# Properties

 Label 8820.2.a.bp Level $8820$ Weight $2$ Character orbit 8820.a Self dual yes Analytic conductor $70.428$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.404.1 Defining polynomial: $$x^{3} - x^{2} - 5 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 1260) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{5} +O(q^{10})$$ $$q + q^{5} + ( -1 + \beta_{2} ) q^{11} + ( 2 - \beta_{2} ) q^{13} + ( \beta_{1} - \beta_{2} ) q^{17} + \beta_{1} q^{19} + ( 2 - \beta_{1} - \beta_{2} ) q^{23} + q^{25} + ( 5 + 2 \beta_{1} - \beta_{2} ) q^{29} + q^{31} + ( -3 + \beta_{1} + \beta_{2} ) q^{37} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{41} + ( -1 - \beta_{1} + \beta_{2} ) q^{43} + ( 4 + 2 \beta_{1} ) q^{47} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -1 + \beta_{2} ) q^{55} + ( -5 - \beta_{2} ) q^{59} + ( -1 - \beta_{1} ) q^{61} + ( 2 - \beta_{2} ) q^{65} + ( -2 + \beta_{2} ) q^{67} + ( 7 - \beta_{2} ) q^{71} + ( 5 + \beta_{1} + \beta_{2} ) q^{73} + ( -6 - 3 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -2 - \beta_{1} + 3 \beta_{2} ) q^{83} + ( \beta_{1} - \beta_{2} ) q^{85} + ( -1 - 2 \beta_{1} + 3 \beta_{2} ) q^{89} + \beta_{1} q^{95} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{5} + O(q^{10})$$ $$3 q + 3 q^{5} - 2 q^{11} + 5 q^{13} - 2 q^{17} - q^{19} + 6 q^{23} + 3 q^{25} + 12 q^{29} + 3 q^{31} - 9 q^{37} + 10 q^{41} - q^{43} + 10 q^{47} + 4 q^{53} - 2 q^{55} - 16 q^{59} - 2 q^{61} + 5 q^{65} - 5 q^{67} + 20 q^{71} + 15 q^{73} - 13 q^{79} - 2 q^{83} - 2 q^{85} + 2 q^{89} - q^{95} + 12 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x - 1$$:

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

## 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
 −1.65544 −0.210756 2.86620
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
 $$n$$: 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

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 8820.2.a.bp 3
3.b odd 2 1 8820.2.a.bo 3
7.b odd 2 1 8820.2.a.bn 3
7.d odd 6 2 1260.2.s.h yes 6
21.c even 2 1 8820.2.a.bq 3
21.g even 6 2 1260.2.s.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.s.g 6 21.g even 6 2
1260.2.s.h yes 6 7.d odd 6 2
8820.2.a.bn 3 7.b odd 2 1
8820.2.a.bo 3 3.b odd 2 1
8820.2.a.bp 3 1.a even 1 1 trivial
8820.2.a.bq 3 21.c even 2 1

## Hecke kernels

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

 $$T_{11}^{3} + 2 T_{11}^{2} - 18 T_{11} + 18$$ $$T_{13}^{3} - 5 T_{13}^{2} - 11 T_{13} - 3$$ $$T_{17}^{3} + 2 T_{17}^{2} - 24 T_{17} + 18$$ $$T_{31} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$T^{3}$$
$11$ $$18 - 18 T + 2 T^{2} + T^{3}$$
$13$ $$-3 - 11 T - 5 T^{2} + T^{3}$$
$17$ $$18 - 24 T + 2 T^{2} + T^{3}$$
$19$ $$-21 - 17 T + T^{2} + T^{3}$$
$23$ $$162 - 36 T - 6 T^{2} + T^{3}$$
$29$ $$378 - 18 T - 12 T^{2} + T^{3}$$
$31$ $$( -1 + T )^{3}$$
$37$ $$-191 - 21 T + 9 T^{2} + T^{3}$$
$41$ $$774 - 78 T - 10 T^{2} + T^{3}$$
$43$ $$-43 - 25 T + T^{2} + T^{3}$$
$47$ $$72 - 36 T - 10 T^{2} + T^{3}$$
$53$ $$-144 - 96 T - 4 T^{2} + T^{3}$$
$59$ $$18 + 66 T + 16 T^{2} + T^{3}$$
$61$ $$4 - 16 T + 2 T^{2} + T^{3}$$
$67$ $$3 - 11 T + 5 T^{2} + T^{3}$$
$71$ $$-198 + 114 T - 20 T^{2} + T^{3}$$
$73$ $$41 + 27 T - 15 T^{2} + T^{3}$$
$79$ $$-1453 - 109 T + 13 T^{2} + T^{3}$$
$83$ $$522 - 156 T + 2 T^{2} + T^{3}$$
$89$ $$126 - 174 T - 2 T^{2} + T^{3}$$
$97$ $$116 - 24 T - 12 T^{2} + T^{3}$$