# Properties

 Label 1176.1.n.b Level $1176$ Weight $1$ Character orbit 1176.n Self dual yes Analytic conductor $0.587$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -24 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1176.n (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.586900454856$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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: $D_6$ Artin field: Galois closure of 6.2.29042496.1

## $q$-expansion

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

## Character values

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

 $$n$$ $$295$$ $$589$$ $$785$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 0
−1.00000 1.00000 1.00000 −1.00000 −1.00000 0 −1.00000 1.00000 1.00000
 $$n$$: 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 $$\Q(\sqrt{-6})$$

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1176, [\chi])$$:

 $$T_{5} + 1$$ $$T_{11} - 1$$

## Hecke characteristic polynomials

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