Embedded newform 138.3.h.a.37.4

• Label: 138.3.h.a.37.4
• 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			37.4
Character	\(\chi\)	\(=\)	138.37
Dual form			138.3.h.a.97.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35693 + 0.398430i) q^{2} +(1.13425 + 1.30900i) q^{3} +(1.68251 - 1.08128i) q^{4} +(5.59508 - 0.804450i) q^{5} +(-2.06064 - 1.32429i) q^{6} +(0.513410 + 0.234466i) q^{7} +(-1.85223 + 2.13758i) q^{8} +(-0.426945 + 2.96946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \)

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
\(5\)	5.59508	−	0.804450i	1.11902	−	0.160890i								0.442110	−	0.896961i	\(-0.354230\pi\)
							0.676905	+	0.736071i	\(0.263321\pi\)
\(7\)	0.513410	+	0.234466i	0.0733443	+	0.0334952i	0.451749	−	0.892145i	\(-0.350800\pi\)
							−0.378405	+	0.925640i	\(0.623527\pi\)
\(11\)	5.36883	−	18.2846i	0.488076	−	1.66223i	−0.235412	−	0.971896i	\(-0.575644\pi\)
							0.723488	−	0.690337i	\(-0.242538\pi\)
\(13\)	4.44482	+	9.73281i	0.341909	+	0.748677i	0.999991	−	0.00426944i	\(-0.00135901\pi\)
							−0.658081	+	0.752947i	\(0.728632\pi\)
\(17\)	−5.60999	+	8.72932i	−0.329999	+	0.513489i	−0.966118	−	0.258103i	\(-0.916903\pi\)
							0.636118	+	0.771592i	\(0.280539\pi\)
\(19\)	19.7451	+	30.7240i	1.03922	+	1.61705i	0.751973	+	0.659194i	\(0.229102\pi\)
							0.287243	+	0.957858i	\(0.407261\pi\)
\(23\)	22.9451	−	1.58855i	0.997612	−	0.0690674i
							\(29\)	−33.0502	−	21.2401i	−1.13966	−	0.732416i	−0.172107	−	0.985078i	\(-0.555058\pi\)
−0.967555	+	0.252662i	\(0.918694\pi\)
\(31\)	13.8156	−	15.9440i	0.445664	−	0.514324i	−0.487819	−	0.872945i	\(-0.662207\pi\)
							0.933483	+	0.358621i	\(0.116753\pi\)
\(37\)	−34.7029	−	4.98952i	−0.937916	−	0.134852i	−0.343629	−	0.939105i	\(-0.611656\pi\)
							−0.594287	+	0.804253i	\(0.702566\pi\)
\(41\)	−3.66477	−	25.4891i	−0.0893847	−	0.621684i	−0.984439	−	0.175727i	\(-0.943772\pi\)
							0.895054	−	0.445957i	\(-0.147137\pi\)
\(43\)	−43.7396	+	37.9006i	−1.01720	+	0.881410i	−0.992978	−	0.118297i	\(-0.962257\pi\)
							−0.0242229	+	0.999707i	\(0.507711\pi\)
\(47\)	29.6457			0.630760			0.315380	−	0.948966i	\(-0.397868\pi\)
							0.315380	+	0.948966i	\(0.397868\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.h.a.37.4	✓	80
3.2	odd	2		414.3.l.b.37.5		80
23.5	odd	22	inner	138.3.h.a.97.4	yes	80
69.5	even	22		414.3.l.b.235.5		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.h.a.37.4	✓	80	1.1	even	1	trivial
138.3.h.a.97.4	yes	80	23.5	odd	22	inner
414.3.l.b.37.5		80	3.2	odd	2
414.3.l.b.235.5		80	69.5	even	22