Embedded newform 37.6.e.a.27.3

• Label: 37.6.e.a.27.3
• Level: $37$
• Weight: $6$
• Character: 37.27
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			27.3
Character	\(\chi\)	\(=\)	37.27
Dual form			37.6.e.a.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.60377 - 4.39004i) q^{2} +(-8.72617 - 15.1142i) q^{3} +(22.5449 + 39.0489i) q^{4} +(-72.9853 + 42.1381i) q^{5} +153.233i q^{6} +(-77.9335 - 134.985i) q^{7} -114.930i q^{8} +(-30.7921 + 53.3336i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 6 q^{2} + 18 q^{3} + 298 q^{4} + 144 q^{5} - 52 q^{7} - 1490 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.60377	−	4.39004i	−1.34417	−	0.776057i	−0.356754	−	0.934199i	\(-0.616116\pi\)
							−0.987416	+	0.158142i	\(0.949450\pi\)
\(3\)	−8.72617	−	15.1142i	−0.559784	−	0.969575i	−0.997514	−	0.0704681i	\(-0.977551\pi\)
							0.437730	−	0.899107i	\(-0.355783\pi\)
\(5\)	−72.9853	+	42.1381i	−1.30560	+	0.753789i	−0.981359	−	0.192185i	\(-0.938443\pi\)
							−0.324243	+	0.945974i	\(0.605109\pi\)
\(7\)	−77.9335	−	134.985i	−0.601145	−	1.04121i	−0.992648	−	0.121037i	\(-0.961378\pi\)
							0.391503	−	0.920177i	\(-0.371955\pi\)
\(11\)	403.781			1.00615			0.503076	−	0.864242i	\(-0.332201\pi\)
							0.503076	+	0.864242i	\(0.332201\pi\)
\(13\)	−257.541	+	148.691i	−0.422657	+	0.244021i	−0.696213	−	0.717835i	\(-0.745133\pi\)
							0.273557	+	0.961856i	\(0.411800\pi\)
\(17\)	−468.276	−	270.359i	−0.392988	−	0.226892i	0.290466	−	0.956885i	\(-0.406190\pi\)
							−0.683454	+	0.729994i	\(0.739523\pi\)
\(19\)	1051.22	−	606.921i	0.668049	−	0.385698i	−0.127288	−	0.991866i	\(-0.540627\pi\)
							0.795337	+	0.606167i	\(0.207294\pi\)
\(23\)			3395.42i			1.33836i	0.743099	+	0.669182i	\(0.233355\pi\)
							−0.743099	+	0.669182i	\(0.766645\pi\)
\(29\)			4535.29i			1.00141i	0.865619	+	0.500703i	\(0.166925\pi\)
							−0.865619	+	0.500703i	\(0.833075\pi\)
\(31\)			9863.11i			1.84336i	0.387954	+	0.921679i	\(0.373182\pi\)
							−0.387954	+	0.921679i	\(0.626818\pi\)
\(37\)	−867.348	−	8282.01i	−0.104157	−	0.994561i
							\(41\)	4604.61	+	7975.41i	0.427792	+	0.740958i	0.996677	−	0.0814595i	\(-0.0259581\pi\)
−0.568884	+	0.822418i	\(0.692625\pi\)
\(43\)		−	14711.9i		−	1.21339i	−0.794936	−	0.606693i	\(-0.792496\pi\)
							0.794936	−	0.606693i	\(-0.207504\pi\)
\(47\)	19308.0			1.27495			0.637473	−	0.770473i	\(-0.279980\pi\)
							0.637473	+	0.770473i	\(0.279980\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.e.a.27.3	yes	32
37.11	even	6	inner	37.6.e.a.11.3	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.e.a.11.3	✓	32	37.11	even	6	inner
37.6.e.a.27.3	yes	32	1.1	even	1	trivial