# Properties

 Label 5184.2.a.bh Level $5184$ Weight $2$ Character orbit 5184.a Self dual yes Analytic conductor $41.394$ Analytic rank $1$ Dimension $2$ CM discriminant -4 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5184 = 2^{6} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5184.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$41.3944484078$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2592) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 2) q^{5}+O(q^{10})$$ q + (b - 2) * q^5 $$q + (\beta - 2) q^{5} + (2 \beta - 3) q^{13} + ( - \beta + 4) q^{17} + ( - 4 \beta + 2) q^{25} + ( - 5 \beta + 2) q^{29} + ( - 6 \beta - 1) q^{37} + 8 q^{41} - 7 q^{49} - 4 q^{53} + (6 \beta - 5) q^{61} + ( - 7 \beta + 12) q^{65} + (8 \beta - 3) q^{73} + (6 \beta - 11) q^{85} + (5 \beta + 8) q^{89} - 18 q^{97}+O(q^{100})$$ q + (b - 2) * q^5 + (2*b - 3) * q^13 + (-b + 4) * q^17 + (-4*b + 2) * q^25 + (-5*b + 2) * q^29 + (-6*b - 1) * q^37 + 8 * q^41 - 7 * q^49 - 4 * q^53 + (6*b - 5) * q^61 + (-7*b + 12) * q^65 + (8*b - 3) * q^73 + (6*b - 11) * q^85 + (5*b + 8) * q^89 - 18 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5}+O(q^{10})$$ 2 * q - 4 * q^5 $$2 q - 4 q^{5} - 6 q^{13} + 8 q^{17} + 4 q^{25} + 4 q^{29} - 2 q^{37} + 16 q^{41} - 14 q^{49} - 8 q^{53} - 10 q^{61} + 24 q^{65} - 6 q^{73} - 22 q^{85} + 16 q^{89} - 36 q^{97}+O(q^{100})$$ 2 * q - 4 * q^5 - 6 * q^13 + 8 * q^17 + 4 * q^25 + 4 * q^29 - 2 * q^37 + 16 * q^41 - 14 * q^49 - 8 * q^53 - 10 * q^61 + 24 * q^65 - 6 * q^73 - 22 * q^85 + 16 * q^89 - 36 * 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
 −1.73205 1.73205
0 0 0 −3.73205 0 0 0 0 0
1.2 0 0 0 −0.267949 0 0 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.a.bh 2
3.b odd 2 1 5184.2.a.ca 2
4.b odd 2 1 CM 5184.2.a.bh 2
8.b even 2 1 2592.2.a.t yes 2
8.d odd 2 1 2592.2.a.t yes 2
12.b even 2 1 5184.2.a.ca 2
24.f even 2 1 2592.2.a.i 2
24.h odd 2 1 2592.2.a.i 2
72.j odd 6 2 2592.2.i.bf 4
72.l even 6 2 2592.2.i.bf 4
72.n even 6 2 2592.2.i.y 4
72.p odd 6 2 2592.2.i.y 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.a.i 2 24.f even 2 1
2592.2.a.i 2 24.h odd 2 1
2592.2.a.t yes 2 8.b even 2 1
2592.2.a.t yes 2 8.d odd 2 1
2592.2.i.y 4 72.n even 6 2
2592.2.i.y 4 72.p odd 6 2
2592.2.i.bf 4 72.j odd 6 2
2592.2.i.bf 4 72.l even 6 2
5184.2.a.bh 2 1.a even 1 1 trivial
5184.2.a.bh 2 4.b odd 2 1 CM
5184.2.a.ca 2 3.b odd 2 1
5184.2.a.ca 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5184))$$:

 $$T_{5}^{2} + 4T_{5} + 1$$ T5^2 + 4*T5 + 1 $$T_{7}$$ T7 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 6T - 3$$
$17$ $$T^{2} - 8T + 13$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 4T - 71$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 2T - 107$$
$41$ $$(T - 8)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$(T + 4)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 10T - 83$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 6T - 183$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 16T - 11$$
$97$ $$(T + 18)^{2}$$