Embedded newform 37.4.h.a.4.5

• Label: 37.4.h.a.4.5
• Level: $37$
• Weight: $4$
• Character: 37.4
• 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			4.5
Character	\(\chi\)	\(=\)	37.4
Dual form			37.4.h.a.28.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.00515467 + 0.000908908i) q^{2} +(-0.799053 - 4.53165i) q^{3} +(-7.51752 - 2.73615i) q^{4} +(-5.10418 - 6.08292i) q^{5} -0.0240855i q^{6} +(2.28847 - 1.92026i) q^{7} +(-0.0725270 - 0.0418735i) q^{8} +(5.47429 - 1.99248i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 3 q^{2} + 9 q^{4} - 27 q^{5} - 108 q^{7} + 144 q^{8} + 42 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.00515467		0.000908908i	0.00182245		0.000321348i	0.174559	−	0.984647i	\(-0.444150\pi\)
							−0.172737	+	0.984968i	\(0.555261\pi\)
\(3\)	−0.799053	−	4.53165i	−0.153778	−	0.872117i	−0.959895	−	0.280361i	\(-0.909546\pi\)
							0.806117	−	0.591756i	\(-0.201565\pi\)
\(5\)	−5.10418	−	6.08292i	−0.456532	−	0.544073i	0.487849	−	0.872928i	\(-0.337782\pi\)
							−0.944380	+	0.328855i	\(0.893337\pi\)
\(7\)	2.28847	−	1.92026i	0.123566	−	0.103684i	−0.578911	−	0.815391i	\(-0.696522\pi\)
							0.702477	+	0.711707i	\(0.252078\pi\)
\(11\)	9.49068	−	16.4383i	0.260141	−	0.450577i	−0.706138	−	0.708074i	\(-0.749564\pi\)
							0.966279	+	0.257497i	\(0.0828977\pi\)
\(13\)	8.64994	−	23.7655i	0.184543	−	0.507028i	−0.812578	−	0.582852i	\(-0.801937\pi\)
							0.997121	+	0.0758242i	\(0.0241588\pi\)
\(17\)	14.3360	+	39.3879i	0.204529	+	0.561939i	0.998969	−	0.0454043i	\(-0.0144576\pi\)
							−0.794440	+	0.607343i	\(0.792235\pi\)
\(19\)	−39.3766	+	6.94316i	−0.475454	+	0.0838353i	−0.406240	−	0.913767i	\(-0.633160\pi\)
							−0.0692139	+	0.997602i	\(0.522049\pi\)
\(23\)	49.1013	−	28.3486i	0.445144	−	0.257004i	−0.260633	−	0.965438i	\(-0.583931\pi\)
							0.705777	+	0.708434i	\(0.250598\pi\)
\(29\)	44.5166	+	25.7017i	0.285052	+	0.164575i	0.635709	−	0.771929i	\(-0.280708\pi\)
							−0.350656	+	0.936504i	\(0.614041\pi\)
\(31\)		−	141.877i		−	0.821997i	−0.911636	−	0.410999i	\(-0.865180\pi\)
							0.911636	−	0.410999i	\(-0.134820\pi\)
\(37\)	−223.283	+	28.2449i	−0.992094	+	0.125498i
							\(41\)	451.662	+	164.391i	1.72043	+	0.626186i	0.997877	−	0.0651284i	\(-0.0207457\pi\)
0.722554	+	0.691314i	\(0.242968\pi\)
\(43\)		−	375.714i		−	1.33246i	−0.745746	−	0.666230i	\(-0.767907\pi\)
							0.745746	−	0.666230i	\(-0.232093\pi\)
\(47\)	207.905	+	360.102i	0.645235	+	1.11758i	0.984247	+	0.176798i	\(0.0565740\pi\)
							−0.339012	+	0.940782i	\(0.610093\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.h.a.4.5	✓	48
37.18	odd	36		1369.4.a.j.1.24		48
37.19	odd	36		1369.4.a.j.1.25		48
37.28	even	18	inner	37.4.h.a.28.5	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.h.a.4.5	✓	48	1.1	even	1	trivial
37.4.h.a.28.5	yes	48	37.28	even	18	inner
1369.4.a.j.1.24		48	37.18	odd	36
1369.4.a.j.1.25		48	37.19	odd	36