Embedded newform 37.4.h.a.3.8

• Label: 37.4.h.a.3.8
• Level: $37$
• Weight: $4$
• Character: 37.3
• Analytic conductor: $2.183$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	37.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.18307067021\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(8\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.8
Character	\(\chi\)	\(=\)	37.3
Dual form			37.4.h.a.25.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.09995 - 3.69437i) q^{2} +(1.24007 - 1.04054i) q^{3} +(-2.64954 - 15.0263i) q^{4} +(-4.06803 + 11.1768i) q^{5} -7.80690i q^{6} +(6.95750 + 2.53232i) q^{7} +(-30.3137 - 17.5016i) q^{8} +(-4.23346 + 24.0091i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 3 q^{2} + 9 q^{4} - 27 q^{5} - 108 q^{7} + 144 q^{8} + 42 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.09995	−	3.69437i	1.09600	−	1.30616i	0.147609	−	0.989046i	\(-0.452842\pi\)
							0.948388	−	0.317113i	\(-0.102713\pi\)
\(3\)	1.24007	−	1.04054i	0.238651	−	0.200252i	−0.515616	−	0.856820i	\(-0.672437\pi\)
							0.754267	+	0.656568i	\(0.227992\pi\)
\(5\)	−4.06803	+	11.1768i	−0.363855	+	0.999685i	0.613798	+	0.789463i	\(0.289641\pi\)
							−0.977654	+	0.210222i	\(0.932581\pi\)
\(7\)	6.95750	+	2.53232i	0.375670	+	0.136733i	0.522953	−	0.852362i	\(-0.324831\pi\)
							−0.147283	+	0.989094i	\(0.547053\pi\)
\(11\)	25.0961	−	43.4677i	0.687888	−	1.19146i	−0.284632	−	0.958637i	\(-0.591872\pi\)
							0.972520	−	0.232819i	\(-0.0747951\pi\)
\(13\)	−55.0895	+	9.71377i	−1.17531	+	0.207240i	−0.727000	−	0.686637i	\(-0.759086\pi\)
							−0.448313	+	0.893876i	\(0.647975\pi\)
\(17\)	25.6114	+	4.51599i	0.365393	+	0.0644287i	0.353331	−	0.935499i	\(-0.385049\pi\)
							0.0120628	+	0.999927i	\(0.496160\pi\)
\(19\)	14.9195	+	17.7804i	0.180146	+	0.214690i	0.848559	−	0.529101i	\(-0.177471\pi\)
							−0.668413	+	0.743790i	\(0.733026\pi\)
\(23\)	−149.328	+	86.2146i	−1.35378	+	0.781608i	−0.988777	−	0.149397i	\(-0.952267\pi\)
							−0.365007	+	0.931005i	\(0.618933\pi\)
\(29\)	−223.710	−	129.159i	−1.43248	−	0.827043i	−0.435170	−	0.900348i	\(-0.643312\pi\)
							−0.997310	+	0.0733055i	\(0.976645\pi\)
\(31\)		−	180.157i		−	1.04378i	−0.853014	−	0.521889i	\(-0.825228\pi\)
							0.853014	−	0.521889i	\(-0.174772\pi\)
\(37\)	161.768	−	156.474i	0.718771	−	0.695247i
							\(41\)	66.4974	+	377.125i	0.253296	+	1.43651i	0.800408	+	0.599455i	\(0.204616\pi\)
−0.547112	+	0.837059i	\(0.684273\pi\)
\(43\)		−	254.195i		−	0.901497i	−0.892651	−	0.450749i	\(-0.851157\pi\)
							0.892651	−	0.450749i	\(-0.148843\pi\)
\(47\)	−3.64201	−	6.30814i	−0.0113030	−	0.0195774i	0.860319	−	0.509757i	\(-0.170265\pi\)
							−0.871622	+	0.490179i	\(0.836931\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.h.a.3.8	✓	48
37.5	odd	36		1369.4.a.j.1.3		48
37.25	even	18	inner	37.4.h.a.25.8	yes	48
37.32	odd	36		1369.4.a.j.1.46		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.h.a.3.8	✓	48	1.1	even	1	trivial
37.4.h.a.25.8	yes	48	37.25	even	18	inner
1369.4.a.j.1.3		48	37.5	odd	36
1369.4.a.j.1.46		48	37.32	odd	36