Embedded newform 38.6.e.b.35.4

• Label: 38.6.e.b.35.4
• Level: $38$
• Weight: $6$
• Character: 38.35
• 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			35.4
Character	\(\chi\)	\(=\)	38.35
Dual form			38.6.e.b.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.06418 + 2.57115i) q^{2} +(18.6743 - 6.79689i) q^{3} +(2.77837 + 15.7569i) q^{4} +(18.3430 - 104.028i) q^{5} +(74.6972 + 27.1876i) q^{6} +(-18.2394 - 31.5915i) q^{7} +(-32.0000 + 55.4256i) q^{8} +(116.383 - 97.6570i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 15 q^{3} + 60 q^{6} - 84 q^{7} - 960 q^{8} - 345 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.06418	+	2.57115i	0.541675	+	0.454519i
							\(3\)	18.6743	−	6.79689i	1.19796	−	0.436021i	0.335446	−	0.942059i	\(-0.391113\pi\)
0.862511	+	0.506039i	\(0.168891\pi\)
\(5\)	18.3430	−	104.028i	0.328130	−	1.86092i	−0.158570	−	0.987348i	\(-0.550688\pi\)
							0.486700	−	0.873569i	\(-0.338200\pi\)
\(7\)	−18.2394	−	31.5915i	−0.140691	−	0.243683i	0.787066	−	0.616868i	\(-0.211599\pi\)
							−0.927757	+	0.373185i	\(0.878266\pi\)
\(11\)	−257.270	+	445.604i	−0.641073	+	1.11037i	0.344121	+	0.938925i	\(0.388177\pi\)
							−0.985194	+	0.171445i	\(0.945156\pi\)
\(13\)	295.437	+	107.530i	0.484848	+	0.176470i	0.572867	−	0.819648i	\(-0.305831\pi\)
							−0.0880184	+	0.996119i	\(0.528053\pi\)
\(17\)	1751.37	+	1469.57i	1.46979	+	1.23330i	0.916330	+	0.400424i	\(0.131137\pi\)
							0.553460	+	0.832876i	\(0.313307\pi\)
\(19\)	−251.836	+	1553.28i	−0.160042	+	0.987110i
							\(23\)	−350.475	−	1987.64i	−0.138146	−	0.783463i	−0.972617	−	0.232412i	\(-0.925338\pi\)
0.834472	−	0.551051i	\(-0.185773\pi\)
\(29\)	−854.251	+	716.802i	−0.188621	+	0.158272i	−0.732208	−	0.681081i	\(-0.761510\pi\)
							0.543587	+	0.839353i	\(0.317066\pi\)
\(31\)	−2588.91	−	4484.13i	−0.483853	−	0.838058i	0.515975	−	0.856604i	\(-0.327430\pi\)
							−0.999828	+	0.0185459i	\(0.994096\pi\)
\(37\)	4294.05			0.515659			0.257829	−	0.966190i	\(-0.416993\pi\)
							0.257829	+	0.966190i	\(0.416993\pi\)
\(41\)	1997.91	−	727.181i	0.185617	−	0.0675589i	−0.247540	−	0.968878i	\(-0.579622\pi\)
							0.433156	+	0.901319i	\(0.357400\pi\)
\(43\)	1179.52	−	6689.41i	0.0972826	−	0.551717i	−0.896741	−	0.442555i	\(-0.854072\pi\)
							0.994024	−	0.109162i	\(-0.0348168\pi\)
\(47\)	−2009.91	+	1686.51i	−0.132718	+	0.111364i	−0.706730	−	0.707483i	\(-0.749831\pi\)
							0.574012	+	0.818847i	\(0.305386\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.b.35.4	yes	30
19.5	even	9		722.6.a.r.1.4		15
19.6	even	9	inner	38.6.e.b.25.4	✓	30
19.14	odd	18		722.6.a.q.1.12		15

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.b.25.4	✓	30	19.6	even	9	inner
38.6.e.b.35.4	yes	30	1.1	even	1	trivial
722.6.a.q.1.12		15	19.14	odd	18
722.6.a.r.1.4		15	19.5	even	9