# Properties

 Label 1161.1.i.c Level $1161$ Weight $1$ Character orbit 1161.i Analytic conductor $0.579$ Analytic rank $0$ Dimension $8$ Projective image $A_{5}$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.12960000.1 Defining polynomial: $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$A_{5}$$ Projective field Galois closure of 5.1.149769.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} + ( -\beta_{3} - \beta_{6} - \beta_{7} ) q^{5} + ( -\beta_{2} - \beta_{4} ) q^{7} + \beta_{7} q^{8} +O(q^{10})$$ $$q + \beta_{7} q^{2} + ( -\beta_{3} - \beta_{6} - \beta_{7} ) q^{5} + ( -\beta_{2} - \beta_{4} ) q^{7} + \beta_{7} q^{8} + ( 1 - \beta_{4} - \beta_{5} ) q^{10} + ( -\beta_{1} + \beta_{6} + \beta_{7} ) q^{11} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{13} + \beta_{1} q^{14} - q^{16} + ( \beta_{1} + \beta_{3} ) q^{17} + \beta_{4} q^{19} + ( -1 - \beta_{2} ) q^{22} + \beta_{6} q^{23} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{25} + ( -\beta_{1} - \beta_{3} ) q^{26} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{34} + \beta_{7} q^{35} -\beta_{6} q^{38} + ( 1 - \beta_{4} - \beta_{5} ) q^{40} + ( -\beta_{1} + \beta_{6} ) q^{41} + ( -1 + \beta_{5} ) q^{43} + \beta_{4} q^{46} -\beta_{4} q^{49} + ( -\beta_{1} - \beta_{3} ) q^{50} -\beta_{6} q^{53} + ( 2 - \beta_{4} - 2 \beta_{5} ) q^{55} + \beta_{1} q^{56} + ( \beta_{1} - \beta_{6} ) q^{59} -\beta_{5} q^{61} - q^{64} + ( \beta_{1} - \beta_{6} - 2 \beta_{7} ) q^{65} + ( -1 + \beta_{5} ) q^{67} - q^{70} + \beta_{3} q^{71} + \beta_{3} q^{77} + ( -\beta_{2} - \beta_{4} - \beta_{5} ) q^{79} + ( \beta_{3} + \beta_{6} + \beta_{7} ) q^{80} -\beta_{2} q^{82} + ( -\beta_{3} - \beta_{7} ) q^{83} + ( -2 - \beta_{2} ) q^{85} + ( -\beta_{3} - \beta_{7} ) q^{86} + ( -1 - \beta_{2} ) q^{88} + ( 1 - \beta_{5} ) q^{91} + \beta_{3} q^{95} + ( 1 + \beta_{2} ) q^{97} + \beta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{7} + O(q^{10})$$ $$8q + 2q^{7} + 2q^{10} + 2q^{13} - 8q^{16} + 2q^{19} - 4q^{22} + 2q^{25} + 2q^{34} + 2q^{40} - 4q^{43} + 2q^{46} - 2q^{49} + 6q^{55} - 4q^{61} - 8q^{64} - 4q^{67} - 8q^{70} - 2q^{79} + 4q^{82} - 12q^{85} - 4q^{88} + 4q^{91} + 4q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 5$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{6} + 8 \nu^{4} - 16 \nu^{2} + 1$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{6} + 8 \nu^{4} - 24 \nu^{2} + 9$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} + 8 \nu^{5} - 24 \nu^{3} + 9 \nu$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{7} + 8 \nu^{5} - 20 \nu^{3} + \nu$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - 2 \beta_{6} + 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{5} + 3 \beta_{4} + 3 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{7} - 5 \beta_{6} + 3 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{2} - 5$$ $$\nu^{7}$$ $$=$$ $$8 \beta_{3} - 13 \beta_{1}$$

## Character values

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

 $$n$$ $$433$$ $$947$$ $$\chi(n)$$ $$-\beta_{5}$$ $$-1$$

## 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
 1.40126 − 0.809017i −0.535233 + 0.309017i 0.535233 − 0.309017i −1.40126 + 0.809017i 0.535233 + 0.309017i −1.40126 − 0.809017i 1.40126 + 0.809017i −0.535233 − 0.309017i
1.00000i 0 0 −0.535233 0.309017i 0 0.809017 + 1.40126i 1.00000i 0 −0.309017 + 0.535233i
350.2 1.00000i 0 0 1.40126 + 0.809017i 0 −0.309017 0.535233i 1.00000i 0 0.809017 1.40126i
350.3 1.00000i 0 0 −1.40126 0.809017i 0 −0.309017 0.535233i 1.00000i 0 0.809017 1.40126i
350.4 1.00000i 0 0 0.535233 + 0.309017i 0 0.809017 + 1.40126i 1.00000i 0 −0.309017 + 0.535233i
1025.1 1.00000i 0 0 −1.40126 + 0.809017i 0 −0.309017 + 0.535233i 1.00000i 0 0.809017 + 1.40126i
1025.2 1.00000i 0 0 0.535233 0.309017i 0 0.809017 1.40126i 1.00000i 0 −0.309017 0.535233i
1025.3 1.00000i 0 0 −0.535233 + 0.309017i 0 0.809017 1.40126i 1.00000i 0 −0.309017 0.535233i
1025.4 1.00000i 0 0 1.40126 0.809017i 0 −0.309017 + 0.535233i 1.00000i 0 0.809017 + 1.40126i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1025.4 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.c 8
3.b odd 2 1 inner 1161.1.i.c 8
9.c even 3 1 3483.1.o.c 8
9.c even 3 1 3483.1.p.c 8
9.d odd 6 1 3483.1.o.c 8
9.d odd 6 1 3483.1.p.c 8
43.c even 3 1 inner 1161.1.i.c 8
129.f odd 6 1 inner 1161.1.i.c 8
387.e even 3 1 3483.1.o.c 8
387.g even 3 1 3483.1.p.c 8
387.o odd 6 1 3483.1.p.c 8
387.p odd 6 1 3483.1.o.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1161.1.i.c 8 1.a even 1 1 trivial
1161.1.i.c 8 3.b odd 2 1 inner
1161.1.i.c 8 43.c even 3 1 inner
1161.1.i.c 8 129.f odd 6 1 inner
3483.1.o.c 8 9.c even 3 1
3483.1.o.c 8 9.d odd 6 1
3483.1.o.c 8 387.e even 3 1
3483.1.o.c 8 387.p odd 6 1
3483.1.p.c 8 9.c even 3 1
3483.1.p.c 8 9.d odd 6 1
3483.1.p.c 8 387.g even 3 1
3483.1.p.c 8 387.o odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{4}$$
$3$ 1
$5$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$7$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
$11$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$13$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
$17$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$19$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
$23$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$29$ $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
$31$ $$( 1 - T^{2} + T^{4} )^{4}$$
$37$ $$( 1 - T^{2} + T^{4} )^{4}$$
$41$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$43$ $$( 1 + T + T^{2} )^{4}$$
$47$ $$( 1 - T )^{8}( 1 + T )^{8}$$
$53$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$59$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$61$ $$( 1 + T )^{8}( 1 - T + T^{2} )^{4}$$
$67$ $$( 1 + T )^{8}( 1 - T + T^{2} )^{4}$$
$71$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$
$73$ $$( 1 - T^{2} + T^{4} )^{4}$$
$79$ $$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$
$83$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$
$89$ $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
$97$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$