# Properties

 Label 1089.1.k.a Level $1089$ Weight $1$ Character orbit 1089.k Analytic conductor $0.543$ Analytic rank $0$ Dimension $8$ Projective image $D_{4}$ CM discriminant -3 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1089.k (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.543481798757$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.64000000.1 Defining polynomial: $$x^{8} - 2 x^{6} + 4 x^{4} - 8 x^{2} + 16$$ 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: $C_5\times {\rm SD}_{16}$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{40} - \cdots)$$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{4} -\beta_{7} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{4} -\beta_{7} q^{7} -\beta_{1} q^{13} + \beta_{4} q^{16} -\beta_{3} q^{19} + ( 1 - \beta_{2} + \beta_{4} - \beta_{6} ) q^{25} + ( -\beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} ) q^{28} -\beta_{5} q^{43} -\beta_{4} q^{49} + \beta_{3} q^{52} + ( \beta_{1} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{61} -\beta_{6} q^{64} + \beta_{7} q^{73} + \beta_{5} q^{76} + \beta_{1} q^{79} + ( -2 + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{91} + 2 \beta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{4} + O(q^{10})$$ $$8q - 2q^{4} - 2q^{16} + 2q^{25} + 2q^{49} - 2q^{64} - 4q^{91} + 4q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{6} + 4 x^{4} - 8 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4}$$$$/4$$ $$\beta_{5}$$ $$=$$ $$\nu^{5}$$$$/4$$ $$\beta_{6}$$ $$=$$ $$\nu^{6}$$$$/8$$ $$\beta_{7}$$ $$=$$ $$\nu^{7}$$$$/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{6}$$ $$\nu^{7}$$ $$=$$ $$8 \beta_{7}$$

## Character values

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

 $$n$$ $$244$$ $$848$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
118.1
 0.831254 + 1.14412i −0.831254 − 1.14412i 0.831254 − 1.14412i −0.831254 + 1.14412i −1.34500 + 0.437016i 1.34500 − 0.437016i −1.34500 − 0.437016i 1.34500 + 0.437016i
0 0 0.309017 0.951057i 0 0 −1.34500 0.437016i 0 0 0
118.2 0 0 0.309017 0.951057i 0 0 1.34500 + 0.437016i 0 0 0
766.1 0 0 0.309017 + 0.951057i 0 0 −1.34500 + 0.437016i 0 0 0
766.2 0 0 0.309017 + 0.951057i 0 0 1.34500 0.437016i 0 0 0
820.1 0 0 −0.809017 + 0.587785i 0 0 −0.831254 1.14412i 0 0 0
820.2 0 0 −0.809017 + 0.587785i 0 0 0.831254 + 1.14412i 0 0 0
838.1 0 0 −0.809017 0.587785i 0 0 −0.831254 + 1.14412i 0 0 0
838.2 0 0 −0.809017 0.587785i 0 0 0.831254 1.14412i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 838.2 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
11.c even 5 3 inner
11.d odd 10 3 inner
33.d even 2 1 inner
33.f even 10 3 inner
33.h odd 10 3 inner

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$16 - 8 T^{2} + 4 T^{4} - 2 T^{6} + T^{8}$$
$11$ $$T^{8}$$
$13$ $$16 - 8 T^{2} + 4 T^{4} - 2 T^{6} + T^{8}$$
$17$ $$T^{8}$$
$19$ $$16 - 8 T^{2} + 4 T^{4} - 2 T^{6} + T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$( 2 + T^{2} )^{4}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$16 - 8 T^{2} + 4 T^{4} - 2 T^{6} + T^{8}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$16 - 8 T^{2} + 4 T^{4} - 2 T^{6} + T^{8}$$
$79$ $$16 - 8 T^{2} + 4 T^{4} - 2 T^{6} + T^{8}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$( 16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4} )^{2}$$