Properties

 Label 8280.2.a.bj 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: 3.3.2597.1 Defining polynomial: $$x^{3} - x^{2} - 9 x + 8$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{5} + ( 1 - \beta_{1} ) q^{7} +O(q^{10})$$ $$q - q^{5} + ( 1 - \beta_{1} ) q^{7} + ( -2 - \beta_{1} ) q^{11} -\beta_{2} q^{13} + ( -3 - \beta_{2} ) q^{17} + ( -4 - \beta_{2} ) q^{19} - q^{23} + q^{25} + ( -5 + \beta_{1} + \beta_{2} ) q^{29} + ( -3 + \beta_{1} ) q^{31} + ( -1 + \beta_{1} ) q^{35} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{37} + ( -3 + \beta_{2} ) q^{41} + 8 q^{43} -2 \beta_{2} q^{47} + ( -2 \beta_{1} + \beta_{2} ) q^{49} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{53} + ( 2 + \beta_{1} ) q^{55} + ( 5 + \beta_{1} + \beta_{2} ) q^{59} + ( 4 - \beta_{1} + 2 \beta_{2} ) q^{61} + \beta_{2} q^{65} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{67} + ( 7 + 2 \beta_{1} - \beta_{2} ) q^{71} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{73} + ( 4 + \beta_{1} + \beta_{2} ) q^{77} -4 \beta_{1} q^{79} + ( 5 + \beta_{1} - \beta_{2} ) q^{83} + ( 3 + \beta_{2} ) q^{85} + ( 4 - 4 \beta_{1} ) q^{89} + ( -2 + 3 \beta_{1} ) q^{91} + ( 4 + \beta_{2} ) q^{95} + ( -2 + 3 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} + 2 q^{7} + O(q^{10})$$ $$3 q - 3 q^{5} + 2 q^{7} - 7 q^{11} - q^{13} - 10 q^{17} - 13 q^{19} - 3 q^{23} + 3 q^{25} - 13 q^{29} - 8 q^{31} - 2 q^{35} + 5 q^{37} - 8 q^{41} + 24 q^{43} - 2 q^{47} - q^{49} - q^{53} + 7 q^{55} + 17 q^{59} + 13 q^{61} + q^{65} + 5 q^{67} + 22 q^{71} + 12 q^{73} + 14 q^{77} - 4 q^{79} + 15 q^{83} + 10 q^{85} + 8 q^{89} - 3 q^{91} + 13 q^{95} - 3 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 6$$

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
 3.07912 0.878468 −2.95759
0 0 0 −1.00000 0 −2.07912 0 0 0
1.2 0 0 0 −1.00000 0 0.121532 0 0 0
1.3 0 0 0 −1.00000 0 3.95759 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.bj 3
3.b odd 2 1 920.2.a.h 3
12.b even 2 1 1840.2.a.s 3
15.d odd 2 1 4600.2.a.x 3
15.e even 4 2 4600.2.e.p 6
24.f even 2 1 7360.2.a.cc 3
24.h odd 2 1 7360.2.a.by 3
60.h even 2 1 9200.2.a.ce 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.h 3 3.b odd 2 1
1840.2.a.s 3 12.b even 2 1
4600.2.a.x 3 15.d odd 2 1
4600.2.e.p 6 15.e even 4 2
7360.2.a.by 3 24.h odd 2 1
7360.2.a.cc 3 24.f even 2 1
8280.2.a.bj 3 1.a even 1 1 trivial
9200.2.a.ce 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8280))$$:

 $$T_{7}^{3} - 2 T_{7}^{2} - 8 T_{7} + 1$$ $$T_{11}^{3} + 7 T_{11}^{2} + 7 T_{11} - 14$$ $$T_{13}^{3} + T_{13}^{2} - 23 T_{13} - 50$$ $$T_{17}^{3} + 10 T_{17}^{2} + 10 T_{17} - 83$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$1 - 8 T - 2 T^{2} + T^{3}$$
$11$ $$-14 + 7 T + 7 T^{2} + T^{3}$$
$13$ $$-50 - 23 T + T^{2} + T^{3}$$
$17$ $$-83 + 10 T + 10 T^{2} + T^{3}$$
$19$ $$-62 + 33 T + 13 T^{2} + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$-76 + 26 T + 13 T^{2} + T^{3}$$
$31$ $$-1 + 12 T + 8 T^{2} + T^{3}$$
$37$ $$496 - 92 T - 5 T^{2} + T^{3}$$
$41$ $$-1 - 2 T + 8 T^{2} + T^{3}$$
$43$ $$( -8 + T )^{3}$$
$47$ $$-400 - 92 T + 2 T^{2} + T^{3}$$
$53$ $$300 - 100 T + T^{2} + T^{3}$$
$59$ $$-36 + 66 T - 17 T^{2} + T^{3}$$
$61$ $$720 - 51 T - 13 T^{2} + T^{3}$$
$67$ $$496 - 92 T - 5 T^{2} + T^{3}$$
$71$ $$225 + 96 T - 22 T^{2} + T^{3}$$
$73$ $$2120 - 176 T - 12 T^{2} + T^{3}$$
$79$ $$-512 - 144 T + 4 T^{2} + T^{3}$$
$83$ $$36 + 40 T - 15 T^{2} + T^{3}$$
$89$ $$64 - 128 T - 8 T^{2} + T^{3}$$
$97$ $$50 - 81 T + 3 T^{2} + T^{3}$$