# Properties

 Label 1161.1.i.a Level $1161$ Weight $1$ Character orbit 1161.i Analytic conductor $0.579$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1161 = 3^{3} \cdot 43$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1161.i (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.579414479667$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{3}$$ Projective field Galois closure of 3.1.49923.1 Artin image $C_3\times S_3$ Artin field Galois closure of 6.0.4043763.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{4} -2 \zeta_{6} q^{7} +O(q^{10})$$ $$q + q^{4} -2 \zeta_{6} q^{7} + \zeta_{6} q^{13} + q^{16} -\zeta_{6}^{2} q^{19} -\zeta_{6} q^{25} -2 \zeta_{6} q^{28} -\zeta_{6}^{2} q^{31} + 2 \zeta_{6}^{2} q^{37} + q^{43} + 3 \zeta_{6}^{2} q^{49} + \zeta_{6} q^{52} + \zeta_{6} q^{61} + q^{64} + 2 \zeta_{6}^{2} q^{67} -2 \zeta_{6} q^{73} -\zeta_{6}^{2} q^{76} + \zeta_{6} q^{79} -2 \zeta_{6}^{2} q^{91} - q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} - 2q^{7} + O(q^{10})$$ $$2q + 2q^{4} - 2q^{7} + q^{13} + 2q^{16} + q^{19} - q^{25} - 2q^{28} + q^{31} - 2q^{37} + 2q^{43} - 3q^{49} + q^{52} + q^{61} + 2q^{64} - 2q^{67} - 2q^{73} + q^{76} + q^{79} + 2q^{91} - 2q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1161\mathbb{Z}\right)^\times$$.

 $$n$$ $$433$$ $$947$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
350.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 1.00000 0 0 −1.00000 1.73205i 0 0 0
1025.1 0 0 1.00000 0 0 −1.00000 + 1.73205i 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 $$\Q(\sqrt{-3})$$
43.c even 3 1 inner
129.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1161.1.i.a 2
3.b odd 2 1 CM 1161.1.i.a 2
9.c even 3 1 3483.1.o.a 2
9.c even 3 1 3483.1.p.a 2
9.d odd 6 1 3483.1.o.a 2
9.d odd 6 1 3483.1.p.a 2
43.c even 3 1 inner 1161.1.i.a 2
129.f odd 6 1 inner 1161.1.i.a 2
387.e even 3 1 3483.1.o.a 2
387.g even 3 1 3483.1.p.a 2
387.o odd 6 1 3483.1.p.a 2
387.p odd 6 1 3483.1.o.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1161.1.i.a 2 1.a even 1 1 trivial
1161.1.i.a 2 3.b odd 2 1 CM
1161.1.i.a 2 43.c even 3 1 inner
1161.1.i.a 2 129.f odd 6 1 inner
3483.1.o.a 2 9.c even 3 1
3483.1.o.a 2 9.d odd 6 1
3483.1.o.a 2 387.e even 3 1
3483.1.o.a 2 387.p odd 6 1
3483.1.p.a 2 9.c even 3 1
3483.1.p.a 2 9.d odd 6 1
3483.1.p.a 2 387.g even 3 1
3483.1.p.a 2 387.o odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{1}^{\mathrm{new}}(1161, [\chi])$$.