Embedded newform 37.6.c.a.10.13

• Label: 37.6.c.a.10.13
• Level: $37$
• Weight: $6$
• Character: 37.10
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(15\) 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			37.6.c.a.26.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.71247 - 6.43018i) q^{2} +(6.84687 + 11.8591i) q^{3} +(-11.5648 - 20.0308i) q^{4} +(46.2802 + 80.1596i) q^{5} +101.675 q^{6} +(3.53945 + 6.13051i) q^{7} +65.8619 q^{8} +(27.7406 - 48.0482i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 4 q^{2} + 16 q^{3} - 230 q^{4} - 51 q^{5} + 60 q^{6} + 50 q^{7} + 24 q^{8} - 1085 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.71247	−	6.43018i	0.656278	−	1.13671i	−0.325294	−	0.945613i	\(-0.605463\pi\)
							0.981572	−	0.191093i	\(-0.0612033\pi\)
\(3\)	6.84687	+	11.8591i	0.439227	+	0.760764i	0.997630	−	0.0688062i	\(-0.0219190\pi\)
							−0.558403	+	0.829570i	\(0.688586\pi\)
\(5\)	46.2802	+	80.1596i	0.827885	+	1.43394i	0.899694	+	0.436520i	\(0.143789\pi\)
							−0.0718096	+	0.997418i	\(0.522877\pi\)
\(7\)	3.53945	+	6.13051i	0.0273018	+	0.0472880i	0.879353	−	0.476170i	\(-0.157975\pi\)
							−0.852052	+	0.523458i	\(0.824642\pi\)
\(11\)	−602.958			−1.50247			−0.751234	−	0.660036i	\(-0.770541\pi\)
							−0.751234	+	0.660036i	\(0.770541\pi\)
\(13\)	−118.470	−	205.196i	−0.194424	−	0.336752i	0.752288	−	0.658835i	\(-0.228950\pi\)
							−0.946711	+	0.322083i	\(0.895617\pi\)
\(17\)	917.936	−	1589.91i	0.770354	−	1.33429i	−0.167016	−	0.985954i	\(-0.553413\pi\)
							0.937369	−	0.348337i	\(-0.113254\pi\)
\(19\)	488.917	+	846.830i	0.310707	+	0.538161i	0.978516	−	0.206172i	\(-0.0661008\pi\)
							−0.667808	+	0.744333i	\(0.732767\pi\)
\(23\)	−1089.35			−0.429386			−0.214693	−	0.976682i	\(-0.568875\pi\)
							−0.214693	+	0.976682i	\(0.568875\pi\)
\(29\)	−6096.32			−1.34609			−0.673043	−	0.739603i	\(-0.735013\pi\)
							−0.673043	+	0.739603i	\(0.735013\pi\)
\(31\)	2228.70			0.416530			0.208265	−	0.978072i	\(-0.433218\pi\)
							0.208265	+	0.978072i	\(0.433218\pi\)
\(37\)	−1480.77	−	8194.59i	−0.177822	−	0.984063i
							\(41\)	5459.67	+	9456.43i	0.507232	+	0.878552i	0.999965	+	0.00837132i	\(0.00266470\pi\)
−0.492733	+	0.870181i	\(0.664002\pi\)
\(43\)	−17280.7			−1.42525			−0.712626	−	0.701545i	\(-0.752494\pi\)
							−0.712626	+	0.701545i	\(0.752494\pi\)
\(47\)	−4139.46			−0.273338			−0.136669	−	0.990617i	\(-0.543640\pi\)
							−0.136669	+	0.990617i	\(0.543640\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.c.a.10.13	✓	30
37.26	even	3	inner	37.6.c.a.26.13	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.c.a.10.13	✓	30	1.1	even	1	trivial
37.6.c.a.26.13	yes	30	37.26	even	3	inner