Embedded newform 38.8.e.b.9.4

• Label: 38.8.e.b.9.4
• Level: $38$
• Weight: $8$
• Character: 38.9
• Analytic conductor: $11.871$
• Analytic rank: $0$
• Dimension: $36$
• 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:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.4
Character	\(\chi\)	\(=\)	38.9
Dual form			38.8.e.b.17.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38919 - 7.87846i) q^{2} +(16.0986 - 13.5083i) q^{3} +(-60.1403 + 21.8893i) q^{4} +(358.329 + 130.421i) q^{5} +(-128.789 - 108.067i) q^{6} +(-386.454 + 669.359i) q^{7} +(256.000 + 443.405i) q^{8} +(-303.078 + 1718.84i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 84 q^{3} + 672 q^{6} - 2988 q^{7} + 9216 q^{8} + 1428 q^{9}+O(q^{10}) \)

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
							\(3\)	16.0986	−	13.5083i	0.344242	−	0.288854i	−0.454231	−	0.890884i	\(-0.650086\pi\)
0.798473	+	0.602030i	\(0.205641\pi\)
\(5\)	358.329	+	130.421i	1.28200	+	0.466608i	0.891092	−	0.453823i	\(-0.149940\pi\)
							0.390904	+	0.920432i	\(0.372163\pi\)
\(7\)	−386.454	+	669.359i	−0.425848	+	0.737591i	−0.996499	−	0.0836016i	\(-0.973358\pi\)
							0.570651	+	0.821193i	\(0.306691\pi\)
\(11\)	1369.41	+	2371.88i	0.310212	+	0.537303i	0.978408	−	0.206682i	\(-0.0662667\pi\)
							−0.668196	+	0.743985i	\(0.732933\pi\)
\(13\)	5548.41	+	4655.67i	0.700433	+	0.587733i	0.921897	−	0.387436i	\(-0.126639\pi\)
							−0.221464	+	0.975169i	\(0.571084\pi\)
\(17\)	−5763.66	−	32687.4i	−0.284530	−	1.61365i	−0.706961	−	0.707253i	\(-0.749934\pi\)
							0.422431	−	0.906395i	\(-0.361177\pi\)
\(19\)	15886.2	+	25327.9i	0.531350	+	0.847152i
							\(23\)	79401.9	−	28899.9i	1.36077	−	0.495278i	0.444475	−	0.895791i	\(-0.353390\pi\)
0.916291	+	0.400513i	\(0.131168\pi\)
\(29\)	11864.5	−	67286.9i	0.0903351	−	0.512316i	−0.905742	−	0.423829i	\(-0.860686\pi\)
							0.996077	−	0.0884868i	\(-0.0282031\pi\)
\(31\)	−81562.6	+	141271.i	−0.491728	+	0.851698i	−0.999955	−	0.00952545i	\(-0.996968\pi\)
							0.508227	+	0.861223i	\(0.330301\pi\)
\(37\)	−574635.			−1.86503			−0.932515	−	0.361131i	\(-0.882391\pi\)
							−0.932515	+	0.361131i	\(0.882391\pi\)
\(41\)	36810.5	−	30887.6i	0.0834119	−	0.0699909i	−0.600128	−	0.799904i	\(-0.704884\pi\)
							0.683540	+	0.729913i	\(0.260439\pi\)
\(43\)	331902.	+	120803.i	0.636606	+	0.231706i	0.640104	−	0.768288i	\(-0.278891\pi\)
							−0.00349791	+	0.999994i	\(0.501113\pi\)
\(47\)	80385.1	−	455887.i	0.112936	−	0.640493i	−0.874815	−	0.484457i	\(-0.839017\pi\)
							0.987751	−	0.156036i	\(-0.0498717\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.b.9.4	✓	36
19.17	even	9	inner	38.8.e.b.17.4	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.b.9.4	✓	36	1.1	even	1	trivial
38.8.e.b.17.4	yes	36	19.17	even	9	inner