# Properties

 Label 2280.2.a.q Level $2280$ Weight $2$ Character orbit 2280.a Self dual yes Analytic conductor $18.206$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + ( -3 + \beta ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + q^{5} + ( -3 + \beta ) q^{7} + q^{9} + ( -1 - 3 \beta ) q^{11} + ( -3 - \beta ) q^{13} + q^{15} + 4 \beta q^{17} + q^{19} + ( -3 + \beta ) q^{21} -6 q^{23} + q^{25} + q^{27} + ( -5 + 3 \beta ) q^{29} -2 q^{31} + ( -1 - 3 \beta ) q^{33} + ( -3 + \beta ) q^{35} + ( 1 - \beta ) q^{37} + ( -3 - \beta ) q^{39} + ( 3 - 5 \beta ) q^{41} + ( -3 + \beta ) q^{43} + q^{45} -10 q^{47} + ( 5 - 6 \beta ) q^{49} + 4 \beta q^{51} + ( -1 - 3 \beta ) q^{55} + q^{57} + ( -6 + 2 \beta ) q^{59} + ( 2 - 2 \beta ) q^{61} + ( -3 + \beta ) q^{63} + ( -3 - \beta ) q^{65} + ( -8 + 4 \beta ) q^{67} -6 q^{69} + ( 2 - 6 \beta ) q^{71} + ( -6 - 4 \beta ) q^{73} + q^{75} + ( -6 + 8 \beta ) q^{77} -8 q^{79} + q^{81} + ( 12 + 2 \beta ) q^{83} + 4 \beta q^{85} + ( -5 + 3 \beta ) q^{87} + ( 1 + 5 \beta ) q^{89} + 6 q^{91} -2 q^{93} + q^{95} + ( -7 - 5 \beta ) q^{97} + ( -1 - 3 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} - 6 q^{13} + 2 q^{15} + 2 q^{19} - 6 q^{21} - 12 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - 2 q^{33} - 6 q^{35} + 2 q^{37} - 6 q^{39} + 6 q^{41} - 6 q^{43} + 2 q^{45} - 20 q^{47} + 10 q^{49} - 2 q^{55} + 2 q^{57} - 12 q^{59} + 4 q^{61} - 6 q^{63} - 6 q^{65} - 16 q^{67} - 12 q^{69} + 4 q^{71} - 12 q^{73} + 2 q^{75} - 12 q^{77} - 16 q^{79} + 2 q^{81} + 24 q^{83} - 10 q^{87} + 2 q^{89} + 12 q^{91} - 4 q^{93} + 2 q^{95} - 14 q^{97} - 2 q^{99} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 1.00000 0 1.00000 0 −4.73205 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 −1.26795 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.q 2
3.b odd 2 1 6840.2.a.v 2
4.b odd 2 1 4560.2.a.bi 2

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

 $$T_{7}^{2} + 6 T_{7} + 6$$ $$T_{11}^{2} + 2 T_{11} - 26$$ $$T_{13}^{2} + 6 T_{13} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$6 + 6 T + T^{2}$$
$11$ $$-26 + 2 T + T^{2}$$
$13$ $$6 + 6 T + T^{2}$$
$17$ $$-48 + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$-2 + 10 T + T^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$-2 - 2 T + T^{2}$$
$41$ $$-66 - 6 T + T^{2}$$
$43$ $$6 + 6 T + T^{2}$$
$47$ $$( 10 + T )^{2}$$
$53$ $$T^{2}$$
$59$ $$24 + 12 T + T^{2}$$
$61$ $$-8 - 4 T + T^{2}$$
$67$ $$16 + 16 T + T^{2}$$
$71$ $$-104 - 4 T + T^{2}$$
$73$ $$-12 + 12 T + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$132 - 24 T + T^{2}$$
$89$ $$-74 - 2 T + T^{2}$$
$97$ $$-26 + 14 T + T^{2}$$