Embedded newform 61.6.g.a.3.14

• Label: 61.6.g.a.3.14
• Level: $61$
• Weight: $6$
• Character: 61.3
• Analytic conductor: $9.783$
• Analytic rank: $0$
• Dimension: $104$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	61.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.78341300859\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			3.14
Character	\(\chi\)	\(=\)	61.3
Dual form			61.6.g.a.41.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.580441 + 0.188597i) q^{2} +(-6.01418 + 18.5098i) q^{3} +(-25.5872 - 18.5902i) q^{4} +(-24.9177 - 18.1038i) q^{5} +(-6.98176 + 9.60957i) q^{6} +(84.3182 - 27.3966i) q^{7} +(-22.8252 - 31.4163i) q^{8} +(-109.850 - 79.8104i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 5 q^{2} + 15 q^{3} + 465 q^{4} - 122 q^{5} + 445 q^{6} + 230 q^{7} - 5 q^{8} - 2841 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.580441	+	0.188597i	0.102608	+	0.0333395i	0.359871	−	0.933002i	\(-0.382821\pi\)
							−0.257263	+	0.966341i	\(0.582821\pi\)
\(3\)	−6.01418	+	18.5098i	−0.385810	+	1.18740i	0.550081	+	0.835111i	\(0.314597\pi\)
							−0.935891	+	0.352290i	\(0.885403\pi\)
\(5\)	−24.9177	−	18.1038i	−0.445741	−	0.323850i	0.342171	−	0.939638i	\(-0.388838\pi\)
							−0.787912	+	0.615788i	\(0.788838\pi\)
\(7\)	84.3182	−	27.3966i	0.650394	−	0.211326i	0.0348059	−	0.999394i	\(-0.488919\pi\)
							0.615588	+	0.788068i	\(0.288919\pi\)
\(11\)		−	336.894i		−	0.839483i	−0.907644	−	0.419742i	\(-0.862121\pi\)
							0.907644	−	0.419742i	\(-0.137879\pi\)
\(13\)	974.467			1.59922			0.799611	−	0.600519i	\(-0.205039\pi\)
							0.799611	+	0.600519i	\(0.205039\pi\)
\(17\)	−917.994	+	1263.51i	−0.770402	+	1.06037i	0.225874	+	0.974156i	\(0.427476\pi\)
							−0.996277	+	0.0862116i	\(0.972524\pi\)
\(19\)	879.015	−	2705.33i	0.558615	−	1.71924i	−0.127586	−	0.991827i	\(-0.540723\pi\)
							0.686201	−	0.727412i	\(-0.259277\pi\)
\(23\)	2452.86	−	3376.07i	0.966837	−	1.33074i	0.0232092	−	0.999731i	\(-0.492612\pi\)
							0.943628	−	0.331007i	\(-0.107388\pi\)
\(29\)		−	6109.92i		−	1.34909i	−0.738235	−	0.674544i	\(-0.764340\pi\)
							0.738235	−	0.674544i	\(-0.235660\pi\)
\(31\)	−4730.81	+	1537.13i	−0.884160	+	0.287281i	−0.715684	−	0.698424i	\(-0.753885\pi\)
							−0.168477	+	0.985706i	\(0.553885\pi\)
\(37\)	4026.69	−	1308.35i	0.483553	−	0.157116i	−0.0570882	−	0.998369i	\(-0.518182\pi\)
							0.540641	+	0.841253i	\(0.318182\pi\)
\(41\)	−4823.76	−	14846.0i	−0.448152	−	1.37927i	−0.878989	−	0.476842i	\(-0.841782\pi\)
							0.430837	−	0.902430i	\(-0.358218\pi\)
\(43\)	10914.6	+	15022.6i	0.900193	+	1.23901i	0.970407	+	0.241477i	\(0.0776318\pi\)
							−0.0702138	+	0.997532i	\(0.522368\pi\)
\(47\)	6985.15			0.461244			0.230622	−	0.973043i	\(-0.425924\pi\)
							0.230622	+	0.973043i	\(0.425924\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.6.g.a.3.14	✓	104
61.41	even	10	inner	61.6.g.a.41.14	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.6.g.a.3.14	✓	104	1.1	even	1	trivial
61.6.g.a.41.14	yes	104	61.41	even	10	inner