# Properties

 Label 1089.1.i.a Level $1089$ Weight $1$ Character orbit 1089.i Analytic conductor $0.543$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -11 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1089.i (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.543481798757$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.26198073.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{6}^{2} q^{3} + \zeta_{6}^{2} q^{4} + ( 1 + \zeta_{6} ) q^{5} -\zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6}^{2} q^{3} + \zeta_{6}^{2} q^{4} + ( 1 + \zeta_{6} ) q^{5} -\zeta_{6} q^{9} + \zeta_{6} q^{12} + ( 1 - \zeta_{6}^{2} ) q^{15} -\zeta_{6} q^{16} + ( -1 + \zeta_{6}^{2} ) q^{20} + ( 1 + \zeta_{6} + \zeta_{6}^{2} ) q^{25} - q^{27} + \zeta_{6}^{2} q^{31} + q^{36} + q^{37} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{45} + ( -1 + \zeta_{6}^{2} ) q^{47} - q^{48} -\zeta_{6}^{2} q^{49} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{53} + ( -1 - \zeta_{6} ) q^{59} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{60} + q^{64} + \zeta_{6}^{2} q^{67} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{71} + ( 1 + \zeta_{6} - \zeta_{6}^{2} ) q^{75} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{80} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{93} + \zeta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{4} + 3 q^{5} - q^{9} + O(q^{10})$$ $$2 q + q^{3} - q^{4} + 3 q^{5} - q^{9} + q^{12} + 3 q^{15} - q^{16} - 3 q^{20} + 2 q^{25} - 2 q^{27} - q^{31} + 2 q^{36} + 2 q^{37} - 3 q^{47} - 2 q^{48} + q^{49} - 3 q^{59} + 2 q^{64} - q^{67} + 4 q^{75} - q^{81} + q^{93} + q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$848$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
122.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i −0.500000 + 0.866025i 1.50000 + 0.866025i 0 0 0 −0.500000 0.866025i 0
848.1 0 0.500000 + 0.866025i −0.500000 0.866025i 1.50000 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.d odd 6 1 inner
99.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.1.i.a 2
3.b odd 2 1 3267.1.i.a 2
9.c even 3 1 3267.1.i.a 2
9.d odd 6 1 inner 1089.1.i.a 2
11.b odd 2 1 CM 1089.1.i.a 2
11.c even 5 4 1089.1.r.a 8
11.d odd 10 4 1089.1.r.a 8
33.d even 2 1 3267.1.i.a 2
33.f even 10 4 3267.1.v.a 8
33.h odd 10 4 3267.1.v.a 8
99.g even 6 1 inner 1089.1.i.a 2
99.h odd 6 1 3267.1.i.a 2
99.m even 15 4 3267.1.v.a 8
99.n odd 30 4 1089.1.r.a 8
99.o odd 30 4 3267.1.v.a 8
99.p even 30 4 1089.1.r.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.1.i.a 2 1.a even 1 1 trivial
1089.1.i.a 2 9.d odd 6 1 inner
1089.1.i.a 2 11.b odd 2 1 CM
1089.1.i.a 2 99.g even 6 1 inner
1089.1.r.a 8 11.c even 5 4
1089.1.r.a 8 11.d odd 10 4
1089.1.r.a 8 99.n odd 30 4
1089.1.r.a 8 99.p even 30 4
3267.1.i.a 2 3.b odd 2 1
3267.1.i.a 2 9.c even 3 1
3267.1.i.a 2 33.d even 2 1
3267.1.i.a 2 99.h odd 6 1
3267.1.v.a 8 33.f even 10 4
3267.1.v.a 8 33.h odd 10 4
3267.1.v.a 8 99.m even 15 4
3267.1.v.a 8 99.o odd 30 4

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1089, [\chi])$$.

## Hecke characteristic polynomials

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