# Properties

 Label 5184.2.a.d Level $5184$ Weight $2$ Character orbit 5184.a Self dual yes Analytic conductor $41.394$ Analytic rank $2$ 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: $$2$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 324) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3 q^{5} - 2 q^{7}+O(q^{10})$$ q - 3 * q^5 - 2 * q^7 $$q - 3 q^{5} - 2 q^{7} - 6 q^{11} - 5 q^{13} - 3 q^{17} + 2 q^{19} - 6 q^{23} + 4 q^{25} - 3 q^{29} + 4 q^{31} + 6 q^{35} - 5 q^{37} - 6 q^{41} - 10 q^{43} - 3 q^{49} + 6 q^{53} + 18 q^{55} - 12 q^{59} - 5 q^{61} + 15 q^{65} + 2 q^{67} - 6 q^{71} - q^{73} + 12 q^{77} + 10 q^{79} + 9 q^{85} - 3 q^{89} + 10 q^{91} - 6 q^{95} - 10 q^{97}+O(q^{100})$$ q - 3 * q^5 - 2 * q^7 - 6 * q^11 - 5 * q^13 - 3 * q^17 + 2 * q^19 - 6 * q^23 + 4 * q^25 - 3 * q^29 + 4 * q^31 + 6 * q^35 - 5 * q^37 - 6 * q^41 - 10 * q^43 - 3 * q^49 + 6 * q^53 + 18 * q^55 - 12 * q^59 - 5 * q^61 + 15 * q^65 + 2 * q^67 - 6 * q^71 - q^73 + 12 * q^77 + 10 * q^79 + 9 * q^85 - 3 * q^89 + 10 * q^91 - 6 * q^95 - 10 * 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
 0
0 0 0 −3.00000 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

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.d 1
3.b odd 2 1 5184.2.a.z 1
4.b odd 2 1 5184.2.a.g 1
8.b even 2 1 1296.2.a.j 1
8.d odd 2 1 324.2.a.d yes 1
12.b even 2 1 5184.2.a.bc 1
24.f even 2 1 324.2.a.b 1
24.h odd 2 1 1296.2.a.a 1
40.e odd 2 1 8100.2.a.a 1
40.k even 4 2 8100.2.d.a 2
72.j odd 6 2 1296.2.i.p 2
72.l even 6 2 324.2.e.d 2
72.n even 6 2 1296.2.i.d 2
72.p odd 6 2 324.2.e.a 2
120.m even 2 1 8100.2.a.f 1
120.q odd 4 2 8100.2.d.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.2.a.b 1 24.f even 2 1
324.2.a.d yes 1 8.d odd 2 1
324.2.e.a 2 72.p odd 6 2
324.2.e.d 2 72.l even 6 2
1296.2.a.a 1 24.h odd 2 1
1296.2.a.j 1 8.b even 2 1
1296.2.i.d 2 72.n even 6 2
1296.2.i.p 2 72.j odd 6 2
5184.2.a.d 1 1.a even 1 1 trivial
5184.2.a.g 1 4.b odd 2 1
5184.2.a.z 1 3.b odd 2 1
5184.2.a.bc 1 12.b even 2 1
8100.2.a.a 1 40.e odd 2 1
8100.2.a.f 1 120.m even 2 1
8100.2.d.a 2 40.k even 4 2
8100.2.d.j 2 120.q odd 4 2

## Hecke kernels

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

 $$T_{5} + 3$$ T5 + 3 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 6$$ T11 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 3$$
$7$ $$T + 2$$
$11$ $$T + 6$$
$13$ $$T + 5$$
$17$ $$T + 3$$
$19$ $$T - 2$$
$23$ $$T + 6$$
$29$ $$T + 3$$
$31$ $$T - 4$$
$37$ $$T + 5$$
$41$ $$T + 6$$
$43$ $$T + 10$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T + 12$$
$61$ $$T + 5$$
$67$ $$T - 2$$
$71$ $$T + 6$$
$73$ $$T + 1$$
$79$ $$T - 10$$
$83$ $$T$$
$89$ $$T + 3$$
$97$ $$T + 10$$