Embedded newform 37.10.c.a.10.13

• Label: 37.10.c.a.10.13
• 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			10.13
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.349842 + 0.605944i) q^{2} +(-41.9286 - 72.6225i) q^{3} +(255.755 + 442.981i) q^{4} +(954.706 + 1653.60i) q^{5} +58.6736 q^{6} +(1656.50 + 2869.15i) q^{7} -716.134 q^{8} +(6325.48 - 10956.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.349842	+	0.605944i	−0.0154610	+	0.0267792i	−0.873652	−	0.486551i	\(-0.838255\pi\)
							0.858191	+	0.513330i	\(0.171588\pi\)
\(3\)	−41.9286	−	72.6225i	−0.298858	−	0.517637i	0.677017	−	0.735967i	\(-0.263272\pi\)
							−0.975875	+	0.218330i	\(0.929939\pi\)
\(5\)	954.706	+	1653.60i	0.683132	+	1.18322i	0.974020	+	0.226462i	\(0.0727159\pi\)
							−0.290888	+	0.956757i	\(0.593951\pi\)
\(7\)	1656.50	+	2869.15i	0.260766	+	0.451661i	0.966446	−	0.256871i	\(-0.0826915\pi\)
							−0.705679	+	0.708531i	\(0.749358\pi\)
\(11\)	−74939.3			−1.54327			−0.771636	−	0.636064i	\(-0.780561\pi\)
							−0.771636	+	0.636064i	\(0.780561\pi\)
\(13\)	−3081.62	−	5337.52i	−0.0299250	−	0.0518316i	0.850675	−	0.525692i	\(-0.176194\pi\)
							−0.880600	+	0.473860i	\(0.842860\pi\)
\(17\)	−50504.5	+	87476.3i	−0.146659	+	0.254021i	−0.929991	−	0.367583i	\(-0.880185\pi\)
							0.783331	+	0.621604i	\(0.213519\pi\)
\(19\)	435480.	+	754274.i	0.766615	+	1.32782i	0.939389	+	0.342854i	\(0.111394\pi\)
							−0.172774	+	0.984962i	\(0.555273\pi\)
\(23\)	290947.			0.216790			0.108395	−	0.994108i	\(-0.465429\pi\)
							0.108395	+	0.994108i	\(0.465429\pi\)
\(29\)	−3.17046e6			−0.832397			−0.416199	−	0.909274i	\(-0.636638\pi\)
							−0.416199	+	0.909274i	\(0.636638\pi\)
\(31\)	49895.1			0.00970353			0.00485177	−	0.999988i	\(-0.498456\pi\)
							0.00485177	+	0.999988i	\(0.498456\pi\)
\(37\)	−7.65334e6	+	8.44915e6i	−0.671341	+	0.741149i
							\(41\)	1.00437e7	+	1.73962e7i	0.555094	+	0.961451i	0.997896	+	0.0648318i	\(0.0206511\pi\)
−0.442802	+	0.896619i	\(0.646016\pi\)
\(43\)	−1.29365e7			−0.577044			−0.288522	−	0.957473i	\(-0.593164\pi\)
							−0.288522	+	0.957473i	\(0.593164\pi\)
\(47\)	5.59858e6			0.167355			0.0836773	−	0.996493i	\(-0.473334\pi\)
							0.0836773	+	0.996493i	\(0.473334\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.13	✓	54
37.26	even	3	inner	37.10.c.a.26.13	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.13	✓	54	1.1	even	1	trivial
37.10.c.a.26.13	yes	54	37.26	even	3	inner