Embedded newform 71.2.c.a.25.2

• Label: 71.2.c.a.25.2
• Level: $71$
• Weight: $2$
• Character: 71.25
• 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			25.2
Root			\(1.27286 + 0.924789i\) of defining polynomial
Character	\(\chi\)	\(=\)	71.25
Dual form			71.2.c.a.54.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27286 + 0.924789i) q^{2} +(-0.850399 + 0.617851i) q^{3} +(0.146911 - 0.452144i) q^{4} +(0.148297 + 0.456410i) q^{5} +(0.511060 - 1.57288i) q^{6} +(-4.12180 + 2.99466i) q^{7} +(-0.741239 - 2.28130i) q^{8} +(-0.585612 + 1.80233i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} - q^{3} - 10 q^{4} - q^{5} - 8 q^{6} - q^{7} - q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27286	+	0.924789i	−0.900050	+	0.653924i	−0.938479	−	0.345337i	\(-0.887765\pi\)
							0.0384291	+	0.999261i	\(0.487765\pi\)
\(3\)	−0.850399	+	0.617851i	−0.490978	+	0.356717i	−0.805560	−	0.592514i	\(-0.798136\pi\)
							0.314582	+	0.949230i	\(0.398136\pi\)
\(5\)	0.148297	+	0.456410i	0.0663202	+	0.204113i	0.978725	−	0.205176i	\(-0.0657768\pi\)
							−0.912405	+	0.409289i	\(0.865777\pi\)
\(7\)	−4.12180	+	2.99466i	−1.55789	+	1.13188i	−0.620184	+	0.784456i	\(0.712942\pi\)
							−0.937710	+	0.347420i	\(0.887058\pi\)
\(11\)	3.85852	−	2.80338i	1.16339	−	0.845250i	0.173184	−	0.984889i	\(-0.444594\pi\)
							0.990202	+	0.139640i	\(0.0445944\pi\)
\(13\)	3.98746	+	2.89706i	1.10592	+	0.803500i	0.982017	−	0.188794i	\(-0.0604579\pi\)
							0.123906	+	0.992294i	\(0.460458\pi\)
\(17\)	1.54622	+	1.12339i	0.375013	+	0.272463i	0.759287	−	0.650756i	\(-0.225548\pi\)
							−0.384273	+	0.923219i	\(0.625548\pi\)
\(19\)	−1.01097	+	0.734509i	−0.231931	+	0.168508i	−0.697681	−	0.716408i	\(-0.745785\pi\)
							0.465750	+	0.884917i	\(0.345785\pi\)
\(23\)	−2.45752			−0.512428			−0.256214	−	0.966620i	\(-0.582475\pi\)
							−0.256214	+	0.966620i	\(0.582475\pi\)
\(29\)	1.32106	+	4.06581i	0.245315	+	0.755003i	0.995584	+	0.0938702i	\(0.0299239\pi\)
							−0.750269	+	0.661132i	\(0.770076\pi\)
\(31\)	−1.22888	+	0.892830i	−0.220713	+	0.160357i	−0.692647	−	0.721276i	\(-0.743556\pi\)
							0.471935	+	0.881633i	\(0.343556\pi\)
\(37\)	−5.25971			−0.864691			−0.432346	−	0.901708i	\(-0.642314\pi\)
							−0.432346	+	0.901708i	\(0.642314\pi\)
\(41\)	−1.67389			−0.261418			−0.130709	−	0.991421i	\(-0.541725\pi\)
							−0.130709	+	0.991421i	\(0.541725\pi\)
\(43\)	1.95712	+	6.02339i	0.298458	+	0.918559i	0.982038	+	0.188683i	\(0.0604219\pi\)
							−0.683580	+	0.729875i	\(0.739578\pi\)
\(47\)	6.28284	+	4.56475i	0.916446	+	0.665837i	0.942637	−	0.333820i	\(-0.108338\pi\)
							−0.0261908	+	0.999657i	\(0.508338\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	71.2.c.a.25.2	✓	20
3.2	odd	2		639.2.f.c.451.4		20
71.14	odd	10		5041.2.a.j.1.8		10
71.54	even	5	inner	71.2.c.a.54.2	yes	20
71.57	even	5		5041.2.a.i.1.8		10
213.125	odd	10		639.2.f.c.622.4		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.2.c.a.25.2	✓	20	1.1	even	1	trivial
71.2.c.a.54.2	yes	20	71.54	even	5	inner
639.2.f.c.451.4		20	3.2	odd	2
639.2.f.c.622.4		20	213.125	odd	10
5041.2.a.i.1.8		10	71.57	even	5
5041.2.a.j.1.8		10	71.14	odd	10