Embedded newform 731.2.u.b.52.1

• Label: 731.2.u.b.52.1
• 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			52.1
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.b.239.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.74998 - 2.19441i) q^{2} +(0.963898 + 0.145284i) q^{3} +(-1.30795 + 5.73049i) q^{4} +(0.755186 + 0.514877i) q^{5} +(-1.36799 - 2.36943i) q^{6} +(-0.932904 + 1.61584i) q^{7} +(9.80631 - 4.72247i) q^{8} +(-1.95873 - 0.604187i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q + 6 q^{2} - 3 q^{3} - 54 q^{4} + q^{5} - 12 q^{6} + 49 q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.74998	−	2.19441i	−1.23742	−	1.55168i	−0.715317	−	0.698800i	\(-0.753718\pi\)
							−0.522106	−	0.852880i	\(-0.674854\pi\)
\(3\)	0.963898	+	0.145284i	0.556507	+	0.0838799i	0.421273	−	0.906934i	\(-0.361583\pi\)
							0.135234	+	0.990814i	\(0.456821\pi\)
\(5\)	0.755186	+	0.514877i	0.337730	+	0.230260i	0.720294	−	0.693669i	\(-0.244007\pi\)
							−0.382565	+	0.923929i	\(0.624959\pi\)
\(7\)	−0.932904	+	1.61584i	−0.352604	+	0.610729i	−0.986705	−	0.162522i	\(-0.948037\pi\)
							0.634101	+	0.773251i	\(0.281371\pi\)
\(11\)	0.805736	+	3.53016i	0.242939	+	1.06438i	0.938327	+	0.345749i	\(0.112375\pi\)
							−0.695388	+	0.718634i	\(0.744768\pi\)
\(13\)	0.195674	−	2.61109i	0.0542702	−	0.724186i	−0.902007	−	0.431722i	\(-0.857906\pi\)
							0.956277	−	0.292463i	\(-0.0944749\pi\)
\(17\)	−0.826239	+	0.563320i	−0.200392	+	0.136625i
							\(19\)	1.88455	−	0.581307i	0.432346	−	0.133361i	−0.0709409	−	0.997481i	\(-0.522600\pi\)
0.503287	+	0.864120i	\(0.332124\pi\)
\(23\)	−5.74938	+	5.33464i	−1.19883	+	1.11235i	−0.207873	+	0.978156i	\(0.566654\pi\)
							−0.990955	+	0.134194i	\(0.957155\pi\)
\(29\)	3.15981	−	0.476264i	0.586761	−	0.0884401i	0.151049	−	0.988526i	\(-0.451735\pi\)
							0.435712	+	0.900086i	\(0.356497\pi\)
\(31\)	−2.59826	+	6.62026i	−0.466661	+	1.18903i	0.482962	+	0.875641i	\(0.339561\pi\)
							−0.949624	+	0.313393i	\(0.898534\pi\)
\(37\)	5.51221	+	9.54742i	0.906201	+	1.56959i	0.819297	+	0.573370i	\(0.194364\pi\)
							0.0869046	+	0.996217i	\(0.472302\pi\)
\(41\)	1.34792	+	1.69024i	0.210510	+	0.263971i	0.875865	−	0.482556i	\(-0.160292\pi\)
							−0.665356	+	0.746527i	\(0.731720\pi\)
\(43\)	−6.45368	−	1.16191i	−0.984177	−	0.177189i
							\(47\)	−2.93627	+	12.8646i	−0.428299	+	1.87650i	0.0507523	+	0.998711i	\(0.483838\pi\)
−0.479051	+	0.877787i	\(0.659019\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.b.52.1	✓	348
43.24	even	21	inner	731.2.u.b.239.1	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.b.52.1	✓	348	1.1	even	1	trivial
731.2.u.b.239.1	yes	348	43.24	even	21	inner