Embedded newform 138.3.h.a.19.6

• Label: 138.3.h.a.19.6
• Level: $138$
• Weight: $3$
• Character: 138.19
• Analytic conductor: $3.760$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	138.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.76022764817\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(8\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			19.6
Character	\(\chi\)	\(=\)	138.19
Dual form			138.3.h.a.109.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.587486 + 1.28641i) q^{2} +(-1.66189 + 0.487975i) q^{3} +(-1.30972 + 1.51150i) q^{4} +(5.01874 - 7.80931i) q^{5} +(-1.60407 - 1.85120i) q^{6} +(5.11630 - 0.735612i) q^{7} +(-2.71386 - 0.796860i) q^{8} +(2.52376 - 1.62192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{15}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.587486	+	1.28641i	0.293743	+	0.643207i
							\(3\)	−1.66189	+	0.487975i	−0.553964	+	0.162658i
\(5\)	5.01874	−	7.80931i	1.00375	−	1.56186i								0.189059	−	0.981966i	\(-0.439456\pi\)
							0.814690	−	0.579897i	\(-0.196907\pi\)
\(7\)	5.11630	−	0.735612i	0.730899	−	0.105087i	0.233190	−	0.972431i	\(-0.425084\pi\)
							0.497710	+	0.867344i	\(0.334175\pi\)
\(11\)	7.49739	+	3.42394i	0.681581	+	0.311267i	0.725950	−	0.687748i	\(-0.241401\pi\)
							−0.0443692	+	0.999015i	\(0.514128\pi\)
\(13\)	−0.170426	+	1.18534i	−0.0131097	+	0.0911798i	−0.995326	−	0.0965739i	\(-0.969212\pi\)
							0.982216	+	0.187754i	\(0.0601207\pi\)
\(17\)	−0.695908	+	0.603008i	−0.0409358	+	0.0354711i	−0.675088	−	0.737737i	\(-0.735895\pi\)
							0.634153	+	0.773208i	\(0.281349\pi\)
\(19\)	5.83348	+	5.05474i	0.307025	+	0.266039i	0.794721	−	0.606975i	\(-0.207617\pi\)
							−0.487696	+	0.873014i	\(0.662163\pi\)
\(23\)	22.4434	+	5.02937i	0.975799	+	0.218668i
							\(29\)	4.59011	+	5.29727i	0.158280	+	0.182664i	0.829350	−	0.558729i	\(-0.188711\pi\)
−0.671071	+	0.741393i	\(0.734165\pi\)
\(31\)	−42.9090	−	12.5992i	−1.38416	−	0.406426i	−0.496945	−	0.867782i	\(-0.665545\pi\)
							−0.887215	+	0.461356i	\(0.847363\pi\)
\(37\)	−37.0310	−	57.6213i	−1.00084	−	1.55733i	−0.818830	−	0.574037i	\(-0.805377\pi\)
							−0.182008	−	0.983297i	\(-0.558260\pi\)
\(41\)	38.4365	+	24.7016i	0.937475	+	0.602479i	0.917678	−	0.397325i	\(-0.130062\pi\)
							0.0197972	+	0.999804i	\(0.493698\pi\)
\(43\)	20.8057	+	70.8578i	0.483854	+	1.64786i	0.733627	+	0.679552i	\(0.237826\pi\)
							−0.249773	+	0.968304i	\(0.580356\pi\)
\(47\)	3.73913			0.0795559			0.0397779	−	0.999209i	\(-0.487335\pi\)
							0.0397779	+	0.999209i	\(0.487335\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.h.a.19.6	✓	80
3.2	odd	2		414.3.l.b.19.1		80
23.17	odd	22	inner	138.3.h.a.109.6	yes	80
69.17	even	22		414.3.l.b.109.1		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.h.a.19.6	✓	80	1.1	even	1	trivial
138.3.h.a.109.6	yes	80	23.17	odd	22	inner
414.3.l.b.19.1		80	3.2	odd	2
414.3.l.b.109.1		80	69.17	even	22