Embedded newform 37.5.g.a.8.8

• Label: 37.5.g.a.8.8
• Level: $37$
• Weight: $5$
• Character: 37.8
• Analytic conductor: $3.825$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.82468863410\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			8.8
Character	\(\chi\)	\(=\)	37.8
Dual form			37.5.g.a.14.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.17412 + 0.582554i) q^{2} +(-5.52639 - 3.19066i) q^{3} +(-9.46898 - 5.46692i) q^{4} +(33.8656 - 9.07426i) q^{5} +(-10.1563 - 10.1563i) q^{6} +(37.3892 - 64.7600i) q^{7} +(-42.8670 - 42.8670i) q^{8} +(-20.1393 - 34.8824i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 10 q^{2} - 6 q^{3} - 24 q^{4} + 98 q^{5} - 36 q^{6} - 2 q^{7} + 240 q^{8} + 306 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.17412	+	0.582554i	0.543530	+	0.145638i	0.520129	−	0.854088i	\(-0.325884\pi\)
							0.0234009	+	0.999726i	\(0.492551\pi\)
\(3\)	−5.52639	−	3.19066i	−0.614043	−	0.354518i	0.160503	−	0.987035i	\(-0.448688\pi\)
							−0.774546	+	0.632517i	\(0.782022\pi\)
\(5\)	33.8656	−	9.07426i	1.35462	−	0.362971i	0.492785	−	0.870151i	\(-0.335979\pi\)
							0.861840	+	0.507181i	\(0.169312\pi\)
\(7\)	37.3892	−	64.7600i	0.763045	−	1.32163i	−0.178229	−	0.983989i	\(-0.557037\pi\)
							0.941274	−	0.337644i	\(-0.109630\pi\)
\(11\)			160.384i			1.32549i	0.748847	+	0.662743i	\(0.230608\pi\)
							−0.748847	+	0.662743i	\(0.769392\pi\)
\(13\)	−68.3982	+	18.3272i	−0.404723	+	0.108445i	−0.455437	−	0.890268i	\(-0.650517\pi\)
							0.0507139	+	0.998713i	\(0.483850\pi\)
\(17\)	−41.1609	+	153.615i	−0.142425	+	0.531539i	0.857431	+	0.514599i	\(0.172059\pi\)
							−0.999857	+	0.0169400i	\(0.994608\pi\)
\(19\)	585.376	−	156.851i	1.62154	−	0.434490i	0.670086	−	0.742283i	\(-0.266257\pi\)
							0.951453	+	0.307793i	\(0.0995905\pi\)
\(23\)	−42.2015	−	42.2015i	−0.0797759	−	0.0797759i	0.666093	−	0.745869i	\(-0.267965\pi\)
							−0.745869	+	0.666093i	\(0.767965\pi\)
\(29\)	−426.675	+	426.675i	−0.507342	+	0.507342i	−0.913710	−	0.406368i	\(-0.866795\pi\)
							0.406368	+	0.913710i	\(0.366795\pi\)
\(31\)	1158.06	−	1158.06i	1.20506	−	1.20506i	0.232450	−	0.972608i	\(-0.425326\pi\)
							0.972608	−	0.232450i	\(-0.0746741\pi\)
\(37\)	−1055.83	+	871.421i	−0.771245	+	0.636539i
							\(41\)	864.000	+	498.831i	0.513980	+	0.296747i	0.734468	−	0.678643i	\(-0.237432\pi\)
−0.220488	+	0.975390i	\(0.570765\pi\)
\(43\)	−582.259	−	582.259i	−0.314905	−	0.314905i	0.531901	−	0.846806i	\(-0.321478\pi\)
							−0.846806	+	0.531901i	\(0.821478\pi\)
\(47\)	3620.75			1.63909			0.819544	−	0.573016i	\(-0.194227\pi\)
							0.819544	+	0.573016i	\(0.194227\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.g.a.8.8	✓	44
37.14	odd	12	inner	37.5.g.a.14.8	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.g.a.8.8	✓	44	1.1	even	1	trivial
37.5.g.a.14.8	yes	44	37.14	odd	12	inner