# Properties

 Label 5184.2.a.br Level $5184$ Weight $2$ Character orbit 5184.a Self dual yes Analytic conductor $41.394$ Analytic rank $0$ Dimension $2$ CM no 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 81) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} + 2 q^{7} +O(q^{10})$$ $$q + \beta q^{5} + 2 q^{7} + 2 \beta q^{11} + q^{13} + 3 \beta q^{17} -2 q^{19} + 2 \beta q^{23} -2 q^{25} -\beta q^{29} + 8 q^{31} + 2 \beta q^{35} + 7 q^{37} -4 \beta q^{41} -2 q^{43} + 4 \beta q^{47} -3 q^{49} + 6 q^{55} -8 \beta q^{59} + 7 q^{61} + \beta q^{65} + 10 q^{67} -6 \beta q^{71} -7 q^{73} + 4 \beta q^{77} + 2 q^{79} + 8 \beta q^{83} + 9 q^{85} -3 \beta q^{89} + 2 q^{91} -2 \beta q^{95} + 2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{7} + O(q^{10})$$ $$2 q + 4 q^{7} + 2 q^{13} - 4 q^{19} - 4 q^{25} + 16 q^{31} + 14 q^{37} - 4 q^{43} - 6 q^{49} + 12 q^{55} + 14 q^{61} + 20 q^{67} - 14 q^{73} + 4 q^{79} + 18 q^{85} + 4 q^{91} + 4 q^{97} + O(q^{100})$$

## 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 −1.73205 0 2.00000 0 0 0
1.2 0 0 0 1.73205 0 2.00000 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.a.br 2
3.b odd 2 1 inner 5184.2.a.br 2
4.b odd 2 1 5184.2.a.bq 2
8.b even 2 1 81.2.a.a 2
8.d odd 2 1 1296.2.a.o 2
12.b even 2 1 5184.2.a.bq 2
24.f even 2 1 1296.2.a.o 2
24.h odd 2 1 81.2.a.a 2
40.f even 2 1 2025.2.a.j 2
40.i odd 4 2 2025.2.b.k 4
56.h odd 2 1 3969.2.a.i 2
72.j odd 6 2 81.2.c.b 4
72.l even 6 2 1296.2.i.s 4
72.n even 6 2 81.2.c.b 4
72.p odd 6 2 1296.2.i.s 4
88.b odd 2 1 9801.2.a.v 2
120.i odd 2 1 2025.2.a.j 2
120.w even 4 2 2025.2.b.k 4
168.i even 2 1 3969.2.a.i 2
216.t even 18 6 729.2.e.o 12
216.x odd 18 6 729.2.e.o 12
264.m even 2 1 9801.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.a.a 2 8.b even 2 1
81.2.a.a 2 24.h odd 2 1
81.2.c.b 4 72.j odd 6 2
81.2.c.b 4 72.n even 6 2
729.2.e.o 12 216.t even 18 6
729.2.e.o 12 216.x odd 18 6
1296.2.a.o 2 8.d odd 2 1
1296.2.a.o 2 24.f even 2 1
1296.2.i.s 4 72.l even 6 2
1296.2.i.s 4 72.p odd 6 2
2025.2.a.j 2 40.f even 2 1
2025.2.a.j 2 120.i odd 2 1
2025.2.b.k 4 40.i odd 4 2
2025.2.b.k 4 120.w even 4 2
3969.2.a.i 2 56.h odd 2 1
3969.2.a.i 2 168.i even 2 1
5184.2.a.bq 2 4.b odd 2 1
5184.2.a.bq 2 12.b even 2 1
5184.2.a.br 2 1.a even 1 1 trivial
5184.2.a.br 2 3.b odd 2 1 inner
9801.2.a.v 2 88.b odd 2 1
9801.2.a.v 2 264.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5184))$$:

 $$T_{5}^{2} - 3$$ $$T_{7} - 2$$ $$T_{11}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-3 + T^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$-12 + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$-27 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$-12 + T^{2}$$
$29$ $$-3 + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$( -7 + T )^{2}$$
$41$ $$-48 + T^{2}$$
$43$ $$( 2 + T )^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$T^{2}$$
$59$ $$-192 + T^{2}$$
$61$ $$( -7 + T )^{2}$$
$67$ $$( -10 + T )^{2}$$
$71$ $$-108 + T^{2}$$
$73$ $$( 7 + T )^{2}$$
$79$ $$( -2 + T )^{2}$$
$83$ $$-192 + T^{2}$$
$89$ $$-27 + T^{2}$$
$97$ $$( -2 + T )^{2}$$