Embedded newform 38.8.e.a.9.3

• Label: 38.8.e.a.9.3
• Level: $38$
• Weight: $8$
• Character: 38.9
• 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			9.3
Character	\(\chi\)	\(=\)	38.9
Dual form			38.8.e.a.17.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38919 + 7.87846i) q^{2} +(-9.71662 + 8.15321i) q^{3} +(-60.1403 + 21.8893i) q^{4} +(424.315 + 154.438i) q^{5} +(-77.7330 - 65.2257i) q^{6} +(847.859 - 1468.54i) q^{7} +(-256.000 - 443.405i) q^{8} +(-351.831 + 1995.33i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 45 q^{3} + 360 q^{6} + 1806 q^{7} - 7680 q^{8} - 3105 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38919	+	7.87846i	0.122788	+	0.696364i
							\(3\)	−9.71662	+	8.15321i	−0.207774	+	0.174343i	−0.740736	−	0.671796i	\(-0.765523\pi\)
0.532962	+	0.846139i	\(0.321079\pi\)
\(5\)	424.315	+	154.438i	1.51807	+	0.552534i	0.960667	−	0.277704i	\(-0.0895736\pi\)
							0.557408	+	0.830238i	\(0.311796\pi\)
\(7\)	847.859	−	1468.54i	0.934288	−	1.61823i	0.158388	−	0.987377i	\(-0.449370\pi\)
							0.775899	−	0.630857i	\(-0.217296\pi\)
\(11\)	−355.352	−	615.488i	−0.0804979	−	0.139426i	0.822966	−	0.568091i	\(-0.192318\pi\)
							−0.903464	+	0.428664i	\(0.858984\pi\)
\(13\)	7814.84	+	6557.43i	0.986548	+	0.827812i	0.985064	−	0.172187i	\(-0.0550833\pi\)
							0.00148360	+	0.999999i	\(0.499528\pi\)
\(17\)	3787.71	+	21481.2i	0.186984	+	1.06044i	0.923379	+	0.383889i	\(0.125416\pi\)
							−0.736395	+	0.676552i	\(0.763473\pi\)
\(19\)	−28741.5	+	8233.82i	−0.961330	+	0.275400i
							\(23\)	43010.3	−	15654.5i	0.737098	−	0.268282i	0.0539315	−	0.998545i	\(-0.482825\pi\)
0.683166	+	0.730263i	\(0.260603\pi\)
\(29\)	9277.13	−	52613.2i	0.0706351	−	0.400592i	−0.928906	−	0.370315i	\(-0.879250\pi\)
							0.999542	−	0.0302771i	\(-0.00963898\pi\)
\(31\)	−32572.1	+	56416.5i	−0.196372	+	0.340126i	−0.947349	−	0.320202i	\(-0.896249\pi\)
							0.750977	+	0.660328i	\(0.229583\pi\)
\(37\)	−262965.			−0.853479			−0.426739	−	0.904375i	\(-0.640338\pi\)
							−0.426739	+	0.904375i	\(0.640338\pi\)
\(41\)	153856.	−	129100.i	0.348635	−	0.292539i	−0.451607	−	0.892217i	\(-0.649149\pi\)
							0.800242	+	0.599678i	\(0.204705\pi\)
\(43\)	478934.	+	174318.i	0.918620	+	0.334350i	0.757689	−	0.652616i	\(-0.226328\pi\)
							0.160930	+	0.986966i	\(0.448551\pi\)
\(47\)	134281.	−	761547.i	0.188657	−	1.06993i	−0.732509	−	0.680757i	\(-0.761651\pi\)
							0.921166	−	0.389170i	\(-0.127238\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.a.9.3	✓	30
19.17	even	9	inner	38.8.e.a.17.3	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.a.9.3	✓	30	1.1	even	1	trivial
38.8.e.a.17.3	yes	30	19.17	even	9	inner