Embedded newform 71.2.d.a.20.5

• Label: 71.2.d.a.20.5
• Level: $71$
• Weight: $2$
• Character: 71.20
• Analytic conductor: $0.567$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	71.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.566937854351\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			20.5
Character	\(\chi\)	\(=\)	71.20
Dual form			71.2.d.a.32.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.471544 + 2.06597i) q^{2} +(-1.23006 + 0.592364i) q^{3} +(-2.24394 + 1.08063i) q^{4} +0.618957 q^{5} +(-1.80383 - 2.26194i) q^{6} +(0.245991 - 1.07776i) q^{7} +(-0.648183 - 0.812795i) q^{8} +(-0.708324 + 0.888211i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 3 q^{2} - 10 q^{3} - 7 q^{4} - 10 q^{5} + 5 q^{6} - 3 q^{7} + 17 q^{8} - 9 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/71\mathbb{Z}\right)^\times\).

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.471544	+	2.06597i	0.333432	+	1.46086i	0.812437	+	0.583049i	\(0.198141\pi\)
							−0.479005	+	0.877812i	\(0.659002\pi\)
\(3\)	−1.23006	+	0.592364i	−0.710174	+	0.342002i	−0.753852	−	0.657044i	\(-0.771807\pi\)
							0.0436785	+	0.999046i	\(0.486092\pi\)
\(5\)	0.618957			0.276806			0.138403	−	0.990376i	\(-0.455803\pi\)
							0.138403	+	0.990376i	\(0.455803\pi\)
\(7\)	0.245991	−	1.07776i	0.0929758	−	0.407354i	−0.906927	−	0.421288i	\(-0.861578\pi\)
							0.999903	+	0.0139339i	\(0.00443544\pi\)
\(11\)	3.58909	−	4.50058i	1.08215	−	1.35698i	0.152599	−	0.988288i	\(-0.451236\pi\)
							0.929553	−	0.368688i	\(-0.120193\pi\)
\(13\)	0.544957	+	0.683355i	0.151144	+	0.189529i	0.851639	−	0.524129i	\(-0.175609\pi\)
							−0.700495	+	0.713657i	\(0.747037\pi\)
\(17\)	2.89753			0.702755			0.351378	−	0.936234i	\(-0.385713\pi\)
							0.351378	+	0.936234i	\(0.385713\pi\)
\(19\)	−3.35869	−	1.61746i	−0.770536	−	0.371071i	0.00694568	−	0.999976i	\(-0.497789\pi\)
							−0.777482	+	0.628905i	\(0.783503\pi\)
\(23\)	0.360987	−	1.58159i	0.0752710	−	0.329784i	−0.923247	−	0.384207i	\(-0.874475\pi\)
							0.998518	+	0.0544232i	\(0.0173320\pi\)
\(29\)	−0.821003	+	0.395374i	−0.152457	+	0.0734192i	−0.508556	−	0.861029i	\(-0.669821\pi\)
							0.356100	+	0.934448i	\(0.384106\pi\)
\(31\)	−5.85273	−	7.33910i	−1.05118	−	1.31814i	−0.946170	−	0.323670i	\(-0.895083\pi\)
							−0.105012	−	0.994471i	\(-0.533488\pi\)
\(37\)	1.46338	+	6.41149i	0.240578	+	1.05404i	0.940492	+	0.339815i	\(0.110364\pi\)
							−0.699914	+	0.714227i	\(0.746778\pi\)
\(41\)	−4.52629	−	5.67579i	−0.706887	−	0.886409i	0.290630	−	0.956836i	\(-0.406135\pi\)
							−0.997517	+	0.0704269i	\(0.977564\pi\)
\(43\)	2.52769	+	11.0745i	0.385469	+	1.68885i	0.680004	+	0.733208i	\(0.261978\pi\)
							−0.294535	+	0.955641i	\(0.595165\pi\)
\(47\)	1.18316	+	0.569778i	0.172581	+	0.0831107i	0.518180	−	0.855272i	\(-0.326610\pi\)
							−0.345599	+	0.938382i	\(0.612324\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	71.2.d.a.20.5	✓	30
3.2	odd	2		639.2.j.c.91.1		30
71.23	odd	14		5041.2.a.m.1.3		15
71.32	even	7	inner	71.2.d.a.32.5	yes	30
71.48	even	7		5041.2.a.l.1.3		15
213.32	odd	14		639.2.j.c.316.1		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.2.d.a.20.5	✓	30	1.1	even	1	trivial
71.2.d.a.32.5	yes	30	71.32	even	7	inner
639.2.j.c.91.1		30	3.2	odd	2
639.2.j.c.316.1		30	213.32	odd	14
5041.2.a.l.1.3		15	71.48	even	7
5041.2.a.m.1.3		15	71.23	odd	14