Embedded newform 37.6.e.a.11.11

• Label: 37.6.e.a.11.11
• Level: $37$
• Weight: $6$
• Character: 37.11
• 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			11.11
Character	\(\chi\)	\(=\)	37.11
Dual form			37.6.e.a.27.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.33309 - 2.50171i) q^{2} +(-3.26958 + 5.66309i) q^{3} +(-3.48290 + 6.03256i) q^{4} +(54.4047 + 31.4106i) q^{5} +32.7182i q^{6} +(22.9091 - 39.6798i) q^{7} +194.962i q^{8} +(100.120 + 173.412i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 6 q^{2} + 18 q^{3} + 298 q^{4} + 144 q^{5} - 52 q^{7} - 1490 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.33309	−	2.50171i	0.765989	−	0.442244i	−0.0654529	−	0.997856i	\(-0.520849\pi\)
							0.831442	+	0.555612i	\(0.187516\pi\)
\(3\)	−3.26958	+	5.66309i	−0.209744	+	0.363287i	−0.951634	−	0.307235i	\(-0.900596\pi\)
							0.741890	+	0.670522i	\(0.233930\pi\)
\(5\)	54.4047	+	31.4106i	0.973220	+	0.561889i	0.900217	−	0.435443i	\(-0.143408\pi\)
							0.0730039	+	0.997332i	\(0.476741\pi\)
\(7\)	22.9091	−	39.6798i	0.176711	−	0.306072i	−0.764041	−	0.645168i	\(-0.776788\pi\)
							0.940752	+	0.339095i	\(0.110121\pi\)
\(11\)	436.765			1.08834			0.544171	−	0.838974i	\(-0.316844\pi\)
							0.544171	+	0.838974i	\(0.316844\pi\)
\(13\)	−639.436	−	369.178i	−1.04939	−	0.605868i	−0.126914	−	0.991914i	\(-0.540507\pi\)
							−0.922480	+	0.386046i	\(0.873841\pi\)
\(17\)	979.861	−	565.723i	0.822322	−	0.474768i	−0.0288943	−	0.999582i	\(-0.509199\pi\)
							0.851217	+	0.524814i	\(0.175865\pi\)
\(19\)	−1731.42	−	999.638i	−1.10032	−	0.635271i	−0.164016	−	0.986458i	\(-0.552445\pi\)
							−0.936305	+	0.351187i	\(0.885778\pi\)
\(23\)			2734.17i			1.07772i	0.842396	+	0.538859i	\(0.181145\pi\)
							−0.842396	+	0.538859i	\(0.818855\pi\)
\(29\)		−	564.726i		−	0.124693i	−0.998055	−	0.0623466i	\(-0.980142\pi\)
							0.998055	−	0.0623466i	\(-0.0198584\pi\)
\(31\)		−	10262.2i		−	1.91794i	−0.283499	−	0.958972i	\(-0.591495\pi\)
							0.283499	−	0.958972i	\(-0.408505\pi\)
\(37\)	6066.81	+	5704.19i	0.728545	+	0.684998i
							\(41\)	3852.70	−	6673.07i	0.357936	−	0.619964i	−0.629680	−	0.776855i	\(-0.716814\pi\)
0.987616	+	0.156891i	\(0.0501472\pi\)
\(43\)		−	16789.4i		−	1.38473i	−0.721550	−	0.692363i	\(-0.756570\pi\)
							0.721550	−	0.692363i	\(-0.243430\pi\)
\(47\)	19433.9			1.28326			0.641630	−	0.767015i	\(-0.278259\pi\)
							0.641630	+	0.767015i	\(0.278259\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.e.a.11.11	✓	32
37.27	even	6	inner	37.6.e.a.27.11	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.e.a.11.11	✓	32	1.1	even	1	trivial
37.6.e.a.27.11	yes	32	37.27	even	6	inner