Embedded newform 37.10.h.a.3.5

• Label: 37.10.h.a.3.5
• 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			3.5
Character	\(\chi\)	\(=\)	37.3
Dual form			37.10.h.a.25.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-22.5902 + 26.9219i) q^{2} +(-90.5435 + 75.9750i) q^{3} +(-125.566 - 712.119i) q^{4} +(323.614 - 889.121i) q^{5} -4153.89i q^{6} +(8597.44 + 3129.21i) q^{7} +(6425.09 + 3709.53i) q^{8} +(-991.995 + 5625.89i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 45 q^{2} + 216 q^{3} + 249 q^{4} + 177 q^{5} - 5820 q^{7} - 74520 q^{8} - 30198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−22.5902	+	26.9219i	−0.998353	+	1.18979i	−0.0165555	+	0.999863i	\(0.505270\pi\)
							−0.981798	+	0.189928i	\(0.939174\pi\)
\(3\)	−90.5435	+	75.9750i	−0.645374	+	0.541533i	−0.905663	−	0.423997i	\(-0.860627\pi\)
							0.260289	+	0.965531i	\(0.416182\pi\)
\(5\)	323.614	−	889.121i	0.231559	−	0.636204i	−0.768434	−	0.639929i	\(-0.778964\pi\)
							0.999993	+	0.00372563i	\(0.00118591\pi\)
\(7\)	8597.44	+	3129.21i	1.35341	+	0.492600i	0.914009	−	0.405693i	\(-0.132970\pi\)
							0.439397	+	0.898293i	\(0.355192\pi\)
\(11\)	−31821.9	+	55117.2i	−0.655329	+	1.13506i	0.326482	+	0.945203i	\(0.394137\pi\)
							−0.981811	+	0.189860i	\(0.939197\pi\)
\(13\)	71150.4	−	12545.7i	0.690927	−	0.121829i	0.182850	−	0.983141i	\(-0.441468\pi\)
							0.508077	+	0.861312i	\(0.330357\pi\)
\(17\)	365586.	+	64462.7i	1.06162	+	0.187193i	0.677076	−	0.735913i	\(-0.263247\pi\)
							0.384546	+	0.923106i	\(0.374358\pi\)
\(19\)	121462.	+	144752.i	0.213820	+	0.254820i	0.862284	−	0.506424i	\(-0.169033\pi\)
							−0.648465	+	0.761245i	\(0.724589\pi\)
\(23\)	382882.	−	221057.i	0.285292	−	0.164713i	−0.350525	−	0.936553i	\(-0.613997\pi\)
							0.635817	+	0.771840i	\(0.280663\pi\)
\(29\)	−3.69920e6	−	2.13573e6i	−0.971217	−	0.560732i	−0.0716099	−	0.997433i	\(-0.522814\pi\)
							−0.899607	+	0.436700i	\(0.856147\pi\)
\(31\)			5.96557e6i			1.16018i	0.814554	+	0.580088i	\(0.196982\pi\)
							−0.814554	+	0.580088i	\(0.803018\pi\)
\(37\)	−9.34272e6	−	6.53264e6i	−0.819531	−	0.573035i
							\(41\)	−699637.	−	3.96784e6i	−0.0386675	−	0.219294i	0.959351	−	0.282216i	\(-0.0910693\pi\)
−0.998018	+	0.0629215i	\(0.979958\pi\)
\(43\)			2.75154e7i			1.22735i	0.789560	+	0.613674i	\(0.210309\pi\)
							−0.789560	+	0.613674i	\(0.789691\pi\)
\(47\)	1.05439e7	+	1.82625e7i	0.315181	+	0.545910i	0.979476	−	0.201561i	\(-0.0646014\pi\)
							−0.664295	+	0.747471i	\(0.731268\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.h.a.3.5	✓	168
37.25	even	18	inner	37.10.h.a.25.5	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.h.a.3.5	✓	168	1.1	even	1	trivial
37.10.h.a.25.5	yes	168	37.25	even	18	inner