# Properties

 Label 5292.2.j.b Level $5292$ Weight $2$ Character orbit 5292.j Analytic conductor $42.257$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5292 = 2^{2} \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5292.j (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$42.2568327497$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 252) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{5} +O(q^{10})$$ $$q -2 \zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{11} + 3 \zeta_{6} q^{13} + 7 q^{17} -5 q^{19} + 4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + ( -1 + \zeta_{6} ) q^{29} -3 \zeta_{6} q^{31} + 11 q^{37} + 9 \zeta_{6} q^{41} + ( -5 + 5 \zeta_{6} ) q^{43} + ( -3 + 3 \zeta_{6} ) q^{47} -3 q^{53} -8 q^{55} + 7 \zeta_{6} q^{59} + ( 3 - 3 \zeta_{6} ) q^{61} + ( 6 - 6 \zeta_{6} ) q^{65} -13 \zeta_{6} q^{67} + 8 q^{71} -7 q^{73} + ( 9 - 9 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{83} -14 \zeta_{6} q^{85} + 15 q^{89} + 10 \zeta_{6} q^{95} + ( -17 + 17 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} + O(q^{10})$$ $$2q - 2q^{5} + 4q^{11} + 3q^{13} + 14q^{17} - 10q^{19} + 4q^{23} + q^{25} - q^{29} - 3q^{31} + 22q^{37} + 9q^{41} - 5q^{43} - 3q^{47} - 6q^{53} - 16q^{55} + 7q^{59} + 3q^{61} + 6q^{65} - 13q^{67} + 16q^{71} - 14q^{73} + 9q^{79} - q^{83} - 14q^{85} + 30q^{89} + 10q^{95} - 17q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$2647$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1765.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −1.00000 + 1.73205i 0 0 0 0 0
3529.1 0 0 0 −1.00000 1.73205i 0 0 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5292.2.j.b 2
3.b odd 2 1 1764.2.j.a 2
7.b odd 2 1 5292.2.j.c 2
7.c even 3 1 5292.2.i.b 2
7.c even 3 1 5292.2.l.b 2
7.d odd 6 1 756.2.i.a 2
7.d odd 6 1 756.2.l.a 2
9.c even 3 1 inner 5292.2.j.b 2
9.d odd 6 1 1764.2.j.a 2
21.c even 2 1 1764.2.j.c 2
21.g even 6 1 252.2.i.a 2
21.g even 6 1 252.2.l.a yes 2
21.h odd 6 1 1764.2.i.b 2
21.h odd 6 1 1764.2.l.b 2
28.f even 6 1 3024.2.q.e 2
28.f even 6 1 3024.2.t.b 2
63.g even 3 1 5292.2.i.b 2
63.h even 3 1 5292.2.l.b 2
63.i even 6 1 252.2.l.a yes 2
63.i even 6 1 2268.2.k.a 2
63.j odd 6 1 1764.2.l.b 2
63.k odd 6 1 756.2.i.a 2
63.k odd 6 1 2268.2.k.b 2
63.l odd 6 1 5292.2.j.c 2
63.n odd 6 1 1764.2.i.b 2
63.o even 6 1 1764.2.j.c 2
63.s even 6 1 252.2.i.a 2
63.s even 6 1 2268.2.k.a 2
63.t odd 6 1 756.2.l.a 2
63.t odd 6 1 2268.2.k.b 2
84.j odd 6 1 1008.2.q.f 2
84.j odd 6 1 1008.2.t.b 2
252.n even 6 1 3024.2.q.e 2
252.r odd 6 1 1008.2.t.b 2
252.bj even 6 1 3024.2.t.b 2
252.bn odd 6 1 1008.2.q.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.a 2 21.g even 6 1
252.2.i.a 2 63.s even 6 1
252.2.l.a yes 2 21.g even 6 1
252.2.l.a yes 2 63.i even 6 1
756.2.i.a 2 7.d odd 6 1
756.2.i.a 2 63.k odd 6 1
756.2.l.a 2 7.d odd 6 1
756.2.l.a 2 63.t odd 6 1
1008.2.q.f 2 84.j odd 6 1
1008.2.q.f 2 252.bn odd 6 1
1008.2.t.b 2 84.j odd 6 1
1008.2.t.b 2 252.r odd 6 1
1764.2.i.b 2 21.h odd 6 1
1764.2.i.b 2 63.n odd 6 1
1764.2.j.a 2 3.b odd 2 1
1764.2.j.a 2 9.d odd 6 1
1764.2.j.c 2 21.c even 2 1
1764.2.j.c 2 63.o even 6 1
1764.2.l.b 2 21.h odd 6 1
1764.2.l.b 2 63.j odd 6 1
2268.2.k.a 2 63.i even 6 1
2268.2.k.a 2 63.s even 6 1
2268.2.k.b 2 63.k odd 6 1
2268.2.k.b 2 63.t odd 6 1
3024.2.q.e 2 28.f even 6 1
3024.2.q.e 2 252.n even 6 1
3024.2.t.b 2 28.f even 6 1
3024.2.t.b 2 252.bj even 6 1
5292.2.i.b 2 7.c even 3 1
5292.2.i.b 2 63.g even 3 1
5292.2.j.b 2 1.a even 1 1 trivial
5292.2.j.b 2 9.c even 3 1 inner
5292.2.j.c 2 7.b odd 2 1
5292.2.j.c 2 63.l odd 6 1
5292.2.l.b 2 7.c even 3 1
5292.2.l.b 2 63.h even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 2 T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(5292, [\chi])$$.