Embedded newform 185.2.n.a.84.8

• Label: 185.2.n.a.84.8
• Level: $185$
• Weight: $2$
• Character: 185.84
• Analytic conductor: $1.477$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 185 = 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	185.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.47723243739\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			84.8
Character	\(\chi\)	\(=\)	185.84
Dual form			185.2.n.a.174.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.398514 - 0.230082i) q^{2} +(2.88437 - 1.66529i) q^{3} +(-0.894124 - 1.54867i) q^{4} +(-2.18772 + 0.462462i) q^{5} -1.53262 q^{6} +(-0.0944407 + 0.0545254i) q^{7} +1.74322i q^{8} +(4.04641 - 7.00858i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 16 q^{4} - 2 q^{5} - 16 q^{6} + 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).

\(n\)	\(76\)	\(112\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.398514	−	0.230082i	−0.281792	−	0.162693i	0.352442	−	0.935834i	\(-0.385351\pi\)
							−0.634234	+	0.773141i	\(0.718685\pi\)
\(3\)	2.88437	−	1.66529i	1.66529	−	0.961458i	0.695171	−	0.718845i	\(-0.255329\pi\)
							0.970123	−	0.242613i	\(-0.0780046\pi\)
\(5\)	−2.18772	+	0.462462i	−0.978379	+	0.206819i
							\(7\)	−0.0944407	+	0.0545254i	−0.0356952	+	0.0206087i	−0.517741	−	0.855537i	\(-0.673227\pi\)
0.482046	+	0.876146i	\(0.339894\pi\)
\(11\)	2.27412			0.685674			0.342837	−	0.939395i	\(-0.388612\pi\)
							0.342837	+	0.939395i	\(0.388612\pi\)
\(13\)	−2.22384	+	1.28394i	−0.616783	+	0.356100i	−0.775616	−	0.631206i	\(-0.782560\pi\)
							0.158832	+	0.987306i	\(0.449227\pi\)
\(17\)	4.60962	+	2.66137i	1.11800	+	0.645476i	0.940889	−	0.338715i	\(-0.109992\pi\)
							0.177109	+	0.984191i	\(0.443326\pi\)
\(19\)	−1.75297	−	3.03624i	−0.402160	−	0.696561i	0.591827	−	0.806065i	\(-0.298407\pi\)
							−0.993986	+	0.109504i	\(0.965074\pi\)
\(23\)			4.47786i			0.933697i	0.884337	+	0.466849i	\(0.154611\pi\)
							−0.884337	+	0.466849i	\(0.845389\pi\)
\(29\)	1.44236			0.267839			0.133920	−	0.990992i	\(-0.457244\pi\)
							0.133920	+	0.990992i	\(0.457244\pi\)
\(31\)	1.87851			0.337390			0.168695	−	0.985668i	\(-0.446045\pi\)
							0.168695	+	0.985668i	\(0.446045\pi\)
\(37\)	−1.93988	+	5.76514i	−0.318915	+	0.947783i
							\(41\)	5.48447	+	9.49938i	0.856530	+	1.48355i	0.875218	+	0.483729i	\(0.160718\pi\)
−0.0186878	+	0.999825i	\(0.505949\pi\)
\(43\)		−	4.53724i		−	0.691922i	−0.938249	−	0.345961i	\(-0.887553\pi\)
							0.938249	−	0.345961i	\(-0.112447\pi\)
\(47\)		−	4.28090i		−	0.624433i	−0.950011	−	0.312217i	\(-0.898929\pi\)
							0.950011	−	0.312217i	\(-0.101071\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	185.2.n.a.84.8	✓	36
5.2	odd	4		925.2.e.f.676.11		36
5.3	odd	4		925.2.e.f.676.8		36
5.4	even	2	inner	185.2.n.a.84.11	yes	36
37.26	even	3	inner	185.2.n.a.174.11	yes	36
185.63	odd	12		925.2.e.f.26.8		36
185.137	odd	12		925.2.e.f.26.11		36
185.174	even	6	inner	185.2.n.a.174.8	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.n.a.84.8	✓	36	1.1	even	1	trivial
185.2.n.a.84.11	yes	36	5.4	even	2	inner
185.2.n.a.174.8	yes	36	185.174	even	6	inner
185.2.n.a.174.11	yes	36	37.26	even	3	inner
925.2.e.f.26.8		36	185.63	odd	12
925.2.e.f.26.11		36	185.137	odd	12
925.2.e.f.676.8		36	5.3	odd	4
925.2.e.f.676.11		36	5.2	odd	4