# Properties

 Label 1161.1.i.b Level $1161$ Weight $1$ Character orbit 1161.i Analytic conductor $0.579$ Analytic rank $0$ Dimension $4$ Projective image $S_{4}$ CM/RM no 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$S_{4}$$ Projective field Galois closure of 4.2.49923.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} - q^{4} -\beta_{1} q^{5} +O(q^{10})$$ $$q -\beta_{3} q^{2} - q^{4} -\beta_{1} q^{5} + ( -2 + 2 \beta_{2} ) q^{10} -\beta_{2} q^{13} - q^{16} + ( -1 + \beta_{2} ) q^{19} + \beta_{1} q^{20} -\beta_{1} q^{23} + \beta_{2} q^{25} + ( -\beta_{1} + \beta_{3} ) q^{26} + ( 1 - \beta_{2} ) q^{31} + \beta_{3} q^{32} + \beta_{1} q^{38} + \beta_{3} q^{41} - q^{43} + ( -2 + 2 \beta_{2} ) q^{46} -\beta_{3} q^{47} + ( 1 - \beta_{2} ) q^{49} + ( \beta_{1} - \beta_{3} ) q^{50} + \beta_{2} q^{52} + \beta_{1} q^{53} + \beta_{2} q^{61} -\beta_{1} q^{62} + q^{64} + \beta_{3} q^{65} + ( \beta_{1} - \beta_{3} ) q^{71} -2 \beta_{2} q^{73} + ( 1 - \beta_{2} ) q^{76} -\beta_{2} q^{79} + \beta_{1} q^{80} + 2 q^{82} + \beta_{3} q^{86} -\beta_{1} q^{89} + \beta_{1} q^{92} -2 q^{94} + ( \beta_{1} - \beta_{3} ) q^{95} + q^{97} -\beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + O(q^{10})$$ $$4q - 4q^{4} - 4q^{10} - 2q^{13} - 4q^{16} - 2q^{19} + 2q^{25} + 2q^{31} - 4q^{43} - 4q^{46} + 2q^{49} + 2q^{52} + 2q^{61} + 4q^{64} - 4q^{73} + 2q^{76} - 2q^{79} + 8q^{82} - 8q^{94} + 4q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$433$$ $$947$$ $$\chi(n)$$ $$-\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
350.1
 1.22474 + 0.707107i −1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
1.41421i 0 −1.00000 −1.22474 0.707107i 0 0 0 0 −1.00000 + 1.73205i
350.2 1.41421i 0 −1.00000 1.22474 + 0.707107i 0 0 0 0 −1.00000 + 1.73205i
1025.1 1.41421i 0 −1.00000 1.22474 0.707107i 0 0 0 0 −1.00000 1.73205i
1025.2 1.41421i 0 −1.00000 −1.22474 + 0.707107i 0 0 0 0 −1.00000 1.73205i
 $$n$$: 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 inner
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.b 4
3.b odd 2 1 inner 1161.1.i.b 4
9.c even 3 1 3483.1.o.b 4
9.c even 3 1 3483.1.p.b 4
9.d odd 6 1 3483.1.o.b 4
9.d odd 6 1 3483.1.p.b 4
43.c even 3 1 inner 1161.1.i.b 4
129.f odd 6 1 inner 1161.1.i.b 4
387.e even 3 1 3483.1.o.b 4
387.g even 3 1 3483.1.p.b 4
387.o odd 6 1 3483.1.p.b 4
387.p odd 6 1 3483.1.o.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1161.1.i.b 4 1.a even 1 1 trivial
1161.1.i.b 4 3.b odd 2 1 inner
1161.1.i.b 4 43.c even 3 1 inner
1161.1.i.b 4 129.f odd 6 1 inner
3483.1.o.b 4 9.c even 3 1
3483.1.o.b 4 9.d odd 6 1
3483.1.o.b 4 387.e even 3 1
3483.1.o.b 4 387.p odd 6 1
3483.1.p.b 4 9.c even 3 1
3483.1.p.b 4 9.d odd 6 1
3483.1.p.b 4 387.g even 3 1
3483.1.p.b 4 387.o odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ 1
$5$ $$1 - T^{4} + T^{8}$$
$7$ $$( 1 - T^{2} + T^{4} )^{2}$$
$11$ $$( 1 - T )^{4}( 1 + T )^{4}$$
$13$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
$17$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
$19$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
$23$ $$1 - T^{4} + T^{8}$$
$29$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
$31$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
$37$ $$( 1 - T^{2} + T^{4} )^{2}$$
$41$ $$( 1 + T^{4} )^{2}$$
$43$ $$( 1 + T )^{4}$$
$47$ $$( 1 + T^{4} )^{2}$$
$53$ $$1 - T^{4} + T^{8}$$
$59$ $$( 1 - T )^{4}( 1 + T )^{4}$$
$61$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
$67$ $$( 1 - T^{2} + T^{4} )^{2}$$
$71$ $$1 - T^{4} + T^{8}$$
$73$ $$( 1 + T + T^{2} )^{4}$$
$79$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
$83$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
$89$ $$1 - T^{4} + T^{8}$$
$97$ $$( 1 - T + T^{2} )^{4}$$