Embedded newform 37.10.h.a.3.16

• Label: 37.10.h.a.3.16
• 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			3.16
Character	\(\chi\)	\(=\)	37.3
Dual form			37.10.h.a.25.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0979000 + 0.116673i) q^{2} +(-67.1156 + 56.3167i) q^{3} +(88.9038 + 504.199i) q^{4} +(-720.391 + 1979.26i) q^{5} -13.3440i q^{6} +(10808.8 + 3934.07i) q^{7} +(-135.063 - 77.9786i) q^{8} +(-2084.98 + 11824.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 45 q^{2} + 216 q^{3} + 249 q^{4} + 177 q^{5} - 5820 q^{7} - 74520 q^{8} - 30198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0979000	+	0.116673i	−0.00432661	+	0.00515625i	−0.768203	−	0.640206i	\(-0.778849\pi\)
							0.763877	+	0.645362i	\(0.223293\pi\)
\(3\)	−67.1156	+	56.3167i	−0.478385	+	0.401413i	−0.849842	−	0.527037i	\(-0.823303\pi\)
							0.371457	+	0.928450i	\(0.378858\pi\)
\(5\)	−720.391	+	1979.26i	−0.515470	+	1.41624i	0.359993	+	0.932955i	\(0.382779\pi\)
							−0.875462	+	0.483286i	\(0.839443\pi\)
\(7\)	10808.8	+	3934.07i	1.70151	+	0.619299i	0.995996	−	0.0893969i	\(-0.0284940\pi\)
							0.705514	+	0.708696i	\(0.250716\pi\)
\(11\)	10236.1	−	17729.5i	0.210799	−	0.365114i	−0.741166	−	0.671322i	\(-0.765727\pi\)
							0.951965	+	0.306208i	\(0.0990602\pi\)
\(13\)	−46902.2	+	8270.12i	−0.455457	+	0.0803094i	−0.396670	−	0.917961i	\(-0.629834\pi\)
							−0.0587871	+	0.998271i	\(0.518723\pi\)
\(17\)	537390.	+	94756.4i	1.56052	+	0.275162i	0.886209	−	0.463285i	\(-0.153329\pi\)
							0.674312	+	0.738447i	\(0.264440\pi\)
\(19\)	280089.	+	333798.i	0.493066	+	0.587614i	0.953995	−	0.299824i	\(-0.0969279\pi\)
							−0.460928	+	0.887437i	\(0.652483\pi\)
\(23\)	−1.01624e6	+	586729.i	−0.757221	+	0.437182i	−0.828297	−	0.560289i	\(-0.810690\pi\)
							0.0710760	+	0.997471i	\(0.477357\pi\)
\(29\)	3.25811e6	+	1.88107e6i	0.855412	+	0.493872i	0.862473	−	0.506103i	\(-0.168914\pi\)
							−0.00706137	+	0.999975i	\(0.502248\pi\)
\(31\)		−	9.14584e6i		−	1.77867i	−0.457253	−	0.889336i	\(-0.651167\pi\)
							0.457253	−	0.889336i	\(-0.348833\pi\)
\(37\)	−6.08572e6	−	9.63980e6i	−0.533832	−	0.845591i
							\(41\)	2.50741e6	+	1.42202e7i	0.138579	+	0.785922i	0.972300	+	0.233736i	\(0.0750951\pi\)
−0.833721	+	0.552186i	\(0.813794\pi\)
\(43\)		−	3.14958e7i		−	1.40490i	−0.711735	−	0.702448i	\(-0.752090\pi\)
							0.711735	−	0.702448i	\(-0.247910\pi\)
\(47\)	3.55193e6	+	6.15212e6i	0.106175	+	0.183901i	0.914218	−	0.405223i	\(-0.132806\pi\)
							−0.808042	+	0.589124i	\(0.799473\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.h.a.3.16	✓	168
37.25	even	18	inner	37.10.h.a.25.16	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.h.a.3.16	✓	168	1.1	even	1	trivial
37.10.h.a.25.16	yes	168	37.25	even	18	inner