Embedded newform 37.10.c.a.10.22

• Label: 37.10.c.a.10.22
• Level: $37$
• Weight: $10$
• Character: 37.10
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(27\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			10.22
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(15.0946 - 26.1447i) q^{2} +(-4.28908 - 7.42890i) q^{3} +(-199.695 - 345.882i) q^{4} +(432.382 + 748.907i) q^{5} -258.968 q^{6} +(1427.57 + 2472.62i) q^{7} +3399.60 q^{8} +(9804.71 - 16982.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	15.0946	−	26.1447i	0.667094	−	1.15544i	−0.311619	−	0.950207i	\(-0.600871\pi\)
							0.978713	−	0.205234i	\(-0.0657955\pi\)
\(3\)	−4.28908	−	7.42890i	−0.0305716	−	0.0529516i	0.850335	−	0.526242i	\(-0.176399\pi\)
							−0.880906	+	0.473290i	\(0.843066\pi\)
\(5\)	432.382	+	748.907i	0.309387	+	0.535874i	0.978228	−	0.207531i	\(-0.0665428\pi\)
							−0.668841	+	0.743405i	\(0.733209\pi\)
\(7\)	1427.57	+	2472.62i	0.224727	+	0.389238i	0.956237	−	0.292592i	\(-0.0945177\pi\)
							−0.731511	+	0.681830i	\(0.761184\pi\)
\(11\)	33339.5			0.686580			0.343290	−	0.939229i	\(-0.388459\pi\)
							0.343290	+	0.939229i	\(0.388459\pi\)
\(13\)	−48686.9	−	84328.2i	−0.472789	−	0.818894i	0.526726	−	0.850035i	\(-0.323419\pi\)
							−0.999515	+	0.0311411i	\(0.990086\pi\)
\(17\)	140981.	−	244187.i	0.409394	−	0.709091i	−0.585428	−	0.810724i	\(-0.699074\pi\)
							0.994822	+	0.101633i	\(0.0324069\pi\)
\(19\)	436799.	+	756559.i	0.768937	+	1.33184i	0.938140	+	0.346257i	\(0.112547\pi\)
							−0.169203	+	0.985581i	\(0.554119\pi\)
\(23\)	−1.03186e6			−0.768857			−0.384429	−	0.923155i	\(-0.625602\pi\)
							−0.384429	+	0.923155i	\(0.625602\pi\)
\(29\)	2.78893e6			0.732229			0.366115	−	0.930570i	\(-0.380688\pi\)
							0.366115	+	0.930570i	\(0.380688\pi\)
\(31\)	−7.65014e6			−1.48779			−0.743896	−	0.668296i	\(-0.767024\pi\)
							−0.743896	+	0.668296i	\(0.767024\pi\)
\(37\)	1.05888e7	−	4.22359e6i	0.928837	−	0.370488i
							\(41\)	−3.44832e6	−	5.97266e6i	−0.190581	−	0.330096i	0.754862	−	0.655884i	\(-0.227704\pi\)
−0.945443	+	0.325788i	\(0.894371\pi\)
\(43\)	2.95985e7			1.32027			0.660133	−	0.751149i	\(-0.270500\pi\)
							0.660133	+	0.751149i	\(0.270500\pi\)
\(47\)	6.54321e6			0.195592			0.0977958	−	0.995206i	\(-0.468821\pi\)
							0.0977958	+	0.995206i	\(0.468821\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.22	✓	54
37.26	even	3	inner	37.10.c.a.26.22	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.22	✓	54	1.1	even	1	trivial
37.10.c.a.26.22	yes	54	37.26	even	3	inner