Embedded newform 138.3.g.a.29.3

• Label: 138.3.g.a.29.3
• 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			29.3
Character	\(\chi\)	\(=\)	138.29
Dual form			138.3.g.a.119.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28641 - 0.587486i) q^{2} +(-0.901649 - 2.86130i) q^{3} +(1.30972 + 1.51150i) q^{4} +(1.85466 + 2.88591i) q^{5} +(-0.521077 + 4.21052i) q^{6} +(-1.93124 + 13.4321i) q^{7} +(-0.796860 - 2.71386i) q^{8} +(-7.37406 + 5.15978i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} + 32 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.901649	−	2.86130i	−0.300550	−	0.953766i
\(5\)	1.85466	+	2.88591i	0.370932	+	0.577182i								0.975670	−	0.219242i	\(-0.0703586\pi\)
							−0.604738	+	0.796424i	\(0.706722\pi\)
\(7\)	−1.93124	+	13.4321i	−0.275892	+	1.91887i	0.105431	+	0.994427i	\(0.466378\pi\)
							−0.381323	+	0.924442i	\(0.624531\pi\)
\(11\)	14.2002	−	6.48500i	1.29092	−	0.589545i	0.352753	−	0.935717i	\(-0.385246\pi\)
							0.938171	+	0.346171i	\(0.112518\pi\)
\(13\)	0.390874	+	2.71859i	0.0300673	+	0.209122i	0.999317	−	0.0369574i	\(-0.0117666\pi\)
							−0.969250	+	0.246080i	\(0.920857\pi\)
\(17\)	2.12790	+	1.84384i	0.125171	+	0.108461i	0.715197	−	0.698923i	\(-0.246337\pi\)
							−0.590026	+	0.807384i	\(0.700882\pi\)
\(19\)	14.7216	+	16.9897i	0.774823	+	0.894193i	0.996724	−	0.0808757i	\(-0.0257717\pi\)
							−0.221901	+	0.975069i	\(0.571226\pi\)
\(23\)	−7.14749	+	21.8612i	−0.310760	+	0.950488i
							\(29\)	−24.4421	−	21.1792i	−0.842832	−	0.730318i	0.122184	−	0.992507i	\(-0.461010\pi\)
−0.965016	+	0.262189i	\(0.915556\pi\)
\(31\)	−8.11856	+	2.38382i	−0.261889	+	0.0768975i	−0.410041	−	0.912067i	\(-0.634486\pi\)
							0.148152	+	0.988965i	\(0.452667\pi\)
\(37\)	59.4748	+	38.2222i	1.60743	+	1.03303i	0.963414	+	0.268017i	\(0.0863685\pi\)
							0.644014	+	0.765014i	\(0.277268\pi\)
\(41\)	−12.8740	−	20.0323i	−0.314000	−	0.488593i	0.648001	−	0.761639i	\(-0.275605\pi\)
							−0.962001	+	0.273046i	\(0.911969\pi\)
\(43\)	7.41890	+	2.17839i	0.172533	+	0.0506601i	0.366858	−	0.930277i	\(-0.380434\pi\)
							−0.194325	+	0.980937i	\(0.562252\pi\)
\(47\)			51.8565i			1.10333i	0.834066	+	0.551665i	\(0.186007\pi\)
							−0.834066	+	0.551665i	\(0.813993\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.g.a.29.3	✓	160
3.2	odd	2	inner	138.3.g.a.29.11	yes	160
23.4	even	11	inner	138.3.g.a.119.11	yes	160
69.50	odd	22	inner	138.3.g.a.119.3	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.g.a.29.3	✓	160	1.1	even	1	trivial
138.3.g.a.29.11	yes	160	3.2	odd	2	inner
138.3.g.a.119.3	yes	160	69.50	odd	22	inner
138.3.g.a.119.11	yes	160	23.4	even	11	inner