Embedded newform 37.6.e.a.11.13

• Label: 37.6.e.a.11.13
• 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			11.13
Character	\(\chi\)	\(=\)	37.11
Dual form			37.6.e.a.27.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.44256 - 3.14226i) q^{2} +(-10.4803 + 18.1524i) q^{3} +(3.74762 - 6.49106i) q^{4} +(-46.0737 - 26.6006i) q^{5} +131.727i q^{6} +(-56.9410 + 98.6246i) q^{7} +154.001i q^{8} +(-98.1728 - 170.040i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 6 q^{2} + 18 q^{3} + 298 q^{4} + 144 q^{5} - 52 q^{7} - 1490 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.44256	−	3.14226i	0.962117	−	0.555479i	0.0652932	−	0.997866i	\(-0.479202\pi\)
							0.896824	+	0.442387i	\(0.145868\pi\)
\(3\)	−10.4803	+	18.1524i	−0.672311	+	1.16448i	0.304937	+	0.952373i	\(0.401365\pi\)
							−0.977247	+	0.212103i	\(0.931969\pi\)
\(5\)	−46.0737	−	26.6006i	−0.824191	−	0.475847i	0.0276688	−	0.999617i	\(-0.491192\pi\)
							−0.851859	+	0.523770i	\(0.824525\pi\)
\(7\)	−56.9410	+	98.6246i	−0.439218	+	0.760747i	−0.997629	−	0.0688168i	\(-0.978078\pi\)
							0.558412	+	0.829564i	\(0.311411\pi\)
\(11\)	−271.894			−0.677514			−0.338757	−	0.940874i	\(-0.610006\pi\)
							−0.338757	+	0.940874i	\(0.610006\pi\)
\(13\)	229.598	+	132.559i	0.376799	+	0.217545i	0.676425	−	0.736512i	\(-0.263528\pi\)
							−0.299625	+	0.954057i	\(0.596862\pi\)
\(17\)	961.054	−	554.865i	0.806539	−	0.465656i	−0.0392134	−	0.999231i	\(-0.512485\pi\)
							0.845753	+	0.533575i	\(0.179152\pi\)
\(19\)	576.750	+	332.987i	0.366525	+	0.211613i	0.671939	−	0.740606i	\(-0.265462\pi\)
							−0.305414	+	0.952220i	\(0.598795\pi\)
\(23\)			991.319i			0.390745i	0.980729	+	0.195373i	\(0.0625917\pi\)
							−0.980729	+	0.195373i	\(0.937408\pi\)
\(29\)		−	2709.22i		−	0.598204i	−0.954221	−	0.299102i	\(-0.903313\pi\)
							0.954221	−	0.299102i	\(-0.0966871\pi\)
\(31\)			9343.34i			1.74622i	0.487527	+	0.873108i	\(0.337899\pi\)
							−0.487527	+	0.873108i	\(0.662101\pi\)
\(37\)	6092.04	−	5677.23i	0.731575	−	0.681761i
							\(41\)	−8722.18	+	15107.3i	−0.810337	+	1.40354i	0.102292	+	0.994754i	\(0.467382\pi\)
−0.912629	+	0.408790i	\(0.865951\pi\)
\(43\)			16703.6i			1.37765i	0.724927	+	0.688826i	\(0.241873\pi\)
							−0.724927	+	0.688826i	\(0.758127\pi\)
\(47\)	−8186.94			−0.540601			−0.270301	−	0.962776i	\(-0.587123\pi\)
							−0.270301	+	0.962776i	\(0.587123\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.e.a.11.13	✓	32
37.27	even	6	inner	37.6.e.a.27.13	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.e.a.11.13	✓	32	1.1	even	1	trivial
37.6.e.a.27.13	yes	32	37.27	even	6	inner