Embedded newform 37.4.h.a.3.7

• Label: 37.4.h.a.3.7
• 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			3.7
Character	\(\chi\)	\(=\)	37.3
Dual form			37.4.h.a.25.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.37200 - 2.82684i) q^{2} +(-1.46213 + 1.22687i) q^{3} +(-0.975449 - 5.53205i) q^{4} +(6.54993 - 17.9958i) q^{5} +7.04333i q^{6} +(-0.795732 - 0.289623i) q^{7} +(7.61432 + 4.39613i) q^{8} +(-4.05590 + 23.0021i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 3 q^{2} + 9 q^{4} - 27 q^{5} - 108 q^{7} + 144 q^{8} + 42 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.37200	−	2.82684i	0.838628	−	0.999438i	−0.161293	−	0.986906i	\(-0.551567\pi\)
							0.999921	−	0.0125315i	\(-0.00398900\pi\)
\(3\)	−1.46213	+	1.22687i	−0.281386	+	0.236111i	−0.772547	−	0.634958i	\(-0.781017\pi\)
							0.491160	+	0.871069i	\(0.336573\pi\)
\(5\)	6.54993	−	17.9958i	0.585844	−	1.60959i	−0.192181	−	0.981359i	\(-0.561556\pi\)
							0.778025	−	0.628233i	\(-0.216222\pi\)
\(7\)	−0.795732	−	0.289623i	−0.0429655	−	0.0156382i	0.320448	−	0.947266i	\(-0.396167\pi\)
							−0.363414	+	0.931628i	\(0.618389\pi\)
\(11\)	−32.8793	+	56.9486i	−0.901226	+	1.56097i	−0.0753209	+	0.997159i	\(0.523998\pi\)
							−0.825905	+	0.563809i	\(0.809335\pi\)
\(13\)	22.6024	−	3.98541i	0.482213	−	0.0850272i	0.0727431	−	0.997351i	\(-0.476825\pi\)
							0.409470	+	0.912323i	\(0.365714\pi\)
\(17\)	42.6177	+	7.51464i	0.608018	+	0.107210i	0.469176	−	0.883105i	\(-0.344551\pi\)
							0.138842	+	0.990315i	\(0.455662\pi\)
\(19\)	−65.7122	−	78.3127i	−0.793442	−	0.945588i	0.206014	−	0.978549i	\(-0.433951\pi\)
							−0.999457	+	0.0329611i	\(0.989506\pi\)
\(23\)	−58.0764	+	33.5304i	−0.526512	+	0.303982i	−0.739595	−	0.673052i	\(-0.764983\pi\)
							0.213083	+	0.977034i	\(0.431649\pi\)
\(29\)	4.90271	+	2.83058i	0.0313935	+	0.0181250i	0.515615	−	0.856821i	\(-0.327564\pi\)
							−0.484221	+	0.874946i	\(0.660897\pi\)
\(31\)			195.420i			1.13221i	0.824333	+	0.566105i	\(0.191550\pi\)
							−0.824333	+	0.566105i	\(0.808450\pi\)
\(37\)	126.854	−	185.906i	0.563639	−	0.826021i
							\(41\)	4.12316	+	23.3836i	0.0157056	+	0.0890708i	0.991653	−	0.128935i	\(-0.0411559\pi\)
−0.975947	+	0.218006i	\(0.930045\pi\)
\(43\)		−	166.655i		−	0.591037i	−0.955337	−	0.295518i	\(-0.904508\pi\)
							0.955337	−	0.295518i	\(-0.0954924\pi\)
\(47\)	49.3091	+	85.4058i	0.153031	+	0.265058i	0.932340	−	0.361582i	\(-0.117763\pi\)
							−0.779309	+	0.626640i	\(0.784430\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.h.a.3.7	✓	48
37.5	odd	36		1369.4.a.j.1.9		48
37.25	even	18	inner	37.4.h.a.25.7	yes	48
37.32	odd	36		1369.4.a.j.1.40		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.h.a.3.7	✓	48	1.1	even	1	trivial
37.4.h.a.25.7	yes	48	37.25	even	18	inner
1369.4.a.j.1.9		48	37.5	odd	36
1369.4.a.j.1.40		48	37.32	odd	36