Embedded newform 37.10.h.a.3.20

• Label: 37.10.h.a.3.20
• Level: $37$
• Weight: $10$
• Character: 37.3
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(28\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.20
Character	\(\chi\)	\(=\)	37.3
Dual form			37.10.h.a.25.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(12.3034 - 14.6627i) q^{2} +(154.895 - 129.973i) q^{3} +(25.2886 + 143.419i) q^{4} +(-383.475 + 1053.59i) q^{5} -3870.29i q^{6} +(1863.39 + 678.219i) q^{7} +(10901.1 + 6293.78i) q^{8} +(3681.76 - 20880.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 45 q^{2} + 216 q^{3} + 249 q^{4} + 177 q^{5} - 5820 q^{7} - 74520 q^{8} - 30198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	12.3034	−	14.6627i	0.543741	−	0.648005i	−0.422281	−	0.906465i	\(-0.638771\pi\)
							0.966022	+	0.258460i	\(0.0832150\pi\)
\(3\)	154.895	−	129.973i	1.10406	−	0.926416i	0.106368	−	0.994327i	\(-0.466078\pi\)
							0.997691	+	0.0679111i	\(0.0216334\pi\)
\(5\)	−383.475	+	1053.59i	−0.274392	+	0.753886i	0.723580	+	0.690240i	\(0.242495\pi\)
							−0.997973	+	0.0636461i	\(0.979727\pi\)
\(7\)	1863.39	+	678.219i	0.293334	+	0.106765i	0.484496	−	0.874794i	\(-0.339003\pi\)
							−0.191161	+	0.981559i	\(0.561225\pi\)
\(11\)	43401.1	−	75172.9i	0.893787	−	1.54808i	0.0584876	−	0.998288i	\(-0.481372\pi\)
							0.835299	−	0.549796i	\(-0.185294\pi\)
\(13\)	135626.	−	23914.5i	1.31703	−	0.232228i	0.529399	−	0.848373i	\(-0.322417\pi\)
							0.787633	+	0.616144i	\(0.211306\pi\)
\(17\)	−272495.	−	48048.2i	−0.791295	−	0.139527i	−0.236629	−	0.971600i	\(-0.576043\pi\)
							−0.554666	+	0.832073i	\(0.687154\pi\)
\(19\)	−30926.5	−	36856.7i	−0.0544427	−	0.0648822i	0.738136	−	0.674652i	\(-0.235706\pi\)
							−0.792579	+	0.609770i	\(0.791262\pi\)
\(23\)	−1.34904e6	+	778868.i	−1.00519	+	0.580348i	−0.909780	−	0.415090i	\(-0.863750\pi\)
							−0.0954120	+	0.995438i	\(0.530417\pi\)
\(29\)	2.24298e6	+	1.29499e6i	0.588891	+	0.339997i	0.764659	−	0.644435i	\(-0.222907\pi\)
							−0.175768	+	0.984432i	\(0.556241\pi\)
\(31\)		−	9.12850e6i		−	1.77530i	−0.460519	−	0.887650i	\(-0.652337\pi\)
							0.460519	−	0.887650i	\(-0.347663\pi\)
\(37\)	−3.42866e6	+	1.08723e7i	−0.300757	+	0.953701i
							\(41\)	866208.	+	4.91251e6i	0.0478735	+	0.271504i	0.999343	−	0.0362385i	\(-0.0115376\pi\)
−0.951470	+	0.307742i	\(0.900426\pi\)
\(43\)			1.56517e7i			0.698157i	0.937094	+	0.349078i	\(0.113505\pi\)
							−0.937094	+	0.349078i	\(0.886495\pi\)
\(47\)	−1.75562e7	−	3.04083e7i	−0.524797	−	0.908975i	−0.999583	−	0.0288737i	\(-0.990808\pi\)
							0.474786	−	0.880101i	\(-0.342525\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.h.a.3.20	✓	168
37.25	even	18	inner	37.10.h.a.25.20	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.h.a.3.20	✓	168	1.1	even	1	trivial
37.10.h.a.25.20	yes	168	37.25	even	18	inner