Embedded newform 185.2.n.a.84.13

• Label: 185.2.n.a.84.13
• 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			84.13
Character	\(\chi\)	\(=\)	185.84
Dual form			185.2.n.a.174.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.843193 + 0.486818i) q^{2} +(1.64972 - 0.952467i) q^{3} +(-0.526017 - 0.911089i) q^{4} +(2.05330 - 0.885408i) q^{5} +1.85471 q^{6} +(-3.89368 + 2.24802i) q^{7} -2.97157i q^{8} +(0.314386 - 0.544533i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 16 q^{4} - 2 q^{5} - 16 q^{6} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.843193	+	0.486818i	0.596227	+	0.344232i	0.767556	−	0.640982i	\(-0.221473\pi\)
							−0.171329	+	0.985214i	\(0.554806\pi\)
\(3\)	1.64972	−	0.952467i	0.952467	−	0.549907i	0.0586204	−	0.998280i	\(-0.481330\pi\)
							0.893846	+	0.448373i	\(0.147996\pi\)
\(5\)	2.05330	−	0.885408i	0.918265	−	0.395966i
							\(7\)	−3.89368	+	2.24802i	−1.47167	+	0.849670i	−0.999493	−	0.0318328i	\(-0.989866\pi\)
−0.472179	+	0.881503i	\(0.656532\pi\)
\(11\)	3.42372			1.03229			0.516145	−	0.856501i	\(-0.327366\pi\)
							0.516145	+	0.856501i	\(0.327366\pi\)
\(13\)	−4.09854	+	2.36630i	−1.13673	+	0.656292i	−0.945619	−	0.325276i	\(-0.894543\pi\)
							−0.191112	+	0.981568i	\(0.561209\pi\)
\(17\)	0.520456	+	0.300485i	0.126229	+	0.0728784i	0.561785	−	0.827283i	\(-0.310115\pi\)
							−0.435556	+	0.900162i	\(0.643448\pi\)
\(19\)	2.78080	+	4.81648i	0.637958	+	1.10498i	0.985880	+	0.167452i	\(0.0535541\pi\)
							−0.347922	+	0.937523i	\(0.613113\pi\)
\(23\)		−	0.727992i		−	0.151797i	−0.997116	−	0.0758984i	\(-0.975818\pi\)
							0.997116	−	0.0758984i	\(-0.0241825\pi\)
\(29\)	5.69641			1.05780			0.528898	−	0.848685i	\(-0.322605\pi\)
							0.528898	+	0.848685i	\(0.322605\pi\)
\(31\)	−6.49908			−1.16727			−0.583635	−	0.812016i	\(-0.698370\pi\)
							−0.583635	+	0.812016i	\(0.698370\pi\)
\(37\)	−3.72728	−	4.80702i	−0.612760	−	0.790269i
							\(41\)	−2.32486	−	4.02677i	−0.363082	−	0.628876i	0.625385	−	0.780316i	\(-0.284942\pi\)
−0.988466	+	0.151441i	\(0.951609\pi\)
\(43\)		−	5.47486i		−	0.834909i	−0.908698	−	0.417454i	\(-0.862922\pi\)
							0.908698	−	0.417454i	\(-0.137078\pi\)
\(47\)		−	2.57812i		−	0.376057i	−0.982164	−	0.188029i	\(-0.939790\pi\)
							0.982164	−	0.188029i	\(-0.0602098\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	185.2.n.a.84.13	yes	36
5.2	odd	4		925.2.e.f.676.6		36
5.3	odd	4		925.2.e.f.676.13		36
5.4	even	2	inner	185.2.n.a.84.6	✓	36
37.26	even	3	inner	185.2.n.a.174.6	yes	36
185.63	odd	12		925.2.e.f.26.13		36
185.137	odd	12		925.2.e.f.26.6		36
185.174	even	6	inner	185.2.n.a.174.13	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.n.a.84.6	✓	36	5.4	even	2	inner
185.2.n.a.84.13	yes	36	1.1	even	1	trivial
185.2.n.a.174.6	yes	36	37.26	even	3	inner
185.2.n.a.174.13	yes	36	185.174	even	6	inner
925.2.e.f.26.6		36	185.137	odd	12
925.2.e.f.26.13		36	185.63	odd	12
925.2.e.f.676.6		36	5.2	odd	4
925.2.e.f.676.13		36	5.3	odd	4