Embedded newform 731.2.u.b.52.20

• Label: 731.2.u.b.52.20
• 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			52.20
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.b.239.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.486709 + 0.610314i) q^{2} +(-1.78702 - 0.269350i) q^{3} +(0.309445 - 1.35577i) q^{4} +(-3.20929 - 2.18806i) q^{5} +(-0.705372 - 1.22174i) q^{6} +(2.22375 - 3.85164i) q^{7} +(2.38468 - 1.14840i) q^{8} +(0.254185 + 0.0784058i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q + 6 q^{2} - 3 q^{3} - 54 q^{4} + q^{5} - 12 q^{6} + 49 q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.486709	+	0.610314i	0.344155	+	0.431557i	0.923543	−	0.383495i	\(-0.125280\pi\)
							−0.579388	+	0.815052i	\(0.696708\pi\)
\(3\)	−1.78702	−	0.269350i	−1.03174	−	0.155510i	−0.388717	−	0.921357i	\(-0.627082\pi\)
							−0.643022	+	0.765848i	\(0.722320\pi\)
\(5\)	−3.20929	−	2.18806i	−1.43524	−	0.978529i	−0.996814	−	0.0797609i	\(-0.974584\pi\)
							−0.438424	−	0.898768i	\(-0.644463\pi\)
\(7\)	2.22375	−	3.85164i	0.840497	−	1.45578i	−0.0489777	−	0.998800i	\(-0.515596\pi\)
							0.889475	−	0.456984i	\(-0.151070\pi\)
\(11\)	0.677982	+	2.97043i	0.204419	+	0.895619i	0.968207	+	0.250151i	\(0.0804803\pi\)
							−0.763788	+	0.645468i	\(0.776663\pi\)
\(13\)	0.0799425	−	1.06676i	0.0221721	−	0.295866i	−0.975084	−	0.221836i	\(-0.928795\pi\)
							0.997256	−	0.0740296i	\(-0.0235859\pi\)
\(17\)	−0.826239	+	0.563320i	−0.200392	+	0.136625i
							\(19\)	−7.05178	+	2.17519i	−1.61779	+	0.499022i	−0.965818	−	0.259221i	\(-0.916534\pi\)
−0.651972	+	0.758243i	\(0.726058\pi\)
\(23\)	3.65973	−	3.39574i	0.763107	−	0.708060i	−0.199220	−	0.979955i	\(-0.563841\pi\)
							0.962327	+	0.271895i	\(0.0876504\pi\)
\(29\)	−4.41798	+	0.665903i	−0.820398	+	0.123655i	−0.545812	−	0.837907i	\(-0.683779\pi\)
							−0.274586	+	0.961563i	\(0.588541\pi\)
\(31\)	1.01664	−	2.59037i	0.182595	−	0.465243i	−0.810042	−	0.586371i	\(-0.800556\pi\)
							0.992637	+	0.121128i	\(0.0386512\pi\)
\(37\)	−2.99388	−	5.18555i	−0.492191	−	0.852499i	0.507769	−	0.861493i	\(-0.330470\pi\)
							−0.999960	+	0.00899410i	\(0.997137\pi\)
\(41\)	0.748064	+	0.938043i	0.116828	+	0.146498i	0.836807	−	0.547499i	\(-0.184420\pi\)
							−0.719979	+	0.693996i	\(0.755848\pi\)
\(43\)	4.37907	+	4.88096i	0.667801	+	0.744339i
							\(47\)	−0.339550	+	1.48767i	−0.0495285	+	0.216999i	−0.993636	−	0.112639i	\(-0.964070\pi\)
0.944107	+	0.329638i	\(0.106927\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.b.52.20	✓	348
43.24	even	21	inner	731.2.u.b.239.20	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.b.52.20	✓	348	1.1	even	1	trivial
731.2.u.b.239.20	yes	348	43.24	even	21	inner