# Properties

 Label 5184.2.c.b Level $5184$ Weight $2$ Character orbit 5184.c 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$41.3944484078$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 144) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{5} + ( 1 - 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{5} + ( 1 - 2 \zeta_{6} ) q^{7} -3 q^{11} + 5 q^{13} + ( 4 - 8 \zeta_{6} ) q^{17} + ( -2 + 4 \zeta_{6} ) q^{19} + 9 q^{23} + 2 q^{25} + ( -1 + 2 \zeta_{6} ) q^{29} + ( -3 + 6 \zeta_{6} ) q^{31} -3 q^{35} -2 q^{37} + ( 3 - 6 \zeta_{6} ) q^{41} + ( 3 - 6 \zeta_{6} ) q^{43} + 3 q^{47} + 4 q^{49} + ( -3 + 6 \zeta_{6} ) q^{55} + 3 q^{59} + q^{61} + ( 5 - 10 \zeta_{6} ) q^{65} + ( -5 + 10 \zeta_{6} ) q^{67} + 12 q^{71} -2 q^{73} + ( -3 + 6 \zeta_{6} ) q^{77} + ( 5 - 10 \zeta_{6} ) q^{79} -15 q^{83} -12 q^{85} + ( -4 + 8 \zeta_{6} ) q^{89} + ( 5 - 10 \zeta_{6} ) q^{91} + 6 q^{95} -5 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 6q^{11} + 10q^{13} + 18q^{23} + 4q^{25} - 6q^{35} - 4q^{37} + 6q^{47} + 8q^{49} + 6q^{59} + 2q^{61} + 24q^{71} - 4q^{73} - 30q^{83} - 24q^{85} + 12q^{95} - 10q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1217$$ $$2431$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5183.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 1.73205i 0 1.73205i 0 0 0
5183.2 0 0 0 1.73205i 0 1.73205i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.c.b 2
3.b odd 2 1 5184.2.c.d 2
4.b odd 2 1 5184.2.c.d 2
8.b even 2 1 1296.2.c.c 2
8.d odd 2 1 1296.2.c.a 2
9.c even 3 1 576.2.s.c 2
9.c even 3 1 1728.2.s.c 2
9.d odd 6 1 576.2.s.b 2
9.d odd 6 1 1728.2.s.d 2
12.b even 2 1 inner 5184.2.c.b 2
24.f even 2 1 1296.2.c.c 2
24.h odd 2 1 1296.2.c.a 2
36.f odd 6 1 576.2.s.b 2
36.f odd 6 1 1728.2.s.d 2
36.h even 6 1 576.2.s.c 2
36.h even 6 1 1728.2.s.c 2
72.j odd 6 1 144.2.s.b 2
72.j odd 6 1 432.2.s.b 2
72.l even 6 1 144.2.s.c yes 2
72.l even 6 1 432.2.s.a 2
72.n even 6 1 144.2.s.c yes 2
72.n even 6 1 432.2.s.a 2
72.p odd 6 1 144.2.s.b 2
72.p odd 6 1 432.2.s.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.b 2 72.j odd 6 1
144.2.s.b 2 72.p odd 6 1
144.2.s.c yes 2 72.l even 6 1
144.2.s.c yes 2 72.n even 6 1
432.2.s.a 2 72.l even 6 1
432.2.s.a 2 72.n even 6 1
432.2.s.b 2 72.j odd 6 1
432.2.s.b 2 72.p odd 6 1
576.2.s.b 2 9.d odd 6 1
576.2.s.b 2 36.f odd 6 1
576.2.s.c 2 9.c even 3 1
576.2.s.c 2 36.h even 6 1
1296.2.c.a 2 8.d odd 2 1
1296.2.c.a 2 24.h odd 2 1
1296.2.c.c 2 8.b even 2 1
1296.2.c.c 2 24.f even 2 1
1728.2.s.c 2 9.c even 3 1
1728.2.s.c 2 36.h even 6 1
1728.2.s.d 2 9.d odd 6 1
1728.2.s.d 2 36.f odd 6 1
5184.2.c.b 2 1.a even 1 1 trivial
5184.2.c.b 2 12.b even 2 1 inner
5184.2.c.d 2 3.b odd 2 1
5184.2.c.d 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 $$S_{2}^{\mathrm{new}}(5184, [\chi])$$:

 $$T_{5}^{2} + 3$$ $$T_{7}^{2} + 3$$ $$T_{11} + 3$$