# Properties

 Label 5184.2.a.bd Level $5184$ Weight $2$ Character orbit 5184.a Self dual yes Analytic conductor $41.394$ Analytic rank $0$ Dimension $1$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 162) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{5} + 4q^{7} + O(q^{10})$$ $$q + 3q^{5} + 4q^{7} + q^{13} - 3q^{17} - 4q^{19} + 4q^{25} - 9q^{29} + 4q^{31} + 12q^{35} + q^{37} + 6q^{41} + 8q^{43} + 12q^{47} + 9q^{49} + 6q^{53} + q^{61} + 3q^{65} - 4q^{67} + 12q^{71} + 11q^{73} + 16q^{79} - 12q^{83} - 9q^{85} - 3q^{89} + 4q^{91} - 12q^{95} + 2q^{97} + O(q^{100})$$

## 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
 0
0 0 0 3.00000 0 4.00000 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

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.bd 1
3.b odd 2 1 5184.2.a.h 1
4.b odd 2 1 5184.2.a.y 1
8.b even 2 1 1296.2.a.c 1
8.d odd 2 1 162.2.a.a 1
12.b even 2 1 5184.2.a.c 1
24.f even 2 1 162.2.a.d yes 1
24.h odd 2 1 1296.2.a.l 1
40.e odd 2 1 4050.2.a.bh 1
40.k even 4 2 4050.2.c.g 2
56.e even 2 1 7938.2.a.n 1
72.j odd 6 2 1296.2.i.b 2
72.l even 6 2 162.2.c.a 2
72.n even 6 2 1296.2.i.n 2
72.p odd 6 2 162.2.c.d 2
120.m even 2 1 4050.2.a.r 1
120.q odd 4 2 4050.2.c.n 2
168.e odd 2 1 7938.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.a.a 1 8.d odd 2 1
162.2.a.d yes 1 24.f even 2 1
162.2.c.a 2 72.l even 6 2
162.2.c.d 2 72.p odd 6 2
1296.2.a.c 1 8.b even 2 1
1296.2.a.l 1 24.h odd 2 1
1296.2.i.b 2 72.j odd 6 2
1296.2.i.n 2 72.n even 6 2
4050.2.a.r 1 120.m even 2 1
4050.2.a.bh 1 40.e odd 2 1
4050.2.c.g 2 40.k even 4 2
4050.2.c.n 2 120.q odd 4 2
5184.2.a.c 1 12.b even 2 1
5184.2.a.h 1 3.b odd 2 1
5184.2.a.y 1 4.b odd 2 1
5184.2.a.bd 1 1.a even 1 1 trivial
7938.2.a.n 1 56.e even 2 1
7938.2.a.s 1 168.e odd 2 1

## Hecke kernels

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

 $$T_{5} - 3$$ $$T_{7} - 4$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 3 T + 5 T^{2}$$
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 + 11 T^{2}$$
$13$ $$1 - T + 13 T^{2}$$
$17$ $$1 + 3 T + 17 T^{2}$$
$19$ $$1 + 4 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 + 9 T + 29 T^{2}$$
$31$ $$1 - 4 T + 31 T^{2}$$
$37$ $$1 - T + 37 T^{2}$$
$41$ $$1 - 6 T + 41 T^{2}$$
$43$ $$1 - 8 T + 43 T^{2}$$
$47$ $$1 - 12 T + 47 T^{2}$$
$53$ $$1 - 6 T + 53 T^{2}$$
$59$ $$1 + 59 T^{2}$$
$61$ $$1 - T + 61 T^{2}$$
$67$ $$1 + 4 T + 67 T^{2}$$
$71$ $$1 - 12 T + 71 T^{2}$$
$73$ $$1 - 11 T + 73 T^{2}$$
$79$ $$1 - 16 T + 79 T^{2}$$
$83$ $$1 + 12 T + 83 T^{2}$$
$89$ $$1 + 3 T + 89 T^{2}$$
$97$ $$1 - 2 T + 97 T^{2}$$