Embedded newform 138.5.g.a.29.6

• Label: 138.5.g.a.29.6
• Level: $138$
• Weight: $5$
• Character: 138.29
• Analytic conductor: $14.265$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	138.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.2650549056\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(32\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			29.6
Character	\(\chi\)	\(=\)	138.29
Dual form			138.5.g.a.119.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57283 - 1.17497i) q^{2} +(-4.43333 + 7.83234i) q^{3} +(5.23889 + 6.04600i) q^{4} +(-6.99614 - 10.8862i) q^{5} +(20.6090 - 14.9422i) q^{6} +(-12.6718 + 88.1344i) q^{7} +(-6.37488 - 21.7108i) q^{8} +(-41.6912 - 69.4467i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 20 q^{3} + 256 q^{4} - 64 q^{6} + 172 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.57283	−	1.17497i	−0.643207	−	0.293743i
							\(3\)	−4.43333	+	7.83234i	−0.492592	+	0.870260i
\(5\)	−6.99614	−	10.8862i	−0.279846	−	0.435448i								0.672668	−	0.739945i	\(-0.265148\pi\)
							−0.952514	+	0.304496i	\(0.901512\pi\)
\(7\)	−12.6718	+	88.1344i	−0.258608	+	1.79866i	0.284159	+	0.958777i	\(0.408286\pi\)
							−0.542767	+	0.839883i	\(0.682623\pi\)
\(11\)	−7.93476	+	3.62368i	−0.0655765	+	0.0299478i	−0.447933	−	0.894067i	\(-0.647840\pi\)
							0.382357	+	0.924015i	\(0.375113\pi\)
\(13\)	27.6059	+	192.003i	0.163348	+	1.13611i	0.892266	+	0.451511i	\(0.149115\pi\)
							−0.728917	+	0.684602i	\(0.759976\pi\)
\(17\)	55.3779	+	47.9852i	0.191619	+	0.166039i	0.745387	−	0.666633i	\(-0.232265\pi\)
							−0.553768	+	0.832671i	\(0.686810\pi\)
\(19\)	−21.0080	−	24.2445i	−0.0581939	−	0.0671593i	0.725906	−	0.687794i	\(-0.241421\pi\)
							−0.784100	+	0.620634i	\(0.786875\pi\)
\(23\)	−418.289	−	323.844i	−0.790717	−	0.612182i
							\(29\)	−907.895	−	786.695i	−1.07954	−	0.935428i	−0.0814194	−	0.996680i	\(-0.525945\pi\)
−0.998122	+	0.0612516i	\(0.980491\pi\)
\(31\)	236.935	−	69.5703i	0.246550	−	0.0723936i	−0.156123	−	0.987738i	\(-0.549900\pi\)
							0.402673	+	0.915344i	\(0.368081\pi\)
\(37\)	−1966.57	−	1263.84i	−1.43650	−	0.923184i	−0.999721	−	0.0236156i	\(-0.992482\pi\)
							−0.436781	−	0.899568i	\(-0.643881\pi\)
\(41\)	−969.804	−	1509.04i	−0.576921	−	0.897707i	0.423044	−	0.906109i	\(-0.360962\pi\)
							−0.999965	+	0.00840265i	\(0.997325\pi\)
\(43\)	330.824	+	97.1387i	0.178921	+	0.0525358i	0.369966	−	0.929045i	\(-0.379369\pi\)
							−0.191045	+	0.981581i	\(0.561188\pi\)
\(47\)			2431.99i			1.10095i	0.834853	+	0.550473i	\(0.185553\pi\)
							−0.834853	+	0.550473i	\(0.814447\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.g.a.29.6	✓	320
3.2	odd	2	inner	138.5.g.a.29.24	yes	320
23.4	even	11	inner	138.5.g.a.119.24	yes	320
69.50	odd	22	inner	138.5.g.a.119.6	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.g.a.29.6	✓	320	1.1	even	1	trivial
138.5.g.a.29.24	yes	320	3.2	odd	2	inner
138.5.g.a.119.6	yes	320	69.50	odd	22	inner
138.5.g.a.119.24	yes	320	23.4	even	11	inner