# Properties

 Label 8820.2.a.bs Level $8820$ Weight $2$ Character orbit 8820.a Self dual yes Analytic conductor $70.428$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8820.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.4280545828$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{5})$$ Defining polynomial: $$x^{4} - 6 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} +O(q^{10})$$ $$q + q^{5} + ( \beta_{1} + \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} + ( -2 + \beta_{3} ) q^{17} + ( -\beta_{1} - 2 \beta_{2} ) q^{19} + ( 2 \beta_{1} + \beta_{2} ) q^{23} + q^{25} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{29} -5 \beta_{1} q^{31} + ( -2 - 2 \beta_{3} ) q^{37} + ( -8 - \beta_{3} ) q^{41} + ( -6 - \beta_{3} ) q^{43} + ( -4 - \beta_{3} ) q^{47} + ( -6 \beta_{1} - \beta_{2} ) q^{53} + ( \beta_{1} + \beta_{2} ) q^{55} + ( 4 - 2 \beta_{3} ) q^{59} + ( -\beta_{1} + 2 \beta_{2} ) q^{61} + ( \beta_{1} - \beta_{2} ) q^{65} + ( -4 + 2 \beta_{3} ) q^{67} + ( -3 \beta_{1} - \beta_{2} ) q^{71} + ( 5 \beta_{1} - \beta_{2} ) q^{73} + ( 2 + 2 \beta_{3} ) q^{79} + ( -4 + 3 \beta_{3} ) q^{83} + ( -2 + \beta_{3} ) q^{85} -6 q^{89} + ( -\beta_{1} - 2 \beta_{2} ) q^{95} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5} + O(q^{10})$$ $$4 q + 4 q^{5} - 8 q^{17} + 4 q^{25} - 8 q^{37} - 32 q^{41} - 24 q^{43} - 16 q^{47} + 16 q^{59} - 16 q^{67} + 8 q^{79} - 16 q^{83} - 8 q^{85} - 24 q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 4 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 8 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} + 4 \beta_{1}$$

## 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
 −2.28825 −0.874032 0.874032 2.28825
0 0 0 1.00000 0 0 0 0 0
1.2 0 0 0 1.00000 0 0 0 0 0
1.3 0 0 0 1.00000 0 0 0 0 0
1.4 0 0 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8820.2.a.bs yes 4
3.b odd 2 1 8820.2.a.br 4
7.b odd 2 1 8820.2.a.br 4
21.c even 2 1 inner 8820.2.a.bs yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8820.2.a.br 4 3.b odd 2 1
8820.2.a.br 4 7.b odd 2 1
8820.2.a.bs yes 4 1.a even 1 1 trivial
8820.2.a.bs yes 4 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8820))$$:

 $$T_{11}^{4} - 24 T_{11}^{2} + 64$$ $$T_{13}^{4} - 24 T_{13}^{2} + 64$$ $$T_{17}^{2} + 4 T_{17} - 16$$ $$T_{31}^{2} - 50$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$T^{4}$$
$11$ $$64 - 24 T^{2} + T^{4}$$
$13$ $$64 - 24 T^{2} + T^{4}$$
$17$ $$( -16 + 4 T + T^{2} )^{2}$$
$19$ $$1444 - 84 T^{2} + T^{4}$$
$23$ $$4 - 36 T^{2} + T^{4}$$
$29$ $$1024 - 96 T^{2} + T^{4}$$
$31$ $$( -50 + T^{2} )^{2}$$
$37$ $$( -76 + 4 T + T^{2} )^{2}$$
$41$ $$( 44 + 16 T + T^{2} )^{2}$$
$43$ $$( 16 + 12 T + T^{2} )^{2}$$
$47$ $$( -4 + 8 T + T^{2} )^{2}$$
$53$ $$3844 - 164 T^{2} + T^{4}$$
$59$ $$( -64 - 8 T + T^{2} )^{2}$$
$61$ $$1444 - 84 T^{2} + T^{4}$$
$67$ $$( -64 + 8 T + T^{2} )^{2}$$
$71$ $$64 - 56 T^{2} + T^{4}$$
$73$ $$1600 - 120 T^{2} + T^{4}$$
$79$ $$( -76 - 4 T + T^{2} )^{2}$$
$83$ $$( -164 + 8 T + T^{2} )^{2}$$
$89$ $$( 6 + T )^{4}$$
$97$ $$5184 - 216 T^{2} + T^{4}$$