Embedded newform 37.4.h.a.4.7

• Label: 37.4.h.a.4.7
• 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			4.7
Character	\(\chi\)	\(=\)	37.4
Dual form			37.4.h.a.28.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.59444 + 0.633796i) q^{2} +(-1.19386 - 6.77069i) q^{3} +(5.00074 + 1.82012i) q^{4} +(7.95665 + 9.48237i) q^{5} -25.0935i q^{6} +(3.25082 - 2.72776i) q^{7} +(-8.46590 - 4.88779i) q^{8} +(-19.0453 + 6.93190i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 3 q^{2} + 9 q^{4} - 27 q^{5} - 108 q^{7} + 144 q^{8} + 42 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.59444	+	0.633796i	1.27083	+	0.224081i	0.768080	−	0.640354i	\(-0.221212\pi\)
							0.502745	+	0.864435i	\(0.332323\pi\)
\(3\)	−1.19386	−	6.77069i	−0.229758	−	1.30302i	−0.853378	−	0.521293i	\(-0.825450\pi\)
							0.623620	−	0.781727i	\(-0.285661\pi\)
\(5\)	7.95665	+	9.48237i	0.711664	+	0.848129i	0.993793	−	0.111248i	\(-0.0354848\pi\)
							−0.282128	+	0.959377i	\(0.591040\pi\)
\(7\)	3.25082	−	2.72776i	0.175528	−	0.147285i	−0.550791	−	0.834643i	\(-0.685674\pi\)
							0.726319	+	0.687358i	\(0.241230\pi\)
\(11\)	−17.6417	+	30.5564i	−0.483562	+	0.837554i	−0.999822	−	0.0188782i	\(-0.993991\pi\)
							0.516260	+	0.856432i	\(0.327324\pi\)
\(13\)	−27.9679	+	76.8410i	−0.596684	+	1.63937i	0.161152	+	0.986930i	\(0.448479\pi\)
							−0.757836	+	0.652445i	\(0.773743\pi\)
\(17\)	−24.0145	−	65.9792i	−0.342610	−	0.941313i	−0.984634	−	0.174629i	\(-0.944128\pi\)
							0.642025	−	0.766684i	\(-0.278095\pi\)
\(19\)	71.4838	−	12.6045i	0.863132	−	0.152194i	0.275480	−	0.961307i	\(-0.411163\pi\)
							0.587653	+	0.809113i	\(0.300052\pi\)
\(23\)	111.455	−	64.3485i	1.01043	−	0.583374i	0.0991143	−	0.995076i	\(-0.468399\pi\)
							0.911318	+	0.411703i	\(0.135066\pi\)
\(29\)	27.5958	+	15.9324i	0.176704	+	0.102020i	0.585743	−	0.810497i	\(-0.300803\pi\)
							−0.409039	+	0.912517i	\(0.634136\pi\)
\(31\)		−	105.884i		−	0.613460i	−0.951797	−	0.306730i	\(-0.900765\pi\)
							0.951797	−	0.306730i	\(-0.0992349\pi\)
\(37\)	38.6053	+	221.726i	0.171532	+	0.985179i
							\(41\)	−463.789	−	168.805i	−1.76663	−	0.642999i	−1.00000		0.000906323i	\(-0.999712\pi\)
−0.766627	−	0.642093i	\(-0.778066\pi\)
\(43\)		−	71.4519i		−	0.253403i	−0.991941	−	0.126701i	\(-0.959561\pi\)
							0.991941	−	0.126701i	\(-0.0404390\pi\)
\(47\)	−6.20823	−	10.7530i	−0.0192673	−	0.0333720i	0.856231	−	0.516593i	\(-0.172800\pi\)
							−0.875498	+	0.483221i	\(0.839467\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.h.a.4.7	✓	48
37.18	odd	36		1369.4.a.j.1.10		48
37.19	odd	36		1369.4.a.j.1.39		48
37.28	even	18	inner	37.4.h.a.28.7	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.h.a.4.7	✓	48	1.1	even	1	trivial
37.4.h.a.28.7	yes	48	37.28	even	18	inner
1369.4.a.j.1.10		48	37.18	odd	36
1369.4.a.j.1.39		48	37.19	odd	36