Embedded newform 172.2.f.b.7.4

• Label: 172.2.f.b.7.4
• Level: $172$
• Weight: $2$
• Character: 172.7
• Analytic conductor: $1.373$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 172 = 2^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	172.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.37342691477\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.4
Character	\(\chi\)	\(=\)	172.7
Dual form			172.2.f.b.123.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.21687 - 0.720574i) q^{2} +(-1.46995 - 2.54603i) q^{3} +(0.961545 + 1.75369i) q^{4} +(-2.46507 + 1.42321i) q^{5} +(-0.0458646 + 4.15740i) q^{6} +(-0.455338 + 0.788668i) q^{7} +(0.0935892 - 2.82688i) q^{8} +(-2.82152 + 4.88701i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 14 q^{4} - 18 q^{5} - 3 q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/172\mathbb{Z}\right)^\times\).

\(n\)	\(87\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.21687	−	0.720574i	−0.860457	−	0.509523i
							\(3\)	−1.46995	−	2.54603i	−0.848677	−	1.46995i	−0.882389	−	0.470520i	\(-0.844066\pi\)
0.0337120	−	0.999432i	\(-0.489267\pi\)
\(5\)	−2.46507	+	1.42321i	−1.10241	+	0.636479i	−0.936854	−	0.349720i	\(-0.886277\pi\)
							−0.165560	+	0.986200i	\(0.552943\pi\)
\(7\)	−0.455338	+	0.788668i	−0.172101	+	0.298088i	−0.939154	−	0.343495i	\(-0.888389\pi\)
							0.767053	+	0.641584i	\(0.221722\pi\)
\(11\)			2.73880i			0.825780i	0.910781	+	0.412890i	\(0.135481\pi\)
							−0.910781	+	0.412890i	\(0.864519\pi\)
\(13\)	2.45686	−	4.25541i	0.681411	−	1.18024i	−0.293140	−	0.956070i	\(-0.594700\pi\)
							0.974550	−	0.224168i	\(-0.0719665\pi\)
\(17\)	−3.33931	+	5.78386i	−0.809902	+	1.40279i	0.103029	+	0.994678i	\(0.467147\pi\)
							−0.912931	+	0.408113i	\(0.866187\pi\)
\(19\)	0.485609	+	0.841100i	0.111406	+	0.192962i	0.916338	−	0.400407i	\(-0.131131\pi\)
							−0.804931	+	0.593368i	\(0.797798\pi\)
\(23\)	−3.03898	+	1.75455i	−0.633670	+	0.365850i	−0.782172	−	0.623062i	\(-0.785888\pi\)
							0.148502	+	0.988912i	\(0.452555\pi\)
\(29\)	−4.41020	−	2.54623i	−0.818954	−	0.472823i	0.0311015	−	0.999516i	\(-0.490098\pi\)
							−0.850056	+	0.526693i	\(0.823432\pi\)
\(31\)	−6.31084	+	3.64356i	−1.13346	+	0.654403i	−0.944803	−	0.327640i	\(-0.893747\pi\)
							−0.188657	+	0.982043i	\(0.560413\pi\)
\(37\)	−7.96035	+	4.59591i	−1.30867	+	0.755563i	−0.981875	−	0.189531i	\(-0.939303\pi\)
							−0.326798	+	0.945094i	\(0.605970\pi\)
\(41\)	−1.25325			−0.195724			−0.0978622	−	0.995200i	\(-0.531200\pi\)
							−0.0978622	+	0.995200i	\(0.531200\pi\)
\(43\)	4.58862	−	4.68450i	0.699758	−	0.714380i
							\(47\)		−	7.73252i		−	1.12790i	−0.825808	−	0.563952i	\(-0.809280\pi\)
0.825808	−	0.563952i	\(-0.190720\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	172.2.f.b.7.4	✓	32
4.3	odd	2	inner	172.2.f.b.7.13	yes	32
43.37	odd	6	inner	172.2.f.b.123.13	yes	32
172.123	even	6	inner	172.2.f.b.123.4	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
172.2.f.b.7.4	✓	32	1.1	even	1	trivial
172.2.f.b.7.13	yes	32	4.3	odd	2	inner
172.2.f.b.123.4	yes	32	172.123	even	6	inner
172.2.f.b.123.13	yes	32	43.37	odd	6	inner