Embedded newform 731.2.k.b.35.10

• Label: 731.2.k.b.35.10
• 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			35.10
Character	\(\chi\)	\(=\)	731.35
Dual form			731.2.k.b.188.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.263173 + 1.15304i) q^{2} +(0.751127 + 3.29090i) q^{3} +(0.541703 + 0.260871i) q^{4} +(0.905581 - 1.13556i) q^{5} -3.99221 q^{6} +4.03940 q^{7} +(-1.91815 + 2.40528i) q^{8} +(-7.56295 + 3.64213i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 180 q + 2 q^{3} - 34 q^{4} + 2 q^{5} + 6 q^{6} - 54 q^{7} + 14 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.263173	+	1.15304i	−0.186092	+	0.815320i	0.792560	+	0.609794i	\(0.208748\pi\)
							−0.978652	+	0.205526i	\(0.934109\pi\)
\(3\)	0.751127	+	3.29090i	0.433664	+	1.90000i	0.435876	+	0.900007i	\(0.356439\pi\)
							−0.00221226	+	0.999998i	\(0.500704\pi\)
\(5\)	0.905581	−	1.13556i	0.404988	−	0.507839i	−0.536955	−	0.843611i	\(-0.680426\pi\)
							0.941943	+	0.335772i	\(0.108997\pi\)
\(7\)	4.03940			1.52675			0.763374	−	0.645956i	\(-0.223541\pi\)
							0.763374	+	0.645956i	\(0.223541\pi\)
\(11\)	4.25295	−	2.04811i	1.28231	−	0.617530i	0.336330	−	0.941744i	\(-0.390814\pi\)
							0.945984	+	0.324215i	\(0.105100\pi\)
\(13\)	1.18254	−	1.48286i	0.327977	−	0.411271i	−0.590315	−	0.807173i	\(-0.700997\pi\)
							0.918293	+	0.395902i	\(0.129568\pi\)
\(17\)	−0.623490	−	0.781831i	−0.151218	−	0.189622i
							\(19\)	−2.52811	−	1.21748i	−0.579989	−	0.279308i	0.120800	−	0.992677i	\(-0.461454\pi\)
−0.700789	+	0.713369i	\(0.747168\pi\)
\(23\)	−6.97179	+	3.35744i	−1.45372	+	0.700074i	−0.983236	−	0.182336i	\(-0.941634\pi\)
							−0.470482	+	0.882409i	\(0.655920\pi\)
\(29\)	−0.278287	+	1.21925i	−0.0516765	+	0.226410i	−0.994171	−	0.107811i	\(-0.965616\pi\)
							0.942495	+	0.334221i	\(0.108473\pi\)
\(31\)	1.31567	−	5.76431i	0.236301	−	1.03530i	−0.707999	−	0.706214i	\(-0.750402\pi\)
							0.944299	−	0.329087i	\(-0.106741\pi\)
\(37\)	0.559724			0.0920181			0.0460090	−	0.998941i	\(-0.485350\pi\)
							0.0460090	+	0.998941i	\(0.485350\pi\)
\(41\)	2.78571	−	12.2050i	0.435055	−	1.90610i	0.0121498	−	0.999926i	\(-0.496133\pi\)
							0.422905	−	0.906174i	\(-0.361010\pi\)
\(43\)	6.32065	+	1.74624i	0.963890	+	0.266299i
							\(47\)	−4.55881	−	2.19541i	−0.664971	−	0.320233i	0.0707737	−	0.997492i	\(-0.477453\pi\)
−0.735744	+	0.677259i	\(0.763167\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.k.b.35.10	✓	180
43.16	even	7	inner	731.2.k.b.188.10	yes	180

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.k.b.35.10	✓	180	1.1	even	1	trivial
731.2.k.b.188.10	yes	180	43.16	even	7	inner