# Properties

 Label 8280.2.a.bl Level $8280$ Weight $2$ Character orbit 8280.a Self dual yes Analytic conductor $66.116$ Analytic rank $0$ Dimension $3$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) 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.

 $$f(q)$$ $$=$$ $$q - q^{5} + ( - \beta_{2} + 2) q^{7}+O(q^{10})$$ q - q^5 + (-b2 + 2) * q^7 $$q - q^{5} + ( - \beta_{2} + 2) q^{7} + ( - \beta_1 - 1) q^{11} + (\beta_1 + 2) q^{13} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{17} + ( - \beta_{2} - 2 \beta_1 + 2) q^{19} + q^{23} + q^{25} + ( - \beta_{2} + 5 \beta_1) q^{29} + (2 \beta_{2} - \beta_1 + 4) q^{31} + (\beta_{2} - 2) q^{35} + ( - 2 \beta_{2} + 6 \beta_1) q^{37} + ( - \beta_{2} - 4 \beta_1 + 3) q^{41} + ( - 4 \beta_1 + 4) q^{43} + (\beta_{2} - \beta_1) q^{47} + ( - 6 \beta_{2} + \beta_1) q^{49} + (4 \beta_{2} - 2) q^{53} + (\beta_1 + 1) q^{55} + (4 \beta_{2} - 6 \beta_1 - 2) q^{59} + (4 \beta_{2} - 3 \beta_1 - 3) q^{61} + ( - \beta_1 - 2) q^{65} + ( - 6 \beta_{2} + 2 \beta_1 - 4) q^{67} + ( - 5 \beta_{2} - 5) q^{71} + ( - \beta_{2} - 3 \beta_1 + 2) q^{73} + (\beta_{2} - \beta_1 - 1) q^{77} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{79} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{83} + (2 \beta_{2} - 3 \beta_1 - 1) q^{85} + (4 \beta_{2} - 4 \beta_1 + 8) q^{89} + ( - 2 \beta_{2} + \beta_1 + 3) q^{91} + (\beta_{2} + 2 \beta_1 - 2) q^{95} + ( - 2 \beta_{2} - \beta_1 + 1) q^{97}+O(q^{100})$$ q - q^5 + (-b2 + 2) * q^7 + (-b1 - 1) * q^11 + (b1 + 2) * q^13 + (-2*b2 + 3*b1 + 1) * q^17 + (-b2 - 2*b1 + 2) * q^19 + q^23 + q^25 + (-b2 + 5*b1) * q^29 + (2*b2 - b1 + 4) * q^31 + (b2 - 2) * q^35 + (-2*b2 + 6*b1) * q^37 + (-b2 - 4*b1 + 3) * q^41 + (-4*b1 + 4) * q^43 + (b2 - b1) * q^47 + (-6*b2 + b1) * q^49 + (4*b2 - 2) * q^53 + (b1 + 1) * q^55 + (4*b2 - 6*b1 - 2) * q^59 + (4*b2 - 3*b1 - 3) * q^61 + (-b1 - 2) * q^65 + (-6*b2 + 2*b1 - 4) * q^67 + (-5*b2 - 5) * q^71 + (-b2 - 3*b1 + 2) * q^73 + (b2 - b1 - 1) * q^77 + (-2*b2 - 2*b1 + 4) * q^79 + (-2*b2 + 2*b1 - 6) * q^83 + (2*b2 - 3*b1 - 1) * q^85 + (4*b2 - 4*b1 + 8) * q^89 + (-2*b2 + b1 + 3) * q^91 + (b2 + 2*b1 - 2) * q^95 + (-2*b2 - b1 + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} + 7 q^{7}+O(q^{10})$$ 3 * q - 3 * q^5 + 7 * q^7 $$3 q - 3 q^{5} + 7 q^{7} - 3 q^{11} + 6 q^{13} + 5 q^{17} + 7 q^{19} + 3 q^{23} + 3 q^{25} + q^{29} + 10 q^{31} - 7 q^{35} + 2 q^{37} + 10 q^{41} + 12 q^{43} - q^{47} + 6 q^{49} - 10 q^{53} + 3 q^{55} - 10 q^{59} - 13 q^{61} - 6 q^{65} - 6 q^{67} - 10 q^{71} + 7 q^{73} - 4 q^{77} + 14 q^{79} - 16 q^{83} - 5 q^{85} + 20 q^{89} + 11 q^{91} - 7 q^{95} + 5 q^{97}+O(q^{100})$$ 3 * q - 3 * q^5 + 7 * q^7 - 3 * q^11 + 6 * q^13 + 5 * q^17 + 7 * q^19 + 3 * q^23 + 3 * q^25 + q^29 + 10 * q^31 - 7 * q^35 + 2 * q^37 + 10 * q^41 + 12 * q^43 - q^47 + 6 * q^49 - 10 * q^53 + 3 * q^55 - 10 * q^59 - 13 * q^61 - 6 * q^65 - 6 * q^67 - 10 * q^71 + 7 * q^73 - 4 * q^77 + 14 * q^79 - 16 * q^83 - 5 * q^85 + 20 * q^89 + 11 * q^91 - 7 * q^95 + 5 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3

