# Properties

 Label 2280.2.a.t Level $2280$ Weight $2$ Character orbit 2280.a Self dual yes Analytic conductor $18.206$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1016.1 Defining polynomial: $$x^{3} - x^{2} - 6 x + 2$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + \beta_{2} q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} - q^{5} + \beta_{2} q^{7} + q^{9} + ( 1 - \beta_{1} - \beta_{2} ) q^{11} + ( 2 + \beta_{2} ) q^{13} - q^{15} + ( 1 + \beta_{1} ) q^{17} - q^{19} + \beta_{2} q^{21} + ( 1 - \beta_{1} ) q^{23} + q^{25} + q^{27} + ( 1 + \beta_{1} - \beta_{2} ) q^{29} + ( 3 + \beta_{1} ) q^{31} + ( 1 - \beta_{1} - \beta_{2} ) q^{33} -\beta_{2} q^{35} + ( 2 + \beta_{2} ) q^{37} + ( 2 + \beta_{2} ) q^{39} + ( -1 - \beta_{1} - \beta_{2} ) q^{41} + ( 4 + \beta_{2} ) q^{43} - q^{45} + ( -1 + \beta_{1} ) q^{47} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{49} + ( 1 + \beta_{1} ) q^{51} + ( 1 + \beta_{1} ) q^{53} + ( -1 + \beta_{1} + \beta_{2} ) q^{55} - q^{57} + ( 4 + 2 \beta_{2} ) q^{59} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{61} + \beta_{2} q^{63} + ( -2 - \beta_{2} ) q^{65} + ( 2 + 2 \beta_{1} ) q^{67} + ( 1 - \beta_{1} ) q^{69} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{71} + 10 q^{73} + q^{75} + ( -7 - \beta_{1} + 4 \beta_{2} ) q^{77} + ( -2 - 2 \beta_{1} ) q^{79} + q^{81} + ( 5 + 3 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -1 - \beta_{1} ) q^{85} + ( 1 + \beta_{1} - \beta_{2} ) q^{87} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{89} + ( 9 - \beta_{1} ) q^{91} + ( 3 + \beta_{1} ) q^{93} + q^{95} + ( 10 + \beta_{2} ) q^{97} + ( 1 - \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - 3 q^{5} + 3 q^{9} + O(q^{10})$$ $$3 q + 3 q^{3} - 3 q^{5} + 3 q^{9} + 4 q^{11} + 6 q^{13} - 3 q^{15} + 2 q^{17} - 3 q^{19} + 4 q^{23} + 3 q^{25} + 3 q^{27} + 2 q^{29} + 8 q^{31} + 4 q^{33} + 6 q^{37} + 6 q^{39} - 2 q^{41} + 12 q^{43} - 3 q^{45} - 4 q^{47} + 7 q^{49} + 2 q^{51} + 2 q^{53} - 4 q^{55} - 3 q^{57} + 12 q^{59} + 14 q^{61} - 6 q^{65} + 4 q^{67} + 4 q^{69} + 8 q^{71} + 30 q^{73} + 3 q^{75} - 20 q^{77} - 4 q^{79} + 3 q^{81} + 12 q^{83} - 2 q^{85} + 2 q^{87} - 2 q^{89} + 28 q^{91} + 8 q^{93} + 3 q^{95} + 30 q^{97} + 4 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{2} + \beta_{1} + 9$$$$)/2$$

## 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
 0.321637 2.85577 −2.17741
0 1.00000 0 −1.00000 0 −4.21819 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 1.29966 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 2.91852 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.t 3
3.b odd 2 1 6840.2.a.bn 3
4.b odd 2 1 4560.2.a.bq 3

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

 $$T_{7}^{3} - 14 T_{7} + 16$$ $$T_{11}^{3} - 4 T_{11}^{2} - 26 T_{11} + 96$$ $$T_{13}^{3} - 6 T_{13}^{2} - 2 T_{13} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$16 - 14 T + T^{3}$$
$11$ $$96 - 26 T - 4 T^{2} + T^{3}$$
$13$ $$36 - 2 T - 6 T^{2} + T^{3}$$
$17$ $$16 - 24 T - 2 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$32 - 20 T - 4 T^{2} + T^{3}$$
$29$ $$156 - 46 T - 2 T^{2} + T^{3}$$
$31$ $$48 - 4 T - 8 T^{2} + T^{3}$$
$37$ $$36 - 2 T - 6 T^{2} + T^{3}$$
$41$ $$36 - 30 T + 2 T^{2} + T^{3}$$
$43$ $$8 + 34 T - 12 T^{2} + T^{3}$$
$47$ $$-32 - 20 T + 4 T^{2} + T^{3}$$
$53$ $$16 - 24 T - 2 T^{2} + T^{3}$$
$59$ $$288 - 8 T - 12 T^{2} + T^{3}$$
$61$ $$576 - 32 T - 14 T^{2} + T^{3}$$
$67$ $$128 - 96 T - 4 T^{2} + T^{3}$$
$71$ $$768 - 104 T - 8 T^{2} + T^{3}$$
$73$ $$( -10 + T )^{3}$$
$79$ $$-128 - 96 T + 4 T^{2} + T^{3}$$
$83$ $$3456 - 284 T - 12 T^{2} + T^{3}$$
$89$ $$-508 - 126 T + 2 T^{2} + T^{3}$$
$97$ $$-844 + 286 T - 30 T^{2} + T^{3}$$