Embedded newform 38.6.e.b.9.2

• Label: 38.6.e.b.9.2
• Level: $38$
• Weight: $6$
• Character: 38.9
• Analytic conductor: $6.095$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.09458515289\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.2
Character	\(\chi\)	\(=\)	38.9
Dual form			38.6.e.b.17.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.694593 + 3.93923i) q^{2} +(-8.20450 + 6.88439i) q^{3} +(-15.0351 + 5.47232i) q^{4} +(-55.8231 - 20.3180i) q^{5} +(-32.8180 - 27.5376i) q^{6} +(115.217 - 199.562i) q^{7} +(-32.0000 - 55.4256i) q^{8} +(-22.2776 + 126.342i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 15 q^{3} + 60 q^{6} - 84 q^{7} - 960 q^{8} - 345 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.694593	+	3.93923i	0.122788	+	0.696364i
							\(3\)	−8.20450	+	6.88439i	−0.526319	+	0.441634i	−0.866828	−	0.498607i	\(-0.833845\pi\)
0.340509	+	0.940241i	\(0.389401\pi\)
\(5\)	−55.8231	−	20.3180i	−0.998595	−	0.363459i	−0.209552	−	0.977797i	\(-0.567201\pi\)
							−0.789042	+	0.614339i	\(0.789423\pi\)
\(7\)	115.217	−	199.562i	0.888734	−	1.53933i	0.0473598	−	0.998878i	\(-0.484919\pi\)
							0.841374	−	0.540454i	\(-0.181747\pi\)
\(11\)	28.7379	+	49.7756i	0.0716100	+	0.124032i	0.899607	−	0.436700i	\(-0.143853\pi\)
							−0.827997	+	0.560732i	\(0.810520\pi\)
\(13\)	−825.682	−	692.829i	−1.35505	−	1.13702i	−0.977478	−	0.211038i	\(-0.932316\pi\)
							−0.377569	−	0.925981i	\(-0.623240\pi\)
\(17\)	−149.904	−	850.150i	−0.125803	−	0.713466i	−0.980828	−	0.194878i	\(-0.937569\pi\)
							0.855024	−	0.518588i	\(-0.173542\pi\)
\(19\)	−1440.26	+	633.833i	−0.915287	+	0.402801i
							\(23\)	−1497.24	+	544.952i	−0.590164	+	0.214802i	−0.619801	−	0.784759i	\(-0.712787\pi\)
0.0296377	+	0.999561i	\(0.490565\pi\)
\(29\)	−1256.82	+	7127.77i	−0.277509	+	1.57383i	0.453367	+	0.891324i	\(0.350223\pi\)
							−0.730876	+	0.682510i	\(0.760888\pi\)
\(31\)	−601.842	+	1042.42i	−0.112481	+	0.194823i	−0.916770	−	0.399416i	\(-0.869213\pi\)
							0.804289	+	0.594238i	\(0.202546\pi\)
\(37\)	14445.2			1.73468			0.867342	−	0.497713i	\(-0.165827\pi\)
							0.867342	+	0.497713i	\(0.165827\pi\)
\(41\)	11228.6	−	9421.90i	1.04319	−	0.875344i	0.0508333	−	0.998707i	\(-0.483812\pi\)
							0.992362	+	0.123363i	\(0.0393678\pi\)
\(43\)	−16318.5	−	5939.44i	−1.34589	−	0.489863i	−0.434226	−	0.900804i	\(-0.642978\pi\)
							−0.911662	+	0.410941i	\(0.865200\pi\)
\(47\)	−785.085	+	4452.44i	−0.0518409	+	0.294004i	−0.999695	−	0.0247007i	\(-0.992137\pi\)
							0.947854	+	0.318705i	\(0.103248\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.b.9.2	✓	30
19.6	even	9		722.6.a.r.1.5		15
19.13	odd	18		722.6.a.q.1.11		15
19.17	even	9	inner	38.6.e.b.17.2	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.b.9.2	✓	30	1.1	even	1	trivial
38.6.e.b.17.2	yes	30	19.17	even	9	inner
722.6.a.q.1.11		15	19.13	odd	18
722.6.a.r.1.5		15	19.6	even	9