Embedded newform 38.3.f.a.21.4

• Label: 38.3.f.a.21.4
• Level: $38$
• Weight: $3$
• Character: 38.21
• Analytic conductor: $1.035$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.03542500457\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			21.4
Character	\(\chi\)	\(=\)	38.21
Dual form			38.3.f.a.29.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39273 + 0.245576i) q^{2} +(1.06027 - 1.26358i) q^{3} +(1.87939 + 0.684040i) q^{4} +(-4.36056 + 1.58711i) q^{5} +(1.78697 - 1.49945i) q^{6} +(-2.18088 - 3.77740i) q^{7} +(2.44949 + 1.41421i) q^{8} +(1.09037 + 6.18380i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{3} + 12 q^{6} - 18 q^{7} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39273	+	0.245576i	0.696364	+	0.122788i
							\(3\)	1.06027	−	1.26358i	0.353423	−	0.421193i	−0.559816	−	0.828617i	\(-0.689128\pi\)
0.913239	+	0.407423i	\(0.133573\pi\)
\(5\)	−4.36056	+	1.58711i	−0.872111	+	0.317422i	−0.739022	−	0.673682i	\(-0.764712\pi\)
							−0.133089	+	0.991104i	\(0.542490\pi\)
\(7\)	−2.18088	−	3.77740i	−0.311555	−	0.539628i	0.667144	−	0.744928i	\(-0.267516\pi\)
							−0.978699	+	0.205300i	\(0.934183\pi\)
\(11\)	−2.86610	+	4.96424i	−0.260555	+	0.451294i	−0.966390	−	0.257082i	\(-0.917239\pi\)
							0.705835	+	0.708377i	\(0.250572\pi\)
\(13\)	−11.9719	−	14.2675i	−0.920913	−	1.09750i	−0.994963	−	0.100246i	\(-0.968037\pi\)
							0.0740501	−	0.997255i	\(-0.476408\pi\)
\(17\)	1.14452	−	6.49087i	0.0673244	−	0.381816i	−0.932464	−	0.361262i	\(-0.882346\pi\)
							0.999789	−	0.0205537i	\(-0.00654290\pi\)
\(19\)	18.9980	+	0.274480i	0.999896	+	0.0144463i
							\(23\)	23.7808	+	8.65552i	1.03395	+	0.376327i	0.802584	−	0.596540i	\(-0.203458\pi\)
0.231366	+	0.972867i	\(0.425680\pi\)
\(29\)	3.49437	−	0.616152i	0.120496	−	0.0212466i	−0.113075	−	0.993586i	\(-0.536070\pi\)
							0.233571	+	0.972340i	\(0.424959\pi\)
\(31\)	25.7126	−	14.8452i	0.829439	−	0.478877i	−0.0242216	−	0.999707i	\(-0.507711\pi\)
							0.853661	+	0.520830i	\(0.174377\pi\)
\(37\)		−	62.3286i		−	1.68456i	−0.539042	−	0.842279i	\(-0.681214\pi\)
							0.539042	−	0.842279i	\(-0.318786\pi\)
\(41\)	−41.6408	+	49.6256i	−1.01563	+	1.21038i	−0.0381677	+	0.999271i	\(0.512152\pi\)
							−0.977462	+	0.211110i	\(0.932292\pi\)
\(43\)	−77.6901	+	28.2769i	−1.80675	+	0.657602i	−0.809205	+	0.587527i	\(0.800102\pi\)
							−0.997542	+	0.0700753i	\(0.977676\pi\)
\(47\)	−2.10783	−	11.9541i	−0.0448475	−	0.254343i	0.954138	−	0.299366i	\(-0.0967750\pi\)
							−0.998986	+	0.0450228i	\(0.985664\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.3.f.a.21.4	✓	24
3.2	odd	2		342.3.z.b.325.2		24
4.3	odd	2		304.3.z.c.97.1		24
19.3	odd	18		722.3.b.f.721.7		24
19.10	odd	18	inner	38.3.f.a.29.4	yes	24
19.16	even	9		722.3.b.f.721.18		24
57.29	even	18		342.3.z.b.181.2		24
76.67	even	18		304.3.z.c.257.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.f.a.21.4	✓	24	1.1	even	1	trivial
38.3.f.a.29.4	yes	24	19.10	odd	18	inner
304.3.z.c.97.1		24	4.3	odd	2
304.3.z.c.257.1		24	76.67	even	18
342.3.z.b.181.2		24	57.29	even	18
342.3.z.b.325.2		24	3.2	odd	2
722.3.b.f.721.7		24	19.3	odd	18
722.3.b.f.721.18		24	19.16	even	9