Embedded newform 38.3.f.a.21.3

• Label: 38.3.f.a.21.3
• 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			21.3
Character	\(\chi\)	\(=\)	38.21
Dual form			38.3.f.a.29.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39273 + 0.245576i) q^{2} +(-3.75536 + 4.47547i) q^{3} +(1.87939 + 0.684040i) q^{4} +(5.30025 - 1.92913i) q^{5} +(-6.32926 + 5.31088i) q^{6} +(-0.990297 - 1.71524i) q^{7} +(2.44949 + 1.41421i) q^{8} +(-4.36422 - 24.7507i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{3} + 12 q^{6} - 18 q^{7} + 6 q^{9}+O(q^{10}) \)

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
							\(3\)	−3.75536	+	4.47547i	−1.25179	+	1.49182i	−0.451420	+	0.892312i	\(0.649082\pi\)
−0.800367	+	0.599510i	\(0.795362\pi\)
\(5\)	5.30025	−	1.92913i	1.06005	−	0.385827i	0.247603	−	0.968862i	\(-0.420357\pi\)
							0.812447	+	0.583035i	\(0.198135\pi\)
\(7\)	−0.990297	−	1.71524i	−0.141471	−	0.245035i	0.786580	−	0.617489i	\(-0.211850\pi\)
							−0.928051	+	0.372454i	\(0.878517\pi\)
\(11\)	1.21127	−	2.09797i	0.110115	−	0.190725i	−0.805701	−	0.592322i	\(-0.798211\pi\)
							0.915817	+	0.401597i	\(0.131545\pi\)
\(13\)	1.83964	+	2.19239i	0.141510	+	0.168646i	0.832145	−	0.554559i	\(-0.187113\pi\)
							−0.690634	+	0.723204i	\(0.742668\pi\)
\(17\)	−2.41571	+	13.7002i	−0.142101	+	0.805894i	0.827549	+	0.561394i	\(0.189735\pi\)
							−0.969649	+	0.244500i	\(0.921376\pi\)
\(19\)	8.50511	−	16.9901i	0.447637	−	0.894215i
							\(23\)	−33.6915	−	12.2627i	−1.46485	−	0.533161i	−0.518151	−	0.855289i	\(-0.673380\pi\)
−0.946696	+	0.322128i	\(0.895602\pi\)
\(29\)	−28.0502	+	4.94600i	−0.967247	+	0.170552i	−0.634891	−	0.772602i	\(-0.718955\pi\)
							−0.332357	+	0.943154i	\(0.607844\pi\)
\(31\)	−1.64455	+	0.949484i	−0.0530501	+	0.0306285i	−0.526290	−	0.850305i	\(-0.676418\pi\)
							0.473240	+	0.880933i	\(0.343084\pi\)
\(37\)		−	15.7293i		−	0.425116i	−0.977148	−	0.212558i	\(-0.931821\pi\)
							0.977148	−	0.212558i	\(-0.0681794\pi\)
\(41\)	−14.3779	+	17.1350i	−0.350682	+	0.417926i	−0.912334	−	0.409447i	\(-0.865722\pi\)
							0.561652	+	0.827374i	\(0.310166\pi\)
\(43\)	30.7652	−	11.1976i	0.715471	−	0.260410i	0.0414690	−	0.999140i	\(-0.486796\pi\)
							0.674002	+	0.738730i	\(0.264574\pi\)
\(47\)	8.75932	+	49.6766i	0.186369	+	1.05695i	0.924184	+	0.381946i	\(0.124746\pi\)
							−0.737816	+	0.675002i	\(0.764143\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.3.f.a.21.3	✓	24
3.2	odd	2		342.3.z.b.325.1		24
4.3	odd	2		304.3.z.c.97.4		24
19.3	odd	18		722.3.b.f.721.1		24
19.10	odd	18	inner	38.3.f.a.29.3	yes	24
19.16	even	9		722.3.b.f.721.24		24
57.29	even	18		342.3.z.b.181.1		24
76.67	even	18		304.3.z.c.257.4		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.f.a.21.3	✓	24	1.1	even	1	trivial
38.3.f.a.29.3	yes	24	19.10	odd	18	inner
304.3.z.c.97.4		24	4.3	odd	2
304.3.z.c.257.4		24	76.67	even	18
342.3.z.b.181.1		24	57.29	even	18
342.3.z.b.325.1		24	3.2	odd	2
722.3.b.f.721.1		24	19.3	odd	18
722.3.b.f.721.24		24	19.16	even	9