Embedded newform 38.9.d.a.31.6

• Label: 38.9.d.a.31.6
• 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			31.6
Character	\(\chi\)	\(=\)	38.31
Dual form			38.9.d.a.27.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.79796 + 5.65685i) q^{2} +(99.7890 - 57.6132i) q^{3} +(64.0000 - 110.851i) q^{4} +(-88.7120 - 153.654i) q^{5} +(-651.819 + 1128.98i) q^{6} +4081.69 q^{7} +1448.15i q^{8} +(3358.06 - 5816.33i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 168 q^{3} + 1536 q^{4} - 558 q^{5} + 896 q^{6} + 11992 q^{7} + 18448 q^{9}+O(q^{10}) \)

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
							\(3\)	99.7890	−	57.6132i	1.23196	−	0.711274i	0.264524	−	0.964379i	\(-0.414785\pi\)
0.967439	+	0.253105i	\(0.0814519\pi\)
\(5\)	−88.7120	−	153.654i	−0.141939	−	0.245846i	0.786288	−	0.617861i	\(-0.212000\pi\)
							−0.928227	+	0.372015i	\(0.878667\pi\)
\(7\)	4081.69			1.70000			0.849999	−	0.526784i	\(-0.176602\pi\)
							0.849999	+	0.526784i	\(0.176602\pi\)
\(11\)	−2976.07			−0.203270			−0.101635	−	0.994822i	\(-0.532407\pi\)
							−0.101635	+	0.994822i	\(0.532407\pi\)
\(13\)	−11706.4	−	6758.68i	−0.409873	−	0.236640i	0.280862	−	0.959748i	\(-0.409380\pi\)
							−0.690735	+	0.723108i	\(0.742713\pi\)
\(17\)	24149.7	+	41828.5i	0.289145	+	0.500814i	0.973606	−	0.228235i	\(-0.0732956\pi\)
							−0.684461	+	0.729050i	\(0.739962\pi\)
\(19\)	15945.7	−	129342.i	0.122357	−	0.992486i
							\(23\)	123129.	−	213265.i	0.439995	−	0.762094i	−0.557693	−	0.830047i	\(-0.688314\pi\)
0.997689	+	0.0679531i	\(0.0216468\pi\)
\(29\)	770786.	+	445013.i	1.08979	+	0.629189i	0.933520	−	0.358526i	\(-0.116721\pi\)
							0.156267	+	0.987715i	\(0.450054\pi\)
\(31\)		−	615162.i		−	0.666105i	−0.942908	−	0.333052i	\(-0.891921\pi\)
							0.942908	−	0.333052i	\(-0.108079\pi\)
\(37\)			763894.i			0.407592i	0.979013	+	0.203796i	\(0.0653280\pi\)
							−0.979013	+	0.203796i	\(0.934672\pi\)
\(41\)	−3.35716e6	+	1.93826e6i	−1.18806	+	0.685924i	−0.957864	−	0.287221i	\(-0.907269\pi\)
							−0.230192	+	0.973145i	\(0.573935\pi\)
\(43\)	−2.49169e6	−	4.31573e6i	−0.728820	−	1.26235i	−0.957382	−	0.288823i	\(-0.906736\pi\)
							0.228563	−	0.973529i	\(-0.426597\pi\)
\(47\)	1.90441e6	−	3.29854e6i	0.390274	−	0.675974i	−0.602212	−	0.798336i	\(-0.705714\pi\)
							0.992485	+	0.122362i	\(0.0390470\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.d.a.31.6	yes	24
19.8	odd	6	inner	38.9.d.a.27.6	✓	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.d.a.27.6	✓	24	19.8	odd	6	inner
38.9.d.a.31.6	yes	24	1.1	even	1	trivial