Embedded newform 37.10.h.a.3.6

• Label: 37.10.h.a.3.6
• 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			3.6
Character	\(\chi\)	\(=\)	37.3
Dual form			37.10.h.a.25.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-22.1667 + 26.4172i) q^{2} +(114.438 - 96.0253i) q^{3} +(-117.600 - 666.941i) q^{4} +(92.1806 - 253.264i) q^{5} +5151.70i q^{6} +(-10964.7 - 3990.81i) q^{7} +(4934.58 + 2848.98i) q^{8} +(457.392 - 2594.00i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 45 q^{2} + 216 q^{3} + 249 q^{4} + 177 q^{5} - 5820 q^{7} - 74520 q^{8} - 30198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−22.1667	+	26.4172i	−0.979637	+	1.16749i	0.00623384	+	0.999981i	\(0.498016\pi\)
							−0.985871	+	0.167506i	\(0.946429\pi\)
\(3\)	114.438	−	96.0253i	0.815692	−	0.684447i	−0.136267	−	0.990672i	\(-0.543510\pi\)
							0.951959	+	0.306225i	\(0.0990660\pi\)
\(5\)	92.1806	−	253.264i	0.0659590	−	0.181221i	−0.902334	−	0.431037i	\(-0.858148\pi\)
							0.968293	+	0.249816i	\(0.0803701\pi\)
\(7\)	−10964.7	−	3990.81i	−1.72605	−	0.628232i	−0.727717	−	0.685878i	\(-0.759418\pi\)
							−0.998337	+	0.0576457i	\(0.981641\pi\)
\(11\)	−30835.8	+	53409.1i	−0.635020	+	1.09989i	0.351491	+	0.936191i	\(0.385675\pi\)
							−0.986511	+	0.163696i	\(0.947658\pi\)
\(13\)	173861.	−	30656.4i	1.68833	−	0.297698i	0.754735	−	0.656030i	\(-0.227765\pi\)
							0.933594	+	0.358332i	\(0.116654\pi\)
\(17\)	106598.	+	18796.0i	0.309547	+	0.0545816i	0.326264	−	0.945279i	\(-0.394210\pi\)
							−0.0167165	+	0.999860i	\(0.505321\pi\)
\(19\)	513265.	+	611685.i	0.903546	+	1.07680i	0.996702	+	0.0811528i	\(0.0258602\pi\)
							−0.0931556	+	0.995652i	\(0.529695\pi\)
\(23\)	320300.	−	184925.i	0.238661	−	0.137791i	−0.375900	−	0.926660i	\(-0.622666\pi\)
							0.614561	+	0.788869i	\(0.289333\pi\)
\(29\)	1.15718e6	+	668100.i	0.303817	+	0.175409i	0.644156	−	0.764894i	\(-0.277209\pi\)
							−0.340340	+	0.940303i	\(0.610542\pi\)
\(31\)		−	6.64232e6i		−	1.29179i	−0.763426	−	0.645896i	\(-0.776484\pi\)
							0.763426	−	0.645896i	\(-0.223516\pi\)
\(37\)	9.56068e6	−	6.20928e6i	0.838650	−	0.544670i
							\(41\)	4.54407e6	+	2.57707e7i	0.251141	+	1.42429i	0.805788	+	0.592204i	\(0.201742\pi\)
−0.554647	+	0.832086i	\(0.687147\pi\)
\(43\)			634604.i			0.0283070i	0.999900	+	0.0141535i	\(0.00450536\pi\)
							−0.999900	+	0.0141535i	\(0.995495\pi\)
\(47\)	−2.70566e7	−	4.68634e7i	−0.808785	−	1.40086i	−0.913706	−	0.406376i	\(-0.866792\pi\)
							0.104922	−	0.994481i	\(-0.466541\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.h.a.3.6	✓	168
37.25	even	18	inner	37.10.h.a.25.6	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.h.a.3.6	✓	168	1.1	even	1	trivial
37.10.h.a.25.6	yes	168	37.25	even	18	inner