Embedded newform 38.9.d.a.31.2

• Label: 38.9.d.a.31.2
• Level: $38$
• Weight: $9$
• Character: 38.31
• Analytic conductor: $15.480$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(15.4803871823\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.2
Character	\(\chi\)	\(=\)	38.31
Dual form			38.9.d.a.27.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.79796 + 5.65685i) q^{2} +(-57.0846 + 32.9578i) q^{3} +(64.0000 - 110.851i) q^{4} +(-412.457 - 714.397i) q^{5} +(372.875 - 645.838i) q^{6} +2843.86 q^{7} +1448.15i q^{8} +(-1108.07 + 1919.23i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 168 q^{3} + 1536 q^{4} - 558 q^{5} + 896 q^{6} + 11992 q^{7} + 18448 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.79796	+	5.65685i	−0.612372	+	0.353553i
							\(3\)	−57.0846	+	32.9578i	−0.704748	+	0.406886i	−0.809113	−	0.587653i	\(-0.800052\pi\)
0.104365	+	0.994539i	\(0.466719\pi\)
\(5\)	−412.457	−	714.397i	−0.659932	−	1.14303i	−0.980633	−	0.195855i	\(-0.937252\pi\)
							0.320701	−	0.947180i	\(-0.396081\pi\)
\(7\)	2843.86			1.18445			0.592224	−	0.805773i	\(-0.298250\pi\)
							0.592224	+	0.805773i	\(0.298250\pi\)
\(11\)	18890.5			1.29025			0.645125	−	0.764077i	\(-0.276805\pi\)
							0.645125	+	0.764077i	\(0.276805\pi\)
\(13\)	−36616.9	−	21140.8i	−1.28206	−	0.740197i	−0.304835	−	0.952405i	\(-0.598601\pi\)
							−0.977224	+	0.212208i	\(0.931935\pi\)
\(17\)	−38539.1	−	66751.7i	−0.461430	−	0.799221i	0.537602	−	0.843199i	\(-0.319330\pi\)
							−0.999032	+	0.0439780i	\(0.985997\pi\)
\(19\)	−5375.39	+	130210.i	−0.0412473	+	0.999149i
							\(23\)	−251139.	+	434986.i	−0.897436	+	1.55441i	−0.0666760	+	0.997775i	\(0.521239\pi\)
−0.830760	+	0.556630i	\(0.812094\pi\)
\(29\)	−316163.	−	182537.i	−0.447012	−	0.258083i	0.259555	−	0.965728i	\(-0.416424\pi\)
							−0.706568	+	0.707646i	\(0.749757\pi\)
\(31\)			1.48070e6i			1.60333i	0.597777	+	0.801663i	\(0.296051\pi\)
							−0.597777	+	0.801663i	\(0.703949\pi\)
\(37\)			1.34309e6i			0.716637i	0.933599	+	0.358319i	\(0.116650\pi\)
							−0.933599	+	0.358319i	\(0.883350\pi\)
\(41\)	−2.99209e6	+	1.72748e6i	−1.05886	+	0.611334i	−0.925117	−	0.379681i	\(-0.876034\pi\)
							−0.133745	+	0.991016i	\(0.542700\pi\)
\(43\)	2.31263e6	+	4.00560e6i	0.676446	+	1.17164i	0.976044	+	0.217574i	\(0.0698142\pi\)
							−0.299598	+	0.954066i	\(0.596852\pi\)
\(47\)	3.04403e6	−	5.27242e6i	0.623818	−	1.08048i	−0.364950	−	0.931027i	\(-0.618914\pi\)
							0.988768	−	0.149458i	\(-0.0477527\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.d.a.31.2	yes	24
19.8	odd	6	inner	38.9.d.a.27.2	✓	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.d.a.27.2	✓	24	19.8	odd	6	inner
38.9.d.a.31.2	yes	24	1.1	even	1	trivial