Embedded newform 138.3.g.a.29.13

• Label: 138.3.g.a.29.13
• Level: $138$
• Weight: $3$
• Character: 138.29
• Analytic conductor: $3.760$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	138.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.76022764817\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(16\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			29.13
Character	\(\chi\)	\(=\)	138.29
Dual form			138.3.g.a.119.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28641 + 0.587486i) q^{2} +(1.01607 - 2.82269i) q^{3} +(1.30972 + 1.51150i) q^{4} +(-3.78542 - 5.89022i) q^{5} +(2.96538 - 3.03422i) q^{6} +(0.377062 - 2.62252i) q^{7} +(0.796860 + 2.71386i) q^{8} +(-6.93519 - 5.73613i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} + 32 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.28641	+	0.587486i	0.643207	+	0.293743i
							\(3\)	1.01607	−	2.82269i	0.338691	−	0.940898i
\(5\)	−3.78542	−	5.89022i	−0.757083	−	1.17804i								−0.979175	−	0.203019i	\(-0.934925\pi\)
							0.222092	−	0.975026i	\(-0.428712\pi\)
\(7\)	0.377062	−	2.62252i	0.0538659	−	0.374646i	−0.945002	−	0.327066i	\(-0.893940\pi\)
							0.998867	−	0.0475798i	\(-0.0151509\pi\)
\(11\)	10.3402	−	4.72220i	0.940016	−	0.429291i	0.114344	−	0.993441i	\(-0.463523\pi\)
							0.825672	+	0.564150i	\(0.190796\pi\)
\(13\)	1.46625	+	10.1980i	0.112788	+	0.784460i	0.965186	+	0.261566i	\(0.0842388\pi\)
							−0.852397	+	0.522895i	\(0.824852\pi\)
\(17\)	−1.15125	−	0.997566i	−0.0677207	−	0.0586803i	0.620343	−	0.784331i	\(-0.286993\pi\)
							−0.688063	+	0.725650i	\(0.741539\pi\)
\(19\)	11.2184	+	12.9467i	0.590441	+	0.681405i	0.969816	−	0.243838i	\(-0.0784064\pi\)
							−0.379375	+	0.925243i	\(0.623861\pi\)
\(23\)	22.1851	−	6.06814i	0.964569	−	0.263832i
							\(29\)	18.2541	+	15.8173i	0.629453	+	0.545424i	0.910101	−	0.414387i	\(-0.136004\pi\)
−0.280648	+	0.959811i	\(0.590549\pi\)
\(31\)	−9.51707	+	2.79446i	−0.307002	+	0.0901440i	−0.431605	−	0.902063i	\(-0.642053\pi\)
							0.124603	+	0.992207i	\(0.460234\pi\)
\(37\)	43.0929	+	27.6941i	1.16467	+	0.748490i	0.972507	−	0.232872i	\(-0.0748125\pi\)
							0.192166	+	0.981362i	\(0.438449\pi\)
\(41\)	−33.4530	−	52.0539i	−0.815927	−	1.26961i	−0.959989	−	0.280038i	\(-0.909653\pi\)
							0.144062	−	0.989569i	\(-0.453983\pi\)
\(43\)	−40.2013	−	11.8042i	−0.934914	−	0.274515i	−0.221421	−	0.975178i	\(-0.571070\pi\)
							−0.713493	+	0.700663i	\(0.752888\pi\)
\(47\)			74.0870i			1.57632i	0.615471	+	0.788160i	\(0.288966\pi\)
							−0.615471	+	0.788160i	\(0.711034\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.g.a.29.13	yes	160
3.2	odd	2	inner	138.3.g.a.29.4	✓	160
23.4	even	11	inner	138.3.g.a.119.4	yes	160
69.50	odd	22	inner	138.3.g.a.119.13	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.g.a.29.4	✓	160	3.2	odd	2	inner
138.3.g.a.29.13	yes	160	1.1	even	1	trivial
138.3.g.a.119.4	yes	160	23.4	even	11	inner
138.3.g.a.119.13	yes	160	69.50	odd	22	inner