Embedded newform 38.3.f.a.21.1

• Label: 38.3.f.a.21.1
• 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			21.1
Character	\(\chi\)	\(=\)	38.21
Dual form			38.3.f.a.29.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39273 - 0.245576i) q^{2} +(-2.33610 + 2.78405i) q^{3} +(1.87939 + 0.684040i) q^{4} +(-7.04907 + 2.56565i) q^{5} +(3.93724 - 3.30374i) q^{6} +(3.79852 + 6.57923i) q^{7} +(-2.44949 - 1.41421i) q^{8} +(-0.730761 - 4.14435i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{3} + 12 q^{6} - 18 q^{7} + 6 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.33610	+	2.78405i	−0.778699	+	0.928017i	−0.998874	−	0.0474433i	\(-0.984893\pi\)
0.220175	+	0.975460i	\(0.429337\pi\)
\(5\)	−7.04907	+	2.56565i	−1.40981	+	0.513130i	−0.931075	−	0.364827i	\(-0.881128\pi\)
							−0.478739	+	0.877957i	\(0.658906\pi\)
\(7\)	3.79852	+	6.57923i	0.542646	+	0.939890i	0.998751	+	0.0499639i	\(0.0159106\pi\)
							−0.456105	+	0.889926i	\(0.650756\pi\)
\(11\)	6.18267	−	10.7087i	0.562061	−	0.973518i	−0.435256	−	0.900307i	\(-0.643342\pi\)
							0.997316	−	0.0732111i	\(-0.0233247\pi\)
\(13\)	−3.05251	−	3.63784i	−0.234808	−	0.279833i	0.635754	−	0.771892i	\(-0.280689\pi\)
							−0.870562	+	0.492058i	\(0.836245\pi\)
\(17\)	−4.55745	+	25.8466i	−0.268085	+	1.52039i	0.492018	+	0.870585i	\(0.336259\pi\)
							−0.760104	+	0.649802i	\(0.774852\pi\)
\(19\)	10.1161	+	16.0830i	0.532428	+	0.846475i
							\(23\)	26.8127	+	9.75904i	1.16577	+	0.424306i	0.851156	−	0.524913i	\(-0.175902\pi\)
0.314616	+	0.949219i	\(0.398124\pi\)
\(29\)	−30.4485	+	5.36890i	−1.04995	+	0.185134i	−0.671893	−	0.740648i	\(-0.734518\pi\)
							−0.378056	+	0.925783i	\(0.623407\pi\)
\(31\)	−4.94060	+	2.85246i	−0.159374	+	0.0920148i	−0.577566	−	0.816344i	\(-0.695997\pi\)
							0.418192	+	0.908359i	\(0.362664\pi\)
\(37\)			2.36636i			0.0639556i	0.999489	+	0.0319778i	\(0.0101806\pi\)
							−0.999489	+	0.0319778i	\(0.989819\pi\)
\(41\)	44.0557	−	52.5036i	1.07453	−	1.28058i	0.116723	−	0.993164i	\(-0.462761\pi\)
							0.957807	−	0.287411i	\(-0.0927946\pi\)
\(43\)	37.3317	−	13.5876i	0.868179	−	0.315991i	0.130749	−	0.991415i	\(-0.458262\pi\)
							0.737429	+	0.675424i	\(0.236039\pi\)
\(47\)	6.27100	+	35.5646i	0.133425	+	0.756693i	0.975943	+	0.218025i	\(0.0699614\pi\)
							−0.842518	+	0.538669i	\(0.818928\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.3.f.a.21.1	✓	24
3.2	odd	2		342.3.z.b.325.4		24
4.3	odd	2		304.3.z.c.97.3		24
19.3	odd	18		722.3.b.f.721.15		24
19.10	odd	18	inner	38.3.f.a.29.1	yes	24
19.16	even	9		722.3.b.f.721.10		24
57.29	even	18		342.3.z.b.181.4		24
76.67	even	18		304.3.z.c.257.3		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.f.a.21.1	✓	24	1.1	even	1	trivial
38.3.f.a.29.1	yes	24	19.10	odd	18	inner
304.3.z.c.97.3		24	4.3	odd	2
304.3.z.c.257.3		24	76.67	even	18
342.3.z.b.181.4		24	57.29	even	18
342.3.z.b.325.4		24	3.2	odd	2
722.3.b.f.721.10		24	19.16	even	9
722.3.b.f.721.15		24	19.3	odd	18