Embedded newform 138.3.g.a.29.4

• Label: 138.3.g.a.29.4
• 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			29.4
Character	\(\chi\)	\(=\)	138.29
Dual form			138.3.g.a.119.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28641 - 0.587486i) q^{2} +(-0.671287 + 2.92393i) q^{3} +(1.30972 + 1.51150i) q^{4} +(3.78542 + 5.89022i) q^{5} +(2.58132 - 3.36701i) q^{6} +(0.377062 - 2.62252i) q^{7} +(-0.796860 - 2.71386i) q^{8} +(-8.09875 - 3.92560i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} + 32 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.671287	+	2.92393i	−0.223762	+	0.974644i
\(5\)	3.78542	+	5.89022i	0.757083	+	1.17804i								0.979175	+	0.203019i	\(0.0650752\pi\)
							−0.222092	+	0.975026i	\(0.571288\pi\)
\(7\)	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	−0.825672	−	0.564150i	\(-0.809204\pi\)
							−0.114344	+	0.993441i	\(0.536477\pi\)
\(13\)	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	0.688063	−	0.725650i	\(-0.258461\pi\)
							−0.620343	+	0.784331i	\(0.713007\pi\)
\(19\)	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	0.280648	−	0.959811i	\(-0.409451\pi\)
−0.910101	+	0.414387i	\(0.863996\pi\)
\(31\)	−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	\(0.0903471\pi\)
							−0.144062	+	0.989569i	\(0.546017\pi\)
\(43\)	−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	\(-0.711034\pi\)
							0.615471	−	0.788160i	\(-0.288966\pi\)

(See \(a_n\) instead)

Twists

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

    

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