Embedded newform 38.8.e.a.5.2

• Label: 38.8.e.a.5.2
• Level: $38$
• Weight: $8$
• Character: 38.5
• 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			5.2
Character	\(\chi\)	\(=\)	38.5
Dual form			38.8.e.a.23.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.51754 + 2.73616i) q^{2} +(-7.18109 + 40.7260i) q^{3} +(49.0268 - 41.1384i) q^{4} +(-24.6197 - 20.6584i) q^{5} +(-57.4487 - 325.808i) q^{6} +(764.198 + 1323.63i) q^{7} +(-256.000 + 443.405i) q^{8} +(448.070 + 163.084i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 45 q^{3} + 360 q^{6} + 1806 q^{7} - 7680 q^{8} - 3105 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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
							\(3\)	−7.18109	+	40.7260i	−0.153556	+	0.870858i	0.806539	+	0.591181i	\(0.201338\pi\)
−0.960094	+	0.279676i	\(0.909773\pi\)
\(5\)	−24.6197	−	20.6584i	−0.0880822	−	0.0739097i	0.597683	−	0.801732i	\(-0.296088\pi\)
							−0.685766	+	0.727823i	\(0.740532\pi\)
\(7\)	764.198	+	1323.63i	0.842098	+	1.45856i	0.888118	+	0.459616i	\(0.152013\pi\)
							−0.0460192	+	0.998941i	\(0.514654\pi\)
\(11\)	−523.733	+	907.132i	−0.118641	+	0.205493i	−0.919229	−	0.393722i	\(-0.871187\pi\)
							0.800588	+	0.599215i	\(0.204521\pi\)
\(13\)	−1069.14	−	6063.39i	−0.134969	−	0.765445i	−0.974882	−	0.222724i	\(-0.928505\pi\)
							0.839913	−	0.542721i	\(-0.182606\pi\)
\(17\)	−32284.3	+	11750.5i	−1.59375	+	0.580077i	−0.978135	−	0.207972i	\(-0.933314\pi\)
							−0.615613	+	0.788049i	\(0.711091\pi\)
\(19\)	−16217.2	+	25117.2i	−0.542422	+	0.840106i
							\(23\)	4865.64	−	4082.75i	0.0833858	−	0.0699690i	−0.600142	−	0.799894i	\(-0.704889\pi\)
0.683528	+	0.729924i	\(0.260445\pi\)
\(29\)	−63919.0	−	23264.6i	−0.486672	−	0.177134i	0.0870177	−	0.996207i	\(-0.472266\pi\)
							−0.573690	+	0.819073i	\(0.694489\pi\)
\(31\)	−18170.9	−	31473.0i	−0.109550	−	0.189746i	0.806038	−	0.591863i	\(-0.201608\pi\)
							−0.915588	+	0.402118i	\(0.868274\pi\)
\(37\)	96004.7			0.311592			0.155796	−	0.987789i	\(-0.450206\pi\)
							0.155796	+	0.987789i	\(0.450206\pi\)
\(41\)	−51287.7	+	290867.i	−0.116217	+	0.659099i	0.869924	+	0.493187i	\(0.164168\pi\)
							−0.986140	+	0.165913i	\(0.946943\pi\)
\(43\)	712452.	+	597818.i	1.36652	+	1.14665i	0.973908	+	0.226943i	\(0.0728732\pi\)
							0.392613	+	0.919704i	\(0.371571\pi\)
\(47\)	−492235.	−	179159.i	−0.691560	−	0.251707i	−0.0277569	−	0.999615i	\(-0.508836\pi\)
							−0.663803	+	0.747908i	\(0.731059\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.a.5.2	✓	30
19.4	even	9	inner	38.8.e.a.23.2	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.a.5.2	✓	30	1.1	even	1	trivial
38.8.e.a.23.2	yes	30	19.4	even	9	inner