Embedded newform 731.2.s.b.214.17

• Label: 731.2.s.b.214.17
• Level: $731$
• Weight: $2$
• Character: 731.214
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $496$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.s (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(62\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			214.17
Character	\(\chi\)	\(=\)	731.214
Dual form			731.2.s.b.386.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34652 - 0.557745i) q^{2} +(0.877192 - 1.31281i) q^{3} +(0.0878103 + 0.0878103i) q^{4} +(0.0560935 + 0.282001i) q^{5} +(-1.91337 + 1.27847i) q^{6} +(-0.596391 + 2.99826i) q^{7} +(1.04623 + 2.52582i) q^{8} +(0.194044 + 0.468464i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{16}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34652	−	0.557745i	−0.952130	−	0.394385i	−0.148099	−	0.988973i	\(-0.547315\pi\)
							−0.804031	+	0.594587i	\(0.797315\pi\)
\(3\)	0.877192	−	1.31281i	0.506447	−	0.757952i	−0.486857	−	0.873482i	\(-0.661857\pi\)
							0.993304	+	0.115530i	\(0.0368567\pi\)
\(5\)	0.0560935	+	0.282001i	0.0250858	+	0.126115i	0.991299	−	0.131629i	\(-0.0420209\pi\)
							−0.966213	+	0.257744i	\(0.917021\pi\)
\(7\)	−0.596391	+	2.99826i	−0.225415	+	1.13324i	0.687845	+	0.725858i	\(0.258557\pi\)
							−0.913259	+	0.407379i	\(0.866443\pi\)
\(11\)	−2.35353	−	3.52231i	−0.709616	−	1.06202i	−0.994630	−	0.103493i	\(-0.966998\pi\)
							0.285014	−	0.958523i	\(-0.408002\pi\)
\(13\)	−0.122886	+	0.122886i	−0.0340824	+	0.0340824i	−0.723943	−	0.689860i	\(-0.757672\pi\)
							0.689860	+	0.723943i	\(0.257672\pi\)
\(17\)	2.79094	−	3.03490i	0.676903	−	0.736072i
							\(19\)	1.01904	+	0.422102i	0.233785	+	0.0968368i	0.496500	−	0.868036i	\(-0.334618\pi\)
−0.262716	+	0.964873i	\(0.584618\pi\)
\(23\)	3.84989	−	2.57241i	0.802758	−	0.536385i	−0.0851712	−	0.996366i	\(-0.527144\pi\)
							0.887929	+	0.459981i	\(0.152144\pi\)
\(29\)	1.90683	−	0.379292i	0.354090	−	0.0704328i	−0.0148407	−	0.999890i	\(-0.504724\pi\)
							0.368930	+	0.929457i	\(0.379724\pi\)
\(31\)	−1.32306	+	1.98009i	−0.237628	+	0.355635i	−0.931047	−	0.364899i	\(-0.881103\pi\)
							0.693419	+	0.720534i	\(0.256103\pi\)
\(37\)	3.20008	−	4.78925i	0.526089	−	0.787348i	−0.469323	−	0.883027i	\(-0.655502\pi\)
							0.995412	+	0.0956784i	\(0.0305020\pi\)
\(41\)	5.53726	+	1.10143i	0.864775	+	0.172014i	0.607496	−	0.794323i	\(-0.292174\pi\)
							0.257279	+	0.966337i	\(0.417174\pi\)
\(43\)	−0.887492	−	6.49710i	−0.135341	−	0.990799i
							\(47\)	−2.87568	+	2.87568i	−0.419462	+	0.419462i	−0.885018	−	0.465556i	\(-0.845854\pi\)
0.465556	+	0.885018i	\(0.345854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.b.214.17	✓	496
17.12	odd	16	inner	731.2.s.b.386.46	yes	496
43.42	odd	2	inner	731.2.s.b.214.46	yes	496
731.386	even	16	inner	731.2.s.b.386.17	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.b.214.17	✓	496	1.1	even	1	trivial
731.2.s.b.214.46	yes	496	43.42	odd	2	inner
731.2.s.b.386.17	yes	496	731.386	even	16	inner
731.2.s.b.386.46	yes	496	17.12	odd	16	inner