# Properties

 Label 2280.2.a.o 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.

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + (\beta - 2) q^{7} + q^{9}+O(q^{10})$$ q + q^3 - q^5 + (b - 2) * q^7 + q^9 $$q + q^{3} - q^{5} + (\beta - 2) q^{7} + q^{9} + \beta q^{11} - 3 \beta q^{13} - q^{15} - 2 \beta q^{17} - q^{19} + (\beta - 2) q^{21} + (2 \beta - 2) q^{23} + q^{25} + q^{27} + ( - 3 \beta - 2) q^{29} + (2 \beta - 6) q^{31} + \beta q^{33} + ( - \beta + 2) q^{35} + 5 \beta q^{37} - 3 \beta q^{39} + ( - 3 \beta - 2) q^{41} + ( - 3 \beta - 6) q^{43} - q^{45} + ( - 2 \beta + 2) q^{47} + ( - 4 \beta - 1) q^{49} - 2 \beta q^{51} + (6 \beta - 4) q^{53} - \beta q^{55} - q^{57} + 2 \beta q^{59} - 8 q^{61} + (\beta - 2) q^{63} + 3 \beta q^{65} + 8 \beta q^{67} + (2 \beta - 2) q^{69} + ( - 2 \beta - 8) q^{71} + (4 \beta + 2) q^{73} + q^{75} + ( - 2 \beta + 2) q^{77} + ( - 4 \beta - 8) q^{79} + q^{81} - 6 q^{83} + 2 \beta q^{85} + ( - 3 \beta - 2) q^{87} + (11 \beta + 2) q^{89} + (6 \beta - 6) q^{91} + (2 \beta - 6) q^{93} + q^{95} - 7 \beta q^{97} + \beta q^{99} +O(q^{100})$$ q + q^3 - q^5 + (b - 2) * q^7 + q^9 + b * q^11 - 3*b * q^13 - q^15 - 2*b * q^17 - q^19 + (b - 2) * q^21 + (2*b - 2) * q^23 + q^25 + q^27 + (-3*b - 2) * q^29 + (2*b - 6) * q^31 + b * q^33 + (-b + 2) * q^35 + 5*b * q^37 - 3*b * q^39 + (-3*b - 2) * q^41 + (-3*b - 6) * q^43 - q^45 + (-2*b + 2) * q^47 + (-4*b - 1) * q^49 - 2*b * q^51 + (6*b - 4) * q^53 - b * q^55 - q^57 + 2*b * q^59 - 8 * q^61 + (b - 2) * q^63 + 3*b * q^65 + 8*b * q^67 + (2*b - 2) * q^69 + (-2*b - 8) * q^71 + (4*b + 2) * q^73 + q^75 + (-2*b + 2) * q^77 + (-4*b - 8) * q^79 + q^81 - 6 * q^83 + 2*b * q^85 + (-3*b - 2) * q^87 + (11*b + 2) * q^89 + (6*b - 6) * q^91 + (2*b - 6) * q^93 + q^95 - 7*b * q^97 + b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^7 + 2 * q^9 $$2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} - 2 q^{15} - 2 q^{19} - 4 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 12 q^{31} + 4 q^{35} - 4 q^{41} - 12 q^{43} - 2 q^{45} + 4 q^{47} - 2 q^{49} - 8 q^{53} - 2 q^{57} - 16 q^{61} - 4 q^{63} - 4 q^{69} - 16 q^{71} + 4 q^{73} + 2 q^{75} + 4 q^{77} - 16 q^{79} + 2 q^{81} - 12 q^{83} - 4 q^{87} + 4 q^{89} - 12 q^{91} - 12 q^{93} + 2 q^{95}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^7 + 2 * q^9 - 2 * q^15 - 2 * q^19 - 4 * q^21 - 4 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 - 12 * q^31 + 4 * q^35 - 4 * q^41 - 12 * q^43 - 2 * q^45 + 4 * q^47 - 2 * q^49 - 8 * q^53 - 2 * q^57 - 16 * q^61 - 4 * q^63 - 4 * q^69 - 16 * q^71 + 4 * q^73 + 2 * q^75 + 4 * q^77 - 16 * q^79 + 2 * q^81 - 12 * q^83 - 4 * q^87 + 4 * q^89 - 12 * q^91 - 12 * q^93 + 2 * q^95

## 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 −3.41421 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −0.585786 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.o 2
3.b odd 2 1 6840.2.a.x 2
4.b odd 2 1 4560.2.a.bg 2

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

 $$T_{7}^{2} + 4T_{7} + 2$$ T7^2 + 4*T7 + 2 $$T_{11}^{2} - 2$$ T11^2 - 2 $$T_{13}^{2} - 18$$ T13^2 - 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 4T + 2$$
$11$ $$T^{2} - 2$$
$13$ $$T^{2} - 18$$
$17$ $$T^{2} - 8$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 4T - 4$$
$29$ $$T^{2} + 4T - 14$$
$31$ $$T^{2} + 12T + 28$$
$37$ $$T^{2} - 50$$
$41$ $$T^{2} + 4T - 14$$
$43$ $$T^{2} + 12T + 18$$
$47$ $$T^{2} - 4T - 4$$
$53$ $$T^{2} + 8T - 56$$
$59$ $$T^{2} - 8$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} - 128$$
$71$ $$T^{2} + 16T + 56$$
$73$ $$T^{2} - 4T - 28$$
$79$ $$T^{2} + 16T + 32$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} - 4T - 238$$
$97$ $$T^{2} - 98$$