# Properties

 Label 1584.1.bf.b Level $1584$ Weight $1$ Character orbit 1584.bf Analytic conductor $0.791$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -11 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1584 = 2^{4} \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1584.bf (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.790518980011$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 99) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.891.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} + \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{6}^{2} q^{3} - \zeta_{6}^{2} q^{5} - \zeta_{6} q^{9} +O(q^{10})$$ q - z^2 * q^3 - z^2 * q^5 - z * q^9 $$q - \zeta_{6}^{2} q^{3} - \zeta_{6}^{2} q^{5} - \zeta_{6} q^{9} + \zeta_{6} q^{11} - \zeta_{6} q^{15} - \zeta_{6}^{2} q^{23} - q^{27} + \zeta_{6}^{2} q^{31} + q^{33} - q^{37} - q^{45} - \zeta_{6} q^{47} + \zeta_{6}^{2} q^{49} - q^{53} + q^{55} + \zeta_{6}^{2} q^{59} + \zeta_{6}^{2} q^{67} - 2 \zeta_{6} q^{69} + q^{71} + \zeta_{6}^{2} q^{81} + q^{89} + \zeta_{6} q^{93} + \zeta_{6} q^{97} - \zeta_{6}^{2} q^{99} +O(q^{100})$$ q - z^2 * q^3 - z^2 * q^5 - z * q^9 + z * q^11 - z * q^15 - z^2 * q^23 - q^27 + z^2 * q^31 + q^33 - q^37 - q^45 - z * q^47 + z^2 * q^49 - q^53 + q^55 + z^2 * q^59 + z^2 * q^67 - 2*z * q^69 + q^71 + z^2 * q^81 + q^89 + z * q^93 + z * q^97 - z^2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{5} - q^{9}+O(q^{10})$$ 2 * q + q^3 + q^5 - q^9 $$2 q + q^{3} + q^{5} - q^{9} + q^{11} - q^{15} + 2 q^{23} - 2 q^{27} - q^{31} + 2 q^{33} - 2 q^{37} - 2 q^{45} - q^{47} - q^{49} - 2 q^{53} + 2 q^{55} - q^{59} - q^{67} - 2 q^{69} + 2 q^{71} - q^{81} + 4 q^{89} + q^{93} + q^{97} + q^{99}+O(q^{100})$$ 2 * q + q^3 + q^5 - q^9 + q^11 - q^15 + 2 * q^23 - 2 * q^27 - q^31 + 2 * q^33 - 2 * q^37 - 2 * q^45 - q^47 - q^49 - 2 * q^53 + 2 * q^55 - q^59 - q^67 - 2 * q^69 + 2 * q^71 - q^81 + 4 * q^89 + q^93 + q^97 + q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$$.

 $$n$$ $$145$$ $$353$$ $$991$$ $$1189$$ $$\chi(n)$$ $$-1$$ $$\zeta_{6}^{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
241.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 0 0 −0.500000 + 0.866025i 0
769.1 0 0.500000 0.866025i 0 0.500000 0.866025i 0 0 0 −0.500000 0.866025i 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
9.c even 3 1 inner
99.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1584.1.bf.b 2
4.b odd 2 1 99.1.h.a 2
9.c even 3 1 inner 1584.1.bf.b 2
11.b odd 2 1 CM 1584.1.bf.b 2
12.b even 2 1 297.1.h.a 2
20.d odd 2 1 2475.1.y.a 2
20.e even 4 2 2475.1.t.a 4
36.f odd 6 1 99.1.h.a 2
36.f odd 6 1 891.1.c.a 1
36.h even 6 1 297.1.h.a 2
36.h even 6 1 891.1.c.b 1
44.c even 2 1 99.1.h.a 2
44.g even 10 4 1089.1.s.a 8
44.h odd 10 4 1089.1.s.a 8
99.h odd 6 1 inner 1584.1.bf.b 2
132.d odd 2 1 297.1.h.a 2
132.n odd 10 4 3267.1.w.a 8
132.o even 10 4 3267.1.w.a 8
180.p odd 6 1 2475.1.y.a 2
180.x even 12 2 2475.1.t.a 4
220.g even 2 1 2475.1.y.a 2
220.i odd 4 2 2475.1.t.a 4
396.k even 6 1 99.1.h.a 2
396.k even 6 1 891.1.c.a 1
396.o odd 6 1 297.1.h.a 2
396.o odd 6 1 891.1.c.b 1
396.ba even 30 4 3267.1.w.a 8
396.bb odd 30 4 3267.1.w.a 8
396.be odd 30 4 1089.1.s.a 8
396.bf even 30 4 1089.1.s.a 8
1980.bk even 6 1 2475.1.y.a 2
1980.cf odd 12 2 2475.1.t.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.1.h.a 2 4.b odd 2 1
99.1.h.a 2 36.f odd 6 1
99.1.h.a 2 44.c even 2 1
99.1.h.a 2 396.k even 6 1
297.1.h.a 2 12.b even 2 1
297.1.h.a 2 36.h even 6 1
297.1.h.a 2 132.d odd 2 1
297.1.h.a 2 396.o odd 6 1
891.1.c.a 1 36.f odd 6 1
891.1.c.a 1 396.k even 6 1
891.1.c.b 1 36.h even 6 1
891.1.c.b 1 396.o odd 6 1
1089.1.s.a 8 44.g even 10 4
1089.1.s.a 8 44.h odd 10 4
1089.1.s.a 8 396.be odd 30 4
1089.1.s.a 8 396.bf even 30 4
1584.1.bf.b 2 1.a even 1 1 trivial
1584.1.bf.b 2 9.c even 3 1 inner
1584.1.bf.b 2 11.b odd 2 1 CM
1584.1.bf.b 2 99.h odd 6 1 inner
2475.1.t.a 4 20.e even 4 2
2475.1.t.a 4 180.x even 12 2
2475.1.t.a 4 220.i odd 4 2
2475.1.t.a 4 1980.cf odd 12 2
2475.1.y.a 2 20.d odd 2 1
2475.1.y.a 2 180.p odd 6 1
2475.1.y.a 2 220.g even 2 1
2475.1.y.a 2 1980.bk even 6 1
3267.1.w.a 8 132.n odd 10 4
3267.1.w.a 8 132.o even 10 4
3267.1.w.a 8 396.ba even 30 4
3267.1.w.a 8 396.bb odd 30 4

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1584, [\chi])$$:

 $$T_{5}^{2} - T_{5} + 1$$ T5^2 - T5 + 1 $$T_{23}^{2} - 2T_{23} + 4$$ T23^2 - 2*T23 + 4

## Hecke characteristic polynomials

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