Embedded newform 38.4.e.b.9.3

• Label: 38.4.e.b.9.3
• Level: $38$
• Weight: $4$
• Character: 38.9
• Analytic conductor: $2.242$
• Analytic rank: $0$
• Dimension: $18$
• 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:	\(18\)
Relative dimension:	\(3\) over \(\Q(\zeta_{9})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial:	x^18 + 165*x^16 - 56*x^15 + 18435*x^14 - 11748*x^13 + 1092662*x^12 - 1833567*x^11 + 46842546*x^10 - 93643115*x^9 + 1273086000*x^8 - 3159018183*x^7 + 24752942108*x^6 - 51265274250*x^5 + 247917053361*x^4 - 421854393530*x^3 + 1634375353605*x^2 - 2066139456366*x + 3892796082289
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.3
Root			\(4.75406 + 8.23428i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.9
Dual form			38.4.e.b.17.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.347296 - 1.96962i) q^{2} +(6.25156 - 5.24568i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(0.852642 + 0.310336i) q^{5} +(-12.5031 - 10.4914i) q^{6} +(-4.68031 + 8.10654i) q^{7} +(4.00000 + 6.92820i) q^{8} +(6.87632 - 38.9975i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{3} - 12 q^{6} - 33 q^{7} + 72 q^{8} + 42 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.347296	−	1.96962i	−0.122788	−	0.696364i
							\(3\)	6.25156	−	5.24568i	1.20311	−	1.00953i	0.203577	−	0.979059i	\(-0.434743\pi\)
0.999536	−	0.0304727i	\(-0.00970127\pi\)
\(5\)	0.852642	+	0.310336i	0.0762626	+	0.0277573i	0.379870	−	0.925040i	\(-0.375969\pi\)
							−0.303607	+	0.952797i	\(0.598191\pi\)
\(7\)	−4.68031	+	8.10654i	−0.252713	+	0.437712i	−0.964272	−	0.264915i	\(-0.914656\pi\)
							0.711559	+	0.702627i	\(0.247990\pi\)
\(11\)	−15.6469	−	27.1013i	−0.428884	−	0.742849i	0.567890	−	0.823104i	\(-0.307760\pi\)
							−0.996774	+	0.0802555i	\(0.974426\pi\)
\(13\)	54.0093	+	45.3192i	1.15227	+	0.966868i	0.999770	−	0.0214354i	\(-0.00682362\pi\)
							0.152499	+	0.988304i	\(0.451268\pi\)
\(17\)	17.1358	+	97.1819i	0.244473	+	1.38647i	0.821714	+	0.569900i	\(0.193018\pi\)
							−0.577241	+	0.816574i	\(0.695871\pi\)
\(19\)	−70.3584	+	43.6887i	−0.849543	+	0.527520i
							\(23\)	150.848	−	54.9041i	1.36756	−	0.497752i	0.449179	−	0.893442i	\(-0.351717\pi\)
0.918384	+	0.395690i	\(0.129494\pi\)
\(29\)	18.8061	−	106.654i	0.120421	−	0.682939i	−0.863502	−	0.504345i	\(-0.831734\pi\)
							0.983923	−	0.178594i	\(-0.0571549\pi\)
\(31\)	−27.0715	+	46.8893i	−0.156845	+	0.271663i	−0.933729	−	0.357980i	\(-0.883466\pi\)
							0.776884	+	0.629643i	\(0.216799\pi\)
\(37\)	−355.461			−1.57939			−0.789696	−	0.613498i	\(-0.789762\pi\)
							−0.789696	+	0.613498i	\(0.789762\pi\)
\(41\)	−166.023	+	139.310i	−0.632401	+	0.530647i	−0.901674	−	0.432417i	\(-0.857661\pi\)
							0.269273	+	0.963064i	\(0.413217\pi\)
\(43\)	−284.608	−	103.589i	−1.00935	−	0.367375i	−0.216170	−	0.976356i	\(-0.569356\pi\)
							−0.793185	+	0.608981i	\(0.791579\pi\)
\(47\)	61.8275	−	350.641i	0.191882	−	1.08822i	−0.724907	−	0.688847i	\(-0.758117\pi\)
							0.916789	−	0.399372i	\(-0.130772\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.4.e.b.9.3	✓	18
19.6	even	9		722.4.a.t.1.9		9
19.13	odd	18		722.4.a.u.1.1		9
19.17	even	9	inner	38.4.e.b.17.3	yes	18

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.e.b.9.3	✓	18	1.1	even	1	trivial
38.4.e.b.17.3	yes	18	19.17	even	9	inner
722.4.a.t.1.9		9	19.6	even	9
722.4.a.u.1.1		9	19.13	odd	18