Embedded newform 185.2.u.a.8.1

• Label: 185.2.u.a.8.1
• Level: $185$
• Weight: $2$
• Character: 185.8
• Analytic conductor: $1.477$
• Analytic rank: $0$
• Dimension: $68$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 185 = 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	185.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.47723243739\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(17\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			8.1
Character	\(\chi\)	\(=\)	185.8
Dual form			185.2.u.a.162.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.38259 + 1.37559i) q^{2} +(0.185681 + 0.692973i) q^{3} +(2.78450 - 4.82289i) q^{4} +(-1.62458 - 1.53647i) q^{5} +(-1.39565 - 1.39565i) q^{6} +(0.502511 + 1.87540i) q^{7} +9.81895i q^{8} +(2.15234 - 1.24266i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 6 q^{2} - 4 q^{3} + 30 q^{4} - 8 q^{6} - 2 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(76\)	\(112\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.38259	+	1.37559i	−1.68475	+	0.972689i	−0.726318	+	0.687359i	\(0.758770\pi\)
							−0.958429	+	0.285330i	\(0.907897\pi\)
\(3\)	0.185681	+	0.692973i	0.107203	+	0.400088i	0.998586	−	0.0531638i	\(-0.0169305\pi\)
							−0.891383	+	0.453252i	\(0.850264\pi\)
\(5\)	−1.62458	−	1.53647i	−0.726533	−	0.687132i
							\(7\)	0.502511	+	1.87540i	0.189931	+	0.708833i	0.993521	+	0.113650i	\(0.0362541\pi\)
−0.803590	+	0.595184i	\(0.797079\pi\)
\(11\)			5.81983i			1.75474i	0.479810	+	0.877372i	\(0.340706\pi\)
							−0.479810	+	0.877372i	\(0.659294\pi\)
\(13\)	0.124686	+	0.0719878i	0.0345818	+	0.0199658i	0.517191	−	0.855870i	\(-0.326978\pi\)
							−0.482609	+	0.875836i	\(0.660311\pi\)
\(17\)	0.235215	+	0.407404i	0.0570480	+	0.0988101i	0.893139	−	0.449781i	\(-0.148498\pi\)
							−0.836091	+	0.548591i	\(0.815165\pi\)
\(19\)	1.51574	+	5.65682i	0.347735	+	1.29776i	0.889385	+	0.457159i	\(0.151133\pi\)
							−0.541650	+	0.840604i	\(0.682200\pi\)
\(23\)			0.125675i			0.0262050i	0.999914	+	0.0131025i	\(0.00417077\pi\)
							−0.999914	+	0.0131025i	\(0.995829\pi\)
\(29\)	−2.22835	−	2.22835i	−0.413795	−	0.413795i	0.469263	−	0.883058i	\(-0.344519\pi\)
							−0.883058	+	0.469263i	\(0.844519\pi\)
\(31\)	−2.45959	+	2.45959i	−0.441755	+	0.441755i	−0.892602	−	0.450846i	\(-0.851122\pi\)
							0.450846	+	0.892602i	\(0.351122\pi\)
\(37\)	3.02888	−	5.27503i	0.497945	−	0.867209i
							\(41\)	6.94440	+	4.00935i	1.08453	+	0.626155i	0.932116	−	0.362161i	\(-0.117961\pi\)
0.152417	+	0.988316i	\(0.451294\pi\)
\(43\)			3.12028i			0.475838i	0.971285	+	0.237919i	\(0.0764654\pi\)
							−0.971285	+	0.237919i	\(0.923535\pi\)
\(47\)	0.406839	−	0.406839i	0.0593436	−	0.0593436i	−0.676812	−	0.736156i	\(-0.736639\pi\)
							0.736156	+	0.676812i	\(0.236639\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	185.2.u.a.8.1	yes	68
5.2	odd	4		185.2.p.a.82.1	✓	68
5.3	odd	4		925.2.t.b.82.17		68
5.4	even	2		925.2.y.b.193.17		68
37.14	odd	12		185.2.p.a.88.1	yes	68
185.14	odd	12		925.2.t.b.643.17		68
185.88	even	12		925.2.y.b.532.17		68
185.162	even	12	inner	185.2.u.a.162.1	yes	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.p.a.82.1	✓	68	5.2	odd	4
185.2.p.a.88.1	yes	68	37.14	odd	12
185.2.u.a.8.1	yes	68	1.1	even	1	trivial
185.2.u.a.162.1	yes	68	185.162	even	12	inner
925.2.t.b.82.17		68	5.3	odd	4
925.2.t.b.643.17		68	185.14	odd	12
925.2.y.b.193.17		68	5.4	even	2
925.2.y.b.532.17		68	185.88	even	12