Properties

 Label 9792.2.a.k Level $9792$ Weight $2$ Character orbit 9792.a Self dual yes Analytic conductor $78.190$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 102) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{5} + O(q^{10})$$ $$q - 2q^{5} - 4q^{11} + 2q^{13} - q^{17} - 4q^{19} - q^{25} - 10q^{29} + 8q^{31} + 2q^{37} - 10q^{41} - 12q^{43} - 7q^{49} + 6q^{53} + 8q^{55} + 12q^{59} + 10q^{61} - 4q^{65} + 12q^{67} + 10q^{73} - 8q^{79} + 4q^{83} + 2q^{85} + 6q^{89} + 8q^{95} - 14q^{97} + O(q^{100})$$

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 −2.00000 0 0 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.k 1
3.b odd 2 1 3264.2.a.m 1
4.b odd 2 1 9792.2.a.l 1
8.b even 2 1 306.2.a.b 1
8.d odd 2 1 2448.2.a.p 1
12.b even 2 1 3264.2.a.bc 1
24.f even 2 1 816.2.a.b 1
24.h odd 2 1 102.2.a.c 1
40.f even 2 1 7650.2.a.ca 1
120.i odd 2 1 2550.2.a.c 1
120.w even 4 2 2550.2.d.m 2
136.h even 2 1 5202.2.a.c 1
168.i even 2 1 4998.2.a.be 1
408.b odd 2 1 1734.2.a.j 1
408.t odd 4 2 1734.2.b.b 2
408.be odd 8 4 1734.2.f.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.c 1 24.h odd 2 1
306.2.a.b 1 8.b even 2 1
816.2.a.b 1 24.f even 2 1
1734.2.a.j 1 408.b odd 2 1
1734.2.b.b 2 408.t odd 4 2
1734.2.f.e 4 408.be odd 8 4
2448.2.a.p 1 8.d odd 2 1
2550.2.a.c 1 120.i odd 2 1
2550.2.d.m 2 120.w even 4 2
3264.2.a.m 1 3.b odd 2 1
3264.2.a.bc 1 12.b even 2 1
4998.2.a.be 1 168.i even 2 1
5202.2.a.c 1 136.h even 2 1
7650.2.a.ca 1 40.f even 2 1
9792.2.a.k 1 1.a even 1 1 trivial
9792.2.a.l 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9792))$$:

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

Hecke characteristic polynomials

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