# Properties

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

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## 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2592) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{5} - 2 q^{7} + O(q^{10})$$ $$q - q^{5} - 2 q^{7} + 2 q^{11} - q^{13} + 3 q^{17} + 2 q^{19} - 6 q^{23} - 4 q^{25} - q^{29} + 8 q^{31} + 2 q^{35} - q^{37} - 2 q^{41} + 10 q^{43} - 4 q^{47} - 3 q^{49} + 10 q^{53} - 2 q^{55} + 4 q^{59} - 9 q^{61} + q^{65} - 14 q^{67} - 10 q^{71} - 9 q^{73} - 4 q^{77} - 10 q^{79} + 12 q^{83} - 3 q^{85} + 11 q^{89} + 2 q^{91} - 2 q^{95} - 2 q^{97} + O(q^{100})$$

## 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
 0
0 0 0 −1.00000 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.j 1
3.b odd 2 1 5184.2.a.t 1
4.b odd 2 1 5184.2.a.m 1
8.b even 2 1 2592.2.a.e yes 1
8.d odd 2 1 2592.2.a.f yes 1
12.b even 2 1 5184.2.a.w 1
24.f even 2 1 2592.2.a.d yes 1
24.h odd 2 1 2592.2.a.c 1
72.j odd 6 2 2592.2.i.o 2
72.l even 6 2 2592.2.i.n 2
72.n even 6 2 2592.2.i.k 2
72.p odd 6 2 2592.2.i.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.a.c 1 24.h odd 2 1
2592.2.a.d yes 1 24.f even 2 1
2592.2.a.e yes 1 8.b even 2 1
2592.2.a.f yes 1 8.d odd 2 1
2592.2.i.j 2 72.p odd 6 2
2592.2.i.k 2 72.n even 6 2
2592.2.i.n 2 72.l even 6 2
2592.2.i.o 2 72.j odd 6 2
5184.2.a.j 1 1.a even 1 1 trivial
5184.2.a.m 1 4.b odd 2 1
5184.2.a.t 1 3.b odd 2 1
5184.2.a.w 1 12.b even 2 1

## Hecke kernels

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

 $$T_{5} + 1$$ $$T_{7} + 2$$ $$T_{11} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$2 + T$$
$11$ $$-2 + T$$
$13$ $$1 + T$$
$17$ $$-3 + T$$
$19$ $$-2 + T$$
$23$ $$6 + T$$
$29$ $$1 + T$$
$31$ $$-8 + T$$
$37$ $$1 + T$$
$41$ $$2 + T$$
$43$ $$-10 + T$$
$47$ $$4 + T$$
$53$ $$-10 + T$$
$59$ $$-4 + T$$
$61$ $$9 + T$$
$67$ $$14 + T$$
$71$ $$10 + T$$
$73$ $$9 + T$$
$79$ $$10 + T$$
$83$ $$-12 + T$$
$89$ $$-11 + T$$
$97$ $$2 + T$$
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