Embedded newform 731.2.p.a.16.17

• Label: 731.2.p.a.16.17
• Level: $731$
• Weight: $2$
• Character: 731.16
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.p (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(64\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			16.17
Character	\(\chi\)	\(=\)	731.16
Dual form			731.2.p.a.594.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.343890 - 1.50668i) q^{2} +(-2.15737 - 0.492406i) q^{3} +(-0.349888 + 0.168497i) q^{4} +(2.57313 - 2.05201i) q^{5} +3.41980i q^{6} -1.34370i q^{7} +(-1.55292 - 1.94730i) q^{8} +(1.70888 + 0.822954i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 12 q^{2} - 72 q^{4} - 8 q^{8} + 62 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.343890	−	1.50668i	−0.243167	−	1.06538i	−0.938115	−	0.346324i	\(-0.887430\pi\)
							0.694948	−	0.719060i	\(-0.255427\pi\)
\(3\)	−2.15737	−	0.492406i	−1.24556	−	0.284291i	−0.451577	−	0.892232i	\(-0.649138\pi\)
							−0.793982	+	0.607941i	\(0.791996\pi\)
\(5\)	2.57313	−	2.05201i	1.15074	−	0.917685i	0.153229	−	0.988191i	\(-0.451033\pi\)
							0.997511	+	0.0705057i	\(0.0224613\pi\)
\(7\)		−	1.34370i		−	0.507870i	−0.967221	−	0.253935i	\(-0.918275\pi\)
							0.967221	−	0.253935i	\(-0.0817249\pi\)
\(11\)	1.74399	−	3.62143i	0.525832	−	1.09190i	−0.453800	−	0.891103i	\(-0.649932\pi\)
							0.979633	−	0.200798i	\(-0.0643536\pi\)
\(13\)	3.19939	+	4.01190i	0.887350	+	1.11270i	0.992978	+	0.118296i	\(0.0377431\pi\)
							−0.105629	+	0.994406i	\(0.533685\pi\)
\(17\)	−2.46141	−	3.30779i	−0.596980	−	0.802256i
							\(19\)	6.58818	−	3.17270i	1.51143	−	0.727867i	0.519480	−	0.854483i	\(-0.326126\pi\)
0.991952	+	0.126616i	\(0.0404115\pi\)
\(23\)	2.59078	−	5.37982i	0.540216	−	1.12177i	−0.434983	−	0.900439i	\(-0.643246\pi\)
							0.975199	−	0.221331i	\(-0.0710400\pi\)
\(29\)	−2.98308	+	0.680868i	−0.553944	+	0.126434i	−0.490324	−	0.871540i	\(-0.663122\pi\)
							−0.0636197	+	0.997974i	\(0.520264\pi\)
\(31\)	0.750690	−	0.171340i	0.134828	−	0.0307736i	−0.154574	−	0.987981i	\(-0.549401\pi\)
							0.289402	+	0.957208i	\(0.406543\pi\)
\(37\)			4.93950i			0.812049i	0.913862	+	0.406024i	\(0.133085\pi\)
							−0.913862	+	0.406024i	\(0.866915\pi\)
\(41\)	−6.19579	+	1.41415i	−0.967620	+	0.220853i	−0.676998	−	0.735985i	\(-0.736719\pi\)
							−0.290622	+	0.956838i	\(0.593862\pi\)
\(43\)	−0.211558	+	6.55402i	−0.0322623	+	0.999479i
							\(47\)	−6.77338	+	3.26189i	−0.987999	+	0.475795i	−0.856849	−	0.515568i	\(-0.827581\pi\)
−0.131150	+	0.991363i	\(0.541867\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.p.a.16.17	✓	384
17.16	even	2	inner	731.2.p.a.16.18	yes	384
43.35	even	7	inner	731.2.p.a.594.18	yes	384
731.594	even	14	inner	731.2.p.a.594.17	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.p.a.16.17	✓	384	1.1	even	1	trivial
731.2.p.a.16.18	yes	384	17.16	even	2	inner
731.2.p.a.594.17	yes	384	731.594	even	14	inner
731.2.p.a.594.18	yes	384	43.35	even	7	inner