# Properties

 Label 1539.1.s.a Level $1539$ Weight $1$ Character orbit 1539.s Analytic conductor $0.768$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1539 = 3^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1539.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.768061054442$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 171) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.22284891.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{7} +O(q^{10})$$ $$q + \zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{7} + ( 1 + \zeta_{6} ) q^{13} -\zeta_{6} q^{16} - q^{19} - q^{25} -\zeta_{6} q^{28} + ( -1 + \zeta_{6}^{2} ) q^{31} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{37} + \zeta_{6} q^{43} + ( -1 + \zeta_{6}^{2} ) q^{52} + q^{61} + q^{64} + ( -1 - \zeta_{6} ) q^{67} -\zeta_{6}^{2} q^{73} -\zeta_{6}^{2} q^{76} + ( 1 - \zeta_{6}^{2} ) q^{79} + ( -1 + \zeta_{6}^{2} ) q^{91} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{4} - q^{7} + O(q^{10})$$ $$2q - q^{4} - q^{7} + 3q^{13} - q^{16} - 2q^{19} - 2q^{25} - q^{28} - 3q^{31} + q^{43} - 3q^{52} + 2q^{61} + 2q^{64} - 3q^{67} + q^{73} + q^{76} + 3q^{79} - 3q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1539\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1217$$ $$\chi(n)$$ $$\zeta_{6}$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
217.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 −0.500000 + 0.866025i 0 0 −0.500000 + 0.866025i 0 0 0
1000.1 0 0 −0.500000 0.866025i 0 0 −0.500000 0.866025i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
171.k even 6 1 inner
171.s odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1539.1.s.a 2
3.b odd 2 1 CM 1539.1.s.a 2
9.c even 3 1 171.1.p.a 2
9.c even 3 1 1539.1.i.a 2
9.d odd 6 1 171.1.p.a 2
9.d odd 6 1 1539.1.i.a 2
19.d odd 6 1 1539.1.i.a 2
36.f odd 6 1 2736.1.cd.a 2
36.h even 6 1 2736.1.cd.a 2
57.f even 6 1 1539.1.i.a 2
171.g even 3 1 3249.1.c.a 2
171.h even 3 1 3249.1.p.b 2
171.i odd 6 1 171.1.p.a 2
171.j odd 6 1 3249.1.p.b 2
171.k even 6 1 inner 1539.1.s.a 2
171.k even 6 1 3249.1.c.a 2
171.l even 6 1 3249.1.p.b 2
171.n odd 6 1 3249.1.c.a 2
171.o odd 6 1 3249.1.p.b 2
171.s odd 6 1 inner 1539.1.s.a 2
171.s odd 6 1 3249.1.c.a 2
171.t even 6 1 171.1.p.a 2
171.v even 9 3 3249.1.ba.a 6
171.w even 9 3 3249.1.ba.b 6
171.x even 18 3 3249.1.ba.b 6
171.z odd 18 3 3249.1.ba.a 6
171.bc odd 18 3 3249.1.ba.b 6
171.bd even 18 3 3249.1.ba.a 6
171.be odd 18 3 3249.1.ba.a 6
171.bf odd 18 3 3249.1.ba.b 6
684.u even 6 1 2736.1.cd.a 2
684.bf odd 6 1 2736.1.cd.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.1.p.a 2 9.c even 3 1
171.1.p.a 2 9.d odd 6 1
171.1.p.a 2 171.i odd 6 1
171.1.p.a 2 171.t even 6 1
1539.1.i.a 2 9.c even 3 1
1539.1.i.a 2 9.d odd 6 1
1539.1.i.a 2 19.d odd 6 1
1539.1.i.a 2 57.f even 6 1
1539.1.s.a 2 1.a even 1 1 trivial
1539.1.s.a 2 3.b odd 2 1 CM
1539.1.s.a 2 171.k even 6 1 inner
1539.1.s.a 2 171.s odd 6 1 inner
2736.1.cd.a 2 36.f odd 6 1
2736.1.cd.a 2 36.h even 6 1
2736.1.cd.a 2 684.u even 6 1
2736.1.cd.a 2 684.bf odd 6 1
3249.1.c.a 2 171.g even 3 1
3249.1.c.a 2 171.k even 6 1
3249.1.c.a 2 171.n odd 6 1
3249.1.c.a 2 171.s odd 6 1
3249.1.p.b 2 171.h even 3 1
3249.1.p.b 2 171.j odd 6 1
3249.1.p.b 2 171.l even 6 1
3249.1.p.b 2 171.o odd 6 1
3249.1.ba.a 6 171.v even 9 3
3249.1.ba.a 6 171.z odd 18 3
3249.1.ba.a 6 171.bd even 18 3
3249.1.ba.a 6 171.be odd 18 3
3249.1.ba.b 6 171.w even 9 3
3249.1.ba.b 6 171.x even 18 3
3249.1.ba.b 6 171.bc odd 18 3
3249.1.ba.b 6 171.bf odd 18 3

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1539, [\chi])$$.

## Hecke characteristic polynomials

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