Embedded newform 38.9.d.a.27.8

• Label: 38.9.d.a.27.8
• Level: $38$
• Weight: $9$
• Character: 38.27
• 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			27.8
Character	\(\chi\)	\(=\)	38.27
Dual form			38.9.d.a.31.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.79796 + 5.65685i) q^{2} +(-36.3650 - 20.9954i) q^{3} +(64.0000 + 110.851i) q^{4} +(106.719 - 184.842i) q^{5} +(-237.535 - 411.423i) q^{6} +680.248 q^{7} +1448.15i q^{8} +(-2398.89 - 4155.00i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 168 q^{3} + 1536 q^{4} - 558 q^{5} + 896 q^{6} + 11992 q^{7} + 18448 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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
							\(3\)	−36.3650	−	20.9954i	−0.448951	−	0.259202i	0.258436	−	0.966028i	\(-0.416793\pi\)
−0.707387	+	0.706826i	\(0.750126\pi\)
\(5\)	106.719	−	184.842i	0.170750	−	0.295747i	−0.767932	−	0.640531i	\(-0.778714\pi\)
							0.938682	+	0.344784i	\(0.112048\pi\)
\(7\)	680.248			0.283318			0.141659	−	0.989915i	\(-0.454756\pi\)
							0.141659	+	0.989915i	\(0.454756\pi\)
\(11\)	377.494			0.0257834			0.0128917	−	0.999917i	\(-0.495896\pi\)
							0.0128917	+	0.999917i	\(0.495896\pi\)
\(13\)	33972.4	−	19613.9i	1.18947	−	0.686739i	0.231281	−	0.972887i	\(-0.425708\pi\)
							0.958185	+	0.286148i	\(0.0923749\pi\)
\(17\)	62282.6	−	107877.i	0.745712	−	1.29161i	−0.204150	−	0.978940i	\(-0.565443\pi\)
							0.949861	−	0.312671i	\(-0.101224\pi\)
\(19\)	124381.	−	38895.5i	0.954423	−	0.298459i
							\(23\)	−94620.8	−	163888.i	−0.338124	−	0.585647i	0.645956	−	0.763374i	\(-0.276459\pi\)
−0.984080	+	0.177727i	\(0.943126\pi\)
\(29\)	98771.1	−	57025.5i	0.139649	−	0.0806264i	−0.428548	−	0.903519i	\(-0.640975\pi\)
							0.568197	+	0.822893i	\(0.307641\pi\)
\(31\)		−	1.11465e6i		−	1.20695i	−0.797381	−	0.603477i	\(-0.793782\pi\)
							0.797381	−	0.603477i	\(-0.206218\pi\)
\(37\)		−	661630.i		−	0.353027i	−0.984298	−	0.176514i	\(-0.943518\pi\)
							0.984298	−	0.176514i	\(-0.0564820\pi\)
\(41\)	−2.38947e6	−	1.37956e6i	−0.845602	−	0.488208i	0.0135627	−	0.999908i	\(-0.495683\pi\)
							−0.859164	+	0.511700i	\(0.829016\pi\)
\(43\)	−1.27537e6	+	2.20901e6i	−0.373047	+	0.646137i	−0.990033	−	0.140838i	\(-0.955020\pi\)
							0.616985	+	0.786975i	\(0.288354\pi\)
\(47\)	3.25239e6	+	5.63330e6i	0.666517	+	1.15444i	0.978872	+	0.204476i	\(0.0655489\pi\)
							−0.312355	+	0.949966i	\(0.601118\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.d.a.27.8	✓	24
19.12	odd	6	inner	38.9.d.a.31.8	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.d.a.27.8	✓	24	1.1	even	1	trivial
38.9.d.a.31.8	yes	24	19.12	odd	6	inner