Embedded newform 38.4.e.a.9.1

• Label: 38.4.e.a.9.1
• Level: $38$
• Weight: $4$
• Character: 38.9
• Analytic conductor: $2.242$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.24207258022\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{9})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - 6*x^11 - 135*x^10 + 730*x^9 + 7953*x^8 - 36258*x^7 - 262940*x^6 + 918855*x^5 + 5157591*x^4 - 11890401*x^3 - 56759508*x^2 + 62864118*x + 272110107
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.1
Root			\(5.30460 - 0.984808i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.9
Dual form			38.4.e.a.17.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.347296 + 1.96962i) q^{2} +(-4.43054 + 3.71766i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(-5.82452 - 2.11995i) q^{5} +(-8.86108 - 7.43533i) q^{6} +(-5.61124 + 9.71895i) q^{7} +(-4.00000 - 6.92820i) q^{8} +(1.12015 - 6.35271i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 9 q^{3} - 18 q^{6} + 21 q^{7} - 48 q^{8} - 27 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.347296	+	1.96962i	0.122788	+	0.696364i
							\(3\)	−4.43054	+	3.71766i	−0.852658	+	0.715465i	−0.960373	−	0.278717i	\(-0.910091\pi\)
0.107715	+	0.994182i	\(0.465646\pi\)
\(5\)	−5.82452	−	2.11995i	−0.520961	−	0.189614i	0.0681370	−	0.997676i	\(-0.478294\pi\)
							−0.589098	+	0.808062i	\(0.700517\pi\)
\(7\)	−5.61124	+	9.71895i	−0.302978	+	0.524774i	−0.976809	−	0.214112i	\(-0.931314\pi\)
							0.673831	+	0.738886i	\(0.264648\pi\)
\(11\)	22.5510	+	39.0594i	0.618125	+	1.07062i	0.989828	+	0.142272i	\(0.0454408\pi\)
							−0.371702	+	0.928352i	\(0.621226\pi\)
\(13\)	17.3540	+	14.5617i	0.370241	+	0.310669i	0.808857	−	0.588005i	\(-0.200087\pi\)
							−0.438616	+	0.898675i	\(0.644531\pi\)
\(17\)	−5.38864	−	30.5605i	−0.0768786	−	0.436000i	−0.998815	−	0.0486599i	\(-0.984505\pi\)
							0.921937	−	0.387340i	\(-0.126606\pi\)
\(19\)	13.8011	+	81.6611i	0.166642	+	0.986017i
							\(23\)	135.607	−	49.3568i	1.22939	−	0.447461i	0.356001	−	0.934486i	\(-0.384140\pi\)
0.873388	+	0.487025i	\(0.161918\pi\)
\(29\)	4.20818	−	23.8658i	0.0269462	−	0.152820i	−0.968366	−	0.249534i	\(-0.919722\pi\)
							0.995312	+	0.0967149i	\(0.0308335\pi\)
\(31\)	−151.827	+	262.972i	−0.879641	+	1.52358i	−0.0279067	+	0.999611i	\(0.508884\pi\)
							−0.851735	+	0.523973i	\(0.824449\pi\)
\(37\)	−121.394			−0.539378			−0.269689	−	0.962947i	\(-0.586921\pi\)
							−0.269689	+	0.962947i	\(0.586921\pi\)
\(41\)	−79.4143	+	66.6365i	−0.302498	+	0.253826i	−0.781383	−	0.624052i	\(-0.785486\pi\)
							0.478885	+	0.877878i	\(0.341041\pi\)
\(43\)	417.583	+	151.988i	1.48095	+	0.539021i	0.951049	−	0.309040i	\(-0.100008\pi\)
							0.529898	+	0.848061i	\(0.322230\pi\)
\(47\)	83.6935	−	474.649i	0.259744	−	1.47308i	−0.523853	−	0.851809i	\(-0.675506\pi\)
							0.783596	−	0.621270i	\(-0.213383\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.4.e.a.9.1	✓	12
19.6	even	9		722.4.a.p.1.2		6
19.13	odd	18		722.4.a.o.1.5		6
19.17	even	9	inner	38.4.e.a.17.1	yes	12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.e.a.9.1	✓	12	1.1	even	1	trivial
38.4.e.a.17.1	yes	12	19.17	even	9	inner
722.4.a.o.1.5		6	19.13	odd	18
722.4.a.p.1.2		6	19.6	even	9