# Properties

 Label 9792.2.a.l 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 102) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{5}+O(q^{10})$$ q - 2 * q^5 $$q - 2 q^{5} + 4 q^{11} + 2 q^{13} - q^{17} + 4 q^{19} - q^{25} - 10 q^{29} - 8 q^{31} + 2 q^{37} - 10 q^{41} + 12 q^{43} - 7 q^{49} + 6 q^{53} - 8 q^{55} - 12 q^{59} + 10 q^{61} - 4 q^{65} - 12 q^{67} + 10 q^{73} + 8 q^{79} - 4 q^{83} + 2 q^{85} + 6 q^{89} - 8 q^{95} - 14 q^{97}+O(q^{100})$$ q - 2 * q^5 + 4 * q^11 + 2 * q^13 - q^17 + 4 * q^19 - q^25 - 10 * q^29 - 8 * q^31 + 2 * q^37 - 10 * q^41 + 12 * q^43 - 7 * q^49 + 6 * q^53 - 8 * q^55 - 12 * q^59 + 10 * q^61 - 4 * q^65 - 12 * q^67 + 10 * q^73 + 8 * q^79 - 4 * q^83 + 2 * q^85 + 6 * q^89 - 8 * q^95 - 14 * 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 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.l 1
3.b odd 2 1 3264.2.a.bc 1
4.b odd 2 1 9792.2.a.k 1
8.b even 2 1 2448.2.a.p 1
8.d odd 2 1 306.2.a.b 1
12.b even 2 1 3264.2.a.m 1
24.f even 2 1 102.2.a.c 1
24.h odd 2 1 816.2.a.b 1
40.e odd 2 1 7650.2.a.ca 1
120.m even 2 1 2550.2.a.c 1
120.q odd 4 2 2550.2.d.m 2
136.e odd 2 1 5202.2.a.c 1
168.e odd 2 1 4998.2.a.be 1
408.h even 2 1 1734.2.a.j 1
408.q even 4 2 1734.2.b.b 2
408.bd even 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.f even 2 1
306.2.a.b 1 8.d odd 2 1
816.2.a.b 1 24.h odd 2 1
1734.2.a.j 1 408.h even 2 1
1734.2.b.b 2 408.q even 4 2
1734.2.f.e 4 408.bd even 8 4
2448.2.a.p 1 8.b even 2 1
2550.2.a.c 1 120.m even 2 1
2550.2.d.m 2 120.q odd 4 2
3264.2.a.m 1 12.b even 2 1
3264.2.a.bc 1 3.b odd 2 1
4998.2.a.be 1 168.e odd 2 1
5202.2.a.c 1 136.e odd 2 1
7650.2.a.ca 1 40.e odd 2 1
9792.2.a.k 1 4.b odd 2 1
9792.2.a.l 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9792))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{7}$$ T7 $$T_{11} - 4$$ T11 - 4 $$T_{13} - 2$$ T13 - 2 $$T_{19} - 4$$ T19 - 4 $$T_{23}$$ T23

## Hecke characteristic polynomials

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