Embedded newform 38.5.f.a.13.2

• Label: 38.5.f.a.13.2
• Level: $38$
• Weight: $5$
• Character: 38.13
• Analytic conductor: $3.928$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.92805859719\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			13.2
Character	\(\chi\)	\(=\)	38.13
Dual form			38.5.f.a.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.81808 - 2.16670i) q^{2} +(-1.13346 + 3.11415i) q^{3} +(-1.38919 + 7.87846i) q^{4} +(-0.544921 - 3.09040i) q^{5} +(8.80815 - 3.20590i) q^{6} +(-41.9404 + 72.6430i) q^{7} +(19.5959 - 11.3137i) q^{8} +(53.6364 + 45.0063i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 12 q^{3} + 48 q^{6} + 90 q^{7} - 84 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.81808	−	2.16670i	−0.454519	−	0.541675i
							\(3\)	−1.13346	+	3.11415i	−0.125940	+	0.346017i	−0.986599	−	0.163165i	\(-0.947830\pi\)
0.860659	+	0.509182i	\(0.170052\pi\)
\(5\)	−0.544921	−	3.09040i	−0.0217968	−	0.123616i	0.971968	−	0.235114i	\(-0.0755463\pi\)
							−0.993765	+	0.111498i	\(0.964435\pi\)
\(7\)	−41.9404	+	72.6430i	−0.855927	+	1.48251i	0.0198550	+	0.999803i	\(0.493680\pi\)
							−0.875782	+	0.482707i	\(0.839654\pi\)
\(11\)	89.6226	+	155.231i	0.740683	+	1.28290i	0.952185	+	0.305523i	\(0.0988311\pi\)
							−0.211502	+	0.977378i	\(0.567836\pi\)
\(13\)	−49.1567	−	135.057i	−0.290868	−	0.799153i	−0.995940	−	0.0900175i	\(-0.971308\pi\)
							0.705072	−	0.709135i	\(-0.250915\pi\)
\(17\)	−278.991	+	234.102i	−0.965368	+	0.810040i	−0.981818	−	0.189825i	\(-0.939208\pi\)
							0.0164500	+	0.999865i	\(0.494764\pi\)
\(19\)	−350.875	−	84.8975i	−0.971953	−	0.235173i
							\(23\)	84.8221	−	481.050i	0.160344	−	0.909357i	−0.793392	−	0.608711i	\(-0.791687\pi\)
0.953736	−	0.300646i	\(-0.0972021\pi\)
\(29\)	−22.2364	+	26.5003i	−0.0264405	+	0.0315105i	−0.779103	−	0.626896i	\(-0.784325\pi\)
							0.752662	+	0.658407i	\(0.228769\pi\)
\(31\)	1155.65	+	667.216i	1.20255	+	0.694294i	0.961122	−	0.276125i	\(-0.0890502\pi\)
							0.241430	+	0.970418i	\(0.422384\pi\)
\(37\)		−	1157.40i		−	0.845434i	−0.906262	−	0.422717i	\(-0.861076\pi\)
							0.906262	−	0.422717i	\(-0.138924\pi\)
\(41\)	−504.497	+	1386.09i	−0.300117	+	0.824565i	0.694362	+	0.719626i	\(0.255687\pi\)
							−0.994479	+	0.104938i	\(0.966535\pi\)
\(43\)	160.766	+	911.748i	0.0869474	+	0.493103i	0.996919	+	0.0784334i	\(0.0249918\pi\)
							−0.909972	+	0.414670i	\(0.863897\pi\)
\(47\)	2338.58	+	1962.31i	1.05866	+	0.888323i	0.993978	−	0.109583i	\(-0.0349515\pi\)
							0.0646843	+	0.997906i	\(0.479396\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.5.f.a.13.2	yes	36
19.3	odd	18	inner	38.5.f.a.3.2	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.f.a.3.2	✓	36	19.3	odd	18	inner
38.5.f.a.13.2	yes	36	1.1	even	1	trivial