Embedded newform 731.2.u.b.52.2

• Label: 731.2.u.b.52.2
• Level: $731$
• Weight: $2$
• Character: 731.52
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $348$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.u (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(348\)
Relative dimension:	\(29\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			52.2
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.b.239.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.67210 - 2.09675i) q^{2} +(2.29142 + 0.345377i) q^{3} +(-1.15539 + 5.06209i) q^{4} +(-2.26489 - 1.54418i) q^{5} +(-3.10732 - 5.38204i) q^{6} +(2.23836 - 3.87695i) q^{7} +(7.71335 - 3.71455i) q^{8} +(2.26462 + 0.698543i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q + 6 q^{2} - 3 q^{3} - 54 q^{4} + q^{5} - 12 q^{6} + 49 q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.67210	−	2.09675i	−1.18235	−	1.48262i	−0.839605	−	0.543198i	\(-0.817213\pi\)
							−0.342749	−	0.939427i	\(-0.611358\pi\)
\(3\)	2.29142	+	0.345377i	1.32295	+	0.199403i	0.772261	−	0.635305i	\(-0.219126\pi\)
							0.550693	+	0.834708i	\(0.314364\pi\)
\(5\)	−2.26489	−	1.54418i	−1.01289	−	0.690577i	−0.0613550	−	0.998116i	\(-0.519542\pi\)
							−0.951535	+	0.307539i	\(0.900495\pi\)
\(7\)	2.23836	−	3.87695i	0.846020	−	1.46535i	−0.0387129	−	0.999250i	\(-0.512326\pi\)
							0.884733	−	0.466099i	\(-0.154341\pi\)
\(11\)	−1.17078	−	5.12952i	−0.353003	−	1.54661i	−0.770207	−	0.637794i	\(-0.779847\pi\)
							0.417204	−	0.908813i	\(-0.363010\pi\)
\(13\)	−0.429945	+	5.73722i	−0.119245	+	1.59122i	0.541561	+	0.840662i	\(0.317834\pi\)
							−0.660806	+	0.750557i	\(0.729785\pi\)
\(17\)	−0.826239	+	0.563320i	−0.200392	+	0.136625i
							\(19\)	5.83270	−	1.79915i	1.33811	−	0.412753i	0.458606	−	0.888640i	\(-0.348349\pi\)
0.879507	+	0.475887i	\(0.157873\pi\)
\(23\)	−0.109336	+	0.101449i	−0.0227981	+	0.0211535i	−0.691490	−	0.722386i	\(-0.743045\pi\)
							0.668691	+	0.743540i	\(0.266855\pi\)
\(29\)	−2.79804	+	0.421737i	−0.519584	+	0.0783146i	−0.403598	−	0.914936i	\(-0.632241\pi\)
							−0.115985	+	0.993251i	\(0.537003\pi\)
\(31\)	−0.128734	+	0.328008i	−0.0231213	+	0.0589120i	−0.941968	−	0.335702i	\(-0.891026\pi\)
							0.918847	+	0.394614i	\(0.129122\pi\)
\(37\)	−4.59639	−	7.96117i	−0.755641	−	1.30881i	−0.945055	−	0.326911i	\(-0.893992\pi\)
							0.189414	−	0.981897i	\(-0.439341\pi\)
\(41\)	−3.62174	−	4.54152i	−0.565621	−	0.709266i	0.413965	−	0.910293i	\(-0.364144\pi\)
							−0.979586	+	0.201027i	\(0.935572\pi\)
\(43\)	3.17489	+	5.73760i	0.484167	+	0.874976i
							\(47\)	−0.748136	+	3.27780i	−0.109127	+	0.478116i	0.890601	+	0.454785i	\(0.150284\pi\)
−0.999728	+	0.0233302i	\(0.992573\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.b.52.2	✓	348
43.24	even	21	inner	731.2.u.b.239.2	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.b.52.2	✓	348	1.1	even	1	trivial
731.2.u.b.239.2	yes	348	43.24	even	21	inner