Embedded newform 37.6.e.a.11.10

• Label: 37.6.e.a.11.10
• Level: $37$
• Weight: $6$
• Character: 37.11
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.10
Character	\(\chi\)	\(=\)	37.11
Dual form			37.6.e.a.27.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.987887 - 0.570357i) q^{2} +(2.15443 - 3.73158i) q^{3} +(-15.3494 + 26.5859i) q^{4} +(19.1329 + 11.0464i) q^{5} -4.91517i q^{6} +(-51.7707 + 89.6695i) q^{7} +71.5213i q^{8} +(112.217 + 194.365i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 6 q^{2} + 18 q^{3} + 298 q^{4} + 144 q^{5} - 52 q^{7} - 1490 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.987887	−	0.570357i	0.174635	−	0.100826i	−0.410134	−	0.912025i	\(-0.634518\pi\)
							0.584770	+	0.811199i	\(0.301185\pi\)
\(3\)	2.15443	−	3.73158i	0.138207	−	0.239381i	−0.788611	−	0.614892i	\(-0.789200\pi\)
							0.926818	+	0.375511i	\(0.122533\pi\)
\(5\)	19.1329	+	11.0464i	0.342260	+	0.197604i	0.661271	−	0.750147i	\(-0.270017\pi\)
							−0.319011	+	0.947751i	\(0.603351\pi\)
\(7\)	−51.7707	+	89.6695i	−0.399336	+	0.691671i	−0.993644	−	0.112567i	\(-0.964093\pi\)
							0.594308	+	0.804238i	\(0.297426\pi\)
\(11\)	−61.8462			−0.154110			−0.0770550	−	0.997027i	\(-0.524552\pi\)
							−0.0770550	+	0.997027i	\(0.524552\pi\)
\(13\)	628.747	+	363.007i	1.03185	+	0.595740i	0.917515	−	0.397702i	\(-0.130192\pi\)
							0.114338	+	0.993442i	\(0.463525\pi\)
\(17\)	−1647.46	+	951.161i	−1.38259	+	0.798237i	−0.992465	−	0.122527i	\(-0.960900\pi\)
							−0.390121	+	0.920763i	\(0.627567\pi\)
\(19\)	1554.54	+	897.511i	0.987908	+	0.570369i	0.904648	−	0.426159i	\(-0.140134\pi\)
							0.0832599	+	0.996528i	\(0.473467\pi\)
\(23\)		−	4567.18i		−	1.80023i	−0.435652	−	0.900115i	\(-0.643482\pi\)
							0.435652	−	0.900115i	\(-0.356518\pi\)
\(29\)		−	282.833i		−	0.0624504i	−0.999512	−	0.0312252i	\(-0.990059\pi\)
							0.999512	−	0.0312252i	\(-0.00994090\pi\)
\(31\)		−	237.227i		−	0.0443363i	−0.999754	−	0.0221681i	\(-0.992943\pi\)
							0.999754	−	0.0221681i	\(-0.00705692\pi\)
\(37\)	8322.11	−	294.049i	0.999376	−	0.0353115i
							\(41\)	2271.52	−	3934.38i	0.211036	−	0.365525i	−0.741003	−	0.671502i	\(-0.765650\pi\)
0.952039	+	0.305977i	\(0.0989829\pi\)
\(43\)			14329.3i			1.18183i	0.806734	+	0.590915i	\(0.201233\pi\)
							−0.806734	+	0.590915i	\(0.798767\pi\)
\(47\)	11508.8			0.759950			0.379975	−	0.924997i	\(-0.375933\pi\)
							0.379975	+	0.924997i	\(0.375933\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.e.a.11.10	✓	32
37.27	even	6	inner	37.6.e.a.27.10	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.e.a.11.10	✓	32	1.1	even	1	trivial
37.6.e.a.27.10	yes	32	37.27	even	6	inner