Embedded newform 71.2.c.a.5.3

• Label: 71.2.c.a.5.3
• Level: $71$
• Weight: $2$
• Character: 71.5
• Analytic conductor: $0.567$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	71.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.566937854351\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(5\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 6*x^19 + 28*x^18 - 91*x^17 + 268*x^16 - 604*x^15 + 1278*x^14 - 1990*x^13 + 3162*x^12 - 4046*x^11 + 6406*x^10 - 8426*x^9 + 12709*x^8 - 13621*x^7 + 17787*x^6 - 13514*x^5 + 9316*x^4 - 1639*x^3 - 228*x^2 + 620*x + 961
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			5.3
Root			\(0.300331 - 0.924322i\) of defining polynomial
Character	\(\chi\)	\(=\)	71.5
Dual form			71.2.c.a.57.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.300331 - 0.924322i) q^{2} +(-0.143416 - 0.441389i) q^{3} +(0.853860 - 0.620366i) q^{4} +(-1.37660 - 1.00016i) q^{5} +(-0.364913 + 0.265125i) q^{6} +(0.656494 + 2.02048i) q^{7} +(-2.40241 - 1.74545i) q^{8} +(2.25280 - 1.63675i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} - q^{3} - 10 q^{4} - q^{5} - 8 q^{6} - q^{7} - q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.300331	−	0.924322i	−0.212366	−	0.653595i	−0.999330	−	0.0365962i	\(-0.988348\pi\)
							0.786964	−	0.616998i	\(-0.211652\pi\)
\(3\)	−0.143416	−	0.441389i	−0.0828012	−	0.254836i	0.901082	−	0.433649i	\(-0.142774\pi\)
							−0.983883	+	0.178813i	\(0.942774\pi\)
\(5\)	−1.37660	−	1.00016i	−0.615635	−	0.447285i	0.235759	−	0.971812i	\(-0.424242\pi\)
							−0.851394	+	0.524526i	\(0.824242\pi\)
\(7\)	0.656494	+	2.02048i	0.248131	+	0.763669i	0.995106	+	0.0988167i	\(0.0315057\pi\)
							−0.746974	+	0.664853i	\(0.768494\pi\)
\(11\)	0.603273	+	1.85668i	0.181894	+	0.559811i	0.999881	−	0.0154251i	\(-0.00491014\pi\)
							−0.817987	+	0.575236i	\(0.804910\pi\)
\(13\)	−1.04920	+	3.22911i	−0.290996	+	0.895594i	0.693541	+	0.720417i	\(0.256050\pi\)
							−0.984537	+	0.175177i	\(0.943950\pi\)
\(17\)	−1.34533	+	4.14050i	−0.326290	+	1.00422i	0.644565	+	0.764550i	\(0.277039\pi\)
							−0.970855	+	0.239668i	\(0.922961\pi\)
\(19\)	1.31744	+	4.05466i	0.302241	+	0.930203i	0.980692	+	0.195558i	\(0.0626516\pi\)
							−0.678451	+	0.734646i	\(0.737348\pi\)
\(23\)	4.94628			1.03137			0.515685	−	0.856778i	\(-0.327537\pi\)
							0.515685	+	0.856778i	\(0.327537\pi\)
\(29\)	−5.88756	−	4.27756i	−1.09329	−	0.794324i	−0.113341	−	0.993556i	\(-0.536155\pi\)
							−0.979952	+	0.199232i	\(0.936155\pi\)
\(31\)	−2.17407	−	6.69111i	−0.390475	−	1.20176i	−0.932430	−	0.361351i	\(-0.882316\pi\)
							0.541955	−	0.840407i	\(-0.317684\pi\)
\(37\)	−5.44096			−0.894488			−0.447244	−	0.894412i	\(-0.647594\pi\)
							−0.447244	+	0.894412i	\(0.647594\pi\)
\(41\)	−0.255249			−0.0398631			−0.0199316	−	0.999801i	\(-0.506345\pi\)
							−0.0199316	+	0.999801i	\(0.506345\pi\)
\(43\)	6.62897	+	4.81623i	1.01091	+	0.734468i	0.964400	−	0.264449i	\(-0.0851902\pi\)
							0.0465096	+	0.998918i	\(0.485190\pi\)
\(47\)	−0.0385134	+	0.118532i	−0.00561776	+	0.0172897i	−0.953826	−	0.300359i	\(-0.902893\pi\)
							0.948208	+	0.317649i	\(0.102893\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	71.2.c.a.5.3	✓	20
3.2	odd	2		639.2.f.c.289.3		20
71.25	even	5		5041.2.a.i.1.4		10
71.46	odd	10		5041.2.a.j.1.4		10
71.57	even	5	inner	71.2.c.a.57.3	yes	20
213.128	odd	10		639.2.f.c.199.3		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.2.c.a.5.3	✓	20	1.1	even	1	trivial
71.2.c.a.57.3	yes	20	71.57	even	5	inner
639.2.f.c.199.3		20	213.128	odd	10
639.2.f.c.289.3		20	3.2	odd	2
5041.2.a.i.1.4		10	71.25	even	5
5041.2.a.j.1.4		10	71.46	odd	10