Embedded newform 731.2.k.b.35.17

• Label: 731.2.k.b.35.17
• Level: $731$
• Weight: $2$
• Character: 731.35
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $180$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.k (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(180\)
Relative dimension:	\(30\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			35.17
Character	\(\chi\)	\(=\)	731.35
Dual form			731.2.k.b.188.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0787180 - 0.344886i) q^{2} +(-0.0523067 - 0.229171i) q^{3} +(1.68919 + 0.813470i) q^{4} +(-0.573925 + 0.719679i) q^{5} -0.0831553 q^{6} +0.480778 q^{7} +(0.854650 - 1.07170i) q^{8} +(2.65312 - 1.27768i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 180 q + 2 q^{3} - 34 q^{4} + 2 q^{5} + 6 q^{6} - 54 q^{7} + 14 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0787180	−	0.344886i	0.0556620	−	0.243871i	−0.939442	−	0.342709i	\(-0.888655\pi\)
							0.995104	+	0.0988380i	\(0.0315126\pi\)
\(3\)	−0.0523067	−	0.229171i	−0.0301993	−	0.132312i	0.957581	−	0.288164i	\(-0.0930448\pi\)
							−0.987780	+	0.155852i	\(0.950188\pi\)
\(5\)	−0.573925	+	0.719679i	−0.256667	+	0.321850i	−0.893424	−	0.449214i	\(-0.851704\pi\)
							0.636757	+	0.771064i	\(0.280275\pi\)
\(7\)	0.480778			0.181717			0.0908584	−	0.995864i	\(-0.471039\pi\)
							0.0908584	+	0.995864i	\(0.471039\pi\)
\(11\)	0.633617	−	0.305134i	0.191043	−	0.0920014i	−0.335917	−	0.941892i	\(-0.609046\pi\)
							0.526960	+	0.849890i	\(0.323332\pi\)
\(13\)	−0.523297	+	0.656194i	−0.145137	+	0.181995i	−0.849086	−	0.528255i	\(-0.822847\pi\)
							0.703950	+	0.710250i	\(0.251418\pi\)
\(17\)	−0.623490	−	0.781831i	−0.151218	−	0.189622i
							\(19\)	−1.88088	−	0.905783i	−0.431503	−	0.207801i	0.205508	−	0.978655i	\(-0.434115\pi\)
−0.637011	+	0.770854i	\(0.719830\pi\)
\(23\)	5.27723	−	2.54138i	1.10038	−	0.529914i	0.206600	−	0.978425i	\(-0.433760\pi\)
							0.893777	+	0.448512i	\(0.148046\pi\)
\(29\)	0.193886	−	0.849472i	0.0360038	−	0.157743i	−0.953730	−	0.300663i	\(-0.902792\pi\)
							0.989734	+	0.142920i	\(0.0456492\pi\)
\(31\)	−1.37139	+	6.00846i	−0.246309	+	1.07915i	0.688845	+	0.724909i	\(0.258118\pi\)
							−0.935154	+	0.354242i	\(0.884739\pi\)
\(37\)	6.86319			1.12830			0.564150	−	0.825672i	\(-0.309204\pi\)
							0.564150	+	0.825672i	\(0.309204\pi\)
\(41\)	1.05147	−	4.60677i	0.164211	−	0.719457i	−0.824029	−	0.566548i	\(-0.808279\pi\)
							0.988240	−	0.152909i	\(-0.0488642\pi\)
\(43\)	−6.54939	−	0.324863i	−0.998772	−	0.0495412i
							\(47\)	8.57319	+	4.12863i	1.25053	+	0.602223i	0.937653	−	0.347572i	\(-0.112994\pi\)
0.312875	+	0.949794i	\(0.398708\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.k.b.35.17	✓	180
43.16	even	7	inner	731.2.k.b.188.17	yes	180

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.k.b.35.17	✓	180	1.1	even	1	trivial
731.2.k.b.188.17	yes	180	43.16	even	7	inner