Embedded newform 731.2.y.a.4.17

• Label: 731.2.y.a.4.17
• Level: $731$
• Weight: $2$
• Character: 731.4
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $768$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.y (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(768\)
Relative dimension:	\(64\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			4.17
Character	\(\chi\)	\(=\)	731.4
Dual form			731.2.y.a.183.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35561 - 1.08107i) q^{2} +(1.45849 + 0.164333i) q^{3} +(0.223944 + 0.981161i) q^{4} +(0.881224 - 0.308354i) q^{5} +(-1.79950 - 1.79950i) q^{6} +(1.96518 + 1.96518i) q^{7} +(-0.747499 + 1.55220i) q^{8} +(-0.824584 - 0.188206i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 14 q^{3} + 104 q^{4} - 6 q^{5} - 36 q^{6} - 28 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35561	−	1.08107i	−0.958564	−	0.764429i	0.0133147	−	0.999911i	\(-0.495762\pi\)
							−0.971879	+	0.235482i	\(0.924333\pi\)
\(3\)	1.45849	+	0.164333i	0.842062	+	0.0948776i	0.522449	−	0.852671i	\(-0.325019\pi\)
							0.319613	+	0.947548i	\(0.396447\pi\)
\(5\)	0.881224	−	0.308354i	0.394096	−	0.137900i	−0.125954	−	0.992036i	\(-0.540199\pi\)
							0.520050	+	0.854136i	\(0.325913\pi\)
\(7\)	1.96518	+	1.96518i	0.742770	+	0.742770i	0.973110	−	0.230340i	\(-0.0739839\pi\)
							−0.230340	+	0.973110i	\(0.573984\pi\)
\(11\)	−4.26521	+	2.68001i	−1.28601	+	0.808054i	−0.989156	−	0.146869i	\(-0.953080\pi\)
							−0.296854	+	0.954923i	\(0.595938\pi\)
\(13\)	1.97914	+	0.953103i	0.548915	+	0.264343i	0.687720	−	0.725976i	\(-0.258612\pi\)
							−0.138805	+	0.990320i	\(0.544326\pi\)
\(17\)	2.12166	+	3.53533i	0.514577	+	0.857444i
							\(19\)	7.69661	−	1.75670i	1.76572	−	0.403015i	0.788466	−	0.615078i	\(-0.210875\pi\)
0.977256	+	0.212063i	\(0.0680183\pi\)
\(23\)	1.03950	−	0.653164i	0.216752	−	0.136194i	−0.419280	−	0.907857i	\(-0.637718\pi\)
							0.636032	+	0.771663i	\(0.280575\pi\)
\(29\)	0.769387	+	6.82850i	0.142872	+	1.26802i	0.836612	+	0.547796i	\(0.184533\pi\)
							−0.693741	+	0.720225i	\(0.744039\pi\)
\(31\)	0.521626	+	4.62956i	0.0936868	+	0.831494i	0.949406	+	0.314053i	\(0.101687\pi\)
							−0.855719	+	0.517441i	\(0.826885\pi\)
\(37\)	3.76353	−	3.76353i	0.618720	−	0.618720i	−0.326483	−	0.945203i	\(-0.605864\pi\)
							0.945203	+	0.326483i	\(0.105864\pi\)
\(41\)	3.93805	−	0.443712i	0.615020	−	0.0692961i	0.201042	−	0.979583i	\(-0.435567\pi\)
							0.413979	+	0.910287i	\(0.364139\pi\)
\(43\)	−5.97142	−	2.70964i	−0.910633	−	0.413217i
							\(47\)	−0.355019	−	1.55544i	−0.0517848	−	0.226884i	0.942413	−	0.334451i	\(-0.108551\pi\)
−0.994198	+	0.107567i	\(0.965694\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.y.a.4.17	✓	768
17.13	even	4	inner	731.2.y.a.47.48	yes	768
43.11	even	7	inner	731.2.y.a.140.48	yes	768
731.183	even	28	inner	731.2.y.a.183.17	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.y.a.4.17	✓	768	1.1	even	1	trivial
731.2.y.a.47.48	yes	768	17.13	even	4	inner
731.2.y.a.140.48	yes	768	43.11	even	7	inner
731.2.y.a.183.17	yes	768	731.183	even	28	inner