Properties

 Label 9792.2.a.y Level $9792$ Weight $2$ Character orbit 9792.a Self dual yes Analytic conductor $78.190$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$9792 = 2^{6} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9792.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$78.1895136592$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 34) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 4q^{7} + O(q^{10})$$ $$q - 4q^{7} + 6q^{11} - 2q^{13} + q^{17} + 4q^{19} - 5q^{25} - 4q^{31} + 4q^{37} - 6q^{41} - 8q^{43} + 9q^{49} - 6q^{53} + 4q^{61} - 8q^{67} + 2q^{73} - 24q^{77} + 8q^{79} + 6q^{89} + 8q^{91} + 14q^{97} + O(q^{100})$$

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 0 0 0 0 −4.00000 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$17$$ $$-1$$

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 9792.2.a.y 1
3.b odd 2 1 1088.2.a.l 1
4.b odd 2 1 9792.2.a.bj 1
8.b even 2 1 306.2.a.a 1
8.d odd 2 1 2448.2.a.k 1
12.b even 2 1 1088.2.a.d 1
24.f even 2 1 272.2.a.d 1
24.h odd 2 1 34.2.a.a 1
40.f even 2 1 7650.2.a.ci 1
120.i odd 2 1 850.2.a.e 1
120.m even 2 1 6800.2.a.b 1
120.w even 4 2 850.2.c.b 2
136.h even 2 1 5202.2.a.d 1
168.i even 2 1 1666.2.a.m 1
264.m even 2 1 4114.2.a.a 1
312.b odd 2 1 5746.2.a.b 1
408.b odd 2 1 578.2.a.a 1
408.h even 2 1 4624.2.a.a 1
408.t odd 4 2 578.2.b.a 2
408.be odd 8 4 578.2.c.e 4
408.bm even 16 8 578.2.d.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.a.a 1 24.h odd 2 1
272.2.a.d 1 24.f even 2 1
306.2.a.a 1 8.b even 2 1
578.2.a.a 1 408.b odd 2 1
578.2.b.a 2 408.t odd 4 2
578.2.c.e 4 408.be odd 8 4
578.2.d.e 8 408.bm even 16 8
850.2.a.e 1 120.i odd 2 1
850.2.c.b 2 120.w even 4 2
1088.2.a.d 1 12.b even 2 1
1088.2.a.l 1 3.b odd 2 1
1666.2.a.m 1 168.i even 2 1
2448.2.a.k 1 8.d odd 2 1
4114.2.a.a 1 264.m even 2 1
4624.2.a.a 1 408.h even 2 1
5202.2.a.d 1 136.h even 2 1
5746.2.a.b 1 312.b odd 2 1
6800.2.a.b 1 120.m even 2 1
7650.2.a.ci 1 40.f even 2 1
9792.2.a.y 1 1.a even 1 1 trivial
9792.2.a.bj 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9792))$$:

 $$T_{5}$$ $$T_{7} + 4$$ $$T_{11} - 6$$ $$T_{13} + 2$$ $$T_{19} - 4$$ $$T_{23}$$

Hecke characteristic polynomials

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