Embedded newform 38.6.c.a.7.2

• Label: 38.6.c.a.7.2
• Level: $38$
• Weight: $6$
• Character: 38.7
• Analytic conductor: $6.095$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	38.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.09458515289\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Defining polynomial:	\( x^{6} + 133x^{4} - 60x^{3} + 17689x^{2} - 3990x + 900 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			7.2
Root			\(0.112825 - 0.195419i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.7
Dual form			38.6.c.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00000 + 3.46410i) q^{2} +(-2.72565 - 4.72096i) q^{3} +(-8.00000 + 13.8564i) q^{4} +(-41.7489 - 72.3112i) q^{5} +(10.9026 - 18.8839i) q^{6} -105.805 q^{7} -64.0000 q^{8} +(106.642 - 184.709i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} - 15 q^{3} - 48 q^{4} + 14 q^{5} + 60 q^{6} - 624 q^{7} - 384 q^{8} - 410 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00000	+	3.46410i	0.353553	+	0.612372i
							\(3\)	−2.72565	−	4.72096i	−0.174851	−	0.302850i	0.765259	−	0.643723i	\(-0.222611\pi\)
−0.940110	+	0.340872i	\(0.889278\pi\)
\(5\)	−41.7489	−	72.3112i	−0.746827	−	1.29354i	−0.949336	−	0.314262i	\(-0.898243\pi\)
							0.202510	−	0.979280i	\(-0.435090\pi\)
\(7\)	−105.805			−0.816135			−0.408067	−	0.912952i	\(-0.633797\pi\)
							−0.408067	+	0.912952i	\(0.633797\pi\)
\(11\)	−245.400			−0.611496			−0.305748	−	0.952112i	\(-0.598906\pi\)
							−0.305748	+	0.952112i	\(0.598906\pi\)
\(13\)	95.4880	−	165.390i	0.156708	−	0.271426i	−0.776972	−	0.629535i	\(-0.783245\pi\)
							0.933680	+	0.358110i	\(0.116579\pi\)
\(17\)	44.7184	+	77.4546i	0.0375288	+	0.0650017i	0.884180	−	0.467147i	\(-0.154718\pi\)
							−0.846651	+	0.532149i	\(0.821385\pi\)
\(19\)	109.271	−	1569.76i	0.0694418	−	0.997586i
							\(23\)	−1915.27	+	3317.34i	−0.754936	+	1.30759i	0.190471	+	0.981693i	\(0.438999\pi\)
−0.945406	+	0.325894i	\(0.894335\pi\)
\(29\)	−1051.03	+	1820.43i	−0.232070	+	0.401957i	−0.958417	−	0.285371i	\(-0.907883\pi\)
							0.726347	+	0.687328i	\(0.241216\pi\)
\(31\)	4690.59			0.876643			0.438322	−	0.898818i	\(-0.355573\pi\)
							0.438322	+	0.898818i	\(0.355573\pi\)
\(37\)	7993.77			0.959948			0.479974	−	0.877283i	\(-0.340646\pi\)
							0.479974	+	0.877283i	\(0.340646\pi\)
\(41\)	−10388.5	−	17993.5i	−0.965150	−	1.67169i	−0.709212	−	0.704996i	\(-0.750949\pi\)
							−0.255938	−	0.966693i	\(-0.582384\pi\)
\(43\)	1105.27	+	1914.39i	0.0911588	+	0.157892i	0.907999	−	0.418972i	\(-0.137610\pi\)
							−0.816840	+	0.576864i	\(0.804276\pi\)
\(47\)	10092.4	−	17480.5i	0.666422	−	1.15428i	−0.312475	−	0.949926i	\(-0.601158\pi\)
							0.978898	−	0.204351i	\(-0.0655085\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.c.a.7.2	✓	6
3.2	odd	2		342.6.g.a.235.3		6
4.3	odd	2		304.6.i.a.273.2		6
19.7	even	3		722.6.a.e.1.2		3
19.11	even	3	inner	38.6.c.a.11.2	yes	6
19.12	odd	6		722.6.a.f.1.2		3
57.11	odd	6		342.6.g.a.163.3		6
76.11	odd	6		304.6.i.a.49.2		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.c.a.7.2	✓	6	1.1	even	1	trivial
38.6.c.a.11.2	yes	6	19.11	even	3	inner
304.6.i.a.49.2		6	76.11	odd	6
304.6.i.a.273.2		6	4.3	odd	2
342.6.g.a.163.3		6	57.11	odd	6
342.6.g.a.235.3		6	3.2	odd	2
722.6.a.e.1.2		3	19.7	even	3
722.6.a.f.1.2		3	19.12	odd	6