Embedded newform 37.4.h.a.3.1

• Label: 37.4.h.a.3.1
• 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			3.1
Character	\(\chi\)	\(=\)	37.3
Dual form			37.4.h.a.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.52321 + 4.19880i) q^{2} +(5.42588 - 4.55286i) q^{3} +(-3.82771 - 21.7080i) q^{4} +(4.86754 - 13.3735i) q^{5} +38.8228i q^{6} +(-13.3663 - 4.86492i) q^{7} +(66.6589 + 38.4856i) q^{8} +(4.02320 - 22.8167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 3 q^{2} + 9 q^{4} - 27 q^{5} - 108 q^{7} + 144 q^{8} + 42 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.52321	+	4.19880i	−1.24564	+	1.48450i	−0.433375	+	0.901214i	\(0.642678\pi\)
							−0.812267	+	0.583285i	\(0.801767\pi\)
\(3\)	5.42588	−	4.55286i	1.04421	−	0.876198i	0.0517385	−	0.998661i	\(-0.483524\pi\)
							0.992473	+	0.122463i	\(0.0390793\pi\)
\(5\)	4.86754	−	13.3735i	0.435366	−	1.19616i	−0.507109	−	0.861882i	\(-0.669286\pi\)
							0.942475	−	0.334276i	\(-0.108492\pi\)
\(7\)	−13.3663	−	4.86492i	−0.721710	−	0.262681i	−0.0450585	−	0.998984i	\(-0.514347\pi\)
							−0.676652	+	0.736303i	\(0.736570\pi\)
\(11\)	2.49528	−	4.32195i	0.0683960	−	0.118465i	−0.829799	−	0.558062i	\(-0.811545\pi\)
							0.898195	+	0.439596i	\(0.144879\pi\)
\(13\)	−12.4077	+	2.18782i	−0.264715	+	0.0466763i	−0.304430	−	0.952535i	\(-0.598466\pi\)
							0.0397154	+	0.999211i	\(0.487355\pi\)
\(17\)	96.2763	+	16.9761i	1.37355	+	0.242195i	0.811233	−	0.584724i	\(-0.198797\pi\)
							0.562322	+	0.826918i	\(0.309908\pi\)
\(19\)	28.6230	+	34.1116i	0.345609	+	0.411880i	0.910648	−	0.413184i	\(-0.135583\pi\)
							−0.565039	+	0.825064i	\(0.691139\pi\)
\(23\)	46.0254	−	26.5728i	0.417259	−	0.240905i	−0.276645	−	0.960972i	\(-0.589223\pi\)
							0.693904	+	0.720068i	\(0.255889\pi\)
\(29\)	−85.0663	−	49.1131i	−0.544704	−	0.314485i	0.202279	−	0.979328i	\(-0.435165\pi\)
							−0.746983	+	0.664843i	\(0.768498\pi\)
\(31\)			151.986i			0.880565i	0.897859	+	0.440283i	\(0.145122\pi\)
							−0.897859	+	0.440283i	\(0.854878\pi\)
\(37\)	−112.591	−	194.875i	−0.500266	−	0.865872i
							\(41\)	66.6183	+	377.811i	0.253757	+	1.43913i	0.799243	+	0.601008i	\(0.205234\pi\)
−0.545486	+	0.838120i	\(0.683655\pi\)
\(43\)		−	141.892i		−	0.503216i	−0.967829	−	0.251608i	\(-0.919041\pi\)
							0.967829	−	0.251608i	\(-0.0809594\pi\)
\(47\)	105.739	+	183.145i	0.328162	+	0.568394i	0.982147	−	0.188114i	\(-0.0602374\pi\)
							−0.653985	+	0.756508i	\(0.726904\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.h.a.3.1	✓	48
37.5	odd	36		1369.4.a.j.1.48		48
37.25	even	18	inner	37.4.h.a.25.1	yes	48
37.32	odd	36		1369.4.a.j.1.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.h.a.3.1	✓	48	1.1	even	1	trivial
37.4.h.a.25.1	yes	48	37.25	even	18	inner
1369.4.a.j.1.1		48	37.32	odd	36
1369.4.a.j.1.48		48	37.5	odd	36