Embedded newform 37.5.g.a.14.3

• Label: 37.5.g.a.14.3
• Level: $37$
• Weight: $5$
• Character: 37.14
• 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			14.3
Character	\(\chi\)	\(=\)	37.14
Dual form			37.5.g.a.8.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.85521 + 1.30095i) q^{2} +(-11.3098 + 6.52971i) q^{3} +(8.02419 - 4.63277i) q^{4} +(-23.1849 - 6.21238i) q^{5} +(46.4166 - 46.4166i) q^{6} +(29.0699 + 50.3505i) q^{7} +(23.9360 - 23.9360i) q^{8} +(44.7743 - 77.5514i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 10 q^{2} - 6 q^{3} - 24 q^{4} + 98 q^{5} - 36 q^{6} - 2 q^{7} + 240 q^{8} + 306 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.85521	+	1.30095i	−1.21380	+	0.325237i	−0.808253	−	0.588836i	\(-0.799587\pi\)
							−0.405550	+	0.914073i	\(0.632920\pi\)
\(3\)	−11.3098	+	6.52971i	−1.25664	+	0.725524i	−0.972421	−	0.233234i	\(-0.925069\pi\)
							−0.284223	+	0.958758i	\(0.591736\pi\)
\(5\)	−23.1849	−	6.21238i	−0.927397	−	0.248495i	−0.236653	−	0.971594i	\(-0.576050\pi\)
							−0.690745	+	0.723099i	\(0.742717\pi\)
\(7\)	29.0699	+	50.3505i	0.593262	+	1.02756i	0.993790	+	0.111276i	\(0.0354937\pi\)
							−0.400527	+	0.916285i	\(0.631173\pi\)
\(11\)		−	128.204i		−	1.05954i	−0.848141	−	0.529770i	\(-0.822278\pi\)
							0.848141	−	0.529770i	\(-0.177722\pi\)
\(13\)	−46.1251	−	12.3592i	−0.272930	−	0.0731313i	0.119758	−	0.992803i	\(-0.461788\pi\)
							−0.392688	+	0.919672i	\(0.628455\pi\)
\(17\)	68.3317	+	255.017i	0.236442	+	0.882413i	0.977494	+	0.210965i	\(0.0676607\pi\)
							−0.741052	+	0.671448i	\(0.765673\pi\)
\(19\)	−259.002	−	69.3992i	−0.717456	−	0.192242i	−0.118420	−	0.992964i	\(-0.537783\pi\)
							−0.599036	+	0.800722i	\(0.704449\pi\)
\(23\)	706.765	−	706.765i	1.33604	−	1.33604i	0.436181	−	0.899859i	\(-0.356331\pi\)
							0.899859	−	0.436181i	\(-0.143669\pi\)
\(29\)	−638.601	−	638.601i	−0.759336	−	0.759336i	0.216866	−	0.976201i	\(-0.430417\pi\)
							−0.976201	+	0.216866i	\(0.930417\pi\)
\(31\)	467.011	+	467.011i	0.485963	+	0.485963i	0.907030	−	0.421067i	\(-0.138344\pi\)
							−0.421067	+	0.907030i	\(0.638344\pi\)
\(37\)	1243.47	−	572.665i	0.908305	−	0.418309i
							\(41\)	1222.30	−	705.695i	0.727126	−	0.419807i	−0.0902436	−	0.995920i	\(-0.528765\pi\)
0.817370	+	0.576113i	\(0.195431\pi\)
\(43\)	1413.24	−	1413.24i	0.764327	−	0.764327i	−0.212775	−	0.977101i	\(-0.568250\pi\)
							0.977101	+	0.212775i	\(0.0682500\pi\)
\(47\)	−3961.42			−1.79331			−0.896654	−	0.442731i	\(-0.854010\pi\)
							−0.896654	+	0.442731i	\(0.854010\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.g.a.14.3	yes	44
37.8	odd	12	inner	37.5.g.a.8.3	✓	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.g.a.8.3	✓	44	37.8	odd	12	inner
37.5.g.a.14.3	yes	44	1.1	even	1	trivial