# Properties

 Label 1176.1.s.c Level $1176$ Weight $1$ Character orbit 1176.s Analytic conductor $0.587$ Analytic rank $0$ Dimension $8$ Projective image $D_{4}$ CM discriminant -56 Inner twists $16$

# 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.4032.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{24}^{10} q^{2} - \zeta_{24}^{5} q^{3} - \zeta_{24}^{8} q^{4} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{5} - \zeta_{24}^{3} q^{6} - \zeta_{24}^{6} q^{8} + \zeta_{24}^{10} q^{9} +O(q^{10})$$ q - z^10 * q^2 - z^5 * q^3 - z^8 * q^4 + (-z^7 - z) * q^5 - z^3 * q^6 - z^6 * q^8 + z^10 * q^9 $$q - \zeta_{24}^{10} q^{2} - \zeta_{24}^{5} q^{3} - \zeta_{24}^{8} q^{4} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{5} - \zeta_{24}^{3} q^{6} - \zeta_{24}^{6} q^{8} + \zeta_{24}^{10} q^{9} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{10} - \zeta_{24} q^{12} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{13} + (\zeta_{24}^{6} - 1) q^{15} - \zeta_{24}^{4} q^{16} + \zeta_{24}^{8} q^{18} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{19} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{20} + \zeta_{24}^{11} q^{24} + \zeta_{24}^{8} q^{25} + (\zeta_{24}^{7} + \zeta_{24}) q^{26} + \zeta_{24}^{3} q^{27} + (\zeta_{24}^{10} + \zeta_{24}^{4}) q^{30} - \zeta_{24}^{2} q^{32} + \zeta_{24}^{6} q^{36} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{38} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{39} + (\zeta_{24}^{7} - \zeta_{24}) q^{40} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{45} + \zeta_{24}^{9} q^{48} + \zeta_{24}^{6} q^{50} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{52} + \zeta_{24} q^{54} + ( - \zeta_{24}^{6} - 1) q^{57} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{59} + (\zeta_{24}^{8} + \zeta_{24}^{2}) q^{60} + (\zeta_{24}^{7} - \zeta_{24}) q^{61} - q^{64} + ( - 2 \zeta_{24}^{10} + \zeta_{24}^{4}) q^{65} - \zeta_{24}^{6} q^{71} + \zeta_{24}^{4} q^{72} + \zeta_{24} q^{75} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{76} + ( - \zeta_{24}^{6} + 1) q^{78} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{80} - \zeta_{24}^{8} q^{81} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{83} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{90} - \zeta_{24}^{2} q^{95} + \zeta_{24}^{7} q^{96} +O(q^{100})$$ q - z^10 * q^2 - z^5 * q^3 - z^8 * q^4 + (-z^7 - z) * q^5 - z^3 * q^6 - z^6 * q^8 + z^10 * q^9 + (z^11 - z^5) * q^10 - z * q^12 + (z^9 + z^3) * q^13 + (z^6 - 1) * q^15 - z^4 * q^16 + z^8 * q^18 + (-z^7 + z) * q^19 + (z^9 - z^3) * q^20 + z^11 * q^24 + z^8 * q^25 + (z^7 + z) * q^26 + z^3 * q^27 + (z^10 + z^4) * q^30 - z^2 * q^32 + z^6 * q^36 + (-z^11 - z^5) * q^38 + (-z^8 + z^2) * q^39 + (z^7 - z) * q^40 + (-z^11 + z^5) * q^45 + z^9 * q^48 + z^6 * q^50 + (-z^11 + z^5) * q^52 + z * q^54 + (-z^6 - 1) * q^57 + (-z^11 - z^5) * q^59 + (z^8 + z^2) * q^60 + (z^7 - z) * q^61 - q^64 + (-2*z^10 + z^4) * q^65 - z^6 * q^71 + z^4 * q^72 + z * q^75 + (-z^9 - z^3) * q^76 + (-z^6 + 1) * q^78 + (z^11 + z^5) * q^80 - z^8 * q^81 + (-z^9 + z^3) * q^83 + (-z^9 + z^3) * q^90 - z^2 * q^95 + z^7 * q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4}+O(q^{10})$$ 8 * q + 4 * q^4 $$8 q + 4 q^{4} - 8 q^{15} - 4 q^{16} - 4 q^{18} - 4 q^{25} + 4 q^{30} + 4 q^{39} - 8 q^{57} - 4 q^{60} - 8 q^{64} + 4 q^{72} + 8 q^{78} + 4 q^{81}+O(q^{100})$$ 8 * q + 4 * q^4 - 8 * q^15 - 4 * q^16 - 4 * q^18 - 4 * q^25 + 4 * q^30 + 4 * q^39 - 8 * q^57 - 4 * q^60 - 8 * q^64 + 4 * q^72 + 8 * q^78 + 4 * q^81

## Character values

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

 $$n$$ $$295$$ $$589$$ $$785$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-\zeta_{24}^{4}$$

