# Properties

 Label 2280.2.a.k 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} + \beta q^{7} + q^{9}+O(q^{10})$$ q - q^3 - q^5 + b * q^7 + q^9 $$q - q^{3} - q^{5} + \beta q^{7} + q^{9} + (\beta - 2) q^{11} + ( - \beta + 2) q^{13} + q^{15} - 2 \beta q^{17} + q^{19} - \beta q^{21} + ( - 2 \beta - 2) q^{23} + q^{25} - q^{27} - 5 \beta q^{29} + ( - 2 \beta - 2) q^{31} + ( - \beta + 2) q^{33} - \beta q^{35} + (3 \beta + 2) q^{37} + (\beta - 2) q^{39} + 7 \beta q^{41} + (5 \beta + 4) q^{43} - q^{45} + (2 \beta - 6) q^{47} - 5 q^{49} + 2 \beta q^{51} + 6 \beta q^{53} + ( - \beta + 2) q^{55} - q^{57} + (2 \beta - 4) q^{59} + ( - 4 \beta + 4) q^{61} + \beta q^{63} + (\beta - 2) q^{65} - 8 \beta q^{67} + (2 \beta + 2) q^{69} + ( - 2 \beta - 12) q^{71} + ( - 4 \beta - 6) q^{73} - q^{75} + ( - 2 \beta + 2) q^{77} + ( - 4 \beta - 8) q^{79} + q^{81} + ( - 4 \beta - 2) q^{83} + 2 \beta q^{85} + 5 \beta q^{87} + ( - 3 \beta - 8) q^{89} + (2 \beta - 2) q^{91} + (2 \beta + 2) q^{93} - q^{95} + (7 \beta + 2) q^{97} + (\beta - 2) q^{99} +O(q^{100})$$ q - q^3 - q^5 + b * q^7 + q^9 + (b - 2) * q^11 + (-b + 2) * q^13 + q^15 - 2*b * q^17 + q^19 - b * q^21 + (-2*b - 2) * q^23 + q^25 - q^27 - 5*b * q^29 + (-2*b - 2) * q^31 + (-b + 2) * q^33 - b * q^35 + (3*b + 2) * q^37 + (b - 2) * q^39 + 7*b * q^41 + (5*b + 4) * q^43 - q^45 + (2*b - 6) * q^47 - 5 * q^49 + 2*b * q^51 + 6*b * q^53 + (-b + 2) * q^55 - q^57 + (2*b - 4) * q^59 + (-4*b + 4) * q^61 + b * q^63 + (b - 2) * q^65 - 8*b * q^67 + (2*b + 2) * q^69 + (-2*b - 12) * q^71 + (-4*b - 6) * q^73 - q^75 + (-2*b + 2) * q^77 + (-4*b - 8) * q^79 + q^81 + (-4*b - 2) * q^83 + 2*b * q^85 + 5*b * q^87 + (-3*b - 8) * q^89 + (2*b - 2) * q^91 + (2*b + 2) * q^93 - q^95 + (7*b + 2) * q^97 + (b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{31} + 4 q^{33} + 4 q^{37} - 4 q^{39} + 8 q^{43} - 2 q^{45} - 12 q^{47} - 10 q^{49} + 4 q^{55} - 2 q^{57} - 8 q^{59} + 8 q^{61} - 4 q^{65} + 4 q^{69} - 24 q^{71} - 12 q^{73} - 2 q^{75} + 4 q^{77} - 16 q^{79} + 2 q^{81} - 4 q^{83} - 16 q^{89} - 4 q^{91} + 4 q^{93} - 2 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^11 + 4 * q^13 + 2 * q^15 + 2 * q^19 - 4 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^31 + 4 * q^33 + 4 * q^37 - 4 * q^39 + 8 * q^43 - 2 * q^45 - 12 * q^47 - 10 * q^49 + 4 * q^55 - 2 * q^57 - 8 * q^59 + 8 * q^61 - 4 * q^65 + 4 * q^69 - 24 * q^71 - 12 * q^73 - 2 * q^75 + 4 * q^77 - 16 * q^79 + 2 * q^81 - 4 * q^83 - 16 * q^89 - 4 * q^91 + 4 * q^93 - 2 * q^95 + 4 * q^97 - 4 * q^99

## 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
 −1.41421 1.41421
0 −1.00000 0 −1.00000 0 −1.41421 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 1.41421 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.k 2
3.b odd 2 1 6840.2.a.bb 2
4.b odd 2 1 4560.2.a.bn 2

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

 $$T_{7}^{2} - 2$$ T7^2 - 2 $$T_{11}^{2} + 4T_{11} + 2$$ T11^2 + 4*T11 + 2 $$T_{13}^{2} - 4T_{13} + 2$$ T13^2 - 4*T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 2$$
$11$ $$T^{2} + 4T + 2$$
$13$ $$T^{2} - 4T + 2$$
$17$ $$T^{2} - 8$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 4T - 4$$
$29$ $$T^{2} - 50$$
$31$ $$T^{2} + 4T - 4$$
$37$ $$T^{2} - 4T - 14$$
$41$ $$T^{2} - 98$$
$43$ $$T^{2} - 8T - 34$$
$47$ $$T^{2} + 12T + 28$$
$53$ $$T^{2} - 72$$
$59$ $$T^{2} + 8T + 8$$
$61$ $$T^{2} - 8T - 16$$
$67$ $$T^{2} - 128$$
$71$ $$T^{2} + 24T + 136$$
$73$ $$T^{2} + 12T + 4$$
$79$ $$T^{2} + 16T + 32$$
$83$ $$T^{2} + 4T - 28$$
$89$ $$T^{2} + 16T + 46$$
$97$ $$T^{2} - 4T - 94$$