Embedded newform 37.8.f.a.7.19

• Label: 37.8.f.a.7.19
• Level: $37$
• Weight: $8$
• Character: 37.7
• Analytic conductor: $11.558$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5582459429\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.19
Character	\(\chi\)	\(=\)	37.7
Dual form			37.8.f.a.16.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(15.5384 - 5.65552i) q^{2} +(-16.3965 - 5.96785i) q^{3} +(111.404 - 93.4791i) q^{4} +(86.0271 + 487.884i) q^{5} -288.528 q^{6} +(270.212 + 1532.45i) q^{7} +(144.088 - 249.568i) q^{8} +(-1442.11 - 1210.07i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	15.5384	−	5.65552i	1.37342	−	0.499882i	0.453240	−	0.891388i	\(-0.350268\pi\)
							0.920176	+	0.391506i	\(0.128046\pi\)
\(3\)	−16.3965	−	5.96785i	−0.350613	−	0.127613i	0.160709	−	0.987002i	\(-0.448622\pi\)
							−0.511322	+	0.859389i	\(0.670844\pi\)
\(5\)	86.0271	+	487.884i	0.307780	+	1.74551i	0.610123	+	0.792307i	\(0.291120\pi\)
							−0.302343	+	0.953199i	\(0.597769\pi\)
\(7\)	270.212	+	1532.45i	0.297757	+	1.68866i	0.655781	+	0.754951i	\(0.272339\pi\)
							−0.358024	+	0.933712i	\(0.616550\pi\)
\(11\)	2185.34	−	3785.12i	0.495045	−	0.857443i	−0.504939	−	0.863155i	\(-0.668485\pi\)
							0.999984	+	0.00571200i	\(0.00181820\pi\)
\(13\)	6485.52	−	5442.00i	0.818734	−	0.687000i	−0.133941	−	0.990989i	\(-0.542763\pi\)
							0.952675	+	0.303990i	\(0.0983189\pi\)
\(17\)	13390.7	+	11236.1i	0.661047	+	0.554684i	0.910400	−	0.413729i	\(-0.135774\pi\)
							−0.249354	+	0.968412i	\(0.580218\pi\)
\(19\)	−22692.7	−	8259.47i	−0.759012	−	0.276258i	−0.0666190	−	0.997778i	\(-0.521221\pi\)
							−0.692393	+	0.721521i	\(0.743443\pi\)
\(23\)	−7086.11	−	12273.5i	−0.121440	−	0.210340i	0.798896	−	0.601469i	\(-0.205418\pi\)
							−0.920336	+	0.391130i	\(0.872084\pi\)
\(29\)	−5320.03	+	9214.56i	−0.0405062	+	0.0701587i	−0.885568	−	0.464510i	\(-0.846230\pi\)
							0.845062	+	0.534669i	\(0.179564\pi\)
\(31\)	228116.			1.37528			0.687639	−	0.726053i	\(-0.258647\pi\)
							0.687639	+	0.726053i	\(0.258647\pi\)
\(37\)	119121.	−	284151.i	0.386619	−	0.922239i
							\(41\)	513959.	−	431263.i	1.16462	−	0.977235i	0.164664	−	0.986350i	\(-0.447346\pi\)
0.999958	+	0.00911506i	\(0.00290145\pi\)
\(43\)	428169.			0.821250			0.410625	−	0.911804i	\(-0.365311\pi\)
							0.410625	+	0.911804i	\(0.365311\pi\)
\(47\)	−523273.	−	906336.i	−0.735167	−	1.27335i	−0.954650	−	0.297730i	\(-0.903770\pi\)
							0.219483	−	0.975616i	\(-0.429563\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.f.a.7.19	✓	132
37.16	even	9	inner	37.8.f.a.16.19	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.f.a.7.19	✓	132	1.1	even	1	trivial
37.8.f.a.16.19	yes	132	37.16	even	9	inner