# Properties

 Label 4560.2.a.bo 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: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + (\beta + 1) q^{7} + q^{9}+O(q^{10})$$ q + q^3 + q^5 + (b + 1) * q^7 + q^9 $$q + q^{3} + q^{5} + (\beta + 1) q^{7} + q^{9} + ( - \beta - 3) q^{11} + ( - \beta - 3) q^{13} + q^{15} - 4 q^{17} + q^{19} + (\beta + 1) q^{21} + (2 \beta - 4) q^{23} + q^{25} + q^{27} + ( - 3 \beta + 1) q^{29} - 6 q^{31} + ( - \beta - 3) q^{33} + (\beta + 1) q^{35} + ( - \beta + 1) q^{37} + ( - \beta - 3) q^{39} + (\beta - 7) q^{41} + (\beta - 3) q^{43} + q^{45} + ( - 2 \beta - 4) q^{47} + (2 \beta + 1) q^{49} - 4 q^{51} + ( - 2 \beta + 6) q^{53} + ( - \beta - 3) q^{55} + q^{57} + (2 \beta - 6) q^{59} + (2 \beta - 6) q^{61} + (\beta + 1) q^{63} + ( - \beta - 3) q^{65} + ( - 4 \beta - 4) q^{67} + (2 \beta - 4) q^{69} + (2 \beta - 2) q^{71} + 10 q^{73} + q^{75} + ( - 4 \beta - 10) q^{77} + (4 \beta + 4) q^{79} + q^{81} - 6 q^{83} - 4 q^{85} + ( - 3 \beta + 1) q^{87} + (3 \beta - 9) q^{89} + ( - 4 \beta - 10) q^{91} - 6 q^{93} + q^{95} + (3 \beta + 5) q^{97} + ( - \beta - 3) q^{99}+O(q^{100})$$ q + q^3 + q^5 + (b + 1) * q^7 + q^9 + (-b - 3) * q^11 + (-b - 3) * q^13 + q^15 - 4 * q^17 + q^19 + (b + 1) * q^21 + (2*b - 4) * q^23 + q^25 + q^27 + (-3*b + 1) * q^29 - 6 * q^31 + (-b - 3) * q^33 + (b + 1) * q^35 + (-b + 1) * q^37 + (-b - 3) * q^39 + (b - 7) * q^41 + (b - 3) * q^43 + q^45 + (-2*b - 4) * q^47 + (2*b + 1) * q^49 - 4 * q^51 + (-2*b + 6) * q^53 + (-b - 3) * q^55 + q^57 + (2*b - 6) * q^59 + (2*b - 6) * q^61 + (b + 1) * q^63 + (-b - 3) * q^65 + (-4*b - 4) * q^67 + (2*b - 4) * q^69 + (2*b - 2) * q^71 + 10 * q^73 + q^75 + (-4*b - 10) * q^77 + (4*b + 4) * q^79 + q^81 - 6 * q^83 - 4 * q^85 + (-3*b + 1) * q^87 + (3*b - 9) * q^89 + (-4*b - 10) * q^91 - 6 * q^93 + q^95 + (3*b + 5) * q^97 + (-b - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 6 q^{13} + 2 q^{15} - 8 q^{17} + 2 q^{19} + 2 q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} + 2 q^{29} - 12 q^{31} - 6 q^{33} + 2 q^{35} + 2 q^{37} - 6 q^{39} - 14 q^{41} - 6 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} - 8 q^{51} + 12 q^{53} - 6 q^{55} + 2 q^{57} - 12 q^{59} - 12 q^{61} + 2 q^{63} - 6 q^{65} - 8 q^{67} - 8 q^{69} - 4 q^{71} + 20 q^{73} + 2 q^{75} - 20 q^{77} + 8 q^{79} + 2 q^{81} - 12 q^{83} - 8 q^{85} + 2 q^{87} - 18 q^{89} - 20 q^{91} - 12 q^{93} + 2 q^{95} + 10 q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 6 * q^11 - 6 * q^13 + 2 * q^15 - 8 * q^17 + 2 * q^19 + 2 * q^21 - 8 * q^23 + 2 * q^25 + 2 * q^27 + 2 * q^29 - 12 * q^31 - 6 * q^33 + 2 * q^35 + 2 * q^37 - 6 * q^39 - 14 * q^41 - 6 * q^43 + 2 * q^45 - 8 * q^47 + 2 * q^49 - 8 * q^51 + 12 * q^53 - 6 * q^55 + 2 * q^57 - 12 * q^59 - 12 * q^61 + 2 * q^63 - 6 * q^65 - 8 * q^67 - 8 * q^69 - 4 * q^71 + 20 * q^73 + 2 * q^75 - 20 * q^77 + 8 * q^79 + 2 * q^81 - 12 * q^83 - 8 * q^85 + 2 * q^87 - 18 * q^89 - 20 * q^91 - 12 * q^93 + 2 * q^95 + 10 * q^97 - 6 * 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
 −2.64575 2.64575
0 1.00000 0 1.00000 0 −1.64575 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 3.64575 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.bo 2
4.b odd 2 1 285.2.a.d 2
12.b even 2 1 855.2.a.g 2
20.d odd 2 1 1425.2.a.p 2
20.e even 4 2 1425.2.c.i 4
60.h even 2 1 4275.2.a.u 2
76.d even 2 1 5415.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.d 2 4.b odd 2 1
855.2.a.g 2 12.b even 2 1
1425.2.a.p 2 20.d odd 2 1
1425.2.c.i 4 20.e even 4 2
4275.2.a.u 2 60.h even 2 1
4560.2.a.bo 2 1.a even 1 1 trivial
5415.2.a.s 2 76.d even 2 1

## Hecke kernels

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

 $$T_{7}^{2} - 2T_{7} - 6$$ T7^2 - 2*T7 - 6 $$T_{11}^{2} + 6T_{11} + 2$$ T11^2 + 6*T11 + 2 $$T_{13}^{2} + 6T_{13} + 2$$ T13^2 + 6*T13 + 2 $$T_{17} + 4$$ T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 2T - 6$$
$11$ $$T^{2} + 6T + 2$$
$13$ $$T^{2} + 6T + 2$$
$17$ $$(T + 4)^{2}$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 8T - 12$$
$29$ $$T^{2} - 2T - 62$$
$31$ $$(T + 6)^{2}$$
$37$ $$T^{2} - 2T - 6$$
$41$ $$T^{2} + 14T + 42$$
$43$ $$T^{2} + 6T + 2$$
$47$ $$T^{2} + 8T - 12$$
$53$ $$T^{2} - 12T + 8$$
$59$ $$T^{2} + 12T + 8$$
$61$ $$T^{2} + 12T + 8$$
$67$ $$T^{2} + 8T - 96$$
$71$ $$T^{2} + 4T - 24$$
$73$ $$(T - 10)^{2}$$
$79$ $$T^{2} - 8T - 96$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 18T + 18$$
$97$ $$T^{2} - 10T - 38$$