# Properties

 Label 8280.2.a.be Level $8280$ Weight $2$ Character orbit 8280.a Self dual yes Analytic conductor $66.116$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8280.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$66.1161328736$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2760) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} - \beta q^{7} +O(q^{10})$$ q + q^5 - b * q^7 $$q + q^{5} - \beta q^{7} + 2 \beta q^{11} + 2 q^{13} + ( - \beta - 2) q^{17} + (2 \beta - 4) q^{19} + q^{23} + q^{25} + ( - \beta - 6) q^{29} + ( - 3 \beta + 4) q^{31} - \beta q^{35} + (\beta - 2) q^{37} + (\beta - 6) q^{41} + ( - 4 \beta + 4) q^{43} - 8 q^{47} + (\beta - 3) q^{49} + ( - \beta + 2) q^{53} + 2 \beta q^{55} + ( - \beta - 4) q^{59} + (6 \beta - 2) q^{61} + 2 q^{65} - \beta q^{67} + ( - 3 \beta - 4) q^{71} + 2 q^{73} + ( - 2 \beta - 8) q^{77} + 4 \beta q^{79} + ( - 3 \beta + 8) q^{83} + ( - \beta - 2) q^{85} + (2 \beta - 2) q^{89} - 2 \beta q^{91} + (2 \beta - 4) q^{95} - 6 q^{97} +O(q^{100})$$ q + q^5 - b * q^7 + 2*b * q^11 + 2 * q^13 + (-b - 2) * q^17 + (2*b - 4) * q^19 + q^23 + q^25 + (-b - 6) * q^29 + (-3*b + 4) * q^31 - b * q^35 + (b - 2) * q^37 + (b - 6) * q^41 + (-4*b + 4) * q^43 - 8 * q^47 + (b - 3) * q^49 + (-b + 2) * q^53 + 2*b * q^55 + (-b - 4) * q^59 + (6*b - 2) * q^61 + 2 * q^65 - b * q^67 + (-3*b - 4) * q^71 + 2 * q^73 + (-2*b - 8) * q^77 + 4*b * q^79 + (-3*b + 8) * q^83 + (-b - 2) * q^85 + (2*b - 2) * q^89 - 2*b * q^91 + (2*b - 4) * q^95 - 6 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 - q^7 $$2 q + 2 q^{5} - q^{7} + 2 q^{11} + 4 q^{13} - 5 q^{17} - 6 q^{19} + 2 q^{23} + 2 q^{25} - 13 q^{29} + 5 q^{31} - q^{35} - 3 q^{37} - 11 q^{41} + 4 q^{43} - 16 q^{47} - 5 q^{49} + 3 q^{53} + 2 q^{55} - 9 q^{59} + 2 q^{61} + 4 q^{65} - q^{67} - 11 q^{71} + 4 q^{73} - 18 q^{77} + 4 q^{79} + 13 q^{83} - 5 q^{85} - 2 q^{89} - 2 q^{91} - 6 q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 - q^7 + 2 * q^11 + 4 * q^13 - 5 * q^17 - 6 * q^19 + 2 * q^23 + 2 * q^25 - 13 * q^29 + 5 * q^31 - q^35 - 3 * q^37 - 11 * q^41 + 4 * q^43 - 16 * q^47 - 5 * q^49 + 3 * q^53 + 2 * q^55 - 9 * q^59 + 2 * q^61 + 4 * q^65 - q^67 - 11 * q^71 + 4 * q^73 - 18 * q^77 + 4 * q^79 + 13 * q^83 - 5 * q^85 - 2 * q^89 - 2 * q^91 - 6 * q^95 - 12 * 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
 2.56155 −1.56155
0 0 0 1.00000 0 −2.56155 0 0 0
1.2 0 0 0 1.00000 0 1.56155 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$-1$$

## 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 8280.2.a.be 2
3.b odd 2 1 2760.2.a.p 2
12.b even 2 1 5520.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.p 2 3.b odd 2 1
5520.2.a.bh 2 12.b even 2 1
8280.2.a.be 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8280))$$:

 $$T_{7}^{2} + T_{7} - 4$$ T7^2 + T7 - 4 $$T_{11}^{2} - 2T_{11} - 16$$ T11^2 - 2*T11 - 16 $$T_{13} - 2$$ T13 - 2 $$T_{17}^{2} + 5T_{17} + 2$$ T17^2 + 5*T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + T - 4$$
$11$ $$T^{2} - 2T - 16$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} + 5T + 2$$
$19$ $$T^{2} + 6T - 8$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} + 13T + 38$$
$31$ $$T^{2} - 5T - 32$$
$37$ $$T^{2} + 3T - 2$$
$41$ $$T^{2} + 11T + 26$$
$43$ $$T^{2} - 4T - 64$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} - 3T - 2$$
$59$ $$T^{2} + 9T + 16$$
$61$ $$T^{2} - 2T - 152$$
$67$ $$T^{2} + T - 4$$
$71$ $$T^{2} + 11T - 8$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} - 4T - 64$$
$83$ $$T^{2} - 13T + 4$$
$89$ $$T^{2} + 2T - 16$$
$97$ $$(T + 6)^{2}$$