Embedded newform 37.6.e.a.11.2

• Label: 37.6.e.a.11.2
• Level: $37$
• Weight: $6$
• Character: 37.11
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.2
Character	\(\chi\)	\(=\)	37.11
Dual form			37.6.e.a.27.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.18598 + 5.30353i) q^{2} +(-6.26292 + 10.8477i) q^{3} +(40.2548 - 69.7233i) q^{4} +(63.9646 + 36.9300i) q^{5} -132.862i q^{6} +(113.255 - 196.163i) q^{7} +514.543i q^{8} +(43.0517 + 74.5678i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 6 q^{2} + 18 q^{3} + 298 q^{4} + 144 q^{5} - 52 q^{7} - 1490 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.18598	+	5.30353i	−1.62387	+	0.937540i	−0.637994	+	0.770041i	\(0.720236\pi\)
							−0.985872	+	0.167498i	\(0.946431\pi\)
\(3\)	−6.26292	+	10.8477i	−0.401766	+	0.695880i	−0.993939	−	0.109931i	\(-0.964937\pi\)
							0.592173	+	0.805811i	\(0.298270\pi\)
\(5\)	63.9646	+	36.9300i	1.14423	+	0.660623i	0.947475	−	0.319829i	\(-0.103625\pi\)
							0.196758	+	0.980452i	\(0.436959\pi\)
\(7\)	113.255	−	196.163i	0.873596	−	1.51311i	0.0153451	−	0.999882i	\(-0.495115\pi\)
							0.858251	−	0.513230i	\(-0.171551\pi\)
\(11\)	134.565			0.335314			0.167657	−	0.985845i	\(-0.446380\pi\)
							0.167657	+	0.985845i	\(0.446380\pi\)
\(13\)	−145.563	−	84.0406i	−0.238886	−	0.137921i	0.375778	−	0.926710i	\(-0.377375\pi\)
							−0.614665	+	0.788788i	\(0.710709\pi\)
\(17\)	444.179	−	256.447i	0.372766	−	0.215216i	−0.301900	−	0.953339i	\(-0.597621\pi\)
							0.674666	+	0.738123i	\(0.264288\pi\)
\(19\)	1944.69	+	1122.77i	1.23585	+	0.713518i	0.968243	−	0.250010i	\(-0.0804340\pi\)
							0.267606	+	0.963528i	\(0.413767\pi\)
\(23\)		−	2768.93i		−	1.09142i	−0.837973	−	0.545711i	\(-0.816260\pi\)
							0.837973	−	0.545711i	\(-0.183740\pi\)
\(29\)			8023.83i			1.77169i	0.463984	+	0.885843i	\(0.346419\pi\)
							−0.463984	+	0.885843i	\(0.653581\pi\)
\(31\)			4034.07i			0.753944i	0.926225	+	0.376972i	\(0.123035\pi\)
							−0.926225	+	0.376972i	\(0.876965\pi\)
\(37\)	8084.81	−	1994.93i	0.970880	−	0.239565i
							\(41\)	−3293.83	+	5705.09i	−0.306014	+	0.530032i	−0.977487	−	0.210997i	\(-0.932329\pi\)
0.671472	+	0.741030i	\(0.265662\pi\)
\(43\)			6097.29i			0.502882i	0.967873	+	0.251441i	\(0.0809044\pi\)
							−0.967873	+	0.251441i	\(0.919096\pi\)
\(47\)	7386.71			0.487760			0.243880	−	0.969805i	\(-0.421580\pi\)
							0.243880	+	0.969805i	\(0.421580\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.e.a.11.2	✓	32
37.27	even	6	inner	37.6.e.a.27.2	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.e.a.11.2	✓	32	1.1	even	1	trivial
37.6.e.a.27.2	yes	32	37.27	even	6	inner