# Properties

 Label 7920.2.a.s Level $7920$ Weight $2$ Character orbit 7920.a Self dual yes Analytic conductor $63.242$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7920.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$63.2415184009$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} - 5q^{7} + O(q^{10})$$ $$q + q^{5} - 5q^{7} + q^{11} + 2q^{13} - 3q^{17} + 7q^{19} - 6q^{23} + q^{25} + 3q^{29} + 7q^{31} - 5q^{35} - 7q^{37} - 6q^{41} - 8q^{43} + 6q^{47} + 18q^{49} + 3q^{53} + q^{55} - 6q^{59} - q^{61} + 2q^{65} - 8q^{67} + 3q^{71} + 2q^{73} - 5q^{77} + 10q^{79} - 6q^{83} - 3q^{85} - 9q^{89} - 10q^{91} + 7q^{95} - 4q^{97} + O(q^{100})$$

## 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 1.00000 0 −5.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

## 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 7920.2.a.s 1
3.b odd 2 1 880.2.a.c 1
4.b odd 2 1 990.2.a.l 1
12.b even 2 1 110.2.a.a 1
15.d odd 2 1 4400.2.a.w 1
15.e even 4 2 4400.2.b.g 2
20.d odd 2 1 4950.2.a.a 1
20.e even 4 2 4950.2.c.a 2
24.f even 2 1 3520.2.a.l 1
24.h odd 2 1 3520.2.a.z 1
33.d even 2 1 9680.2.a.j 1
60.h even 2 1 550.2.a.i 1
60.l odd 4 2 550.2.b.b 2
84.h odd 2 1 5390.2.a.h 1
132.d odd 2 1 1210.2.a.k 1
660.g odd 2 1 6050.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.a 1 12.b even 2 1
550.2.a.i 1 60.h even 2 1
550.2.b.b 2 60.l odd 4 2
880.2.a.c 1 3.b odd 2 1
990.2.a.l 1 4.b odd 2 1
1210.2.a.k 1 132.d odd 2 1
3520.2.a.l 1 24.f even 2 1
3520.2.a.z 1 24.h odd 2 1
4400.2.a.w 1 15.d odd 2 1
4400.2.b.g 2 15.e even 4 2
4950.2.a.a 1 20.d odd 2 1
4950.2.c.a 2 20.e even 4 2
5390.2.a.h 1 84.h odd 2 1
6050.2.a.i 1 660.g odd 2 1
7920.2.a.s 1 1.a even 1 1 trivial
9680.2.a.j 1 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7920))$$:

 $$T_{7} + 5$$ $$T_{13} - 2$$ $$T_{17} + 3$$ $$T_{19} - 7$$ $$T_{23} + 6$$ $$T_{29} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$5 + T$$
$11$ $$-1 + T$$
$13$ $$-2 + T$$
$17$ $$3 + T$$
$19$ $$-7 + T$$
$23$ $$6 + T$$
$29$ $$-3 + T$$
$31$ $$-7 + T$$
$37$ $$7 + T$$
$41$ $$6 + T$$
$43$ $$8 + T$$
$47$ $$-6 + T$$
$53$ $$-3 + T$$
$59$ $$6 + T$$
$61$ $$1 + T$$
$67$ $$8 + T$$
$71$ $$-3 + T$$
$73$ $$-2 + T$$
$79$ $$-10 + T$$
$83$ $$6 + T$$
$89$ $$9 + T$$
$97$ $$4 + T$$