# Properties

 Label 1152.1.n.a Level $1152$ Weight $1$ Character orbit 1152.n Analytic conductor $0.575$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1152.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.574922894553$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.20155392.5

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{3} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{3} + \zeta_{6}^{2} q^{9} -\zeta_{6} q^{11} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{17} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{19} + \zeta_{6} q^{25} + q^{27} + \zeta_{6}^{2} q^{33} + ( -1 - \zeta_{6} ) q^{41} + ( -1 + \zeta_{6}^{2} ) q^{43} -\zeta_{6}^{2} q^{49} + ( -1 + \zeta_{6}^{2} ) q^{51} + ( -1 + \zeta_{6}^{2} ) q^{57} -\zeta_{6}^{2} q^{59} + ( 1 + \zeta_{6} ) q^{67} + q^{73} -\zeta_{6}^{2} q^{75} -\zeta_{6} q^{81} + 2 \zeta_{6} q^{83} + \zeta_{6} q^{97} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - q^{9} + O(q^{10})$$ $$2q - q^{3} - q^{9} - q^{11} + q^{25} + 2q^{27} - q^{33} - 3q^{41} - 3q^{43} + q^{49} - 3q^{51} - 3q^{57} + q^{59} + 3q^{67} + 2q^{73} + q^{75} - q^{81} + 2q^{83} + q^{97} + 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 + 0.866025i 0 0 0 0 0 −0.500000 0.866025i 0
833.1 0 −0.500000 0.866025i 0 0 0 0 0 −0.500000 + 0.866025i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
36.h even 6 1 inner
72.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.1.n.a 2
3.b odd 2 1 3456.1.n.b 2
4.b odd 2 1 1152.1.n.b yes 2
8.b even 2 1 1152.1.n.b yes 2
8.d odd 2 1 CM 1152.1.n.a 2
9.c even 3 1 3456.1.n.a 2
9.d odd 6 1 1152.1.n.b yes 2
12.b even 2 1 3456.1.n.a 2
16.e even 4 2 2304.1.q.c 4
16.f odd 4 2 2304.1.q.c 4
24.f even 2 1 3456.1.n.b 2
24.h odd 2 1 3456.1.n.a 2
36.f odd 6 1 3456.1.n.b 2
36.h even 6 1 inner 1152.1.n.a 2
72.j odd 6 1 inner 1152.1.n.a 2
72.l even 6 1 1152.1.n.b yes 2
72.n even 6 1 3456.1.n.b 2
72.p odd 6 1 3456.1.n.a 2
144.u even 12 2 2304.1.q.c 4
144.w odd 12 2 2304.1.q.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.1.n.a 2 1.a even 1 1 trivial
1152.1.n.a 2 8.d odd 2 1 CM
1152.1.n.a 2 36.h even 6 1 inner
1152.1.n.a 2 72.j odd 6 1 inner
1152.1.n.b yes 2 4.b odd 2 1
1152.1.n.b yes 2 8.b even 2 1
1152.1.n.b yes 2 9.d odd 6 1
1152.1.n.b yes 2 72.l even 6 1
2304.1.q.c 4 16.e even 4 2
2304.1.q.c 4 16.f odd 4 2
2304.1.q.c 4 144.u even 12 2
2304.1.q.c 4 144.w odd 12 2
3456.1.n.a 2 9.c even 3 1
3456.1.n.a 2 12.b even 2 1
3456.1.n.a 2 24.h odd 2 1
3456.1.n.a 2 72.p odd 6 1
3456.1.n.b 2 3.b odd 2 1
3456.1.n.b 2 24.f even 2 1
3456.1.n.b 2 36.f odd 6 1
3456.1.n.b 2 72.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} + T_{11} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1152, [\chi])$$.

## Hecke characteristic polynomials

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