Embedded newform 38.6.e.b.17.3

• Label: 38.6.e.b.17.3
• 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			17.3
Character	\(\chi\)	\(=\)	38.17
Dual form			38.6.e.b.9.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.694593 - 3.93923i) q^{2} +(0.893216 + 0.749497i) q^{3} +(-15.0351 - 5.47232i) q^{4} +(61.2040 - 22.2764i) q^{5} +(3.57286 - 2.99799i) q^{6} +(-47.2621 - 81.8604i) q^{7} +(-32.0000 + 55.4256i) q^{8} +(-41.9604 - 237.969i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 15 q^{3} + 60 q^{6} - 84 q^{7} - 960 q^{8} - 345 q^{9}+O(q^{10}) \)

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
							\(3\)	0.893216	+	0.749497i	0.0572998	+	0.0480803i	0.670988	−	0.741468i	\(-0.265870\pi\)
−0.613688	+	0.789549i	\(0.710315\pi\)
\(5\)	61.2040	−	22.2764i	1.09485	−	0.398493i	0.269434	−	0.963019i	\(-0.413163\pi\)
							0.825415	+	0.564526i	\(0.190941\pi\)
\(7\)	−47.2621	−	81.8604i	−0.364559	−	0.631435i	0.624146	−	0.781308i	\(-0.285447\pi\)
							−0.988705	+	0.149872i	\(0.952114\pi\)
\(11\)	229.131	−	396.867i	0.570955	−	0.988924i	−0.425513	−	0.904952i	\(-0.639906\pi\)
							0.996468	−	0.0839713i	\(-0.0267604\pi\)
\(13\)	−754.074	+	632.744i	−1.23753	+	1.03841i	−0.239817	+	0.970818i	\(0.577088\pi\)
							−0.997713	+	0.0675931i	\(0.978468\pi\)
\(17\)	368.946	−	2092.40i	0.309628	−	1.75599i	−0.291248	−	0.956647i	\(-0.594071\pi\)
							0.600877	−	0.799342i	\(-0.294818\pi\)
\(19\)	1046.65	+	1175.00i	0.665145	+	0.746715i
							\(23\)	3613.00	+	1315.02i	1.42412	+	0.518339i	0.935241	−	0.354010i	\(-0.115182\pi\)
0.488883	+	0.872349i	\(0.337404\pi\)
\(29\)	1273.95	+	7224.92i	0.281292	+	1.59528i	0.718239	+	0.695797i	\(0.244949\pi\)
							−0.436947	+	0.899487i	\(0.643940\pi\)
\(31\)	588.529	+	1019.36i	0.109993	+	0.190513i	0.915767	−	0.401710i	\(-0.131584\pi\)
							−0.805774	+	0.592223i	\(0.798251\pi\)
\(37\)	1760.46			0.211409			0.105704	−	0.994398i	\(-0.466290\pi\)
							0.105704	+	0.994398i	\(0.466290\pi\)
\(41\)	1877.07	+	1575.05i	0.174389	+	0.146330i	0.725805	−	0.687901i	\(-0.241468\pi\)
							−0.551415	+	0.834231i	\(0.685912\pi\)
\(43\)	−1997.13	+	726.896i	−0.164716	+	0.0599516i	−0.423062	−	0.906101i	\(-0.639045\pi\)
							0.258346	+	0.966052i	\(0.416822\pi\)
\(47\)	2211.05	+	12539.5i	0.146001	+	0.828010i	0.966558	+	0.256447i	\(0.0825519\pi\)
							−0.820558	+	0.571564i	\(0.806337\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.b.17.3	yes	30
19.3	odd	18		722.6.a.q.1.8		15
19.9	even	9	inner	38.6.e.b.9.3	✓	30
19.16	even	9		722.6.a.r.1.8		15

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.b.9.3	✓	30	19.9	even	9	inner
38.6.e.b.17.3	yes	30	1.1	even	1	trivial
722.6.a.q.1.8		15	19.3	odd	18
722.6.a.r.1.8		15	19.16	even	9