# Properties

 Label 1176.1.s.a Level $1176$ Weight $1$ Character orbit 1176.s Analytic conductor $0.587$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -24 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1176.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.586900454856$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.1176.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6}^{2} q^{5} - q^{6} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{6}^{2} q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6}^{2} q^{5} - q^{6} + q^{8} + \zeta_{6}^{2} q^{9} -\zeta_{6} q^{10} + \zeta_{6} q^{11} -\zeta_{6}^{2} q^{12} - q^{15} + \zeta_{6}^{2} q^{16} -\zeta_{6} q^{18} + q^{20} - q^{22} + \zeta_{6} q^{24} - q^{27} - q^{29} -\zeta_{6}^{2} q^{30} -\zeta_{6} q^{31} -\zeta_{6} q^{32} + \zeta_{6}^{2} q^{33} + q^{36} + \zeta_{6}^{2} q^{40} -\zeta_{6}^{2} q^{44} -\zeta_{6} q^{45} - q^{48} + \zeta_{6} q^{53} -\zeta_{6}^{2} q^{54} - q^{55} -\zeta_{6}^{2} q^{58} -\zeta_{6} q^{59} + \zeta_{6} q^{60} + q^{62} + q^{64} -\zeta_{6} q^{66} + \zeta_{6}^{2} q^{72} + 2 \zeta_{6} q^{73} -\zeta_{6}^{2} q^{79} -\zeta_{6} q^{80} -\zeta_{6} q^{81} + q^{83} -\zeta_{6} q^{87} + \zeta_{6} q^{88} + q^{90} -\zeta_{6}^{2} q^{93} -\zeta_{6}^{2} q^{96} + q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{3} - q^{4} - q^{5} - 2 q^{6} + 2 q^{8} - q^{9} + O(q^{10})$$ $$2 q - q^{2} + q^{3} - q^{4} - q^{5} - 2 q^{6} + 2 q^{8} - q^{9} - q^{10} + q^{11} + q^{12} - 2 q^{15} - q^{16} - q^{18} + 2 q^{20} - 2 q^{22} + q^{24} - 2 q^{27} - 2 q^{29} + q^{30} - q^{31} - q^{32} - q^{33} + 2 q^{36} - q^{40} + q^{44} - q^{45} - 2 q^{48} + q^{53} + q^{54} - 2 q^{55} + q^{58} - q^{59} + q^{60} + 2 q^{62} + 2 q^{64} - q^{66} - q^{72} + 2 q^{73} + q^{79} - q^{80} - q^{81} + 2 q^{83} - q^{87} + q^{88} + 2 q^{90} + q^{93} + q^{96} + 2 q^{97} - 2 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$$.

 $$n$$ $$295$$ $$589$$ $$785$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$\zeta_{6}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
557.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 0 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
1157.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −1.00000 0 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
 $$n$$: 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
7.c even 3 1 inner
168.s odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1176.1.s.a 2
3.b odd 2 1 1176.1.s.b 2
7.b odd 2 1 168.1.s.a 2
7.c even 3 1 1176.1.n.c 1
7.c even 3 1 inner 1176.1.s.a 2
7.d odd 6 1 168.1.s.a 2
7.d odd 6 1 1176.1.n.d 1
8.b even 2 1 1176.1.s.b 2
21.c even 2 1 168.1.s.b yes 2
21.g even 6 1 168.1.s.b yes 2
21.g even 6 1 1176.1.n.a 1
21.h odd 6 1 1176.1.n.b 1
21.h odd 6 1 1176.1.s.b 2
24.h odd 2 1 CM 1176.1.s.a 2
28.d even 2 1 672.1.ba.b 2
28.f even 6 1 672.1.ba.b 2
56.e even 2 1 672.1.ba.a 2
56.h odd 2 1 168.1.s.b yes 2
56.j odd 6 1 168.1.s.b yes 2
56.j odd 6 1 1176.1.n.a 1
56.m even 6 1 672.1.ba.a 2
56.p even 6 1 1176.1.n.b 1
56.p even 6 1 1176.1.s.b 2
84.h odd 2 1 672.1.ba.a 2
84.j odd 6 1 672.1.ba.a 2
168.e odd 2 1 672.1.ba.b 2
168.i even 2 1 168.1.s.a 2
168.s odd 6 1 1176.1.n.c 1
168.s odd 6 1 inner 1176.1.s.a 2
168.ba even 6 1 168.1.s.a 2
168.ba even 6 1 1176.1.n.d 1
168.be odd 6 1 672.1.ba.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.1.s.a 2 7.b odd 2 1
168.1.s.a 2 7.d odd 6 1
168.1.s.a 2 168.i even 2 1
168.1.s.a 2 168.ba even 6 1
168.1.s.b yes 2 21.c even 2 1
168.1.s.b yes 2 21.g even 6 1
168.1.s.b yes 2 56.h odd 2 1
168.1.s.b yes 2 56.j odd 6 1
672.1.ba.a 2 56.e even 2 1
672.1.ba.a 2 56.m even 6 1
672.1.ba.a 2 84.h odd 2 1
672.1.ba.a 2 84.j odd 6 1
672.1.ba.b 2 28.d even 2 1
672.1.ba.b 2 28.f even 6 1
672.1.ba.b 2 168.e odd 2 1
672.1.ba.b 2 168.be odd 6 1
1176.1.n.a 1 21.g even 6 1
1176.1.n.a 1 56.j odd 6 1
1176.1.n.b 1 21.h odd 6 1
1176.1.n.b 1 56.p even 6 1
1176.1.n.c 1 7.c even 3 1
1176.1.n.c 1 168.s odd 6 1
1176.1.n.d 1 7.d odd 6 1
1176.1.n.d 1 168.ba even 6 1
1176.1.s.a 2 1.a even 1 1 trivial
1176.1.s.a 2 7.c even 3 1 inner
1176.1.s.a 2 24.h odd 2 1 CM
1176.1.s.a 2 168.s odd 6 1 inner
1176.1.s.b 2 3.b odd 2 1
1176.1.s.b 2 8.b even 2 1
1176.1.s.b 2 21.h odd 6 1
1176.1.s.b 2 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + T_{5} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1176, [\chi])$$.

## Hecke characteristic polynomials

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