Embedded newform 138.3.h.a.19.2

• Label: 138.3.h.a.19.2
• 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			19.2
Character	\(\chi\)	\(=\)	138.19
Dual form			138.3.h.a.109.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.587486 - 1.28641i) q^{2} +(-1.66189 + 0.487975i) q^{3} +(-1.30972 + 1.51150i) q^{4} +(0.791659 - 1.23185i) q^{5} +(1.60407 + 1.85120i) q^{6} +(8.65930 - 1.24502i) q^{7} +(2.71386 + 0.796860i) q^{8} +(2.52376 - 1.62192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \)

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
\(5\)	0.791659	−	1.23185i	0.158332	−	0.246369i								−0.753020	−	0.657998i	\(-0.771404\pi\)
							0.911352	+	0.411629i	\(0.135040\pi\)
\(7\)	8.65930	−	1.24502i	1.23704	−	0.177860i	0.507414	−	0.861703i	\(-0.330602\pi\)
							0.729630	+	0.683843i	\(0.239693\pi\)
\(11\)	−15.4560	−	7.05850i	−1.40509	−	0.641682i	−0.438667	−	0.898650i	\(-0.644549\pi\)
							−0.966420	+	0.256968i	\(0.917277\pi\)
\(13\)	3.51415	−	24.4415i	0.270319	−	1.88011i	−0.174744	−	0.984614i	\(-0.555910\pi\)
							0.445063	−	0.895499i	\(-0.353181\pi\)
\(17\)	7.39872	−	6.41103i	0.435219	−	0.377119i	−0.409517	−	0.912302i	\(-0.634303\pi\)
							0.844736	+	0.535183i	\(0.179757\pi\)
\(19\)	2.05634	+	1.78183i	0.108229	+	0.0937806i	0.707294	−	0.706919i	\(-0.249916\pi\)
							−0.599066	+	0.800700i	\(0.704461\pi\)
\(23\)	12.7043	−	19.1729i	0.552360	−	0.833606i
							\(29\)	−20.5481	−	23.7138i	−0.708557	−	0.817718i	0.281325	−	0.959613i	\(-0.409226\pi\)
−0.989882	+	0.141894i	\(0.954681\pi\)
\(31\)	28.1592	+	8.26828i	0.908361	+	0.266719i	0.702351	−	0.711831i	\(-0.252134\pi\)
							0.206010	+	0.978550i	\(0.433952\pi\)
\(37\)	−9.86724	−	15.3537i	−0.266682	−	0.414966i	0.681925	−	0.731422i	\(-0.261143\pi\)
							−0.948607	+	0.316457i	\(0.897507\pi\)
\(41\)	27.7213	+	17.8154i	0.676130	+	0.434522i	0.833131	−	0.553076i	\(-0.186546\pi\)
							−0.157001	+	0.987598i	\(0.550183\pi\)
\(43\)	22.5341	+	76.7441i	0.524049	+	1.78475i	0.614583	+	0.788852i	\(0.289324\pi\)
							−0.0905344	+	0.995893i	\(0.528858\pi\)
\(47\)	−69.2488			−1.47338			−0.736689	−	0.676232i	\(-0.763612\pi\)
							−0.736689	+	0.676232i	\(0.763612\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.h.a.19.2	✓	80
3.2	odd	2		414.3.l.b.19.6		80
23.17	odd	22	inner	138.3.h.a.109.2	yes	80
69.17	even	22		414.3.l.b.109.6		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.h.a.19.2	✓	80	1.1	even	1	trivial
138.3.h.a.109.2	yes	80	23.17	odd	22	inner
414.3.l.b.19.6		80	3.2	odd	2
414.3.l.b.109.6		80	69.17	even	22