Embedded newform 38.8.e.b.17.4

• Label: 38.8.e.b.17.4
• Level: $38$
• Weight: $8$
• Character: 38.17
• 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			17.4
Character	\(\chi\)	\(=\)	38.17
Dual form			38.8.e.b.9.4

$q$-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 \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.b.9.4	✓	36	19.9	even	9	inner
38.8.e.b.17.4	yes	36	1.1	even	1	trivial