# Properties

 Label 7920.2.a.ch Level $7920$ Weight $2$ Character orbit 7920.a Self dual yes Analytic conductor $63.242$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 55) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + 2 q^{7} +O(q^{10})$$ $$q + q^{5} + 2 q^{7} + q^{11} + ( -4 + \beta ) q^{13} + ( -4 - \beta ) q^{17} -\beta q^{23} + q^{25} + ( -2 + 2 \beta ) q^{29} + 2 q^{35} + ( -2 - 2 \beta ) q^{37} -6 q^{41} + 6 q^{43} + \beta q^{47} -3 q^{49} + ( -6 - 2 \beta ) q^{53} + q^{55} + ( -4 + 2 \beta ) q^{59} + ( 2 - 4 \beta ) q^{61} + ( -4 + \beta ) q^{65} + ( -4 - 3 \beta ) q^{67} + 4 \beta q^{71} + ( -4 + \beta ) q^{73} + 2 q^{77} -4 q^{79} -6 q^{83} + ( -4 - \beta ) q^{85} + ( 2 + 4 \beta ) q^{89} + ( -8 + 2 \beta ) q^{91} + ( -2 + 2 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + 4q^{7} + O(q^{10})$$ $$2q + 2q^{5} + 4q^{7} + 2q^{11} - 8q^{13} - 8q^{17} + 2q^{25} - 4q^{29} + 4q^{35} - 4q^{37} - 12q^{41} + 12q^{43} - 6q^{49} - 12q^{53} + 2q^{55} - 8q^{59} + 4q^{61} - 8q^{65} - 8q^{67} - 8q^{73} + 4q^{77} - 8q^{79} - 12q^{83} - 8q^{85} + 4q^{89} - 16q^{91} - 4q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 0 0 1.00000 0 2.00000 0 0 0
1.2 0 0 0 1.00000 0 2.00000 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.ch 2
3.b odd 2 1 880.2.a.m 2
4.b odd 2 1 495.2.a.b 2
12.b even 2 1 55.2.a.b 2
15.d odd 2 1 4400.2.a.bn 2
15.e even 4 2 4400.2.b.q 4
20.d odd 2 1 2475.2.a.x 2
20.e even 4 2 2475.2.c.l 4
24.f even 2 1 3520.2.a.bn 2
24.h odd 2 1 3520.2.a.bo 2
33.d even 2 1 9680.2.a.bn 2
44.c even 2 1 5445.2.a.y 2
60.h even 2 1 275.2.a.c 2
60.l odd 4 2 275.2.b.d 4
84.h odd 2 1 2695.2.a.f 2
132.d odd 2 1 605.2.a.d 2
132.n odd 10 4 605.2.g.l 8
132.o even 10 4 605.2.g.f 8
156.h even 2 1 9295.2.a.g 2
660.g odd 2 1 3025.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 12.b even 2 1
275.2.a.c 2 60.h even 2 1
275.2.b.d 4 60.l odd 4 2
495.2.a.b 2 4.b odd 2 1
605.2.a.d 2 132.d odd 2 1
605.2.g.f 8 132.o even 10 4
605.2.g.l 8 132.n odd 10 4
880.2.a.m 2 3.b odd 2 1
2475.2.a.x 2 20.d odd 2 1
2475.2.c.l 4 20.e even 4 2
2695.2.a.f 2 84.h odd 2 1
3025.2.a.o 2 660.g odd 2 1
3520.2.a.bn 2 24.f even 2 1
3520.2.a.bo 2 24.h odd 2 1
4400.2.a.bn 2 15.d odd 2 1
4400.2.b.q 4 15.e even 4 2
5445.2.a.y 2 44.c even 2 1
7920.2.a.ch 2 1.a even 1 1 trivial
9295.2.a.g 2 156.h even 2 1
9680.2.a.bn 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7920))$$:

 $$T_{7} - 2$$ $$T_{13}^{2} + 8 T_{13} + 8$$ $$T_{17}^{2} + 8 T_{17} + 8$$ $$T_{19}$$ $$T_{23}^{2} - 8$$ $$T_{29}^{2} + 4 T_{29} - 28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$8 + 8 T + T^{2}$$
$17$ $$8 + 8 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$-8 + T^{2}$$
$29$ $$-28 + 4 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$-28 + 4 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( -6 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$4 + 12 T + T^{2}$$
$59$ $$-16 + 8 T + T^{2}$$
$61$ $$-124 - 4 T + T^{2}$$
$67$ $$-56 + 8 T + T^{2}$$
$71$ $$-128 + T^{2}$$
$73$ $$8 + 8 T + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$-124 - 4 T + T^{2}$$
$97$ $$-28 + 4 T + T^{2}$$