Embedded newform 37.4.h.a.21.2

• Label: 37.4.h.a.21.2
• Level: $37$
• Weight: $4$
• Character: 37.21
• 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			21.2
Character	\(\chi\)	\(=\)	37.21
Dual form			37.4.h.a.30.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.05871 + 2.90878i) q^{2} +(4.78213 - 1.74055i) q^{3} +(-1.21176 - 1.01679i) q^{4} +(7.19342 + 1.26839i) q^{5} +15.7529i q^{6} +(-0.149662 + 0.848773i) q^{7} +(-17.2054 + 9.93356i) q^{8} +(-0.843972 + 0.708177i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 3 q^{2} + 9 q^{4} - 27 q^{5} - 108 q^{7} + 144 q^{8} + 42 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.05871	+	2.90878i	−0.374310	+	1.02841i	0.599367	+	0.800474i	\(0.295419\pi\)
							−0.973677	+	0.227933i	\(0.926803\pi\)
\(3\)	4.78213	−	1.74055i	0.920321	−	0.334969i	0.161955	−	0.986798i	\(-0.448220\pi\)
							0.758366	+	0.651829i	\(0.225998\pi\)
\(5\)	7.19342	+	1.26839i	0.643399	+	0.113449i	0.485822	−	0.874058i	\(-0.338520\pi\)
							0.157578	+	0.987507i	\(0.449632\pi\)
\(7\)	−0.149662	+	0.848773i	−0.00808097	+	0.0458294i	−0.988582	−	0.150686i	\(-0.951852\pi\)
							0.980501	+	0.196516i	\(0.0629627\pi\)
\(11\)	−0.165014	−	0.285812i	−0.00452304	−	0.00783414i	0.863755	−	0.503912i	\(-0.168106\pi\)
							−0.868278	+	0.496078i	\(0.834773\pi\)
\(13\)	35.7188	−	42.5680i	0.762047	−	0.908172i	−0.235928	−	0.971770i	\(-0.575813\pi\)
							0.997976	+	0.0635980i	\(0.0202576\pi\)
\(17\)	−23.8628	−	28.4385i	−0.340445	−	0.405727i	0.568472	−	0.822702i	\(-0.307535\pi\)
							−0.908918	+	0.416975i	\(0.863090\pi\)
\(19\)	−32.7694	−	90.0333i	−0.395675	−	1.08711i	−0.964369	−	0.264559i	\(-0.914774\pi\)
							0.568695	−	0.822549i	\(-0.307449\pi\)
\(23\)	75.6068	+	43.6516i	0.685439	+	0.395738i	0.801901	−	0.597457i	\(-0.203822\pi\)
							−0.116462	+	0.993195i	\(0.537155\pi\)
\(29\)	−12.2756	+	7.08733i	−0.0786043	+	0.0453822i	−0.538787	−	0.842442i	\(-0.681117\pi\)
							0.460183	+	0.887824i	\(0.347784\pi\)
\(31\)			7.96974i			0.0461744i	0.999733	+	0.0230872i	\(0.00734954\pi\)
							−0.999733	+	0.0230872i	\(0.992650\pi\)
\(37\)	−142.615	−	174.109i	−0.633670	−	0.773603i
							\(41\)	226.019	+	189.652i	0.860931	+	0.722407i	0.962168	−	0.272455i	\(-0.0878357\pi\)
−0.101237	+	0.994862i	\(0.532280\pi\)
\(43\)		−	192.160i		−	0.681493i	−0.940155	−	0.340746i	\(-0.889320\pi\)
							0.940155	−	0.340746i	\(-0.110680\pi\)
\(47\)	−305.598	+	529.311i	−0.948426	+	1.64272i	−0.199684	+	0.979860i	\(0.563991\pi\)
							−0.748742	+	0.662861i	\(0.769342\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.h.a.21.2	✓	48
37.17	odd	36		1369.4.a.j.1.13		48
37.20	odd	36		1369.4.a.j.1.36		48
37.30	even	18	inner	37.4.h.a.30.2	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.h.a.21.2	✓	48	1.1	even	1	trivial
37.4.h.a.30.2	yes	48	37.30	even	18	inner
1369.4.a.j.1.13		48	37.17	odd	36
1369.4.a.j.1.36		48	37.20	odd	36