Embedded newform 731.2.k.b.35.1

• Label: 731.2.k.b.35.1
• 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			35.1
Character	\(\chi\)	\(=\)	731.35
Dual form			731.2.k.b.188.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.609444 + 2.67015i) q^{2} +(0.133331 + 0.584162i) q^{3} +(-4.95634 - 2.38685i) q^{4} +(-0.259403 + 0.325281i) q^{5} -1.64106 q^{6} +1.45101 q^{7} +(5.97861 - 7.49695i) q^{8} +(2.37944 - 1.14588i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 180 q + 2 q^{3} - 34 q^{4} + 2 q^{5} + 6 q^{6} - 54 q^{7} + 14 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.609444	+	2.67015i	−0.430942	+	1.88808i	0.0279858	+	0.999608i	\(0.491091\pi\)
							−0.458928	+	0.888473i	\(0.651766\pi\)
\(3\)	0.133331	+	0.584162i	0.0769788	+	0.337266i	0.998722	−	0.0505338i	\(-0.0160923\pi\)
							−0.921744	+	0.387800i	\(0.873235\pi\)
\(5\)	−0.259403	+	0.325281i	−0.116009	+	0.145470i	−0.836445	−	0.548051i	\(-0.815370\pi\)
							0.720437	+	0.693521i	\(0.243941\pi\)
\(7\)	1.45101			0.548430			0.274215	−	0.961668i	\(-0.411582\pi\)
							0.274215	+	0.961668i	\(0.411582\pi\)
\(11\)	0.108279	−	0.0521442i	0.0326472	−	0.0157221i	−0.417489	−	0.908682i	\(-0.637090\pi\)
							0.450136	+	0.892960i	\(0.351375\pi\)
\(13\)	2.97196	−	3.72672i	0.824273	−	1.03361i	−0.174528	−	0.984652i	\(-0.555840\pi\)
							0.998801	−	0.0489536i	\(-0.0155887\pi\)
\(17\)	−0.623490	−	0.781831i	−0.151218	−	0.189622i
							\(19\)	2.34699	+	1.13025i	0.538436	+	0.259297i	0.683279	−	0.730158i	\(-0.260553\pi\)
−0.144843	+	0.989455i	\(0.546268\pi\)
\(23\)	1.63882	−	0.789212i	0.341717	−	0.164562i	−0.255152	−	0.966901i	\(-0.582126\pi\)
							0.596869	+	0.802339i	\(0.296411\pi\)
\(29\)	2.05735	−	9.01385i	0.382041	−	1.67383i	−0.309037	−	0.951050i	\(-0.600007\pi\)
							0.691078	−	0.722781i	\(-0.257136\pi\)
\(31\)	0.588771	−	2.57957i	0.105746	−	0.463305i	−0.894133	−	0.447801i	\(-0.852207\pi\)
							0.999880	−	0.0155046i	\(-0.00493546\pi\)
\(37\)	−5.70112			−0.937258			−0.468629	−	0.883395i	\(-0.655252\pi\)
							−0.468629	+	0.883395i	\(0.655252\pi\)
\(41\)	1.09114	−	4.78058i	0.170407	−	0.746601i	−0.815425	−	0.578863i	\(-0.803497\pi\)
							0.985832	−	0.167738i	\(-0.0536462\pi\)
\(43\)	3.02842	+	5.81624i	0.461830	+	0.886968i
							\(47\)	3.74381	+	1.80293i	0.546091	+	0.262984i	0.686525	−	0.727106i	\(-0.259135\pi\)
−0.140433	+	0.990090i	\(0.544850\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.k.b.35.1	✓	180
43.16	even	7	inner	731.2.k.b.188.1	yes	180

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.k.b.35.1	✓	180	1.1	even	1	trivial
731.2.k.b.188.1	yes	180	43.16	even	7	inner