# Properties

 Label 168.1.s.a Level $168$ Weight $1$ Character orbit 168.s Analytic conductor $0.084$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -24 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 168.s (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.0838429221223$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.1176.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.677376.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} - \zeta_{6}^{2} q^{5} + q^{6} - \zeta_{6} q^{7} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ q + z^2 * q^2 - z * q^3 - z * q^4 - z^2 * q^5 + q^6 - z * q^7 + q^8 + z^2 * q^9 $$q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} - \zeta_{6}^{2} q^{5} + q^{6} - \zeta_{6} q^{7} + q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{10} + \zeta_{6} q^{11} + \zeta_{6}^{2} q^{12} + q^{14} - q^{15} + \zeta_{6}^{2} q^{16} - \zeta_{6} q^{18} - q^{20} + \zeta_{6}^{2} q^{21} - q^{22} - \zeta_{6} q^{24} + q^{27} + \zeta_{6}^{2} q^{28} - q^{29} - \zeta_{6}^{2} q^{30} + \zeta_{6} q^{31} - \zeta_{6} q^{32} - \zeta_{6}^{2} q^{33} - q^{35} + q^{36} - \zeta_{6}^{2} q^{40} - \zeta_{6} q^{42} - \zeta_{6}^{2} q^{44} + \zeta_{6} q^{45} + q^{48} + \zeta_{6}^{2} q^{49} + \zeta_{6} q^{53} + \zeta_{6}^{2} q^{54} + q^{55} - \zeta_{6} q^{56} - \zeta_{6}^{2} q^{58} + \zeta_{6} q^{59} + \zeta_{6} q^{60} - q^{62} + q^{63} + q^{64} + \zeta_{6} q^{66} - \zeta_{6}^{2} q^{70} + \zeta_{6}^{2} q^{72} - \zeta_{6} q^{73} - \zeta_{6}^{2} q^{77} - \zeta_{6}^{2} q^{79} + \zeta_{6} q^{80} - \zeta_{6} q^{81} - q^{83} + q^{84} + \zeta_{6} q^{87} + \zeta_{6} q^{88} - q^{90} - \zeta_{6}^{2} q^{93} + \zeta_{6}^{2} q^{96} - q^{97} - \zeta_{6} q^{98} - q^{99} +O(q^{100})$$ q + z^2 * q^2 - z * q^3 - z * q^4 - z^2 * q^5 + q^6 - z * q^7 + q^8 + z^2 * q^9 + z * q^10 + z * q^11 + z^2 * q^12 + q^14 - q^15 + z^2 * q^16 - z * q^18 - q^20 + z^2 * q^21 - q^22 - z * q^24 + q^27 + z^2 * q^28 - q^29 - z^2 * q^30 + z * q^31 - z * q^32 - z^2 * q^33 - q^35 + q^36 - z^2 * q^40 - z * q^42 - z^2 * q^44 + z * q^45 + q^48 + z^2 * q^49 + z * q^53 + z^2 * q^54 + q^55 - z * q^56 - z^2 * q^58 + z * q^59 + z * q^60 - q^62 + q^63 + q^64 + z * q^66 - z^2 * q^70 + z^2 * q^72 - z * q^73 - z^2 * q^77 - z^2 * q^79 + z * q^80 - z * q^81 - q^83 + q^84 + z * q^87 + z * q^88 - q^90 - z^2 * q^93 + z^2 * q^96 - q^97 - z * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{3} - q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 - q^3 - q^4 + q^5 + 2 * q^6 - q^7 + 2 * q^8 - q^9 $$2 q - q^{2} - q^{3} - q^{4} + q^{5} + 2 q^{6} - q^{7} + 2 q^{8} - q^{9} + q^{10} + q^{11} - q^{12} + 2 q^{14} - 2 q^{15} - q^{16} - q^{18} - 2 q^{20} - q^{21} - 2 q^{22} - q^{24} + 2 q^{27} - q^{28} - 2 q^{29} + q^{30} + q^{31} - q^{32} + q^{33} - 2 q^{35} + 2 q^{36} + q^{40} - q^{42} + q^{44} + q^{45} + 2 q^{48} - q^{49} + q^{53} - q^{54} + 2 q^{55} - q^{56} + q^{58} + q^{59} + q^{60} - 2 q^{62} + 2 q^{63} + 2 q^{64} + q^{66} + q^{70} - q^{72} - 2 q^{73} + q^{77} + q^{79} + q^{80} - q^{81} - 2 q^{83} + 2 q^{84} + q^{87} + q^{88} - 2 q^{90} + q^{93} - q^{96} - 2 q^{97} - q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^3 - q^4 + q^5 + 2 * q^6 - q^7 + 2 * q^8 - q^9 + q^10 + q^11 - q^12 + 2 * q^14 - 2 * q^15 - q^16 - q^18 - 2 * q^20 - q^21 - 2 * q^22 - q^24 + 2 * q^27 - q^28 - 2 * q^29 + q^30 + q^31 - q^32 + q^33 - 2 * q^35 + 2 * q^36 + q^40 - q^42 + q^44 + q^45 + 2 * q^48 - q^49 + q^53 - q^54 + 2 * q^55 - q^56 + q^58 + q^59 + q^60 - 2 * q^62 + 2 * q^63 + 2 * q^64 + q^66 + q^70 - q^72 - 2 * q^73 + q^77 + q^79 + q^80 - q^81 - 2 * q^83 + 2 * q^84 + q^87 + q^88 - 2 * q^90 + q^93 - q^96 - 2 * q^97 - q^98 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$$.

