# Properties

 Label 5184.2.a.bu 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: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 288) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + ( - \beta - 1) q^{7}+O(q^{10})$$ q + q^5 + (-b - 1) * q^7 $$q + q^{5} + ( - \beta - 1) q^{7} + (\beta - 1) q^{11} + ( - 2 \beta - 1) q^{13} + 2 \beta q^{17} + 4 q^{19} + (\beta + 3) q^{23} - 4 q^{25} + (2 \beta - 5) q^{29} + (\beta - 5) q^{31} + ( - \beta - 1) q^{35} + (2 \beta - 4) q^{37} + ( - 2 \beta - 7) q^{41} + ( - 3 \beta + 5) q^{43} + (3 \beta - 1) q^{47} + 2 \beta q^{49} + ( - 2 \beta - 4) q^{53} + (\beta - 1) q^{55} + (3 \beta + 7) q^{59} + (2 \beta + 3) q^{61} + ( - 2 \beta - 1) q^{65} + (3 \beta + 5) q^{67} + ( - 4 \beta + 2) q^{71} - 2 \beta q^{73} - 5 q^{77} + ( - \beta - 11) q^{79} + (\beta - 3) q^{83} + 2 \beta q^{85} + ( - 2 \beta - 8) q^{89} + (3 \beta + 13) q^{91} + 4 q^{95} + ( - 2 \beta + 1) q^{97} +O(q^{100})$$ q + q^5 + (-b - 1) * q^7 + (b - 1) * q^11 + (-2*b - 1) * q^13 + 2*b * q^17 + 4 * q^19 + (b + 3) * q^23 - 4 * q^25 + (2*b - 5) * q^29 + (b - 5) * q^31 + (-b - 1) * q^35 + (2*b - 4) * q^37 + (-2*b - 7) * q^41 + (-3*b + 5) * q^43 + (3*b - 1) * q^47 + 2*b * q^49 + (-2*b - 4) * q^53 + (b - 1) * q^55 + (3*b + 7) * q^59 + (2*b + 3) * q^61 + (-2*b - 1) * q^65 + (3*b + 5) * q^67 + (-4*b + 2) * q^71 - 2*b * q^73 - 5 * q^77 + (-b - 11) * q^79 + (b - 3) * q^83 + 2*b * q^85 + (-2*b - 8) * q^89 + (3*b + 13) * q^91 + 4 * q^95 + (-2*b + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 - 2 * q^7 $$2 q + 2 q^{5} - 2 q^{7} - 2 q^{11} - 2 q^{13} + 8 q^{19} + 6 q^{23} - 8 q^{25} - 10 q^{29} - 10 q^{31} - 2 q^{35} - 8 q^{37} - 14 q^{41} + 10 q^{43} - 2 q^{47} - 8 q^{53} - 2 q^{55} + 14 q^{59} + 6 q^{61} - 2 q^{65} + 10 q^{67} + 4 q^{71} - 10 q^{77} - 22 q^{79} - 6 q^{83} - 16 q^{89} + 26 q^{91} + 8 q^{95} + 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^11 - 2 * q^13 + 8 * q^19 + 6 * q^23 - 8 * q^25 - 10 * q^29 - 10 * q^31 - 2 * q^35 - 8 * q^37 - 14 * q^41 + 10 * q^43 - 2 * q^47 - 8 * q^53 - 2 * q^55 + 14 * q^59 + 6 * q^61 - 2 * q^65 + 10 * q^67 + 4 * q^71 - 10 * q^77 - 22 * q^79 - 6 * q^83 - 16 * q^89 + 26 * q^91 + 8 * q^95 + 2 * 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
 2.44949 −2.44949
0 0 0 1.00000 0 −3.44949 0 0 0
1.2 0 0 0 1.00000 0 1.44949 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.bu 2
3.b odd 2 1 5184.2.a.bj 2
4.b odd 2 1 5184.2.a.by 2
8.b even 2 1 2592.2.a.j 2
8.d odd 2 1 2592.2.a.n 2
9.c even 3 2 576.2.i.m 4
9.d odd 6 2 1728.2.i.m 4
12.b even 2 1 5184.2.a.bn 2
24.f even 2 1 2592.2.a.s 2
24.h odd 2 1 2592.2.a.o 2
36.f odd 6 2 576.2.i.i 4
36.h even 6 2 1728.2.i.k 4
72.j odd 6 2 864.2.i.e 4
72.l even 6 2 864.2.i.c 4
72.n even 6 2 288.2.i.c 4
72.p odd 6 2 288.2.i.e yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.c 4 72.n even 6 2
288.2.i.e yes 4 72.p odd 6 2
576.2.i.i 4 36.f odd 6 2
576.2.i.m 4 9.c even 3 2
864.2.i.c 4 72.l even 6 2
864.2.i.e 4 72.j odd 6 2
1728.2.i.k 4 36.h even 6 2
1728.2.i.m 4 9.d odd 6 2
2592.2.a.j 2 8.b even 2 1
2592.2.a.n 2 8.d odd 2 1
2592.2.a.o 2 24.h odd 2 1
2592.2.a.s 2 24.f even 2 1
5184.2.a.bj 2 3.b odd 2 1
5184.2.a.bn 2 12.b even 2 1
5184.2.a.bu 2 1.a even 1 1 trivial
5184.2.a.by 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5184))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 2T - 5$$
$11$ $$T^{2} + 2T - 5$$
$13$ $$T^{2} + 2T - 23$$
$17$ $$T^{2} - 24$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} - 6T + 3$$
$29$ $$T^{2} + 10T + 1$$
$31$ $$T^{2} + 10T + 19$$
$37$ $$T^{2} + 8T - 8$$
$41$ $$T^{2} + 14T + 25$$
$43$ $$T^{2} - 10T - 29$$
$47$ $$T^{2} + 2T - 53$$
$53$ $$T^{2} + 8T - 8$$
$59$ $$T^{2} - 14T - 5$$
$61$ $$T^{2} - 6T - 15$$
$67$ $$T^{2} - 10T - 29$$
$71$ $$T^{2} - 4T - 92$$
$73$ $$T^{2} - 24$$
$79$ $$T^{2} + 22T + 115$$
$83$ $$T^{2} + 6T + 3$$
$89$ $$T^{2} + 16T + 40$$
$97$ $$T^{2} - 2T - 23$$