# Properties

 Label 8820.2.a.be Level $8820$ Weight $2$ Character orbit 8820.a Self dual yes Analytic conductor $70.428$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 420) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5}+O(q^{10})$$ q - q^5 $$q - q^{5} + (\beta - 1) q^{11} - \beta q^{13} + (\beta - 1) q^{17} + (2 \beta + 3) q^{19} + ( - \beta + 1) q^{23} + q^{25} + ( - \beta - 5) q^{29} + ( - 2 \beta + 1) q^{31} + (\beta - 2) q^{37} + ( - 3 \beta + 3) q^{41} + ( - 3 \beta + 2) q^{43} - 6 q^{47} + ( - 2 \beta + 2) q^{53} + ( - \beta + 1) q^{55} + ( - 3 \beta - 3) q^{59} + 8 q^{61} + \beta q^{65} + (\beta - 2) q^{67} + ( - \beta - 11) q^{71} + 5 \beta q^{73} + (2 \beta - 3) q^{79} + (3 \beta + 3) q^{83} + ( - \beta + 1) q^{85} + (5 \beta + 1) q^{89} + ( - 2 \beta - 3) q^{95} + 8 q^{97} +O(q^{100})$$ q - q^5 + (b - 1) * q^11 - b * q^13 + (b - 1) * q^17 + (2*b + 3) * q^19 + (-b + 1) * q^23 + q^25 + (-b - 5) * q^29 + (-2*b + 1) * q^31 + (b - 2) * q^37 + (-3*b + 3) * q^41 + (-3*b + 2) * q^43 - 6 * q^47 + (-2*b + 2) * q^53 + (-b + 1) * q^55 + (-3*b - 3) * q^59 + 8 * q^61 + b * q^65 + (b - 2) * q^67 + (-b - 11) * q^71 + 5*b * q^73 + (2*b - 3) * q^79 + (3*b + 3) * q^83 + (-b + 1) * q^85 + (5*b + 1) * q^89 + (-2*b - 3) * q^95 + 8 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^5 $$2 q - 2 q^{5} - 2 q^{11} - 2 q^{17} + 6 q^{19} + 2 q^{23} + 2 q^{25} - 10 q^{29} + 2 q^{31} - 4 q^{37} + 6 q^{41} + 4 q^{43} - 12 q^{47} + 4 q^{53} + 2 q^{55} - 6 q^{59} + 16 q^{61} - 4 q^{67} - 22 q^{71} - 6 q^{79} + 6 q^{83} + 2 q^{85} + 2 q^{89} - 6 q^{95} + 16 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 2 * q^11 - 2 * q^17 + 6 * q^19 + 2 * q^23 + 2 * q^25 - 10 * q^29 + 2 * q^31 - 4 * q^37 + 6 * q^41 + 4 * q^43 - 12 * q^47 + 4 * q^53 + 2 * q^55 - 6 * q^59 + 16 * q^61 - 4 * q^67 - 22 * q^71 - 6 * q^79 + 6 * q^83 + 2 * q^85 + 2 * q^89 - 6 * q^95 + 16 * q^97

## 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
 −2.64575 2.64575
0 0 0 −1.00000 0 0 0 0 0
1.2 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.be 2
3.b odd 2 1 2940.2.a.s 2
7.b odd 2 1 8820.2.a.bj 2
7.c even 3 2 1260.2.s.f 4
21.c even 2 1 2940.2.a.m 2
21.g even 6 2 2940.2.q.t 4
21.h odd 6 2 420.2.q.c 4
84.n even 6 2 1680.2.bg.q 4
105.o odd 6 2 2100.2.q.h 4
105.x even 12 4 2100.2.bc.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.c 4 21.h odd 6 2
1260.2.s.f 4 7.c even 3 2
1680.2.bg.q 4 84.n even 6 2
2100.2.q.h 4 105.o odd 6 2
2100.2.bc.e 8 105.x even 12 4
2940.2.a.m 2 21.c even 2 1
2940.2.a.s 2 3.b odd 2 1
2940.2.q.t 4 21.g even 6 2
8820.2.a.be 2 1.a even 1 1 trivial
8820.2.a.bj 2 7.b odd 2 1

## Hecke kernels

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

 $$T_{11}^{2} + 2T_{11} - 6$$ T11^2 + 2*T11 - 6 $$T_{13}^{2} - 7$$ T13^2 - 7 $$T_{17}^{2} + 2T_{17} - 6$$ T17^2 + 2*T17 - 6 $$T_{31}^{2} - 2T_{31} - 27$$ T31^2 - 2*T31 - 27

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 2T - 6$$
$13$ $$T^{2} - 7$$
$17$ $$T^{2} + 2T - 6$$
$19$ $$T^{2} - 6T - 19$$
$23$ $$T^{2} - 2T - 6$$
$29$ $$T^{2} + 10T + 18$$
$31$ $$T^{2} - 2T - 27$$
$37$ $$T^{2} + 4T - 3$$
$41$ $$T^{2} - 6T - 54$$
$43$ $$T^{2} - 4T - 59$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2} - 4T - 24$$
$59$ $$T^{2} + 6T - 54$$
$61$ $$(T - 8)^{2}$$
$67$ $$T^{2} + 4T - 3$$
$71$ $$T^{2} + 22T + 114$$
$73$ $$T^{2} - 175$$
$79$ $$T^{2} + 6T - 19$$
$83$ $$T^{2} - 6T - 54$$
$89$ $$T^{2} - 2T - 174$$
$97$ $$(T - 8)^{2}$$