Embedded newform 37.6.e.a.27.4

• Label: 37.6.e.a.27.4
• Level: $37$
• Weight: $6$
• Character: 37.27
• 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			27.4
Character	\(\chi\)	\(=\)	37.27
Dual form			37.6.e.a.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.08219 - 3.51156i) q^{2} +(3.15654 + 5.46728i) q^{3} +(8.66206 + 15.0031i) q^{4} +(30.9238 - 17.8538i) q^{5} -44.3374i q^{6} +(-47.9780 - 83.1003i) q^{7} +103.070i q^{8} +(101.573 - 175.929i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 6 q^{2} + 18 q^{3} + 298 q^{4} + 144 q^{5} - 52 q^{7} - 1490 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.08219	−	3.51156i	−1.07519	−	0.620761i	−0.145595	−	0.989344i	\(-0.546510\pi\)
							−0.929595	+	0.368583i	\(0.879843\pi\)
\(3\)	3.15654	+	5.46728i	0.202492	+	0.350726i	0.949331	−	0.314279i	\(-0.101763\pi\)
							−0.746839	+	0.665005i	\(0.768429\pi\)
\(5\)	30.9238	−	17.8538i	0.553181	−	0.319379i	−0.197223	−	0.980359i	\(-0.563192\pi\)
							0.750404	+	0.660979i	\(0.229859\pi\)
\(7\)	−47.9780	−	83.1003i	−0.370081	−	0.640999i	0.619497	−	0.784999i	\(-0.287337\pi\)
							−0.989578	+	0.144000i	\(0.954003\pi\)
\(11\)	−604.561			−1.50646			−0.753231	−	0.657756i	\(-0.771506\pi\)
							−0.753231	+	0.657756i	\(0.771506\pi\)
\(13\)	−940.081	+	542.756i	−1.54279	+	0.890730i	−0.544128	+	0.839002i	\(0.683140\pi\)
							−0.998661	+	0.0517279i	\(0.983527\pi\)
\(17\)	−927.402	−	535.436i	−0.778297	−	0.449350i	0.0575291	−	0.998344i	\(-0.481678\pi\)
							−0.835827	+	0.548994i	\(0.815011\pi\)
\(19\)	305.683	−	176.486i	0.194262	−	0.112157i	−0.399714	−	0.916640i	\(-0.630891\pi\)
							0.593976	+	0.804483i	\(0.297557\pi\)
\(23\)		−	4776.67i		−	1.88280i	−0.337285	−	0.941402i	\(-0.609509\pi\)
							0.337285	−	0.941402i	\(-0.390491\pi\)
\(29\)			646.267i			0.142698i	0.997451	+	0.0713489i	\(0.0227304\pi\)
							−0.997451	+	0.0713489i	\(0.977270\pi\)
\(31\)			105.959i			0.0198031i	0.999951	+	0.00990157i	\(0.00315182\pi\)
							−0.999951	+	0.00990157i	\(0.996848\pi\)
\(37\)	−1790.65	+	8132.50i	−0.215033	+	0.976607i
							\(41\)	−6892.19	−	11937.6i	−0.640321	−	1.10907i	−0.985361	−	0.170480i	\(-0.945468\pi\)
0.345040	−	0.938588i	\(-0.387865\pi\)
\(43\)			3812.18i			0.314414i	0.987566	+	0.157207i	\(0.0502490\pi\)
							−0.987566	+	0.157207i	\(0.949751\pi\)
\(47\)	15052.1			0.993924			0.496962	−	0.867772i	\(-0.334449\pi\)
							0.496962	+	0.867772i	\(0.334449\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.e.a.27.4	yes	32
37.11	even	6	inner	37.6.e.a.11.4	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.e.a.11.4	✓	32	37.11	even	6	inner
37.6.e.a.27.4	yes	32	1.1	even	1	trivial