Embedded newform 71.2.c.a.57.2

• Label: 71.2.c.a.57.2
• Level: $71$
• Weight: $2$
• Character: 71.57
• 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			57.2
Root			\(0.561200 + 1.72720i\) of defining polynomial
Character	\(\chi\)	\(=\)	71.57
Dual form			71.2.c.a.5.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.561200 + 1.72720i) q^{2} +(0.906993 - 2.79144i) q^{3} +(-1.05022 - 0.763032i) q^{4} +(1.21045 - 0.879441i) q^{5} +(4.31235 + 3.13311i) q^{6} +(-0.388444 + 1.19551i) q^{7} +(-1.03119 + 0.749203i) q^{8} +(-4.54244 - 3.30028i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} - q^{3} - 10 q^{4} - q^{5} - 8 q^{6} - q^{7} - q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.561200	+	1.72720i	−0.396828	+	1.22131i	0.530701	+	0.847559i	\(0.321929\pi\)
							−0.927529	+	0.373752i	\(0.878071\pi\)
\(3\)	0.906993	−	2.79144i	0.523653	−	1.61164i	−0.243312	−	0.969948i	\(-0.578234\pi\)
							0.766964	−	0.641690i	\(-0.221766\pi\)
\(5\)	1.21045	−	0.879441i	0.541328	−	0.393298i	−0.283250	−	0.959046i	\(-0.591413\pi\)
							0.824578	+	0.565748i	\(0.191413\pi\)
\(7\)	−0.388444	+	1.19551i	−0.146818	+	0.451860i	−0.997240	−	0.0742410i	\(-0.976347\pi\)
							0.850422	+	0.526101i	\(0.176347\pi\)
\(11\)	−1.22818	+	3.77994i	−0.370309	+	1.13969i	0.576280	+	0.817252i	\(0.304504\pi\)
							−0.946589	+	0.322442i	\(0.895496\pi\)
\(13\)	−0.173990	−	0.535487i	−0.0482562	−	0.148517i	0.924025	−	0.382332i	\(-0.124879\pi\)
							−0.972281	+	0.233815i	\(0.924879\pi\)
\(17\)	1.87505	+	5.77080i	0.454766	+	1.39962i	0.871410	+	0.490555i	\(0.163206\pi\)
							−0.416645	+	0.909069i	\(0.636794\pi\)
\(19\)	2.31993	−	7.14000i	0.532228	−	1.63803i	−0.217335	−	0.976097i	\(-0.569737\pi\)
							0.749564	−	0.661932i	\(-0.230263\pi\)
\(23\)	−8.89324			−1.85437			−0.927184	−	0.374606i	\(-0.877778\pi\)
							−0.927184	+	0.374606i	\(0.877778\pi\)
\(29\)	1.96757	−	1.42952i	0.365368	−	0.265456i	−0.389919	−	0.920849i	\(-0.627497\pi\)
							0.755288	+	0.655393i	\(0.227497\pi\)
\(31\)	0.162981	−	0.501605i	0.0292723	−	0.0900909i	−0.935353	−	0.353716i	\(-0.884918\pi\)
							0.964625	+	0.263625i	\(0.0849181\pi\)
\(37\)	1.78389			0.293270			0.146635	−	0.989191i	\(-0.453156\pi\)
							0.146635	+	0.989191i	\(0.453156\pi\)
\(41\)	2.18573			0.341353			0.170677	−	0.985327i	\(-0.445405\pi\)
							0.170677	+	0.985327i	\(0.445405\pi\)
\(43\)	0.171513	−	0.124611i	0.0261554	−	0.0190030i	−0.574631	−	0.818413i	\(-0.694854\pi\)
							0.600786	+	0.799410i	\(0.294854\pi\)
\(47\)	−3.23735	−	9.96355i	−0.472216	−	1.45333i	−0.849675	−	0.527306i	\(-0.823202\pi\)
							0.377459	−	0.926026i	\(-0.376798\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	71.2.c.a.57.2	yes	20
3.2	odd	2		639.2.f.c.199.4		20
71.5	even	5	inner	71.2.c.a.5.2	✓	20
71.17	odd	10		5041.2.a.j.1.2		10
71.54	even	5		5041.2.a.i.1.2		10
213.5	odd	10		639.2.f.c.289.4		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.2.c.a.5.2	✓	20	71.5	even	5	inner
71.2.c.a.57.2	yes	20	1.1	even	1	trivial
639.2.f.c.199.4		20	3.2	odd	2
639.2.f.c.289.4		20	213.5	odd	10
5041.2.a.i.1.2		10	71.54	even	5
5041.2.a.j.1.2		10	71.17	odd	10