Embedded newform 38.4.e.a.25.1

• Label: 38.4.e.a.25.1
• Level: $38$
• Weight: $4$
• Character: 38.25
• 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			25.1
Root			\(-5.28151 + 0.642788i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.25
Dual form			38.4.e.a.35.1

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.e.a.25.1	✓	12	1.1	even	1	trivial
38.4.e.a.35.1	yes	12	19.16	even	9	inner
722.4.a.o.1.1		6	19.15	odd	18
722.4.a.p.1.6		6	19.4	even	9