Embedded newform 61.7.j.a.8.4

• Label: 61.7.j.a.8.4
• Level: $61$
• Weight: $7$
• Character: 61.8
• Analytic conductor: $14.033$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	61.j (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.0332991008\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(30\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			8.4
Character	\(\chi\)	\(=\)	61.8
Dual form			61.7.j.a.23.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.0990 - 1.91629i) q^{2} +(10.2537 + 14.1131i) q^{3} +(81.8450 + 26.5930i) q^{4} +(10.5714 + 3.43485i) q^{5} +(-97.0148 - 190.402i) q^{6} +(-33.3080 - 210.298i) q^{7} +(-240.743 - 122.664i) q^{8} +(131.234 - 403.897i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 16 q^{2} - 10 q^{3} - 10 q^{4} - 230 q^{5} + 1384 q^{6} + 248 q^{7} + 632 q^{8} + 13602 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/61\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.0990	−	1.91629i	−1.51237	−	0.239536i	−0.655550	−	0.755152i	\(-0.727563\pi\)
							−0.856820	+	0.515616i	\(0.827563\pi\)
\(3\)	10.2537	+	14.1131i	0.379768	+	0.522706i	0.955523	−	0.294916i	\(-0.0952918\pi\)
							−0.575755	+	0.817622i	\(0.695292\pi\)
\(5\)	10.5714	+	3.43485i	0.0845711	+	0.0274788i	0.350997	−	0.936377i	\(-0.385843\pi\)
							−0.266426	+	0.963855i	\(0.585843\pi\)
\(7\)	−33.3080	−	210.298i	−0.0971079	−	0.613115i	−0.987463	−	0.157849i	\(-0.949544\pi\)
							0.890355	−	0.455266i	\(-0.150456\pi\)
\(11\)	−1774.53	+	1774.53i	−1.33323	+	1.33323i	−0.430765	+	0.902464i	\(0.641756\pi\)
							−0.902464	+	0.430765i	\(0.858244\pi\)
\(13\)	1268.39			0.577330			0.288665	−	0.957430i	\(-0.406789\pi\)
							0.288665	+	0.957430i	\(0.406789\pi\)
\(17\)	210.353	+	412.841i	0.0428156	+	0.0840304i	0.911423	−	0.411470i	\(-0.134985\pi\)
							−0.868608	+	0.495501i	\(0.834985\pi\)
\(19\)	−1492.28	−	2053.95i	−0.217566	−	0.299453i	0.686258	−	0.727358i	\(-0.259252\pi\)
							−0.903824	+	0.427905i	\(0.859252\pi\)
\(23\)	18876.1	−	9617.85i	1.55142	−	0.790486i	0.552347	−	0.833614i	\(-0.313732\pi\)
							0.999070	+	0.0431279i	\(0.0137323\pi\)
\(29\)	23267.1	−	23267.1i	0.954001	−	0.954001i	−0.0449868	−	0.998988i	\(-0.514325\pi\)
							0.998988	+	0.0449868i	\(0.0143246\pi\)
\(31\)	20503.5	−	3247.43i	0.688244	−	0.109007i	0.197490	−	0.980305i	\(-0.436721\pi\)
							0.490755	+	0.871298i	\(0.336721\pi\)
\(37\)	−31409.6	+	4974.80i	−0.620095	+	0.0982133i	−0.458574	−	0.888656i	\(-0.651640\pi\)
							−0.161520	+	0.986869i	\(0.551640\pi\)
\(41\)	28107.5	−	38686.6i	0.407822	−	0.561319i	−0.554864	−	0.831941i	\(-0.687230\pi\)
							0.962685	+	0.270623i	\(0.0872296\pi\)
\(43\)	37117.0	+	18912.1i	0.466839	+	0.237867i	0.671555	−	0.740955i	\(-0.265627\pi\)
							−0.204715	+	0.978822i	\(0.565627\pi\)
\(47\)	130556.			1.25749			0.628743	−	0.777613i	\(-0.283570\pi\)
							0.628743	+	0.777613i	\(0.283570\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.7.j.a.8.4	✓	240
61.23	odd	20	inner	61.7.j.a.23.4	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.7.j.a.8.4	✓	240	1.1	even	1	trivial
61.7.j.a.23.4	yes	240	61.23	odd	20	inner