# Properties

 Label 7920.2.a.cg Level $7920$ Weight $2$ Character orbit 7920.a Self dual yes Analytic conductor $63.242$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) 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.

 $$f(q)$$ $$=$$ $$q + q^{5} + ( 2 + \beta ) q^{7} +O(q^{10})$$ $$q + q^{5} + ( 2 + \beta ) q^{7} - q^{11} + 2 \beta q^{13} + ( 4 + \beta ) q^{17} + ( 4 - \beta ) q^{19} -4 q^{23} + q^{25} + ( 2 - \beta ) q^{29} + ( 2 + \beta ) q^{35} + ( 6 - 2 \beta ) q^{37} + ( -2 + \beta ) q^{41} + ( 6 - \beta ) q^{43} -4 q^{47} + ( 5 + 4 \beta ) q^{49} + ( 2 + 4 \beta ) q^{53} - q^{55} -4 q^{59} + ( -6 + 2 \beta ) q^{61} + 2 \beta q^{65} -2 \beta q^{67} + ( 8 + 2 \beta ) q^{71} -4 \beta q^{73} + ( -2 - \beta ) q^{77} + 3 \beta q^{79} -10 q^{83} + ( 4 + \beta ) q^{85} + ( 2 + 2 \beta ) q^{89} + ( 16 + 4 \beta ) q^{91} + ( 4 - \beta ) q^{95} + ( 6 - 2 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + 4q^{7} + O(q^{10})$$ $$2q + 2q^{5} + 4q^{7} - 2q^{11} + 8q^{17} + 8q^{19} - 8q^{23} + 2q^{25} + 4q^{29} + 4q^{35} + 12q^{37} - 4q^{41} + 12q^{43} - 8q^{47} + 10q^{49} + 4q^{53} - 2q^{55} - 8q^{59} - 12q^{61} + 16q^{71} - 4q^{77} - 20q^{83} + 8q^{85} + 4q^{89} + 32q^{91} + 8q^{95} + 12q^{97} + O(q^{100})$$

## 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 −0.828427 0 0 0
1.2 0 0 0 1.00000 0 4.82843 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.cg 2
3.b odd 2 1 2640.2.a.bb 2
4.b odd 2 1 495.2.a.d 2
12.b even 2 1 165.2.a.a 2
20.d odd 2 1 2475.2.a.m 2
20.e even 4 2 2475.2.c.m 4
44.c even 2 1 5445.2.a.m 2
60.h even 2 1 825.2.a.g 2
60.l odd 4 2 825.2.c.e 4
84.h odd 2 1 8085.2.a.ba 2
132.d odd 2 1 1815.2.a.k 2
660.g odd 2 1 9075.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.a 2 12.b even 2 1
495.2.a.d 2 4.b odd 2 1
825.2.a.g 2 60.h even 2 1
825.2.c.e 4 60.l odd 4 2
1815.2.a.k 2 132.d odd 2 1
2475.2.a.m 2 20.d odd 2 1
2475.2.c.m 4 20.e even 4 2
2640.2.a.bb 2 3.b odd 2 1
5445.2.a.m 2 44.c even 2 1
7920.2.a.cg 2 1.a even 1 1 trivial
8085.2.a.ba 2 84.h odd 2 1
9075.2.a.v 2 660.g odd 2 1

## Hecke kernels

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

 $$T_{7}^{2} - 4 T_{7} - 4$$ $$T_{13}^{2} - 32$$ $$T_{17}^{2} - 8 T_{17} + 8$$ $$T_{19}^{2} - 8 T_{19} + 8$$ $$T_{23} + 4$$ $$T_{29}^{2} - 4 T_{29} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$-4 - 4 T + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-32 + T^{2}$$
$17$ $$8 - 8 T + T^{2}$$
$19$ $$8 - 8 T + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$-4 - 4 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$4 - 12 T + T^{2}$$
$41$ $$-4 + 4 T + T^{2}$$
$43$ $$28 - 12 T + T^{2}$$
$47$ $$( 4 + T )^{2}$$
$53$ $$-124 - 4 T + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$4 + 12 T + T^{2}$$
$67$ $$-32 + T^{2}$$
$71$ $$32 - 16 T + T^{2}$$
$73$ $$-128 + T^{2}$$
$79$ $$-72 + T^{2}$$
$83$ $$( 10 + T )^{2}$$
$89$ $$-28 - 4 T + T^{2}$$
$97$ $$4 - 12 T + T^{2}$$