Embedded newform 37.6.e.a.11.15

• Label: 37.6.e.a.11.15
• 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			11.15
Character	\(\chi\)	\(=\)	37.11
Dual form			37.6.e.a.27.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.27772 - 4.77914i) q^{2} +(2.34354 - 4.05913i) q^{3} +(29.6804 - 51.4080i) q^{4} +(-63.0654 - 36.4108i) q^{5} -44.8004i q^{6} +(27.0685 - 46.8840i) q^{7} -261.522i q^{8} +(110.516 + 191.419i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 6 q^{2} + 18 q^{3} + 298 q^{4} + 144 q^{5} - 52 q^{7} - 1490 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.27772	−	4.77914i	1.46331	−	0.844841i	0.464145	−	0.885759i	\(-0.346362\pi\)
							0.999162	+	0.0409182i	\(0.0130283\pi\)
\(3\)	2.34354	−	4.05913i	0.150338	−	0.260393i	−0.781014	−	0.624514i	\(-0.785297\pi\)
							0.931352	+	0.364121i	\(0.118630\pi\)
\(5\)	−63.0654	−	36.4108i	−1.12815	−	0.651337i	−0.184680	−	0.982799i	\(-0.559125\pi\)
							−0.943469	+	0.331462i	\(0.892458\pi\)
\(7\)	27.0685	−	46.8840i	0.208795	−	0.361643i	−0.742541	−	0.669801i	\(-0.766379\pi\)
							0.951335	+	0.308158i	\(0.0997127\pi\)
\(11\)	796.399			1.98449			0.992246	−	0.124293i	\(-0.0396662\pi\)
							0.992246	+	0.124293i	\(0.0396662\pi\)
\(13\)	−54.9193	−	31.7077i	−0.0901295	−	0.0520363i	0.454258	−	0.890870i	\(-0.349904\pi\)
							−0.544387	+	0.838834i	\(0.683238\pi\)
\(17\)	−1508.29	+	870.813i	−1.26579	+	0.730807i	−0.974189	−	0.225732i	\(-0.927523\pi\)
							−0.291605	+	0.956539i	\(0.594189\pi\)
\(19\)	331.088	+	191.154i	0.210406	+	0.121478i	0.601500	−	0.798873i	\(-0.294570\pi\)
							−0.391094	+	0.920351i	\(0.627903\pi\)
\(23\)		−	1722.90i		−	0.679111i	−0.940586	−	0.339555i	\(-0.889723\pi\)
							0.940586	−	0.339555i	\(-0.110277\pi\)
\(29\)			1160.91i			0.256332i	0.991753	+	0.128166i	\(0.0409090\pi\)
							−0.991753	+	0.128166i	\(0.959091\pi\)
\(31\)			4372.44i			0.817184i	0.912717	+	0.408592i	\(0.133980\pi\)
							−0.912717	+	0.408592i	\(0.866020\pi\)
\(37\)	−8322.38	+	286.161i	−0.999409	+	0.0343641i
							\(41\)	−6299.95	+	10911.8i	−0.585298	+	1.01377i	0.409540	+	0.912292i	\(0.365689\pi\)
−0.994838	+	0.101474i	\(0.967644\pi\)
\(43\)		−	20904.9i		−	1.72416i	−0.506773	−	0.862080i	\(-0.669162\pi\)
							0.506773	−	0.862080i	\(-0.330838\pi\)
\(47\)	−3047.96			−0.201263			−0.100632	−	0.994924i	\(-0.532086\pi\)
							−0.100632	+	0.994924i	\(0.532086\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.e.a.11.15	✓	32
37.27	even	6	inner	37.6.e.a.27.15	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.e.a.11.15	✓	32	1.1	even	1	trivial
37.6.e.a.27.15	yes	32	37.27	even	6	inner