## 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
 2.11491 −1.86081 −0.254102
0 0 0 −1.00000 0 0.527166 0 0 0
1.2 0 0 0 −1.00000 0 1.53740 0 0 0
1.3 0 0 0 −1.00000 0 4.93543 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.bl 3
3.b odd 2 1 920.2.a.i 3
12.b even 2 1 1840.2.a.q 3
15.d odd 2 1 4600.2.a.v 3
15.e even 4 2 4600.2.e.q 6
24.f even 2 1 7360.2.a.cf 3
24.h odd 2 1 7360.2.a.bw 3
60.h even 2 1 9200.2.a.ci 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.i 3 3.b odd 2 1
1840.2.a.q 3 12.b even 2 1
4600.2.a.v 3 15.d odd 2 1
4600.2.e.q 6 15.e even 4 2
7360.2.a.bw 3 24.h odd 2 1
7360.2.a.cf 3 24.f even 2 1
8280.2.a.bl 3 1.a even 1 1 trivial
9200.2.a.ci 3 60.h even 2 1

## Hecke kernels

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

 $$T_{7}^{3} - 7T_{7}^{2} + 11T_{7} - 4$$ T7^3 - 7*T7^2 + 11*T7 - 4 $$T_{11}^{3} + 3T_{11}^{2} - T_{11} - 2$$ T11^3 + 3*T11^2 - T11 - 2 $$T_{13}^{3} - 6T_{13}^{2} + 8T_{13} - 1$$ T13^3 - 6*T13^2 + 8*T13 - 1 $$T_{17}^{3} - 5T_{17}^{2} - 31T_{17} + 148$$ T17^3 - 5*T17^2 - 31*T17 + 148

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} - 7 T^{2} + 11 T - 4$$
$11$ $$T^{3} + 3T^{2} - T - 2$$
$13$ $$T^{3} - 6 T^{2} + 8 T - 1$$
$17$ $$T^{3} - 5 T^{2} - 31 T + 148$$
$19$ $$T^{3} - 7 T^{2} - 11 T + 106$$
$23$ $$(T - 1)^{3}$$
$29$ $$T^{3} - T^{2} - 90 T + 148$$
$31$ $$T^{3} - 10 T^{2} + 14 T + 53$$
$37$ $$T^{3} - 2 T^{2} - 128 T + 512$$
$41$ $$T^{3} - 10 T^{2} - 48 T + 481$$
$43$ $$T^{3} - 12 T^{2} - 16 T + 256$$
$47$ $$T^{3} + T^{2} - 6T - 4$$
$53$ $$T^{3} + 10 T^{2} - 52 T - 8$$
$59$ $$T^{3} + 10 T^{2} - 124 T - 1184$$
$61$ $$T^{3} + 13 T^{2} - 29 T - 214$$
$67$ $$T^{3} + 6 T^{2} - 160 T - 1184$$
$71$ $$T^{3} + 10 T^{2} - 100 T - 875$$
$73$ $$T^{3} - 7 T^{2} - 34 T + 236$$
$79$ $$T^{3} - 14 T^{2} + 16 T + 224$$
$83$ $$T^{3} + 16 T^{2} + 60 T + 32$$
$89$ $$T^{3} - 20 T^{2} + 32 T + 256$$
$97$ $$T^{3} - 5 T^{2} - 23 T + 56$$