# Properties

 Label 4560.2.a.bm Level $4560$ Weight $2$ Character orbit 4560.a Self dual yes Analytic conductor $36.412$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.4117833217$$ 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: $$2$$ Twist minimal: no (minimal twist has level 2280) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. 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 $$q + q^{3} - q^{5} + \beta q^{7} + q^{9} - \beta q^{11} + (\beta - 2) q^{13} - q^{15} + ( - 2 \beta - 2) q^{17} + q^{19} + \beta q^{21} - 4 q^{23} + q^{25} + q^{27} + ( - \beta + 2) q^{29} - \beta q^{33} - \beta q^{35} + ( - \beta - 2) q^{37} + (\beta - 2) q^{39} + ( - \beta + 2) q^{41} - \beta q^{43} - q^{45} - 4 q^{47} + q^{49} + ( - 2 \beta - 2) q^{51} - 2 q^{53} + \beta q^{55} + q^{57} - 8 q^{59} + (4 \beta - 2) q^{61} + \beta q^{63} + ( - \beta + 2) q^{65} + (2 \beta - 4) q^{67} - 4 q^{69} - 8 q^{71} + (2 \beta - 6) q^{73} + q^{75} - 8 q^{77} + ( - 2 \beta - 8) q^{79} + q^{81} - 8 q^{83} + (2 \beta + 2) q^{85} + ( - \beta + 2) q^{87} + ( - \beta + 10) q^{89} + ( - 2 \beta + 8) q^{91} - q^{95} + ( - \beta - 2) q^{97} - \beta q^{99} +O(q^{100})$$ q + q^3 - q^5 + b * q^7 + q^9 - b * q^11 + (b - 2) * q^13 - q^15 + (-2*b - 2) * q^17 + q^19 + b * q^21 - 4 * q^23 + q^25 + q^27 + (-b + 2) * q^29 - b * q^33 - b * q^35 + (-b - 2) * q^37 + (b - 2) * q^39 + (-b + 2) * q^41 - b * q^43 - q^45 - 4 * q^47 + q^49 + (-2*b - 2) * q^51 - 2 * q^53 + b * q^55 + q^57 - 8 * q^59 + (4*b - 2) * q^61 + b * q^63 + (-b + 2) * q^65 + (2*b - 4) * q^67 - 4 * q^69 - 8 * q^71 + (2*b - 6) * q^73 + q^75 - 8 * q^77 + (-2*b - 8) * q^79 + q^81 - 8 * q^83 + (2*b + 2) * q^85 + (-b + 2) * q^87 + (-b + 10) * q^89 + (-2*b + 8) * q^91 - q^95 + (-b - 2) * q^97 - b * 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 $$2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{13} - 2 q^{15} - 4 q^{17} + 2 q^{19} - 8 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} - 4 q^{37} - 4 q^{39} + 4 q^{41} - 2 q^{45} - 8 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} + 2 q^{57} - 16 q^{59} - 4 q^{61} + 4 q^{65} - 8 q^{67} - 8 q^{69} - 16 q^{71} - 12 q^{73} + 2 q^{75} - 16 q^{77} - 16 q^{79} + 2 q^{81} - 16 q^{83} + 4 q^{85} + 4 q^{87} + 20 q^{89} + 16 q^{91} - 2 q^{95} - 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^13 - 2 * q^15 - 4 * q^17 + 2 * q^19 - 8 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 - 4 * q^37 - 4 * q^39 + 4 * q^41 - 2 * q^45 - 8 * q^47 + 2 * q^49 - 4 * q^51 - 4 * q^53 + 2 * q^57 - 16 * q^59 - 4 * q^61 + 4 * q^65 - 8 * q^67 - 8 * q^69 - 16 * q^71 - 12 * q^73 + 2 * q^75 - 16 * q^77 - 16 * q^79 + 2 * q^81 - 16 * q^83 + 4 * q^85 + 4 * q^87 + 20 * q^89 + 16 * q^91 - 2 * q^95 - 4 * q^97

## 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 −2.82843 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 2.82843 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 4560.2.a.bm 2
4.b odd 2 1 2280.2.a.l 2
12.b even 2 1 6840.2.a.ba 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.l 2 4.b odd 2 1
4560.2.a.bm 2 1.a even 1 1 trivial
6840.2.a.ba 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4560))$$:

 $$T_{7}^{2} - 8$$ T7^2 - 8 $$T_{11}^{2} - 8$$ T11^2 - 8 $$T_{13}^{2} + 4T_{13} - 4$$ T13^2 + 4*T13 - 4 $$T_{17}^{2} + 4T_{17} - 28$$ T17^2 + 4*T17 - 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 8$$
$11$ $$T^{2} - 8$$
$13$ $$T^{2} + 4T - 4$$
$17$ $$T^{2} + 4T - 28$$
$19$ $$(T - 1)^{2}$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} - 4T - 4$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 4T - 4$$
$41$ $$T^{2} - 4T - 4$$
$43$ $$T^{2} - 8$$
$47$ $$(T + 4)^{2}$$
$53$ $$(T + 2)^{2}$$
$59$ $$(T + 8)^{2}$$
$61$ $$T^{2} + 4T - 124$$
$67$ $$T^{2} + 8T - 16$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 12T + 4$$
$79$ $$T^{2} + 16T + 32$$
$83$ $$(T + 8)^{2}$$
$89$ $$T^{2} - 20T + 92$$
$97$ $$T^{2} + 4T - 4$$