Embedded newform 731.2.p.a.16.18

• Label: 731.2.p.a.16.18
• 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			16.18
Character	\(\chi\)	\(=\)	731.16
Dual form			731.2.p.a.594.18

$q$-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	0.793982	−	0.607941i	\(-0.208004\pi\)
							0.451577	+	0.892232i	\(0.350862\pi\)
\(5\)	−2.57313	+	2.05201i	−1.15074	+	0.917685i	−0.997511	−	0.0705057i	\(-0.977539\pi\)
							−0.153229	+	0.988191i	\(0.548967\pi\)
\(7\)			1.34370i			0.507870i	0.967221	+	0.253935i	\(0.0817249\pi\)
							−0.967221	+	0.253935i	\(0.918275\pi\)
\(11\)	−1.74399	+	3.62143i	−0.525832	+	1.09190i	0.453800	+	0.891103i	\(0.350068\pi\)
							−0.979633	+	0.200798i	\(0.935646\pi\)
\(13\)	3.19939	+	4.01190i	0.887350	+	1.11270i	0.992978	+	0.118296i	\(0.0377431\pi\)
							−0.105629	+	0.994406i	\(0.533685\pi\)
\(17\)	3.77257	+	1.66365i	0.914982	+	0.403494i
							\(19\)	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	\(0.356754\pi\)
							−0.975199	+	0.221331i	\(0.928960\pi\)
\(29\)	2.98308	−	0.680868i	0.553944	−	0.126434i	0.0636197	−	0.997974i	\(-0.479736\pi\)
							0.490324	+	0.871540i	\(0.336878\pi\)
\(31\)	−0.750690	+	0.171340i	−0.134828	+	0.0307736i	−0.289402	−	0.957208i	\(-0.593457\pi\)
							0.154574	+	0.987981i	\(0.450599\pi\)
\(37\)		−	4.93950i		−	0.812049i	−0.913862	−	0.406024i	\(-0.866915\pi\)
							0.913862	−	0.406024i	\(-0.133085\pi\)
\(41\)	6.19579	−	1.41415i	0.967620	−	0.220853i	0.290622	−	0.956838i	\(-0.406138\pi\)
							0.676998	+	0.735985i	\(0.263281\pi\)
\(43\)	−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.16.18	yes	384
17.16	even	2	inner	731.2.p.a.16.17	✓	384
43.35	even	7	inner	731.2.p.a.594.17	yes	384
731.594	even	14	inner	731.2.p.a.594.18	yes	384

    

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