Embedded newform 138.3.h.a.79.1

• Label: 138.3.h.a.79.1
• Level: $138$
• Weight: $3$
• Character: 138.79
• 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			79.1
Character	\(\chi\)	\(=\)	138.79
Dual form			138.3.h.a.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.926113 - 1.06879i) q^{2} +(-1.45709 - 0.936417i) q^{3} +(-0.284630 + 1.97964i) q^{4} +(1.12201 + 0.512407i) q^{5} +(0.348599 + 2.42456i) q^{6} +(2.20763 - 7.51851i) q^{7} +(2.37942 - 1.52916i) q^{8} +(1.24625 + 2.72890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.926113	−	1.06879i	−0.463056	−	0.534396i
							\(3\)	−1.45709	−	0.936417i	−0.485698	−	0.312139i
\(5\)	1.12201	+	0.512407i	0.224403	+	0.102481i								0.524444	−	0.851445i	\(-0.324273\pi\)
							−0.300041	+	0.953926i	\(0.597000\pi\)
\(7\)	2.20763	−	7.51851i	0.315376	−	1.07407i	−0.637434	−	0.770505i	\(-0.720004\pi\)
							0.952810	−	0.303568i	\(-0.0981778\pi\)
\(11\)	−15.5963	−	13.5143i	−1.41785	−	1.22857i	−0.935849	−	0.352401i	\(-0.885365\pi\)
							−0.482000	−	0.876171i	\(-0.660089\pi\)
\(13\)	−14.5601	+	4.27524i	−1.12001	+	0.328864i	−0.788774	−	0.614684i	\(-0.789284\pi\)
							−0.331235	+	0.943548i	\(0.607465\pi\)
\(17\)	−19.6785	+	2.82935i	−1.15756	+	0.166432i	−0.694230	−	0.719753i	\(-0.744255\pi\)
							−0.463331	+	0.886185i	\(0.653346\pi\)
\(19\)	32.9814	+	4.74201i	1.73586	+	0.249580i	0.936350	−	0.351067i	\(-0.114181\pi\)
							0.799514	+	0.600647i	\(0.205090\pi\)
\(23\)	−20.9466	−	9.49950i	−0.910721	−	0.413022i
							\(29\)	−3.31586	−	23.0623i	−0.114340	−	0.795252i	−0.963614	−	0.267299i	\(-0.913869\pi\)
0.849273	−	0.527953i	\(-0.177040\pi\)
\(31\)	14.5557	−	9.35437i	0.469538	−	0.301754i	−0.284392	−	0.958708i	\(-0.591792\pi\)
							0.753931	+	0.656954i	\(0.228156\pi\)
\(37\)	30.6081	−	13.9782i	0.827245	−	0.377790i	0.0436385	−	0.999047i	\(-0.486105\pi\)
							0.783607	+	0.621257i	\(0.213378\pi\)
\(41\)	25.2644	−	55.3213i	0.616205	−	1.34930i	−0.302043	−	0.953294i	\(-0.597669\pi\)
							0.918248	−	0.396006i	\(-0.129604\pi\)
\(43\)	−28.8157	+	44.8382i	−0.670133	+	1.04275i	0.325143	+	0.945665i	\(0.394588\pi\)
							−0.995276	+	0.0970831i	\(0.969049\pi\)
\(47\)	57.1118			1.21515			0.607573	−	0.794264i	\(-0.292143\pi\)
							0.607573	+	0.794264i	\(0.292143\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.h.a.79.1	yes	80
3.2	odd	2		414.3.l.b.217.6		80
23.7	odd	22	inner	138.3.h.a.7.1	✓	80
69.53	even	22		414.3.l.b.145.6		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.h.a.7.1	✓	80	23.7	odd	22	inner
138.3.h.a.79.1	yes	80	1.1	even	1	trivial
414.3.l.b.145.6		80	69.53	even	22
414.3.l.b.217.6		80	3.2	odd	2