Embedded newform 38.9.f.a.3.13

• Label: 38.9.f.a.3.13
• Level: $38$
• Weight: $9$
• Character: 38.3
• Analytic conductor: $15.480$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.4803871823\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(14\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.13
Character	\(\chi\)	\(=\)	38.3
Dual form			38.9.f.a.13.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.27231 - 8.66680i) q^{2} +(31.3279 + 86.0726i) q^{3} +(-22.2270 - 126.055i) q^{4} +(-59.6685 + 338.397i) q^{5} +(973.800 + 354.434i) q^{6} +(321.792 + 557.360i) q^{7} +(-1254.14 - 724.077i) q^{8} +(-1401.04 + 1175.61i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 168 q^{3} - 2688 q^{6} - 6570 q^{7} - 2856 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.27231	−	8.66680i	0.454519	−	0.541675i
							\(3\)	31.3279	+	86.0726i	0.386764	+	1.06262i	0.968449	+	0.249211i	\(0.0801713\pi\)
−0.581685	+	0.813414i	\(0.697607\pi\)
\(5\)	−59.6685	+	338.397i	−0.0954696	+	0.541435i	0.899133	+	0.437676i	\(0.144198\pi\)
							−0.994602	+	0.103759i	\(0.966913\pi\)
\(7\)	321.792	+	557.360i	0.134024	+	0.232137i	0.925224	−	0.379421i	\(-0.123877\pi\)
							−0.791200	+	0.611557i	\(0.790543\pi\)
\(11\)	−5147.62	+	8915.93i	−0.351589	+	0.608970i	−0.986528	−	0.163592i	\(-0.947692\pi\)
							0.634939	+	0.772562i	\(0.281025\pi\)
\(13\)	−15403.8	+	42321.7i	−0.539331	+	1.48180i	0.308339	+	0.951276i	\(0.400227\pi\)
							−0.847670	+	0.530523i	\(0.821995\pi\)
\(17\)	−13983.5	−	11733.6i	−0.167425	−	0.140487i	0.555225	−	0.831700i	\(-0.312632\pi\)
							−0.722651	+	0.691213i	\(0.757076\pi\)
\(19\)	−130006.	+	9055.84i	−0.997583	+	0.0694887i
							\(23\)	58830.6	+	333645.i	0.210229	+	1.19227i	0.888997	+	0.457913i	\(0.151403\pi\)
−0.678768	+	0.734352i	\(0.737486\pi\)
\(29\)	33301.0	+	39686.5i	0.0470831	+	0.0561114i	0.789072	−	0.614301i	\(-0.210562\pi\)
							−0.741989	+	0.670412i	\(0.766117\pi\)
\(31\)	391605.	−	226094.i	0.424035	−	0.244817i	−0.272767	−	0.962080i	\(-0.587939\pi\)
							0.696802	+	0.717263i	\(0.254605\pi\)
\(37\)			203893.i			0.108791i	0.998519	+	0.0543957i	\(0.0173232\pi\)
							−0.998519	+	0.0543957i	\(0.982677\pi\)
\(41\)	−971938.	−	2.67038e6i	−0.343956	−	0.945012i	−0.984234	−	0.176869i	\(-0.943403\pi\)
							0.640278	−	0.768143i	\(-0.278819\pi\)
\(43\)	280409.	−	1.59028e6i	0.0820198	−	0.465157i	−0.915940	−	0.401315i	\(-0.868553\pi\)
							0.997960	−	0.0638425i	\(-0.0203355\pi\)
\(47\)	3.21073e6	−	2.69412e6i	0.657979	−	0.552110i	−0.251501	−	0.967857i	\(-0.580924\pi\)
							0.909480	+	0.415747i	\(0.136480\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.f.a.3.13	✓	84
19.13	odd	18	inner	38.9.f.a.13.13	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.f.a.3.13	✓	84	1.1	even	1	trivial
38.9.f.a.13.13	yes	84	19.13	odd	18	inner