Embedded newform 185.2.n.a.174.2

• Label: 185.2.n.a.174.2
• Level: $185$
• Weight: $2$
• Character: 185.174
• 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			174.2
Character	\(\chi\)	\(=\)	185.174
Dual form			185.2.n.a.84.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99875 + 1.15398i) q^{2} +(-1.81123 - 1.04571i) q^{3} +(1.66334 - 2.88098i) q^{4} +(-0.0902553 + 2.23425i) q^{5} +4.82692 q^{6} +(3.28655 + 1.89749i) q^{7} +3.06190i q^{8} +(0.687027 + 1.18997i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 16 q^{4} - 2 q^{5} - 16 q^{6} + 14 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99875	+	1.15398i	−1.41333	+	0.815986i	−0.995701	−	0.0926310i	\(-0.970472\pi\)
							−0.417629	+	0.908617i	\(0.637139\pi\)
\(3\)	−1.81123	−	1.04571i	−1.04571	−	0.603742i	−0.124266	−	0.992249i	\(-0.539658\pi\)
							−0.921446	+	0.388507i	\(0.872991\pi\)
\(5\)	−0.0902553	+	2.23425i	−0.0403634	+	0.999185i
							\(7\)	3.28655	+	1.89749i	1.24220	+	0.717183i	0.969541	−	0.244928i	\(-0.0787644\pi\)
0.272657	+	0.962111i	\(0.412098\pi\)
\(11\)	−3.98792			−1.20240			−0.601202	−	0.799097i	\(-0.705311\pi\)
							−0.601202	+	0.799097i	\(0.705311\pi\)
\(13\)	−5.14127	−	2.96831i	−1.42593	−	0.823261i	−0.429134	−	0.903241i	\(-0.641181\pi\)
							−0.996797	+	0.0799795i	\(0.974515\pi\)
\(17\)	−2.60081	+	1.50158i	−0.630789	+	0.364186i	−0.781057	−	0.624459i	\(-0.785319\pi\)
							0.150269	+	0.988645i	\(0.451986\pi\)
\(19\)	−0.477453	+	0.826973i	−0.109535	+	0.189721i	−0.915582	−	0.402131i	\(-0.868270\pi\)
							0.806047	+	0.591852i	\(0.201603\pi\)
\(23\)			0.297301i			0.0619915i	0.999520	+	0.0309958i	\(0.00986784\pi\)
							−0.999520	+	0.0309958i	\(0.990132\pi\)
\(29\)	−6.45700			−1.19903			−0.599517	−	0.800362i	\(-0.704641\pi\)
							−0.599517	+	0.800362i	\(0.704641\pi\)
\(31\)	−1.69659			−0.304716			−0.152358	−	0.988325i	\(-0.548687\pi\)
							−0.152358	+	0.988325i	\(0.548687\pi\)
\(37\)	−3.31614	−	5.09934i	−0.545170	−	0.838326i
							\(41\)	3.89586	−	6.74782i	0.608430	−	1.05383i	−0.383069	−	0.923720i	\(-0.625133\pi\)
0.991499	−	0.130113i	\(-0.0415339\pi\)
\(43\)			4.65534i			0.709933i	0.934879	+	0.354966i	\(0.115508\pi\)
							−0.934879	+	0.354966i	\(0.884492\pi\)
\(47\)			3.36817i			0.491298i	0.969359	+	0.245649i	\(0.0790010\pi\)
							−0.969359	+	0.245649i	\(0.920999\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	185.2.n.a.174.2	yes	36
5.2	odd	4		925.2.e.f.26.2		36
5.3	odd	4		925.2.e.f.26.17		36
5.4	even	2	inner	185.2.n.a.174.17	yes	36
37.10	even	3	inner	185.2.n.a.84.17	yes	36
185.47	odd	12		925.2.e.f.676.2		36
185.84	even	6	inner	185.2.n.a.84.2	✓	36
185.158	odd	12		925.2.e.f.676.17		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.n.a.84.2	✓	36	185.84	even	6	inner
185.2.n.a.84.17	yes	36	37.10	even	3	inner
185.2.n.a.174.2	yes	36	1.1	even	1	trivial
185.2.n.a.174.17	yes	36	5.4	even	2	inner
925.2.e.f.26.2		36	5.2	odd	4
925.2.e.f.26.17		36	5.3	odd	4
925.2.e.f.676.2		36	185.47	odd	12
925.2.e.f.676.17		36	185.158	odd	12