Embedded newform 38.4.e.a.35.1

• Label: 38.4.e.a.35.1
• Level: $38$
• Weight: $4$
• Character: 38.35
• 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			35.1
Root			\(-5.28151 - 0.642788i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.35
Dual form			38.4.e.a.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.53209 + 1.28558i) q^{2} +(-6.18284 + 2.25037i) q^{3} +(0.694593 + 3.93923i) q^{4} +(-1.97089 + 11.1775i) q^{5} +(-12.3657 - 4.50074i) q^{6} +(4.35993 + 7.55162i) q^{7} +(-4.00000 + 6.92820i) q^{8} +(12.4801 - 10.4721i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 9 q^{3} - 18 q^{6} + 21 q^{7} - 48 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.53209	+	1.28558i	0.541675	+	0.454519i
							\(3\)	−6.18284	+	2.25037i	−1.18989	+	0.433084i	−0.859684	−	0.510826i	\(-0.829340\pi\)
−0.330204	+	0.943910i	\(0.607117\pi\)
\(5\)	−1.97089	+	11.1775i	−0.176281	+	0.999742i	0.760373	+	0.649487i	\(0.225016\pi\)
							−0.936654	+	0.350255i	\(0.886095\pi\)
\(7\)	4.35993	+	7.55162i	0.235414	+	0.407749i	0.959393	−	0.282073i	\(-0.0910221\pi\)
							−0.723979	+	0.689822i	\(0.757689\pi\)
\(11\)	25.8541	−	44.7806i	0.708663	−	1.22744i	−0.256690	−	0.966494i	\(-0.582632\pi\)
							0.965353	−	0.260947i	\(-0.0840349\pi\)
\(13\)	26.0508	+	9.48171i	0.555784	+	0.202289i	0.604614	−	0.796518i	\(-0.293327\pi\)
							−0.0488306	+	0.998807i	\(0.515549\pi\)
\(17\)	89.9509	+	75.4778i	1.28331	+	1.07683i	0.992779	+	0.119961i	\(0.0382770\pi\)
							0.290533	+	0.956865i	\(0.406167\pi\)
\(19\)	−66.4025	+	49.4945i	−0.801778	+	0.597622i
							\(23\)	−17.2918	−	98.0667i	−0.156765	−	0.889057i	−0.957155	−	0.289577i	\(-0.906485\pi\)
0.800390	−	0.599480i	\(-0.204626\pi\)
\(29\)	−106.910	+	89.7079i	−0.684573	+	0.574425i	−0.917339	−	0.398108i	\(-0.869667\pi\)
							0.232765	+	0.972533i	\(0.425223\pi\)
\(31\)	1.55152	+	2.68730i	0.00898905	+	0.0155695i	0.870485	−	0.492195i	\(-0.163805\pi\)
							−0.861496	+	0.507765i	\(0.830472\pi\)
\(37\)	−91.5180			−0.406634			−0.203317	−	0.979113i	\(-0.565172\pi\)
							−0.203317	+	0.979113i	\(0.565172\pi\)
\(41\)	234.973	−	85.5233i	0.895041	−	0.325768i	0.146777	−	0.989170i	\(-0.453110\pi\)
							0.748264	+	0.663401i	\(0.230888\pi\)
\(43\)	−71.8763	+	407.631i	−0.254908	+	1.44565i	0.541401	+	0.840764i	\(0.317894\pi\)
							−0.796309	+	0.604890i	\(0.793217\pi\)
\(47\)	468.670	−	393.260i	1.45452	−	1.22049i	0.525324	−	0.850902i	\(-0.323944\pi\)
							0.929197	−	0.369586i	\(-0.120500\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.4.e.a.35.1	yes	12
19.5	even	9		722.4.a.p.1.6		6
19.6	even	9	inner	38.4.e.a.25.1	✓	12
19.14	odd	18		722.4.a.o.1.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.e.a.25.1	✓	12	19.6	even	9	inner
38.4.e.a.35.1	yes	12	1.1	even	1	trivial
722.4.a.o.1.1		6	19.14	odd	18
722.4.a.p.1.6		6	19.5	even	9