Embedded newform 185.2.n.a.84.1

• Label: 185.2.n.a.84.1
• 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			84.1
Character	\(\chi\)	\(=\)	185.84
Dual form			185.2.n.a.174.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.33663 - 1.34906i) q^{2} +(2.35586 - 1.36016i) q^{3} +(2.63990 + 4.57244i) q^{4} +(0.261363 - 2.22074i) q^{5} -7.33972 q^{6} +(-1.64348 + 0.948866i) q^{7} -8.84926i q^{8} +(2.20006 - 3.81062i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 16 q^{4} - 2 q^{5} - 16 q^{6} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.33663	−	1.34906i	−1.65225	−	0.953926i	−0.976145	−	0.217119i	\(-0.930334\pi\)
							−0.676103	−	0.736807i	\(-0.736333\pi\)
\(3\)	2.35586	−	1.36016i	1.36016	−	0.785288i	0.370514	−	0.928827i	\(-0.379181\pi\)
							0.989645	+	0.143539i	\(0.0458481\pi\)
\(5\)	0.261363	−	2.22074i	0.116885	−	0.993145i
							\(7\)	−1.64348	+	0.948866i	−0.621178	+	0.358638i	−0.777328	−	0.629096i	\(-0.783425\pi\)
0.156149	+	0.987733i	\(0.450092\pi\)
\(11\)	3.17880			0.958444			0.479222	−	0.877694i	\(-0.340919\pi\)
							0.479222	+	0.877694i	\(0.340919\pi\)
\(13\)	2.13674	−	1.23364i	0.592624	−	0.342151i	−0.173511	−	0.984832i	\(-0.555511\pi\)
							0.766134	+	0.642681i	\(0.222178\pi\)
\(17\)	−4.72509	−	2.72803i	−1.14600	−	0.661645i	−0.198093	−	0.980183i	\(-0.563475\pi\)
							−0.947910	+	0.318538i	\(0.896808\pi\)
\(19\)	−0.821035	−	1.42207i	−0.188358	−	0.326246i	0.756345	−	0.654173i	\(-0.226983\pi\)
							−0.944703	+	0.327927i	\(0.893650\pi\)
\(23\)			7.20428i			1.50220i	0.660190	+	0.751098i	\(0.270476\pi\)
							−0.660190	+	0.751098i	\(0.729524\pi\)
\(29\)	0.245379			0.0455657			0.0227829	−	0.999740i	\(-0.492747\pi\)
							0.0227829	+	0.999740i	\(0.492747\pi\)
\(31\)	1.81454			0.325901			0.162950	−	0.986634i	\(-0.447899\pi\)
							0.162950	+	0.986634i	\(0.447899\pi\)
\(37\)	4.84059	+	3.68356i	0.795789	+	0.605574i
							\(41\)	−2.88091	−	4.98987i	−0.449922	−	0.779287i	0.548459	−	0.836178i	\(-0.315215\pi\)
−0.998380	+	0.0568903i	\(0.981881\pi\)
\(43\)			1.43884i			0.219420i	0.993964	+	0.109710i	\(0.0349923\pi\)
							−0.993964	+	0.109710i	\(0.965008\pi\)
\(47\)			4.32913i			0.631469i	0.948848	+	0.315734i	\(0.102251\pi\)
							−0.948848	+	0.315734i	\(0.897749\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	185.2.n.a.84.1	✓	36
5.2	odd	4		925.2.e.f.676.18		36
5.3	odd	4		925.2.e.f.676.1		36
5.4	even	2	inner	185.2.n.a.84.18	yes	36
37.26	even	3	inner	185.2.n.a.174.18	yes	36
185.63	odd	12		925.2.e.f.26.1		36
185.137	odd	12		925.2.e.f.26.18		36
185.174	even	6	inner	185.2.n.a.174.1	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.n.a.84.1	✓	36	1.1	even	1	trivial
185.2.n.a.84.18	yes	36	5.4	even	2	inner
185.2.n.a.174.1	yes	36	185.174	even	6	inner
185.2.n.a.174.18	yes	36	37.26	even	3	inner
925.2.e.f.26.1		36	185.63	odd	12
925.2.e.f.26.18		36	185.137	odd	12
925.2.e.f.676.1		36	5.3	odd	4
925.2.e.f.676.18		36	5.2	odd	4