## 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
 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i
−0.866025 + 0.500000i −0.965926 + 0.258819i 0.500000 0.866025i 0.707107 + 1.22474i 0.707107 0.707107i 0 1.00000i 0.866025 0.500000i −1.22474 0.707107i
557.2 −0.866025 + 0.500000i 0.965926 0.258819i 0.500000 0.866025i −0.707107 1.22474i −0.707107 + 0.707107i 0 1.00000i 0.866025 0.500000i 1.22474 + 0.707107i
557.3 0.866025 0.500000i −0.258819 0.965926i 0.500000 0.866025i −0.707107 1.22474i −0.707107 0.707107i 0 1.00000i −0.866025 + 0.500000i −1.22474 0.707107i
557.4 0.866025 0.500000i 0.258819 + 0.965926i 0.500000 0.866025i 0.707107 + 1.22474i 0.707107 + 0.707107i 0 1.00000i −0.866025 + 0.500000i 1.22474 + 0.707107i
1157.1 −0.866025 0.500000i −0.965926 0.258819i 0.500000 + 0.866025i 0.707107 1.22474i 0.707107 + 0.707107i 0 1.00000i 0.866025 + 0.500000i −1.22474 + 0.707107i
1157.2 −0.866025 0.500000i 0.965926 + 0.258819i 0.500000 + 0.866025i −0.707107 + 1.22474i −0.707107 0.707107i 0 1.00000i 0.866025 + 0.500000i 1.22474 0.707107i
1157.3 0.866025 + 0.500000i −0.258819 + 0.965926i 0.500000 + 0.866025i −0.707107 + 1.22474i −0.707107 + 0.707107i 0 1.00000i −0.866025 0.500000i −1.22474 + 0.707107i
1157.4 0.866025 + 0.500000i 0.258819 0.965926i 0.500000 + 0.866025i 0.707107 1.22474i 0.707107 0.707107i 0 1.00000i −0.866025 0.500000i 1.22474 0.707107i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1157.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner
24.h odd 2 1 inner
56.j odd 6 1 inner
56.p even 6 1 inner
168.i even 2 1 inner
168.s odd 6 1 inner
168.ba even 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.c 8
3.b odd 2 1 inner 1176.1.s.c 8
7.b odd 2 1 inner 1176.1.s.c 8
7.c even 3 1 1176.1.n.e 4
7.c even 3 1 inner 1176.1.s.c 8
7.d odd 6 1 1176.1.n.e 4
7.d odd 6 1 inner 1176.1.s.c 8
8.b even 2 1 inner 1176.1.s.c 8
21.c even 2 1 inner 1176.1.s.c 8
21.g even 6 1 1176.1.n.e 4
21.g even 6 1 inner 1176.1.s.c 8
21.h odd 6 1 1176.1.n.e 4
21.h odd 6 1 inner 1176.1.s.c 8
24.h odd 2 1 inner 1176.1.s.c 8
56.h odd 2 1 CM 1176.1.s.c 8
56.j odd 6 1 1176.1.n.e 4
56.j odd 6 1 inner 1176.1.s.c 8
56.p even 6 1 1176.1.n.e 4
56.p even 6 1 inner 1176.1.s.c 8
168.i even 2 1 inner 1176.1.s.c 8
168.s odd 6 1 1176.1.n.e 4
168.s odd 6 1 inner 1176.1.s.c 8
168.ba even 6 1 1176.1.n.e 4
168.ba even 6 1 inner 1176.1.s.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.1.n.e 4 7.c even 3 1
1176.1.n.e 4 7.d odd 6 1
1176.1.n.e 4 21.g even 6 1
1176.1.n.e 4 21.h odd 6 1
1176.1.n.e 4 56.j odd 6 1
1176.1.n.e 4 56.p even 6 1
1176.1.n.e 4 168.s odd 6 1
1176.1.n.e 4 168.ba even 6 1
1176.1.s.c 8 1.a even 1 1 trivial
1176.1.s.c 8 3.b odd 2 1 inner
1176.1.s.c 8 7.b odd 2 1 inner
1176.1.s.c 8 7.c even 3 1 inner
1176.1.s.c 8 7.d odd 6 1 inner
1176.1.s.c 8 8.b even 2 1 inner
1176.1.s.c 8 21.c even 2 1 inner
1176.1.s.c 8 21.g even 6 1 inner
1176.1.s.c 8 21.h odd 6 1 inner
1176.1.s.c 8 24.h odd 2 1 inner
1176.1.s.c 8 56.h odd 2 1 CM
1176.1.s.c 8 56.j odd 6 1 inner
1176.1.s.c 8 56.p even 6 1 inner
1176.1.s.c 8 168.i even 2 1 inner
1176.1.s.c 8 168.s odd 6 1 inner
1176.1.s.c 8 168.ba even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 2T_{5}^{2} + 4$$ acting on $$S_{1}^{\mathrm{new}}(1176, [\chi])$$.

## Hecke characteristic polynomials

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