Embedded newform 138.3.h.a.37.2

• Label: 138.3.h.a.37.2
• Level: $138$
• Weight: $3$
• Character: 138.37
• 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			37.2
Character	\(\chi\)	\(=\)	138.37
Dual form			138.3.h.a.97.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35693 + 0.398430i) q^{2} +(-1.13425 - 1.30900i) q^{3} +(1.68251 - 1.08128i) q^{4} +(4.69745 - 0.675391i) q^{5} +(2.06064 + 1.32429i) q^{6} +(-5.04392 - 2.30348i) q^{7} +(-1.85223 + 2.13758i) q^{8} +(-0.426945 + 2.96946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{21}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35693	+	0.398430i	−0.678464	+	0.199215i
							\(3\)	−1.13425	−	1.30900i	−0.378084	−	0.436332i
\(5\)	4.69745	−	0.675391i	0.939489	−	0.135078i								0.344477	−	0.938795i	\(-0.388056\pi\)
							0.595012	+	0.803717i	\(0.297147\pi\)
\(7\)	−5.04392	−	2.30348i	−0.720560	−	0.329069i	0.0211620	−	0.999776i	\(-0.493263\pi\)
							−0.741722	+	0.670707i	\(0.765991\pi\)
\(11\)	2.53846	−	8.64519i	0.230769	−	0.785926i	−0.759948	−	0.649984i	\(-0.774776\pi\)
							0.990717	−	0.135942i	\(-0.0434062\pi\)
\(13\)	−1.54682	−	3.38705i	−0.118986	−	0.260543i	0.840762	−	0.541404i	\(-0.182107\pi\)
							−0.959748	+	0.280862i	\(0.909380\pi\)
\(17\)	17.0353	−	26.5074i	1.00207	−	1.55926i	0.184990	−	0.982740i	\(-0.440775\pi\)
							0.817085	−	0.576518i	\(-0.195589\pi\)
\(19\)	−11.2282	−	17.4715i	−0.590960	−	0.919551i	−0.999975	−	0.00702370i	\(-0.997764\pi\)
							0.409016	−	0.912527i	\(-0.365872\pi\)
\(23\)	5.12153	−	22.4225i	0.222675	−	0.974893i
							\(29\)	24.0465	+	15.4538i	0.829190	+	0.532888i	0.885020	−	0.465552i	\(-0.154144\pi\)
−0.0558304	+	0.998440i	\(0.517781\pi\)
\(31\)	−19.2802	+	22.2506i	−0.621943	+	0.717761i	−0.976075	−	0.217435i	\(-0.930231\pi\)
							0.354131	+	0.935196i	\(0.384777\pi\)
\(37\)	3.54762	+	0.510070i	0.0958816	+	0.0137857i	0.190089	−	0.981767i	\(-0.439122\pi\)
							−0.0942070	+	0.995553i	\(0.530032\pi\)
\(41\)	7.36875	+	51.2508i	0.179726	+	1.25002i	0.857397	+	0.514655i	\(0.172080\pi\)
							−0.677672	+	0.735365i	\(0.737011\pi\)
\(43\)	20.7297	−	17.9624i	0.482085	−	0.417729i	−0.379617	−	0.925144i	\(-0.623944\pi\)
							0.861702	+	0.507415i	\(0.169399\pi\)
\(47\)	−6.19684			−0.131848			−0.0659238	−	0.997825i	\(-0.520999\pi\)
							−0.0659238	+	0.997825i	\(0.520999\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.h.a.37.2	✓	80
3.2	odd	2		414.3.l.b.37.6		80
23.5	odd	22	inner	138.3.h.a.97.2	yes	80
69.5	even	22		414.3.l.b.235.6		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.h.a.37.2	✓	80	1.1	even	1	trivial
138.3.h.a.97.2	yes	80	23.5	odd	22	inner
414.3.l.b.37.6		80	3.2	odd	2
414.3.l.b.235.6		80	69.5	even	22