Embedded newform 38.5.f.a.15.5

• Label: 38.5.f.a.15.5
• Level: $38$
• Weight: $5$
• Character: 38.15
• 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			15.5
Character	\(\chi\)	\(=\)	38.15
Dual form			38.5.f.a.33.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.967379 - 2.65785i) q^{2} +(3.41796 - 0.602679i) q^{3} +(-6.12836 - 5.14230i) q^{4} +(10.5708 - 8.86998i) q^{5} +(1.70463 - 9.66745i) q^{6} +(40.2843 - 69.7745i) q^{7} +(-19.5959 + 11.3137i) q^{8} +(-64.7959 + 23.5838i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 12 q^{3} + 48 q^{6} + 90 q^{7} - 84 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.967379	−	2.65785i	0.241845	−	0.664463i
							\(3\)	3.41796	−	0.602679i	0.379773	−	0.0669643i	0.0194970	−	0.999810i	\(-0.493794\pi\)
0.360276	+	0.932846i	\(0.382682\pi\)
\(5\)	10.5708	−	8.86998i	0.422833	−	0.354799i	−0.406406	−	0.913692i	\(-0.633218\pi\)
							0.829240	+	0.558893i	\(0.188774\pi\)
\(7\)	40.2843	−	69.7745i	0.822129	−	1.42397i	−0.0819647	−	0.996635i	\(-0.526119\pi\)
							0.904094	−	0.427334i	\(-0.140547\pi\)
\(11\)	−1.61150	−	2.79119i	−0.0133181	−	0.0230677i	0.859290	−	0.511490i	\(-0.170906\pi\)
							−0.872608	+	0.488422i	\(0.837573\pi\)
\(13\)	122.860	+	21.6635i	0.726981	+	0.128186i	0.524877	−	0.851178i	\(-0.324111\pi\)
							0.202104	+	0.979364i	\(0.435222\pi\)
\(17\)	108.347	+	39.4352i	0.374904	+	0.136454i	0.522598	−	0.852579i	\(-0.324963\pi\)
							−0.147694	+	0.989033i	\(0.547185\pi\)
\(19\)	−118.180	+	341.108i	−0.327369	+	0.944897i
							\(23\)	773.335	+	648.905i	1.46188	+	1.22666i	0.923294	+	0.384094i	\(0.125486\pi\)
0.538587	+	0.842570i	\(0.318958\pi\)
\(29\)	−53.8463	−	147.942i	−0.0640265	−	0.175911i	0.903554	−	0.428474i	\(-0.140949\pi\)
							−0.967580	+	0.252563i	\(0.918727\pi\)
\(31\)	−547.381	−	316.030i	−0.569595	−	0.328856i	0.187393	−	0.982285i	\(-0.439996\pi\)
							−0.756987	+	0.653429i	\(0.773330\pi\)
\(37\)		−	1482.75i		−	1.08309i	−0.840671	−	0.541546i	\(-0.817839\pi\)
							0.840671	−	0.541546i	\(-0.182161\pi\)
\(41\)	−1339.58	+	236.204i	−0.796895	+	0.140514i	−0.557248	−	0.830346i	\(-0.688143\pi\)
							−0.239647	+	0.970860i	\(0.577032\pi\)
\(43\)	−96.9537	+	81.3538i	−0.0524357	+	0.0439988i	−0.668629	−	0.743596i	\(-0.733119\pi\)
							0.616193	+	0.787595i	\(0.288674\pi\)
\(47\)	1998.22	−	727.292i	0.904580	−	0.329240i	0.152494	−	0.988304i	\(-0.451270\pi\)
							0.752087	+	0.659064i	\(0.229047\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.5.f.a.15.5	✓	36
19.14	odd	18	inner	38.5.f.a.33.5	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.f.a.15.5	✓	36	1.1	even	1	trivial
38.5.f.a.33.5	yes	36	19.14	odd	18	inner