# Properties

 Label 2280.2.a.s Level $2280$ Weight $2$ Character orbit 2280.a Self dual yes Analytic conductor $18.206$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2280.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.2058916609$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} + ( -1 + \beta_{1} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + q^{5} + ( -1 + \beta_{1} ) q^{7} + q^{9} + ( -1 - \beta_{2} ) q^{11} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{13} - q^{15} + ( -2 - \beta_{1} - \beta_{2} ) q^{17} - q^{19} + ( 1 - \beta_{1} ) q^{21} + ( -4 - \beta_{1} - \beta_{2} ) q^{23} + q^{25} - q^{27} + ( 1 - \beta_{2} ) q^{29} + ( -\beta_{1} + 3 \beta_{2} ) q^{31} + ( 1 + \beta_{2} ) q^{33} + ( -1 + \beta_{1} ) q^{35} + ( -3 + \beta_{1} ) q^{37} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{39} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{41} + ( -5 - \beta_{1} - 2 \beta_{2} ) q^{43} + q^{45} + ( -4 + \beta_{1} + \beta_{2} ) q^{47} + ( 3 + \beta_{1} - \beta_{2} ) q^{49} + ( 2 + \beta_{1} + \beta_{2} ) q^{51} + ( 2 + \beta_{1} + \beta_{2} ) q^{53} + ( -1 - \beta_{2} ) q^{55} + q^{57} + ( -6 - 2 \beta_{2} ) q^{59} + ( -4 + 3 \beta_{1} - 3 \beta_{2} ) q^{61} + ( -1 + \beta_{1} ) q^{63} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{65} + 4 \beta_{2} q^{67} + ( 4 + \beta_{1} + \beta_{2} ) q^{69} + ( -2 - 2 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{73} - q^{75} + ( -\beta_{1} + 3 \beta_{2} ) q^{77} + q^{81} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{83} + ( -2 - \beta_{1} - \beta_{2} ) q^{85} + ( -1 + \beta_{2} ) q^{87} + ( -1 + 2 \beta_{1} - 5 \beta_{2} ) q^{89} + ( -8 - \beta_{1} - 5 \beta_{2} ) q^{91} + ( \beta_{1} - 3 \beta_{2} ) q^{93} - q^{95} + ( -3 + \beta_{1} - 4 \beta_{2} ) q^{97} + ( -1 - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} + O(q^{10})$$ $$3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 3 q^{15} - 6 q^{17} - 3 q^{19} + 2 q^{21} - 12 q^{23} + 3 q^{25} - 3 q^{27} + 4 q^{29} - 4 q^{31} + 2 q^{33} - 2 q^{35} - 8 q^{37} - 14 q^{43} + 3 q^{45} - 12 q^{47} + 11 q^{49} + 6 q^{51} + 6 q^{53} - 2 q^{55} + 3 q^{57} - 16 q^{59} - 6 q^{61} - 2 q^{63} - 4 q^{67} + 12 q^{69} - 12 q^{71} - 2 q^{73} - 3 q^{75} - 4 q^{77} + 3 q^{81} - 4 q^{83} - 6 q^{85} - 4 q^{87} + 4 q^{89} - 20 q^{91} + 4 q^{93} - 3 q^{95} - 4 q^{97} - 2 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 3$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 6$$$$)/2$$

## 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
 0.470683 −1.81361 2.34292
0 −1.00000 0 1.00000 0 −3.30777 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −2.52444 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 3.83221 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$19$$ $$1$$

## 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 2280.2.a.s 3
3.b odd 2 1 6840.2.a.bf 3
4.b odd 2 1 4560.2.a.bv 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.s 3 1.a even 1 1 trivial
4560.2.a.bv 3 4.b odd 2 1
6840.2.a.bf 3 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2280))$$:

 $$T_{7}^{3} + 2 T_{7}^{2} - 14 T_{7} - 32$$ $$T_{11}^{3} + 2 T_{11}^{2} - 6 T_{11} - 8$$ $$T_{13}^{3} - 34 T_{13} - 76$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$-32 - 14 T + 2 T^{2} + T^{3}$$
$11$ $$-8 - 6 T + 2 T^{2} + T^{3}$$
$13$ $$-76 - 34 T + T^{3}$$
$17$ $$-64 - 16 T + 6 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$-64 + 20 T + 12 T^{2} + T^{3}$$
$29$ $$4 - 2 T - 4 T^{2} + T^{3}$$
$31$ $$-256 - 60 T + 4 T^{2} + T^{3}$$
$37$ $$-44 + 6 T + 8 T^{2} + T^{3}$$
$41$ $$124 - 58 T + T^{3}$$
$43$ $$-296 + 10 T + 14 T^{2} + T^{3}$$
$47$ $$-32 + 20 T + 12 T^{2} + T^{3}$$
$53$ $$64 - 16 T - 6 T^{2} + T^{3}$$
$59$ $$-32 + 56 T + 16 T^{2} + T^{3}$$
$61$ $$-176 - 144 T + 6 T^{2} + T^{3}$$
$67$ $$64 - 112 T + 4 T^{2} + T^{3}$$
$71$ $$-1088 - 88 T + 12 T^{2} + T^{3}$$
$73$ $$-8 - 68 T + 2 T^{2} + T^{3}$$
$79$ $$T^{3}$$
$83$ $$-688 - 156 T + 4 T^{2} + T^{3}$$
$89$ $$1228 - 186 T - 4 T^{2} + T^{3}$$
$97$ $$124 - 106 T + 4 T^{2} + T^{3}$$