Embedded newform 731.2.u.b.52.17

• Label: 731.2.u.b.52.17
• 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			52.17
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.b.239.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.384419 + 0.482046i) q^{2} +(-0.444042 - 0.0669286i) q^{3} +(0.360451 - 1.57924i) q^{4} +(0.0789639 + 0.0538367i) q^{5} +(-0.138436 - 0.239777i) q^{6} +(-1.46965 + 2.54550i) q^{7} +(2.01084 - 0.968367i) q^{8} +(-2.67402 - 0.824827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q + 6 q^{2} - 3 q^{3} - 54 q^{4} + q^{5} - 12 q^{6} + 49 q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.384419	+	0.482046i	0.271825	+	0.340858i	0.898942	−	0.438067i	\(-0.144337\pi\)
							−0.627117	+	0.778925i	\(0.715765\pi\)
\(3\)	−0.444042	−	0.0669286i	−0.256368	−	0.0386412i	0.0196008	−	0.999808i	\(-0.493760\pi\)
							−0.275969	+	0.961167i	\(0.588999\pi\)
\(5\)	0.0789639	+	0.0538367i	0.0353137	+	0.0240765i	0.580848	−	0.814012i	\(-0.302721\pi\)
							−0.545535	+	0.838088i	\(0.683673\pi\)
\(7\)	−1.46965	+	2.54550i	−0.555474	+	0.962110i	0.442392	+	0.896822i	\(0.354130\pi\)
							−0.997866	+	0.0652879i	\(0.979203\pi\)
\(11\)	−0.553527	−	2.42516i	−0.166895	−	0.731214i	−0.987226	−	0.159325i	\(-0.949068\pi\)
							0.820331	−	0.571889i	\(-0.193789\pi\)
\(13\)	0.380309	−	5.07487i	0.105479	−	1.40751i	−0.653760	−	0.756702i	\(-0.726809\pi\)
							0.759238	−	0.650813i	\(-0.225572\pi\)
\(17\)	−0.826239	+	0.563320i	−0.200392	+	0.136625i
							\(19\)	2.10108	−	0.648099i	0.482022	−	0.148684i	−0.0442150	−	0.999022i	\(-0.514079\pi\)
0.526237	+	0.850338i	\(0.323602\pi\)
\(23\)	2.22129	−	2.06106i	0.463171	−	0.429760i	−0.413753	−	0.910389i	\(-0.635782\pi\)
							0.876924	+	0.480629i	\(0.159592\pi\)
\(29\)	−5.59161	+	0.842800i	−1.03834	+	0.156504i	−0.646015	−	0.763324i	\(-0.723566\pi\)
							−0.392321	+	0.919828i	\(0.628328\pi\)
\(31\)	−1.91568	+	4.88108i	−0.344067	+	0.876668i	0.649178	+	0.760636i	\(0.275113\pi\)
							−0.993245	+	0.116032i	\(0.962982\pi\)
\(37\)	−2.54842	−	4.41400i	−0.418958	−	0.725657i	0.576877	−	0.816831i	\(-0.304271\pi\)
							−0.995835	+	0.0911745i	\(0.970938\pi\)
\(41\)	−4.00626	−	5.02370i	−0.625673	−	0.784570i	0.363457	−	0.931611i	\(-0.381596\pi\)
							−0.989130	+	0.147041i	\(0.953025\pi\)
\(43\)	5.75885	−	3.13619i	0.878216	−	0.478264i
							\(47\)	2.29911	−	10.0731i	0.335360	−	1.46931i	−0.473231	−	0.880938i	\(-0.656912\pi\)
0.808591	−	0.588371i	\(-0.200231\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.b.52.17	✓	348
43.24	even	21	inner	731.2.u.b.239.17	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.b.52.17	✓	348	1.1	even	1	trivial
731.2.u.b.239.17	yes	348	43.24	even	21	inner