Embedded newform 138.3.h.a.19.1

• Label: 138.3.h.a.19.1
• 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			19.1
Character	\(\chi\)	\(=\)	138.19
Dual form			138.3.h.a.109.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.587486 - 1.28641i) q^{2} +(-1.66189 + 0.487975i) q^{3} +(-1.30972 + 1.51150i) q^{4} +(-0.634924 + 0.987962i) q^{5} +(1.60407 + 1.85120i) q^{6} +(1.01096 - 0.145354i) q^{7} +(2.71386 + 0.796860i) q^{8} +(2.52376 - 1.62192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \)

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
\(5\)	−0.634924	+	0.987962i	−0.126985	+	0.197592i								−0.898925	−	0.438103i	\(-0.855650\pi\)
							0.771940	+	0.635696i	\(0.219287\pi\)
\(7\)	1.01096	−	0.145354i	0.144423	−	0.0207648i	−0.0697243	−	0.997566i	\(-0.522212\pi\)
							0.214147	+	0.976801i	\(0.431303\pi\)
\(11\)	17.3309	+	7.91477i	1.57554	+	0.719525i	0.995473	−	0.0950445i	\(-0.0302993\pi\)
							0.580066	+	0.814569i	\(0.303027\pi\)
\(13\)	−3.12343	+	21.7239i	−0.240264	+	1.67107i	0.410554	+	0.911836i	\(0.365335\pi\)
							−0.650818	+	0.759234i	\(0.725574\pi\)
\(17\)	25.1171	−	21.7641i	1.47748	−	1.28024i	0.600580	−	0.799565i	\(-0.294936\pi\)
							0.876898	−	0.480677i	\(-0.159609\pi\)
\(19\)	8.79808	+	7.62358i	0.463057	+	0.401241i	0.854901	−	0.518790i	\(-0.173618\pi\)
							−0.391845	+	0.920031i	\(0.628163\pi\)
\(23\)	−22.4075	−	5.18677i	−0.974240	−	0.225512i
							\(29\)	14.2328	+	16.4256i	0.490788	+	0.566399i	0.946076	−	0.323945i	\(-0.105009\pi\)
−0.455288	+	0.890344i	\(0.650464\pi\)
\(31\)	−43.6134	−	12.8060i	−1.40688	−	0.413098i	−0.511841	−	0.859080i	\(-0.671036\pi\)
							−0.895042	+	0.445982i	\(0.852855\pi\)
\(37\)	5.86914	+	9.13256i	0.158625	+	0.246826i	0.911465	−	0.411379i	\(-0.134953\pi\)
							−0.752839	+	0.658205i	\(0.771316\pi\)
\(41\)	56.2999	+	36.1817i	1.37317	+	0.882481i	0.998993	−	0.0448768i	\(-0.0142895\pi\)
							0.374175	+	0.927358i	\(0.377926\pi\)
\(43\)	0.861110	+	2.93267i	0.0200258	+	0.0682016i	0.968900	−	0.247454i	\(-0.0795938\pi\)
							−0.948874	+	0.315655i	\(0.897776\pi\)
\(47\)	−49.5387			−1.05402			−0.527008	−	0.849861i	\(-0.676686\pi\)
							−0.527008	+	0.849861i	\(0.676686\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.h.a.19.1	✓	80
3.2	odd	2		414.3.l.b.19.7		80
23.17	odd	22	inner	138.3.h.a.109.1	yes	80
69.17	even	22		414.3.l.b.109.7		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.h.a.19.1	✓	80	1.1	even	1	trivial
138.3.h.a.109.1	yes	80	23.17	odd	22	inner
414.3.l.b.19.7		80	3.2	odd	2
414.3.l.b.109.7		80	69.17	even	22