Embedded newform 38.9.f.a.3.12

• Label: 38.9.f.a.3.12
• Level: $38$
• Weight: $9$
• Character: 38.3
• Analytic conductor: $15.480$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.4803871823\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(14\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.12
Character	\(\chi\)	\(=\)	38.3
Dual form			38.9.f.a.13.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.27231 - 8.66680i) q^{2} +(11.8688 + 32.6091i) q^{3} +(-22.2270 - 126.055i) q^{4} +(32.5420 - 184.555i) q^{5} +(368.930 + 134.280i) q^{6} +(1327.43 + 2299.17i) q^{7} +(-1254.14 - 724.077i) q^{8} +(4103.53 - 3443.27i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 168 q^{3} - 2688 q^{6} - 6570 q^{7} - 2856 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.27231	−	8.66680i	0.454519	−	0.541675i
							\(3\)	11.8688	+	32.6091i	0.146528	+	0.402582i	0.991144	−	0.132790i	\(-0.0423936\pi\)
−0.844616	+	0.535372i	\(0.820171\pi\)
\(5\)	32.5420	−	184.555i	0.0520672	−	0.295288i	−0.947644	−	0.319329i	\(-0.896542\pi\)
							0.999711	+	0.0240417i	\(0.00765345\pi\)
\(7\)	1327.43	+	2299.17i	0.552865	+	0.957590i	0.998066	+	0.0621598i	\(0.0197988\pi\)
							−0.445201	+	0.895431i	\(0.646868\pi\)
\(11\)	2001.03	−	3465.88i	0.136673	−	0.236725i	−0.789562	−	0.613670i	\(-0.789692\pi\)
							0.926235	+	0.376946i	\(0.123026\pi\)
\(13\)	7958.91	−	21866.9i	0.278663	−	0.765622i	−0.718851	−	0.695164i	\(-0.755332\pi\)
							0.997515	−	0.0704577i	\(-0.0224460\pi\)
\(17\)	112553.	+	94443.4i	1.34760	+	1.13077i	0.979601	+	0.200950i	\(0.0644029\pi\)
							0.368003	+	0.929824i	\(0.380042\pi\)
\(19\)	89446.3	−	94778.3i	0.686354	−	0.727268i
							\(23\)	−94957.2	−	538529.i	−0.339326	−	1.92441i	−0.379440	−	0.925216i	\(-0.623883\pi\)
0.0401144	−	0.999195i	\(-0.487228\pi\)
\(29\)	−498882.	−	594544.i	−0.705352	−	0.840606i	0.287769	−	0.957700i	\(-0.407087\pi\)
							−0.993121	+	0.117094i	\(0.962642\pi\)
\(31\)	−318206.	+	183716.i	−0.344557	+	0.198930i	−0.662285	−	0.749252i	\(-0.730413\pi\)
							0.317728	+	0.948182i	\(0.397080\pi\)
\(37\)			1.45754e6i			0.777700i	0.921301	+	0.388850i	\(0.127128\pi\)
							−0.921301	+	0.388850i	\(0.872872\pi\)
\(41\)	1.69562e6	+	4.65867e6i	0.600057	+	1.64864i	0.751156	+	0.660125i	\(0.229497\pi\)
							−0.151098	+	0.988519i	\(0.548281\pi\)
\(43\)	−33640.5	+	190785.i	−0.00983987	+	0.0558046i	−0.989332	−	0.145676i	\(-0.953464\pi\)
							0.979492	+	0.201481i	\(0.0645754\pi\)
\(47\)	−2.92336e6	+	2.45299e6i	−0.599089	+	0.502695i	−0.891153	−	0.453703i	\(-0.850103\pi\)
							0.292064	+	0.956399i	\(0.405658\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.f.a.3.12	✓	84
19.13	odd	18	inner	38.9.f.a.13.12	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.f.a.3.12	✓	84	1.1	even	1	trivial
38.9.f.a.13.12	yes	84	19.13	odd	18	inner