Embedded newform 38.6.e.b.17.2

• Label: 38.6.e.b.17.2
• Level: $38$
• Weight: $6$
• Character: 38.17
• 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			17.2
Character	\(\chi\)	\(=\)	38.17
Dual form			38.6.e.b.9.2

$q$-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 \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.b.9.2	✓	30	19.9	even	9	inner
38.6.e.b.17.2	yes	30	1.1	even	1	trivial
722.6.a.q.1.11		15	19.3	odd	18
722.6.a.r.1.5		15	19.16	even	9