl 1
7.b odd 2 1 9408.2.a.ck 1
8.b even 2 1 2352.2.a.r 1
8.d odd 2 1 1176.2.a.b 1
24.f even 2 1 3528.2.a.m 1
24.h odd 2 1 7056.2.a.ba 1
28.d even 2 1 9408.2.a.u 1
56.e even 2 1 1176.2.a.h yes 1
56.h odd 2 1 2352.2.a.h 1
56.j odd 6 2 2352.2.q.t 2
56.k odd 6 2 1176.2.q.h 2
56.m even 6 2 1176.2.q.c 2
56.p even 6 2 2352.2.q.h 2
168.e odd 2 1 3528.2.a.n 1
168.i even 2 1 7056.2.a.bc 1
168.v even 6 2 3528.2.s.m 2
168.be odd 6 2 3528.2.s.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.b 1 8.d odd 2 1
1176.2.a.h yes 1 56.e even 2 1
1176.2.q.c 2 56.m even 6 2
1176.2.q.h 2 56.k odd 6 2
2352.2.a.h 1 56.h odd 2 1
2352.2.a.r 1 8.b even 2 1
2352.2.q.h 2 56.p even 6 2
2352.2.q.t 2 56.j odd 6 2
3528.2.a.m 1 24.f even 2 1
3528.2.a.n 1 168.e odd 2 1
3528.2.s.m 2 168.v even 6 2
3528.2.s.n 2 168.be odd 6 2
7056.2.a.ba 1 24.h odd 2 1
7056.2.a.bc 1 168.i even 2 1
9408.2.a.u 1 28.d even 2 1
9408.2.a.v 1 1.a even 1 1 trivial
9408.2.a.ck 1 7.b odd 2 1
9408.2.a.cl 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9408))$$:

 $$T_{5}$$ T5 $$T_{11}$$ T11 $$T_{13} - 4$$ T13 - 4 $$T_{17} + 4$$ T17 + 4 $$T_{19} - 4$$ T19 - 4 $$T_{31} - 8$$ T31 - 8

Hecke characteristic polynomials

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