Embedded newform 38.6.e.b.5.3

• Label: 38.6.e.b.5.3
• Level: $38$
• Weight: $6$
• Character: 38.5
• Analytic conductor: $6.095$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.09458515289\)
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			5.3
Character	\(\chi\)	\(=\)	38.5
Dual form			38.6.e.b.23.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.75877 + 1.36808i) q^{2} +(-0.196437 + 1.11405i) q^{3} +(12.2567 - 10.2846i) q^{4} +(8.74989 + 7.34203i) q^{5} +(-0.785748 - 4.45620i) q^{6} +(-7.74142 - 13.4085i) q^{7} +(-32.0000 + 55.4256i) q^{8} +(227.143 + 82.6732i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 15 q^{3} + 60 q^{6} - 84 q^{7} - 960 q^{8} - 345 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.75877	+	1.36808i	−0.664463	+	0.241845i
							\(3\)	−0.196437	+	1.11405i	−0.0126014	+	0.0714663i	−0.990460	−	0.137801i	\(-0.955997\pi\)
0.977859	+	0.209267i	\(0.0671078\pi\)
\(5\)	8.74989	+	7.34203i	0.156523	+	0.131338i	0.717686	−	0.696367i	\(-0.245201\pi\)
							−0.561163	+	0.827705i	\(0.689646\pi\)
\(7\)	−7.74142	−	13.4085i	−0.0597139	−	0.103428i	0.834623	−	0.550822i	\(-0.185686\pi\)
							−0.894337	+	0.447394i	\(0.852352\pi\)
\(11\)	−298.390	+	516.826i	−0.743536	+	1.28784i	0.207340	+	0.978269i	\(0.433519\pi\)
							−0.950876	+	0.309573i	\(0.899814\pi\)
\(13\)	176.976	+	1003.68i	0.290439	+	1.64716i	0.685182	+	0.728372i	\(0.259723\pi\)
							−0.394742	+	0.918792i	\(0.629166\pi\)
\(17\)	−244.634	+	89.0396i	−0.205303	+	0.0747241i	−0.442625	−	0.896707i	\(-0.645953\pi\)
							0.237322	+	0.971431i	\(0.423730\pi\)
\(19\)	1392.58	+	732.684i	0.884984	+	0.465621i
							\(23\)	144.742	−	121.453i	0.0570526	−	0.0478728i	−0.613815	−	0.789450i	\(-0.710366\pi\)
0.670868	+	0.741577i	\(0.265922\pi\)
\(29\)	2818.93	+	1026.01i	0.622428	+	0.226545i	0.633932	−	0.773389i	\(-0.281440\pi\)
							−0.0115043	+	0.999934i	\(0.503662\pi\)
\(31\)	−2442.48	−	4230.50i	−0.456485	−	0.790655i	0.542287	−	0.840193i	\(-0.317558\pi\)
							−0.998772	+	0.0495380i	\(0.984225\pi\)
\(37\)	3790.08			0.455139			0.227570	−	0.973762i	\(-0.426922\pi\)
							0.227570	+	0.973762i	\(0.426922\pi\)
\(41\)	−3040.19	+	17241.8i	−0.282450	+	1.60185i	0.431805	+	0.901967i	\(0.357877\pi\)
							−0.714255	+	0.699886i	\(0.753234\pi\)
\(43\)	2760.32	+	2316.18i	0.227661	+	0.191030i	0.749482	−	0.662025i	\(-0.230303\pi\)
							−0.521821	+	0.853055i	\(0.674747\pi\)
\(47\)	−11856.8	−	4315.52i	−0.782930	−	0.284963i	−0.0805360	−	0.996752i	\(-0.525663\pi\)
							−0.702394	+	0.711789i	\(0.747885\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.b.5.3	✓	30
19.2	odd	18		722.6.a.q.1.9		15
19.4	even	9	inner	38.6.e.b.23.3	yes	30
19.17	even	9		722.6.a.r.1.7		15

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.b.5.3	✓	30	1.1	even	1	trivial
38.6.e.b.23.3	yes	30	19.4	even	9	inner
722.6.a.q.1.9		15	19.2	odd	18
722.6.a.r.1.7		15	19.17	even	9