Embedded newform 38.3.f.a.33.3

• Label: 38.3.f.a.33.3
• Level: $38$
• Weight: $3$
• Character: 38.33
• 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			33.3
Character	\(\chi\)	\(=\)	38.33
Dual form			38.3.f.a.15.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.483690 + 1.32893i) q^{2} +(-0.173101 - 0.0305223i) q^{3} +(-1.53209 + 1.28558i) q^{4} +(6.04513 + 5.07246i) q^{5} +(-0.0431650 - 0.244801i) q^{6} +(-3.82954 - 6.63297i) q^{7} +(-2.44949 - 1.41421i) q^{8} +(-8.42820 - 3.06761i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{3} + 12 q^{6} - 18 q^{7} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{7}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.483690	+	1.32893i	0.241845	+	0.664463i
							\(3\)	−0.173101	−	0.0305223i	−0.0577002	−	0.0101741i	0.144724	−	0.989472i	\(-0.453771\pi\)
−0.202424	+	0.979298i	\(0.564882\pi\)
\(5\)	6.04513	+	5.07246i	1.20903	+	1.01449i	0.999325	+	0.0367272i	\(0.0116933\pi\)
							0.209700	+	0.977766i	\(0.432751\pi\)
\(7\)	−3.82954	−	6.63297i	−0.547078	−	0.947566i	−0.998473	−	0.0552423i	\(-0.982407\pi\)
							0.451395	−	0.892324i	\(-0.350926\pi\)
\(11\)	7.09172	−	12.2832i	0.644702	−	1.11666i	−0.339668	−	0.940545i	\(-0.610315\pi\)
							0.984370	−	0.176111i	\(-0.0563518\pi\)
\(13\)	6.69619	−	1.18072i	0.515092	−	0.0908246i	0.0899409	−	0.995947i	\(-0.471332\pi\)
							0.425151	+	0.905123i	\(0.360221\pi\)
\(17\)	−16.4686	+	5.99408i	−0.968741	+	0.352593i	−0.777453	−	0.628941i	\(-0.783489\pi\)
							−0.191288	+	0.981534i	\(0.561266\pi\)
\(19\)	−4.02423	+	18.5689i	−0.211802	+	0.977313i
							\(23\)	4.48261	−	3.76136i	0.194896	−	0.163537i	−0.540117	−	0.841590i	\(-0.681620\pi\)
0.735014	+	0.678052i	\(0.237176\pi\)
\(29\)	3.32345	−	9.13111i	0.114602	−	0.314866i	−0.869110	−	0.494619i	\(-0.835308\pi\)
							0.983712	+	0.179753i	\(0.0575299\pi\)
\(31\)	−11.0964	+	6.40652i	−0.357949	+	0.206662i	−0.668181	−	0.743999i	\(-0.732927\pi\)
							0.310232	+	0.950661i	\(0.399593\pi\)
\(37\)			64.6675i			1.74777i	0.486133	+	0.873885i	\(0.338407\pi\)
							−0.486133	+	0.873885i	\(0.661593\pi\)
\(41\)	60.3857	+	10.6476i	1.47282	+	0.259698i	0.851706	−	0.524020i	\(-0.175568\pi\)
							0.621116	+	0.783718i	\(0.286679\pi\)
\(43\)	−42.7426	−	35.8653i	−0.994014	−	0.834077i	−0.00787020	−	0.999969i	\(-0.502505\pi\)
							−0.986144	+	0.165892i	\(0.946950\pi\)
\(47\)	−45.2870	−	16.4831i	−0.963554	−	0.350705i	−0.188128	−	0.982144i	\(-0.560242\pi\)
							−0.775425	+	0.631440i	\(0.782464\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.3.f.a.33.3	yes	24
3.2	odd	2		342.3.z.b.109.1		24
4.3	odd	2		304.3.z.c.33.3		24
19.2	odd	18		722.3.b.f.721.19		24
19.15	odd	18	inner	38.3.f.a.15.3	✓	24
19.17	even	9		722.3.b.f.721.6		24
57.53	even	18		342.3.z.b.91.1		24
76.15	even	18		304.3.z.c.129.3		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.f.a.15.3	✓	24	19.15	odd	18	inner
38.3.f.a.33.3	yes	24	1.1	even	1	trivial
304.3.z.c.33.3		24	4.3	odd	2
304.3.z.c.129.3		24	76.15	even	18
342.3.z.b.91.1		24	57.53	even	18
342.3.z.b.109.1		24	3.2	odd	2
722.3.b.f.721.6		24	19.17	even	9
722.3.b.f.721.19		24	19.2	odd	18