Embedded newform 37.8.f.a.16.8

• Label: 37.8.f.a.16.8
• Level: $37$
• Weight: $8$
• Character: 37.16
• 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			16.8
Character	\(\chi\)	\(=\)	37.16
Dual form			37.8.f.a.7.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.31992 - 1.93629i) q^{2} +(16.2628 - 5.91916i) q^{3} +(-73.5013 - 61.6750i) q^{4} +(5.57004 - 31.5893i) q^{5} -97.9779 q^{6} +(125.391 - 711.126i) q^{7} +(633.927 + 1097.99i) q^{8} +(-1445.90 + 1213.25i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.31992	−	1.93629i	−0.470219	−	0.171146i	0.0960325	−	0.995378i	\(-0.469385\pi\)
							−0.566252	+	0.824232i	\(0.691607\pi\)
\(3\)	16.2628	−	5.91916i	0.347752	−	0.126571i	−0.162238	−	0.986752i	\(-0.551871\pi\)
							0.509990	+	0.860180i	\(0.329649\pi\)
\(5\)	5.57004	−	31.5893i	0.0199280	−	0.113017i	−0.973221	−	0.229871i	\(-0.926170\pi\)
							0.993149	+	0.116854i	\(0.0372808\pi\)
\(7\)	125.391	−	711.126i	0.138173	−	0.783616i	−0.834425	−	0.551122i	\(-0.814200\pi\)
							0.972598	−	0.232495i	\(-0.0746888\pi\)
\(11\)	−1429.88	−	2476.62i	−0.323911	−	0.561030i	0.657381	−	0.753559i	\(-0.271664\pi\)
							−0.981291	+	0.192529i	\(0.938331\pi\)
\(13\)	4088.44	+	3430.61i	0.516126	+	0.433081i	0.863279	−	0.504727i	\(-0.168407\pi\)
							−0.347153	+	0.937809i	\(0.612851\pi\)
\(17\)	−16632.8	+	13956.6i	−0.821095	+	0.688981i	−0.953228	−	0.302251i	\(-0.902262\pi\)
							0.132133	+	0.991232i	\(0.457817\pi\)
\(19\)	−48474.3	+	17643.2i	−1.62134	+	0.590120i	−0.983637	−	0.180162i	\(-0.942338\pi\)
							−0.637704	+	0.770282i	\(0.720116\pi\)
\(23\)	−54890.2	+	95072.6i	−0.940692	+	1.62933i	−0.176536	+	0.984294i	\(0.556489\pi\)
							−0.764156	+	0.645032i	\(0.776844\pi\)
\(29\)	−107242.	−	185749.i	−0.816532	−	1.41428i	−0.908223	−	0.418488i	\(-0.862560\pi\)
							0.0916904	−	0.995788i	\(-0.470773\pi\)
\(31\)	206272.			1.24358			0.621792	−	0.783182i	\(-0.286405\pi\)
							0.621792	+	0.783182i	\(0.286405\pi\)
\(37\)	−93172.5	−	293685.i	−0.302400	−	0.953181i
							\(41\)	55805.3	+	46826.2i	0.126454	+	0.106107i	0.703821	−	0.710377i	\(-0.251476\pi\)
−0.577368	+	0.816484i	\(0.695920\pi\)
\(43\)	−643506.			−1.23428			−0.617139	−	0.786854i	\(-0.711708\pi\)
							−0.617139	+	0.786854i	\(0.711708\pi\)
\(47\)	261769.	−	453398.i	0.367770	−	0.636997i	−0.621446	−	0.783457i	\(-0.713455\pi\)
							0.989217	+	0.146460i	\(0.0467880\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.f.a.16.8	yes	132
37.7	even	9	inner	37.8.f.a.7.8	✓	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.f.a.7.8	✓	132	37.7	even	9	inner
37.8.f.a.16.8	yes	132	1.1	even	1	trivial