# Properties

 Label 5184.2.a.bz Level $5184$ Weight $2$ Character orbit 5184.a Self dual yes Analytic conductor $41.394$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# 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 648) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta + 2) q^{5} - 2 \beta q^{7} +O(q^{10})$$ q + (b + 2) * q^5 - 2*b * q^7 $$q + (\beta + 2) q^{5} - 2 \beta q^{7} - 2 q^{11} + (2 \beta - 1) q^{13} + (\beta - 4) q^{17} + ( - 2 \beta - 4) q^{19} + ( - 4 \beta + 2) q^{23} + (4 \beta + 2) q^{25} + ( - \beta + 6) q^{29} + (4 \beta + 4) q^{31} + ( - 4 \beta - 6) q^{35} + (2 \beta - 3) q^{37} - 4 \beta q^{41} + (2 \beta - 8) q^{43} + 4 \beta q^{47} + 5 q^{49} + ( - 4 \beta - 4) q^{53} + ( - 2 \beta - 4) q^{55} - 8 q^{59} + ( - 2 \beta - 7) q^{61} + (3 \beta + 4) q^{65} + (2 \beta - 4) q^{67} - 2 q^{71} + q^{73} + 4 \beta q^{77} + (2 \beta - 4) q^{79} + (4 \beta - 4) q^{83} + ( - 2 \beta - 5) q^{85} + 3 \beta q^{89} + (2 \beta - 12) q^{91} + ( - 8 \beta - 14) q^{95} + ( - 8 \beta + 2) q^{97} +O(q^{100})$$ q + (b + 2) * q^5 - 2*b * q^7 - 2 * q^11 + (2*b - 1) * q^13 + (b - 4) * q^17 + (-2*b - 4) * q^19 + (-4*b + 2) * q^23 + (4*b + 2) * q^25 + (-b + 6) * q^29 + (4*b + 4) * q^31 + (-4*b - 6) * q^35 + (2*b - 3) * q^37 - 4*b * q^41 + (2*b - 8) * q^43 + 4*b * q^47 + 5 * q^49 + (-4*b - 4) * q^53 + (-2*b - 4) * q^55 - 8 * q^59 + (-2*b - 7) * q^61 + (3*b + 4) * q^65 + (2*b - 4) * q^67 - 2 * q^71 + q^73 + 4*b * q^77 + (2*b - 4) * q^79 + (4*b - 4) * q^83 + (-2*b - 5) * q^85 + 3*b * q^89 + (2*b - 12) * q^91 + (-8*b - 14) * q^95 + (-8*b + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5}+O(q^{10})$$ 2 * q + 4 * q^5 $$2 q + 4 q^{5} - 4 q^{11} - 2 q^{13} - 8 q^{17} - 8 q^{19} + 4 q^{23} + 4 q^{25} + 12 q^{29} + 8 q^{31} - 12 q^{35} - 6 q^{37} - 16 q^{43} + 10 q^{49} - 8 q^{53} - 8 q^{55} - 16 q^{59} - 14 q^{61} + 8 q^{65} - 8 q^{67} - 4 q^{71} + 2 q^{73} - 8 q^{79} - 8 q^{83} - 10 q^{85} - 24 q^{91} - 28 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 4 * q^5 - 4 * q^11 - 2 * q^13 - 8 * q^17 - 8 * q^19 + 4 * q^23 + 4 * q^25 + 12 * q^29 + 8 * q^31 - 12 * q^35 - 6 * q^37 - 16 * q^43 + 10 * q^49 - 8 * q^53 - 8 * q^55 - 16 * q^59 - 14 * q^61 + 8 * q^65 - 8 * q^67 - 4 * q^71 + 2 * q^73 - 8 * q^79 - 8 * q^83 - 10 * q^85 - 24 * q^91 - 28 * 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
 −1.73205 1.73205
0 0 0 0.267949 0 3.46410 0 0 0
1.2 0 0 0 3.73205 0 −3.46410 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

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 5184.2.a.bz 2
3.b odd 2 1 5184.2.a.bi 2
4.b odd 2 1 5184.2.a.cb 2
8.b even 2 1 1296.2.a.m 2
8.d odd 2 1 648.2.a.e 2
12.b even 2 1 5184.2.a.bg 2
24.f even 2 1 648.2.a.h yes 2
24.h odd 2 1 1296.2.a.q 2
72.j odd 6 2 1296.2.i.r 4
72.l even 6 2 648.2.i.i 4
72.n even 6 2 1296.2.i.t 4
72.p odd 6 2 648.2.i.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.2.a.e 2 8.d odd 2 1
648.2.a.h yes 2 24.f even 2 1
648.2.i.i 4 72.l even 6 2
648.2.i.j 4 72.p odd 6 2
1296.2.a.m 2 8.b even 2 1
1296.2.a.q 2 24.h odd 2 1
1296.2.i.r 4 72.j odd 6 2
1296.2.i.t 4 72.n even 6 2
5184.2.a.bg 2 12.b even 2 1
5184.2.a.bi 2 3.b odd 2 1
5184.2.a.bz 2 1.a even 1 1 trivial
5184.2.a.cb 2 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5184))$$:

 $$T_{5}^{2} - 4T_{5} + 1$$ T5^2 - 4*T5 + 1 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 1$$
$7$ $$T^{2} - 12$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 2T - 11$$
$17$ $$T^{2} + 8T + 13$$
$19$ $$T^{2} + 8T + 4$$
$23$ $$T^{2} - 4T - 44$$
$29$ $$T^{2} - 12T + 33$$
$31$ $$T^{2} - 8T - 32$$
$37$ $$T^{2} + 6T - 3$$
$41$ $$T^{2} - 48$$
$43$ $$T^{2} + 16T + 52$$
$47$ $$T^{2} - 48$$
$53$ $$T^{2} + 8T - 32$$
$59$ $$(T + 8)^{2}$$
$61$ $$T^{2} + 14T + 37$$
$67$ $$T^{2} + 8T + 4$$
$71$ $$(T + 2)^{2}$$
$73$ $$(T - 1)^{2}$$
$79$ $$T^{2} + 8T + 4$$
$83$ $$T^{2} + 8T - 32$$
$89$ $$T^{2} - 27$$
$97$ $$T^{2} - 4T - 188$$