Embedded newform 38.10.e.a.9.3

• Label: 38.10.e.a.9.3
• Level: $38$
• Weight: $10$
• Character: 38.9
• Analytic conductor: $19.571$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.5713617742\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.3
Character	\(\chi\)	\(=\)	38.9
Dual form			38.10.e.a.17.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.77837 - 15.7569i) q^{2} +(-69.9035 + 58.6560i) q^{3} +(-240.561 + 87.5572i) q^{4} +(-954.597 - 347.445i) q^{5} +(1118.46 + 938.495i) q^{6} +(-2075.12 + 3594.21i) q^{7} +(2048.00 + 3547.24i) q^{8} +(-1971.95 + 11183.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 252 q^{3} + 4032 q^{6} + 14373 q^{7} + 86016 q^{8} + 12852 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.77837	−	15.7569i	−0.122788	−	0.696364i
							\(3\)	−69.9035	+	58.6560i	−0.498257	+	0.418087i	−0.856974	−	0.515359i	\(-0.827659\pi\)
0.358718	+	0.933446i	\(0.383214\pi\)
\(5\)	−954.597	−	347.445i	−0.683054	−	0.248611i	−0.0228959	−	0.999738i	\(-0.507289\pi\)
							−0.660158	+	0.751126i	\(0.729511\pi\)
\(7\)	−2075.12	+	3594.21i	−0.326664	+	0.565799i	−0.981848	−	0.189671i	\(-0.939258\pi\)
							0.655184	+	0.755470i	\(0.272591\pi\)
\(11\)	−30352.7	−	52572.3i	−0.625071	−	1.08266i	−0.988527	−	0.151044i	\(-0.951737\pi\)
							0.363456	−	0.931611i	\(-0.381597\pi\)
\(13\)	55055.6	+	46197.1i	0.534634	+	0.448611i	0.869698	−	0.493584i	\(-0.164314\pi\)
							−0.335064	+	0.942195i	\(0.608758\pi\)
\(17\)	36320.4	+	205983.i	0.105470	+	0.598152i	0.991031	+	0.133629i	\(0.0426630\pi\)
							−0.885561	+	0.464523i	\(0.846226\pi\)
\(19\)	289267.	−	488889.i	0.509223	−	0.860635i
							\(23\)	1.88544e6	−	686243.i	1.40487	−	0.511331i	0.475251	−	0.879850i	\(-0.342357\pi\)
0.929620	+	0.368519i	\(0.120135\pi\)
\(29\)	980593.	−	5.56122e6i	0.257453	−	1.46009i	−0.532244	−	0.846591i	\(-0.678651\pi\)
							0.789697	−	0.613497i	\(-0.210238\pi\)
\(31\)	−1.46639e6	+	2.53987e6i	−0.285182	+	0.493950i	−0.972653	−	0.232261i	\(-0.925388\pi\)
							0.687471	+	0.726212i	\(0.258721\pi\)
\(37\)	1.21571e6			0.106640			0.0533202	−	0.998577i	\(-0.483020\pi\)
							0.0533202	+	0.998577i	\(0.483020\pi\)
\(41\)	1.56089e7	−	1.30975e7i	0.862673	−	0.723868i	−0.0998695	−	0.995001i	\(-0.531843\pi\)
							0.962542	+	0.271132i	\(0.0873981\pi\)
\(43\)	3.43010e7	+	1.24845e7i	1.53003	+	0.556884i	0.963628	−	0.267248i	\(-0.0861142\pi\)
							0.566398	+	0.824132i	\(0.308336\pi\)
\(47\)	−8.73956e6	+	4.95645e7i	−0.261246	+	1.48160i	0.518271	+	0.855216i	\(0.326576\pi\)
							−0.779517	+	0.626381i	\(0.784535\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.10.e.a.9.3	✓	42
19.17	even	9	inner	38.10.e.a.17.3	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.e.a.9.3	✓	42	1.1	even	1	trivial
38.10.e.a.17.3	yes	42	19.17	even	9	inner