Embedded newform 38.6.e.b.5.2

• Label: 38.6.e.b.5.2
• 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			5.2
Character	\(\chi\)	\(=\)	38.5
Dual form			38.6.e.b.23.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.75877 + 1.36808i) q^{2} +(-1.36581 + 7.74591i) q^{3} +(12.2567 - 10.2846i) q^{4} +(-39.1860 - 32.8809i) q^{5} +(-5.46326 - 30.9837i) q^{6} +(-9.18601 - 15.9106i) q^{7} +(-32.0000 + 55.4256i) q^{8} +(170.212 + 61.9519i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 15 q^{3} + 60 q^{6} - 84 q^{7} - 960 q^{8} - 345 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.36581	+	7.74591i	−0.0876170	+	0.496901i	0.909144	+	0.416482i	\(0.136737\pi\)
−0.996761	+	0.0804191i	\(0.974374\pi\)
\(5\)	−39.1860	−	32.8809i	−0.700980	−	0.588192i	0.221072	−	0.975257i	\(-0.429044\pi\)
							−0.922052	+	0.387066i	\(0.873489\pi\)
\(7\)	−9.18601	−	15.9106i	−0.0708568	−	0.122728i	0.828420	−	0.560107i	\(-0.189240\pi\)
							−0.899277	+	0.437379i	\(0.855907\pi\)
\(11\)	375.072	−	649.643i	0.934614	−	1.61880i	0.159294	−	0.987231i	\(-0.449078\pi\)
							0.775321	−	0.631568i	\(-0.217588\pi\)
\(13\)	−90.9619	−	515.871i	−0.149280	−	0.846608i	−0.963831	−	0.266516i	\(-0.914128\pi\)
							0.814551	−	0.580092i	\(-0.196984\pi\)
\(17\)	1764.94	−	642.386i	1.48118	−	0.539106i	0.530070	−	0.847954i	\(-0.322166\pi\)
							0.951111	+	0.308848i	\(0.0999436\pi\)
\(19\)	−1488.16	+	511.338i	−0.945729	+	0.324955i
							\(23\)	−374.217	+	314.005i	−0.147504	+	0.123770i	−0.713554	−	0.700601i	\(-0.752915\pi\)
0.566050	+	0.824371i	\(0.308471\pi\)
\(29\)	−1228.35	−	447.083i	−0.271223	−	0.0987173i	0.202828	−	0.979214i	\(-0.434987\pi\)
							−0.474052	+	0.880497i	\(0.657209\pi\)
\(31\)	−3275.09	−	5672.62i	−0.612095	−	1.06018i	−0.990887	−	0.134698i	\(-0.956994\pi\)
							0.378792	−	0.925482i	\(-0.376340\pi\)
\(37\)	−8297.48			−0.996419			−0.498209	−	0.867057i	\(-0.666009\pi\)
							−0.498209	+	0.867057i	\(0.666009\pi\)
\(41\)	1497.47	−	8492.59i	0.139123	−	0.789007i	−0.832776	−	0.553610i	\(-0.813250\pi\)
							0.971899	−	0.235397i	\(-0.0756390\pi\)
\(43\)	10710.7	+	8987.33i	0.883377	+	0.741241i	0.966871	−	0.255267i	\(-0.0821633\pi\)
							−0.0834935	+	0.996508i	\(0.526608\pi\)
\(47\)	15515.2	+	5647.07i	1.02450	+	0.372888i	0.798985	−	0.601351i	\(-0.205371\pi\)
							0.225517	+	0.974239i	\(0.427593\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.b.5.2	✓	30
19.2	odd	18		722.6.a.q.1.10		15
19.4	even	9	inner	38.6.e.b.23.2	yes	30
19.17	even	9		722.6.a.r.1.6		15

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.b.5.2	✓	30	1.1	even	1	trivial
38.6.e.b.23.2	yes	30	19.4	even	9	inner
722.6.a.q.1.10		15	19.2	odd	18
722.6.a.r.1.6		15	19.17	even	9