# Properties

 Label 2280.2.a.r Level $2280$ Weight $2$ Character orbit 2280.a Self dual yes Analytic conductor $18.206$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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## 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: 3.3.568.1 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.

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} - q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} + ( 1 - \beta_{2} ) q^{11} + ( -2 + \beta_{1} ) q^{13} + q^{15} + 2 \beta_{1} q^{17} + q^{19} + ( 1 + \beta_{2} ) q^{21} + ( -3 - \beta_{1} - \beta_{2} ) q^{23} + q^{25} - q^{27} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{29} + ( -2 - 2 \beta_{1} ) q^{31} + ( -1 + \beta_{2} ) q^{33} + ( 1 + \beta_{2} ) q^{35} + ( -2 - 3 \beta_{1} ) q^{37} + ( 2 - \beta_{1} ) q^{39} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{41} + 3 \beta_{1} q^{43} - q^{45} + ( -5 + \beta_{1} + \beta_{2} ) q^{47} + ( 6 + \beta_{1} - \beta_{2} ) q^{49} -2 \beta_{1} q^{51} + ( 7 - \beta_{1} - \beta_{2} ) q^{53} + ( -1 + \beta_{2} ) q^{55} - q^{57} + ( 7 + 3 \beta_{1} - \beta_{2} ) q^{59} + ( 1 + 3 \beta_{1} + \beta_{2} ) q^{61} + ( -1 - \beta_{2} ) q^{63} + ( 2 - \beta_{1} ) q^{65} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 3 + \beta_{1} + \beta_{2} ) q^{69} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{73} - q^{75} + ( 11 + \beta_{1} - 3 \beta_{2} ) q^{77} + ( -3 - \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{83} -2 \beta_{1} q^{85} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{87} + ( 8 + 3 \beta_{1} ) q^{89} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{91} + ( 2 + 2 \beta_{1} ) q^{93} - q^{95} + ( -2 - 3 \beta_{1} ) q^{97} + ( 1 - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{3} - 3q^{5} - 2q^{7} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{3} - 3q^{5} - 2q^{7} + 3q^{9} + 4q^{11} - 6q^{13} + 3q^{15} + 3q^{19} + 2q^{21} - 8q^{23} + 3q^{25} - 3q^{27} + 10q^{29} - 6q^{31} - 4q^{33} + 2q^{35} - 6q^{37} + 6q^{39} + 4q^{41} - 3q^{45} - 16q^{47} + 19q^{49} + 22q^{53} - 4q^{55} - 3q^{57} + 22q^{59} + 2q^{61} - 2q^{63} + 6q^{65} + 4q^{67} + 8q^{69} - 8q^{71} + 10q^{73} - 3q^{75} + 36q^{77} - 10q^{79} + 3q^{81} + 2q^{83} - 10q^{87} + 24q^{89} + 8q^{91} + 6q^{93} - 3q^{95} - 6q^{97} + 4q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu - 4$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 3 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 3 \beta_{1} + 9$$$$)/2$$

## 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
 3.12489 −0.363328 −1.76156
0 −1.00000 0 −1.00000 0 −3.60975 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −2.77801 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 4.38776 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.r 3
3.b odd 2 1 6840.2.a.bk 3
4.b odd 2 1 4560.2.a.bt 3

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

 $$T_{7}^{3} + 2 T_{7}^{2} - 18 T_{7} - 44$$ $$T_{11}^{3} - 4 T_{11}^{2} - 14 T_{11} - 8$$ $$T_{13}^{3} + 6 T_{13}^{2} + 2 T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$-44 - 18 T + 2 T^{2} + T^{3}$$
$11$ $$-8 - 14 T - 4 T^{2} + T^{3}$$
$13$ $$-4 + 2 T + 6 T^{2} + T^{3}$$
$17$ $$64 - 40 T + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$-16 - 4 T + 8 T^{2} + T^{3}$$
$29$ $$268 - 18 T - 10 T^{2} + T^{3}$$
$31$ $$-136 - 28 T + 6 T^{2} + T^{3}$$
$37$ $$-388 - 78 T + 6 T^{2} + T^{3}$$
$41$ $$400 - 90 T - 4 T^{2} + T^{3}$$
$43$ $$216 - 90 T + T^{3}$$
$47$ $$-16 + 60 T + 16 T^{2} + T^{3}$$
$53$ $$-176 + 136 T - 22 T^{2} + T^{3}$$
$59$ $$976 + 40 T - 22 T^{2} + T^{3}$$
$61$ $$-160 - 96 T - 2 T^{2} + T^{3}$$
$67$ $$-128 - 96 T - 4 T^{2} + T^{3}$$
$71$ $$-1600 - 184 T + 8 T^{2} + T^{3}$$
$73$ $$488 - 100 T - 10 T^{2} + T^{3}$$
$79$ $$-64 + 10 T^{2} + T^{3}$$
$83$ $$328 - 100 T - 2 T^{2} + T^{3}$$
$89$ $$424 + 102 T - 24 T^{2} + T^{3}$$
$97$ $$-388 - 78 T + 6 T^{2} + T^{3}$$
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