# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.574922894553$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ 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 6.2.13436928.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{12} q^{3} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{12} q^{3} + \zeta_{12}^{2} q^{9} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{11} + q^{17} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{19} + \zeta_{12}^{4} q^{25} -\zeta_{12}^{3} q^{27} + ( 1 - \zeta_{12}^{4} ) q^{33} + \zeta_{12}^{2} q^{41} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{43} -\zeta_{12}^{2} q^{49} -\zeta_{12} q^{51} + ( -1 - \zeta_{12}^{2} ) q^{57} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{59} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{67} + q^{73} -\zeta_{12}^{5} q^{75} + \zeta_{12}^{4} q^{81} -2 q^{89} + \zeta_{12}^{4} q^{97} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{9} + O(q^{10})$$ $$4q + 2q^{9} + 4q^{17} - 2q^{25} + 6q^{33} + 2q^{41} - 2q^{49} - 6q^{57} + 4q^{73} - 2q^{81} - 8q^{89} - 2q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
319.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.866025 + 0.500000i 0 0 0 0 0 0.500000 0.866025i 0
319.2 0 0.866025 0.500000i 0 0 0 0 0 0.500000 0.866025i 0
1087.1 0 −0.866025 0.500000i 0 0 0 0 0 0.500000 + 0.866025i 0
1087.2 0 0.866025 + 0.500000i 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 $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
72.n even 6 1 inner
72.p 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.t.a 4
3.b odd 2 1 3456.1.t.a 4
4.b odd 2 1 inner 1152.1.t.a 4
8.b even 2 1 inner 1152.1.t.a 4
8.d odd 2 1 CM 1152.1.t.a 4
9.c even 3 1 inner 1152.1.t.a 4
9.d odd 6 1 3456.1.t.a 4
12.b even 2 1 3456.1.t.a 4
16.e even 4 1 2304.1.o.a 2
16.e even 4 1 2304.1.o.b 2
16.f odd 4 1 2304.1.o.a 2
16.f odd 4 1 2304.1.o.b 2
24.f even 2 1 3456.1.t.a 4
24.h odd 2 1 3456.1.t.a 4
36.f odd 6 1 inner 1152.1.t.a 4
36.h even 6 1 3456.1.t.a 4
72.j odd 6 1 3456.1.t.a 4
72.l even 6 1 3456.1.t.a 4
72.n even 6 1 inner 1152.1.t.a 4
72.p odd 6 1 inner 1152.1.t.a 4
144.v odd 12 1 2304.1.o.a 2
144.v odd 12 1 2304.1.o.b 2
144.x even 12 1 2304.1.o.a 2
144.x even 12 1 2304.1.o.b 2

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1152, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$9 + 3 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -1 + T )^{4}$$
$19$ $$( -3 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 1 - T + T^{2} )^{2}$$
$43$ $$9 + 3 T^{2} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$9 + 3 T^{2} + T^{4}$$
$61$ $$T^{4}$$
$67$ $$9 + 3 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -1 + T )^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 2 + T )^{4}$$
$97$ $$( 1 + T + T^{2} )^{2}$$