# Properties

 Label 1089.1.c.a Level $1089$ Weight $1$ Character orbit 1089.c Analytic conductor $0.543$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.543481798757$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.3993.1 Artin image: $SD_{16}$ Artin field: Galois closure of 8.2.14206147659.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{4} -\beta q^{7} +O(q^{10})$$ $$q + q^{4} -\beta q^{7} + \beta q^{13} + q^{16} -\beta q^{19} - q^{25} -\beta q^{28} + \beta q^{43} - q^{49} + \beta q^{52} + \beta q^{61} + q^{64} + \beta q^{73} -\beta q^{76} -\beta q^{79} + 2 q^{91} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + O(q^{10})$$ $$2q + 2q^{4} + 2q^{16} - 2q^{25} - 2q^{49} + 2q^{64} + 4q^{91} - 4q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$244$$ $$848$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
604.1
 1.41421i − 1.41421i
0 0 1.00000 0 0 1.41421i 0 0 0
604.2 0 0 1.00000 0 0 1.41421i 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 $$\Q(\sqrt{-3})$$
11.b odd 2 1 inner
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.1.c.a 2
3.b odd 2 1 CM 1089.1.c.a 2
11.b odd 2 1 inner 1089.1.c.a 2
11.c even 5 4 1089.1.k.a 8
11.d odd 10 4 1089.1.k.a 8
33.d even 2 1 inner 1089.1.c.a 2
33.f even 10 4 1089.1.k.a 8
33.h odd 10 4 1089.1.k.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.1.c.a 2 1.a even 1 1 trivial
1089.1.c.a 2 3.b odd 2 1 CM
1089.1.c.a 2 11.b odd 2 1 inner
1089.1.c.a 2 33.d even 2 1 inner
1089.1.k.a 8 11.c even 5 4
1089.1.k.a 8 11.d odd 10 4
1089.1.k.a 8 33.f even 10 4
1089.1.k.a 8 33.h odd 10 4

## Hecke kernels

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

## Hecke characteristic polynomials

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