Embedded newform 37.4.h.a.3.6

• Label: 37.4.h.a.3.6
• 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			3.6
Character	\(\chi\)	\(=\)	37.3
Dual form			37.4.h.a.25.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.20911 - 1.44096i) q^{2} +(6.02403 - 5.05476i) q^{3} +(0.774762 + 4.39389i) q^{4} +(-1.51038 + 4.14973i) q^{5} -14.7922i q^{6} +(-23.8352 - 8.67530i) q^{7} +(20.3005 + 11.7205i) q^{8} +(6.04983 - 34.3103i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 3 q^{2} + 9 q^{4} - 27 q^{5} - 108 q^{7} + 144 q^{8} + 42 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.20911	−	1.44096i	0.427485	−	0.509457i	−0.508710	−	0.860938i	\(-0.669877\pi\)
							0.936195	+	0.351481i	\(0.114322\pi\)
\(3\)	6.02403	−	5.05476i	1.15933	−	0.972790i	0.159429	−	0.987209i	\(-0.449035\pi\)
							0.999896	+	0.0144197i	\(0.00459010\pi\)
\(5\)	−1.51038	+	4.14973i	−0.135092	+	0.371163i	−0.988731	−	0.149702i	\(-0.952168\pi\)
							0.853639	+	0.520866i	\(0.174391\pi\)
\(7\)	−23.8352	−	8.67530i	−1.28698	−	0.468422i	−0.394245	−	0.919005i	\(-0.628994\pi\)
							−0.892734	+	0.450583i	\(0.851216\pi\)
\(11\)	−11.9966	+	20.7788i	−0.328829	+	0.569549i	−0.982280	−	0.187420i	\(-0.939987\pi\)
							0.653451	+	0.756969i	\(0.273321\pi\)
\(13\)	19.1309	−	3.37329i	0.408150	−	0.0719679i	0.0341963	−	0.999415i	\(-0.489113\pi\)
							0.373954	+	0.927447i	\(0.378002\pi\)
\(17\)	−89.7905	−	15.8325i	−1.28102	−	0.225879i	−0.508609	−	0.860998i	\(-0.669840\pi\)
							−0.772415	+	0.635119i	\(0.780951\pi\)
\(19\)	53.3575	+	63.5890i	0.644265	+	0.767806i	0.985037	−	0.172340i	\(-0.0551329\pi\)
							−0.340772	+	0.940146i	\(0.610688\pi\)
\(23\)	93.7406	−	54.1211i	0.849837	−	0.490654i	−0.0107585	−	0.999942i	\(-0.503425\pi\)
							0.860596	+	0.509288i	\(0.170091\pi\)
\(29\)	−143.078	−	82.6060i	−0.916169	−	0.528950i	−0.0337579	−	0.999430i	\(-0.510748\pi\)
							−0.882411	+	0.470480i	\(0.844081\pi\)
\(31\)		−	269.867i		−	1.56354i	−0.623570	−	0.781768i	\(-0.714318\pi\)
							0.623570	−	0.781768i	\(-0.285682\pi\)
\(37\)	28.3586	−	223.268i	0.126004	−	0.992030i
							\(41\)	−77.6817	−	440.555i	−0.295899	−	1.67812i	−0.663528	−	0.748152i	\(-0.730942\pi\)
0.367629	−	0.929972i	\(-0.380170\pi\)
\(43\)			352.808i			1.25123i	0.780133	+	0.625613i	\(0.215151\pi\)
							−0.780133	+	0.625613i	\(0.784849\pi\)
\(47\)	72.7985	+	126.091i	0.225931	+	0.391324i	0.956598	−	0.291410i	\(-0.0941243\pi\)
							−0.730667	+	0.682734i	\(0.760791\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.h.a.3.6	✓	48
37.5	odd	36		1369.4.a.j.1.17		48
37.25	even	18	inner	37.4.h.a.25.6	yes	48
37.32	odd	36		1369.4.a.j.1.32		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.h.a.3.6	✓	48	1.1	even	1	trivial
37.4.h.a.25.6	yes	48	37.25	even	18	inner
1369.4.a.j.1.17		48	37.5	odd	36
1369.4.a.j.1.32		48	37.32	odd	36