Properties

 Label 8280.2.a.bn Level $8280$ Weight $2$ Character orbit 8280.a Self dual yes Analytic conductor $66.116$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1436.1 Defining polynomial: $$x^{3} - 11 x - 12$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + ( -1 + \beta_{2} ) q^{7} +O(q^{10})$$ $$q + q^{5} + ( -1 + \beta_{2} ) q^{7} + ( -1 - \beta_{1} - \beta_{2} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} ) q^{13} + ( -2 + \beta_{1} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} ) q^{19} - q^{23} + q^{25} + \beta_{1} q^{29} + ( 1 - \beta_{2} ) q^{31} + ( -1 + \beta_{2} ) q^{35} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{37} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -3 - \beta_{1} + 3 \beta_{2} ) q^{47} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{49} + ( -2 + 3 \beta_{1} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} ) q^{55} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{59} + ( 1 + \beta_{1} + \beta_{2} ) q^{61} + ( -1 + \beta_{1} + \beta_{2} ) q^{65} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{67} + ( -6 + \beta_{1} ) q^{71} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{73} + ( -5 + 3 \beta_{1} + \beta_{2} ) q^{77} -8 q^{79} + ( -8 - \beta_{1} ) q^{83} + ( -2 + \beta_{1} ) q^{85} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 7 - 3 \beta_{1} - 3 \beta_{2} ) q^{91} + ( 1 - \beta_{1} - \beta_{2} ) q^{95} + ( 4 + 4 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{5} - 2 q^{7} + O(q^{10})$$ $$3 q + 3 q^{5} - 2 q^{7} - 4 q^{11} - 2 q^{13} - 6 q^{17} + 2 q^{19} - 3 q^{23} + 3 q^{25} + 2 q^{31} - 2 q^{35} + 8 q^{37} - 2 q^{41} + 10 q^{43} - 6 q^{47} + 5 q^{49} - 6 q^{53} - 4 q^{55} - 8 q^{59} + 4 q^{61} - 2 q^{65} + 8 q^{67} - 18 q^{71} + 8 q^{73} - 14 q^{77} - 24 q^{79} - 24 q^{83} - 6 q^{85} - 4 q^{89} + 18 q^{91} + 2 q^{95} + 12 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 11 x - 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 7$$

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
 −1.28282 3.76644 −2.48361
0 0 0 1.00000 0 −3.78872 0 0 0
1.2 0 0 0 1.00000 0 −1.34683 0 0 0
1.3 0 0 0 1.00000 0 3.13555 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.bn 3
3.b odd 2 1 2760.2.a.s 3
12.b even 2 1 5520.2.a.bw 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.s 3 3.b odd 2 1
5520.2.a.bw 3 12.b even 2 1
8280.2.a.bn 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8280))$$:

 $$T_{7}^{3} + 2 T_{7}^{2} - 11 T_{7} - 16$$ $$T_{11}^{3} + 4 T_{11}^{2} - 10 T_{11} - 36$$ $$T_{13}^{3} + 2 T_{13}^{2} - 14 T_{13} + 8$$ $$T_{17}^{3} + 6 T_{17}^{2} + T_{17} - 26$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$-16 - 11 T + 2 T^{2} + T^{3}$$
$11$ $$-36 - 10 T + 4 T^{2} + T^{3}$$
$13$ $$8 - 14 T + 2 T^{2} + T^{3}$$
$17$ $$-26 + T + 6 T^{2} + T^{3}$$
$19$ $$-8 - 14 T - 2 T^{2} + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$-12 - 11 T + T^{3}$$
$31$ $$16 - 11 T - 2 T^{2} + T^{3}$$
$37$ $$214 - 51 T - 8 T^{2} + T^{3}$$
$41$ $$-82 - 99 T + 2 T^{2} + T^{3}$$
$43$ $$24 - 28 T - 10 T^{2} + T^{3}$$
$47$ $$-936 - 134 T + 6 T^{2} + T^{3}$$
$53$ $$-514 - 87 T + 6 T^{2} + T^{3}$$
$59$ $$72 - 55 T + 8 T^{2} + T^{3}$$
$61$ $$36 - 10 T - 4 T^{2} + T^{3}$$
$67$ $$214 - 51 T - 8 T^{2} + T^{3}$$
$71$ $$138 + 97 T + 18 T^{2} + T^{3}$$
$73$ $$484 - 66 T - 8 T^{2} + T^{3}$$
$79$ $$( 8 + T )^{3}$$
$83$ $$436 + 181 T + 24 T^{2} + T^{3}$$
$89$ $$-576 - 120 T + 4 T^{2} + T^{3}$$
$97$ $$-128 - 128 T - 12 T^{2} + T^{3}$$