 $$n$$ $$73$$ $$85$$ $$113$$ $$127$$ $$\chi(n)$$ $$\zeta_{6}^{2}$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 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 −0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 0.500000 0.866025i
149.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i 1.00000 −0.500000 0.866025i 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 168.1.s.a 2
3.b odd 2 1 168.1.s.b yes 2
4.b odd 2 1 672.1.ba.b 2
7.b odd 2 1 1176.1.s.a 2
7.c even 3 1 inner 168.1.s.a 2
7.c even 3 1 1176.1.n.d 1
7.d odd 6 1 1176.1.n.c 1
7.d odd 6 1 1176.1.s.a 2
8.b even 2 1 168.1.s.b yes 2
8.d odd 2 1 672.1.ba.a 2
12.b even 2 1 672.1.ba.a 2
21.c even 2 1 1176.1.s.b 2
21.g even 6 1 1176.1.n.b 1
21.g even 6 1 1176.1.s.b 2
21.h odd 6 1 168.1.s.b yes 2
21.h odd 6 1 1176.1.n.a 1
24.f even 2 1 672.1.ba.b 2
24.h odd 2 1 CM 168.1.s.a 2
28.g odd 6 1 672.1.ba.b 2
56.h odd 2 1 1176.1.s.b 2
56.j odd 6 1 1176.1.n.b 1
56.j odd 6 1 1176.1.s.b 2
56.k odd 6 1 672.1.ba.a 2
56.p even 6 1 168.1.s.b yes 2
56.p even 6 1 1176.1.n.a 1
84.n even 6 1 672.1.ba.a 2
168.i even 2 1 1176.1.s.a 2
168.s odd 6 1 inner 168.1.s.a 2
168.s odd 6 1 1176.1.n.d 1
168.v even 6 1 672.1.ba.b 2
168.ba even 6 1 1176.1.n.c 1
168.ba even 6 1 1176.1.s.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.1.s.a 2 1.a even 1 1 trivial
168.1.s.a 2 7.c even 3 1 inner
168.1.s.a 2 24.h odd 2 1 CM
168.1.s.a 2 168.s odd 6 1 inner
168.1.s.b yes 2 3.b odd 2 1
168.1.s.b yes 2 8.b even 2 1
168.1.s.b yes 2 21.h odd 6 1
168.1.s.b yes 2 56.p even 6 1
672.1.ba.a 2 8.d odd 2 1
672.1.ba.a 2 12.b even 2 1
672.1.ba.a 2 56.k odd 6 1
672.1.ba.a 2 84.n even 6 1
672.1.ba.b 2 4.b odd 2 1
672.1.ba.b 2 24.f even 2 1
672.1.ba.b 2 28.g odd 6 1
672.1.ba.b 2 168.v even 6 1
1176.1.n.a 1 21.h odd 6 1
1176.1.n.a 1 56.p even 6 1
1176.1.n.b 1 21.g even 6 1
1176.1.n.b 1 56.j odd 6 1
1176.1.n.c 1 7.d odd 6 1
1176.1.n.c 1 168.ba even 6 1
1176.1.n.d 1 7.c even 3 1
1176.1.n.d 1 168.s odd 6 1
1176.1.s.a 2 7.b odd 2 1
1176.1.s.a 2 7.d odd 6 1
1176.1.s.a 2 168.i even 2 1
1176.1.s.a 2 168.ba even 6 1
1176.1.s.b 2 21.c even 2 1
1176.1.s.b 2 21.g even 6 1
1176.1.s.b 2 56.h odd 2 1
1176.1.s.b 2 56.j odd 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 $$S_{1}^{\mathrm{new}}(168, [\chi])$$.

## Hecke characteristic polynomials

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