Embedded newform 71.5.e.a.14.19

• Label: 71.5.e.a.14.19
• Level: $71$
• Weight: $5$
• Character: 71.14
• Analytic conductor: $7.339$
• Analytic rank: $0$
• Dimension: $92$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 71 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	71.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.33926737895\)
Analytic rank:	\(0\)
Dimension:	\(92\)
Relative dimension:	\(23\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			14.19
Character	\(\chi\)	\(=\)	71.14
Dual form			71.5.e.a.66.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30403 - 4.01340i) q^{2} +(-0.422249 + 1.29955i) q^{3} +(-1.46264 - 1.06267i) q^{4} +(-37.2304 + 27.0495i) q^{5} +(4.66499 + 3.38931i) q^{6} +(56.2511 + 18.2771i) q^{7} +(48.4518 - 35.2023i) q^{8} +(64.0198 + 46.5131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q + 6 q^{2} - 13 q^{3} - 170 q^{4} - 9 q^{5} - 70 q^{6} - 5 q^{7} + 329 q^{8} - 742 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(7\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30403	−	4.01340i	0.326009	−	1.00335i	−0.644975	−	0.764204i	\(-0.723132\pi\)
							0.970983	−	0.239147i	\(-0.0768679\pi\)
\(3\)	−0.422249	+	1.29955i	−0.0469165	+	0.144394i	−0.971770	−	0.235928i	\(-0.924187\pi\)
							0.924854	+	0.380322i	\(0.124187\pi\)
\(5\)	−37.2304	+	27.0495i	−1.48922	+	1.08198i	−0.514779	+	0.857323i	\(0.672126\pi\)
							−0.974438	+	0.224656i	\(0.927874\pi\)
\(7\)	56.2511	+	18.2771i	1.14798	+	0.373002i	0.820382	−	0.571815i	\(-0.193761\pi\)
							0.327599	+	0.944817i	\(0.393761\pi\)
\(11\)	91.9776	+	29.8853i	0.760146	+	0.246986i	0.663341	−	0.748317i	\(-0.269138\pi\)
							0.0968044	+	0.995303i	\(0.469138\pi\)
\(13\)	−244.332	+	79.3883i	−1.44575	+	0.469753i	−0.923686	−	0.383151i	\(-0.874839\pi\)
							−0.522067	+	0.852905i	\(0.674839\pi\)
\(17\)	263.435	−	85.5953i	0.911541	−	0.296177i	0.184549	−	0.982823i	\(-0.440918\pi\)
							0.726992	+	0.686646i	\(0.240918\pi\)
\(19\)	−132.220	+	406.931i	−0.366260	+	1.12723i	0.582928	+	0.812524i	\(0.301907\pi\)
							−0.949188	+	0.314709i	\(0.898093\pi\)
\(23\)			764.871i			1.44588i	0.690911	+	0.722940i	\(0.257210\pi\)
							−0.690911	+	0.722940i	\(0.742790\pi\)
\(29\)	815.306	−	592.355i	0.969449	−	0.704346i	0.0141228	−	0.999900i	\(-0.495504\pi\)
							0.955326	+	0.295555i	\(0.0955044\pi\)
\(31\)	−759.202	−	246.680i	−0.790013	−	0.256691i	−0.113903	−	0.993492i	\(-0.536335\pi\)
							−0.676109	+	0.736801i	\(0.736335\pi\)
\(37\)	−1085.36			−0.792809			−0.396405	−	0.918076i	\(-0.629742\pi\)
							−0.396405	+	0.918076i	\(0.629742\pi\)
\(41\)		−	189.553i		−	0.112762i	−0.998409	−	0.0563811i	\(-0.982044\pi\)
							0.998409	−	0.0563811i	\(-0.0179562\pi\)
\(43\)	231.695	−	168.336i	0.125308	−	0.0910416i	−0.523366	−	0.852108i	\(-0.675324\pi\)
							0.648674	+	0.761066i	\(0.275324\pi\)
\(47\)	2948.78	−	958.118i	1.33490	−	0.433734i	0.447311	−	0.894378i	\(-0.352382\pi\)
							0.887585	+	0.460644i	\(0.152382\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	71.5.e.a.14.19	✓	92
71.66	odd	10	inner	71.5.e.a.66.19	yes	92

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.5.e.a.14.19	✓	92	1.1	even	1	trivial
71.5.e.a.66.19	yes	92	71.66	odd	10	inner