Embedded newform 38.10.e.a.9.5

• Label: 38.10.e.a.9.5
• Level: $38$
• Weight: $10$
• Character: 38.9
• Analytic conductor: $19.571$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.5713617742\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.5
Character	\(\chi\)	\(=\)	38.9
Dual form			38.10.e.a.17.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.77837 - 15.7569i) q^{2} +(74.8003 - 62.7649i) q^{3} +(-240.561 + 87.5572i) q^{4} +(1601.32 + 582.833i) q^{5} +(-1196.81 - 1004.24i) q^{6} +(2055.81 - 3560.77i) q^{7} +(2048.00 + 3547.24i) q^{8} +(-1762.26 + 9994.30i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 252 q^{3} + 4032 q^{6} + 14373 q^{7} + 86016 q^{8} + 12852 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.77837	−	15.7569i	−0.122788	−	0.696364i
							\(3\)	74.8003	−	62.7649i	0.533160	−	0.447375i	−0.336031	−	0.941851i	\(-0.609085\pi\)
0.869191	+	0.494476i	\(0.164640\pi\)
\(5\)	1601.32	+	582.833i	1.14581	+	0.417041i	0.844009	−	0.536330i	\(-0.180190\pi\)
							0.301802	+	0.953371i	\(0.402412\pi\)
\(7\)	2055.81	−	3560.77i	0.323625	−	0.560535i	−0.657608	−	0.753360i	\(-0.728432\pi\)
							0.981233	+	0.192825i	\(0.0617651\pi\)
\(11\)	29167.3	+	50519.2i	0.600660	+	1.04037i	0.992721	+	0.120434i	\(0.0384287\pi\)
							−0.392061	+	0.919939i	\(0.628238\pi\)
\(13\)	85526.6	+	71765.4i	0.830532	+	0.696899i	0.955413	−	0.295273i	\(-0.0954106\pi\)
							−0.124881	+	0.992172i	\(0.539855\pi\)
\(17\)	51435.9	+	291707.i	0.149364	+	0.847085i	0.963759	+	0.266774i	\(0.0859578\pi\)
							−0.814395	+	0.580311i	\(0.802931\pi\)
\(19\)	72881.4	−	563361.i	0.128300	−	0.991735i
							\(23\)	−589167.	+	214439.i	−0.438999	+	0.159783i	−0.552058	−	0.833806i	\(-0.686157\pi\)
0.113059	+	0.993588i	\(0.463935\pi\)
\(29\)	982446.	−	5.57173e6i	0.257939	−	1.46285i	−0.530474	−	0.847701i	\(-0.677986\pi\)
							0.788413	−	0.615146i	\(-0.210903\pi\)
\(31\)	4.95106e6	−	8.57549e6i	0.962877	−	1.66775i	0.247662	−	0.968847i	\(-0.420338\pi\)
							0.715215	−	0.698905i	\(-0.246329\pi\)
\(37\)	4.69775e6			0.412081			0.206040	−	0.978544i	\(-0.433942\pi\)
							0.206040	+	0.978544i	\(0.433942\pi\)
\(41\)	−2.20058e7	+	1.84651e7i	−1.21621	+	1.02052i	−0.217199	+	0.976127i	\(0.569692\pi\)
							−0.999014	+	0.0443965i	\(0.985864\pi\)
\(43\)	924985.	+	336667.i	0.0412597	+	0.0150173i	0.362568	−	0.931957i	\(-0.381900\pi\)
							−0.321308	+	0.946975i	\(0.604122\pi\)
\(47\)	1.54451e6	−	8.75933e6i	0.0461689	−	0.261837i	−0.952982	−	0.303025i	\(-0.902003\pi\)
							0.999151	+	0.0411885i	\(0.0131144\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.10.e.a.9.5	✓	42
19.17	even	9	inner	38.10.e.a.17.5	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.e.a.9.5	✓	42	1.1	even	1	trivial
38.10.e.a.17.5	yes	42	19.17	even	9	inner