Embedded newform 731.2.s.b.214.12

• Label: 731.2.s.b.214.12
• Level: $731$
• Weight: $2$
• Character: 731.214
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• 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			214.12
Character	\(\chi\)	\(=\)	731.214
Dual form			731.2.s.b.386.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.90221 - 0.787922i) q^{2} +(-1.35593 + 2.02929i) q^{3} +(1.58338 + 1.58338i) q^{4} +(-0.106216 - 0.533986i) q^{5} +(4.17819 - 2.79178i) q^{6} +(-0.224543 + 1.12885i) q^{7} +(-0.188497 - 0.455072i) q^{8} +(-1.13143 - 2.73151i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.90221	−	0.787922i	−1.34507	−	0.557145i	−0.410152	−	0.912017i	\(-0.634524\pi\)
							−0.934915	+	0.354872i	\(0.884524\pi\)
\(3\)	−1.35593	+	2.02929i	−0.782846	+	1.17161i	0.198639	+	0.980073i	\(0.436348\pi\)
							−0.981485	+	0.191539i	\(0.938652\pi\)
\(5\)	−0.106216	−	0.533986i	−0.0475014	−	0.238806i	0.949739	−	0.313042i	\(-0.101348\pi\)
							−0.997241	+	0.0742363i	\(0.976348\pi\)
\(7\)	−0.224543	+	1.12885i	−0.0848694	+	0.426667i	0.914868	+	0.403753i	\(0.132294\pi\)
							−0.999737	+	0.0229142i	\(0.992706\pi\)
\(11\)	1.31316	+	1.96528i	0.395931	+	0.592553i	0.974858	−	0.222826i	\(-0.0715283\pi\)
							−0.578927	+	0.815380i	\(0.696528\pi\)
\(13\)	−0.145649	+	0.145649i	−0.0403957	+	0.0403957i	−0.727016	−	0.686620i	\(-0.759093\pi\)
							0.686620	+	0.727016i	\(0.259093\pi\)
\(17\)	3.49790	+	2.18282i	0.848365	+	0.529411i
							\(19\)	2.58047	+	1.06887i	0.592000	+	0.245214i	0.658511	−	0.752571i	\(-0.271187\pi\)
−0.0665107	+	0.997786i	\(0.521187\pi\)
\(23\)	−1.15641	+	0.772685i	−0.241127	+	0.161116i	−0.670262	−	0.742124i	\(-0.733818\pi\)
							0.429135	+	0.903240i	\(0.358818\pi\)
\(29\)	8.05610	−	1.60246i	1.49598	−	0.297569i	0.621802	−	0.783174i	\(-0.286401\pi\)
							0.874178	+	0.485605i	\(0.161401\pi\)
\(31\)	0.628901	−	0.941216i	0.112954	−	0.169047i	−0.770685	−	0.637216i	\(-0.780086\pi\)
							0.883639	+	0.468168i	\(0.155086\pi\)
\(37\)	0.373764	−	0.559377i	0.0614464	−	0.0919610i	−0.799474	−	0.600701i	\(-0.794888\pi\)
							0.860920	+	0.508740i	\(0.169888\pi\)
\(41\)	−9.61292	−	1.91213i	−1.50129	−	0.298624i	−0.625079	−	0.780561i	\(-0.714933\pi\)
							−0.876206	+	0.481937i	\(0.839933\pi\)
\(43\)	6.55222	+	0.261469i	0.999205	+	0.0398737i
							\(47\)	−7.86596	+	7.86596i	−1.14737	+	1.14737i	−0.160300	+	0.987068i	\(0.551246\pi\)
−0.987068	+	0.160300i	\(0.948754\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.b.214.12	✓	496
17.12	odd	16	inner	731.2.s.b.386.51	yes	496
43.42	odd	2	inner	731.2.s.b.214.51	yes	496
731.386	even	16	inner	731.2.s.b.386.12	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.b.214.12	✓	496	1.1	even	1	trivial
731.2.s.b.214.51	yes	496	43.42	odd	2	inner
731.2.s.b.386.12	yes	496	731.386	even	16	inner
731.2.s.b.386.51	yes	496	17.12	odd	16	inner