Embedded newform 185.2.n.a.84.12

• Label: 185.2.n.a.84.12
• Level: $185$
• Weight: $2$
• Character: 185.84
• Analytic conductor: $1.477$
• Analytic rank: $0$
• Dimension: $36$
• 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			84.12
Character	\(\chi\)	\(=\)	185.84
Dual form			185.2.n.a.174.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.704704 + 0.406861i) q^{2} +(0.142682 - 0.0823772i) q^{3} +(-0.668928 - 1.15862i) q^{4} +(-0.681051 - 2.12983i) q^{5} +0.134064 q^{6} +(2.76887 - 1.59861i) q^{7} -2.71609i q^{8} +(-1.48643 + 2.57457i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 16 q^{4} - 2 q^{5} - 16 q^{6} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.704704	+	0.406861i	0.498301	+	0.287694i	0.728012	−	0.685565i	\(-0.240445\pi\)
							−0.229711	+	0.973259i	\(0.573778\pi\)
\(3\)	0.142682	−	0.0823772i	0.0823772	−	0.0475605i	−0.458246	−	0.888826i	\(-0.651522\pi\)
							0.540623	+	0.841265i	\(0.318189\pi\)
\(5\)	−0.681051	−	2.12983i	−0.304575	−	0.952488i
							\(7\)	2.76887	−	1.59861i	1.04654	−	0.604218i	0.124858	−	0.992175i	\(-0.460152\pi\)
0.921677	+	0.387957i	\(0.126819\pi\)
\(11\)	0.642046			0.193584			0.0967921	−	0.995305i	\(-0.469142\pi\)
							0.0967921	+	0.995305i	\(0.469142\pi\)
\(13\)	−0.0879391	+	0.0507716i	−0.0243899	+	0.0140815i	−0.512145	−	0.858899i	\(-0.671149\pi\)
							0.487755	+	0.872980i	\(0.337816\pi\)
\(17\)	5.39880	+	3.11700i	1.30940	+	0.755983i	0.981996	−	0.188902i	\(-0.0604927\pi\)
							0.327404	+	0.944884i	\(0.393826\pi\)
\(19\)	−1.58053	−	2.73756i	−0.362599	−	0.628040i	0.625789	−	0.779993i	\(-0.284777\pi\)
							−0.988388	+	0.151953i	\(0.951444\pi\)
\(23\)			6.40020i			1.33453i	0.744818	+	0.667267i	\(0.232536\pi\)
							−0.744818	+	0.667267i	\(0.767464\pi\)
\(29\)	−0.922601			−0.171323			−0.0856613	−	0.996324i	\(-0.527300\pi\)
							−0.0856613	+	0.996324i	\(0.527300\pi\)
\(31\)	2.12639			0.381911			0.190956	−	0.981599i	\(-0.438841\pi\)
							0.190956	+	0.981599i	\(0.438841\pi\)
\(37\)	5.74093	−	2.01041i	0.943802	−	0.330510i
							\(41\)	−3.24434	−	5.61936i	−0.506681	−	0.877597i	−0.999970	−	0.00773187i	\(-0.997539\pi\)
0.493289	−	0.869865i	\(-0.335794\pi\)
\(43\)		−	4.73587i		−	0.722214i	−0.932524	−	0.361107i	\(-0.882399\pi\)
							0.932524	−	0.361107i	\(-0.117601\pi\)
\(47\)			11.7571i			1.71495i	0.514525	+	0.857475i	\(0.327968\pi\)
							−0.514525	+	0.857475i	\(0.672032\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	185.2.n.a.84.12	yes	36
5.2	odd	4		925.2.e.f.676.7		36
5.3	odd	4		925.2.e.f.676.12		36
5.4	even	2	inner	185.2.n.a.84.7	✓	36
37.26	even	3	inner	185.2.n.a.174.7	yes	36
185.63	odd	12		925.2.e.f.26.12		36
185.137	odd	12		925.2.e.f.26.7		36
185.174	even	6	inner	185.2.n.a.174.12	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.n.a.84.7	✓	36	5.4	even	2	inner
185.2.n.a.84.12	yes	36	1.1	even	1	trivial
185.2.n.a.174.7	yes	36	37.26	even	3	inner
185.2.n.a.174.12	yes	36	185.174	even	6	inner
925.2.e.f.26.7		36	185.137	odd	12
925.2.e.f.26.12		36	185.63	odd	12
925.2.e.f.676.7		36	5.2	odd	4
925.2.e.f.676.12		36	5.3	odd	4