Embedded newform 38.8.e.a.23.3

• Label: 38.8.e.a.23.3
• Level: $38$
• Weight: $8$
• Character: 38.23
• Analytic conductor: $11.871$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8706309684\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			23.3
Character	\(\chi\)	\(=\)	38.23
Dual form			38.8.e.a.5.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.51754 - 2.73616i) q^{2} +(5.10191 + 28.9344i) q^{3} +(49.0268 + 41.1384i) q^{4} +(128.722 - 108.010i) q^{5} +(40.8153 - 231.475i) q^{6} +(-158.544 + 274.607i) q^{7} +(-256.000 - 443.405i) q^{8} +(1243.94 - 452.756i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 45 q^{3} + 360 q^{6} + 1806 q^{7} - 7680 q^{8} - 3105 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.51754	−	2.73616i	−0.664463	−	0.241845i
							\(3\)	5.10191	+	28.9344i	0.109096	+	0.618714i	0.989505	+	0.144499i	\(0.0461570\pi\)
−0.880409	+	0.474215i	\(0.842732\pi\)
\(5\)	128.722	−	108.010i	0.460529	−	0.386430i	−0.382796	−	0.923833i	\(-0.625039\pi\)
							0.843326	+	0.537403i	\(0.180594\pi\)
\(7\)	−158.544	+	274.607i	−0.174706	+	0.302600i	−0.940059	−	0.341010i	\(-0.889231\pi\)
							0.765353	+	0.643610i	\(0.222564\pi\)
\(11\)	−2151.37	−	3726.28i	−0.487350	−	0.844115i	0.512544	−	0.858661i	\(-0.328703\pi\)
							−0.999894	+	0.0145461i	\(0.995370\pi\)
\(13\)	−2490.06	+	14121.8i	−0.314346	+	1.78275i	0.261516	+	0.965199i	\(0.415778\pi\)
							−0.575862	+	0.817547i	\(0.695333\pi\)
\(17\)	27814.0	+	10123.5i	1.37307	+	0.499756i	0.920070	−	0.391755i	\(-0.128132\pi\)
							0.452999	+	0.891511i	\(0.350354\pi\)
\(19\)	28760.2	+	8168.36i	0.961954	+	0.273210i
							\(23\)	57504.3	+	48251.8i	0.985491	+	0.826925i	0.984909	−	0.173074i	\(-0.0553701\pi\)
0.000582588		1.00000i	\(0.499815\pi\)
\(29\)	−227783.	+	82906.3i	−1.73432	+	0.631240i	−0.998923	−	0.0463986i	\(-0.985226\pi\)
							−0.735395	+	0.677639i	\(0.763003\pi\)
\(31\)	51635.6	−	89435.4i	0.311303	−	0.539192i	−0.667342	−	0.744752i	\(-0.732568\pi\)
							0.978645	+	0.205559i	\(0.0659013\pi\)
\(37\)	−67418.1			−0.218812			−0.109406	−	0.993997i	\(-0.534895\pi\)
							−0.109406	+	0.993997i	\(0.534895\pi\)
\(41\)	−31149.2	−	176656.i	−0.0705836	−	0.400299i	−0.999546	−	0.0301256i	\(-0.990409\pi\)
							0.928963	−	0.370174i	\(-0.120702\pi\)
\(43\)	9915.02	−	8319.69i	0.0190175	−	0.0159576i	−0.633229	−	0.773964i	\(-0.718271\pi\)
							0.652247	+	0.758007i	\(0.273827\pi\)
\(47\)	845916.	−	307888.i	1.18846	−	0.432564i	0.329277	−	0.944233i	\(-0.393195\pi\)
							0.859182	+	0.511669i	\(0.170973\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.a.23.3	yes	30
19.5	even	9	inner	38.8.e.a.5.3	✓	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.a.5.3	✓	30	19.5	even	9	inner
38.8.e.a.23.3	yes	30	1.1	even	1	trivial