Embedded newform 9.15.d.a.5.13

• Label: 9.15.d.a.5.13
• Level: $9$
• Weight: $15$
• Character: 9.5
• Analytic conductor: $11.190$
• Analytic rank: $0$
• Dimension: $26$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	9.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.1896071337\)
Analytic rank:	\(0\)
Dimension:	\(26\)
Relative dimension:	\(13\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			5.13
Character	\(\chi\)	\(=\)	9.5
Dual form			9.15.d.a.2.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(209.734 - 121.090i) q^{2} +(-1566.90 + 1525.71i) q^{3} +(21133.6 - 36604.4i) q^{4} +(-69749.9 - 40270.1i) q^{5} +(-143885. + 509729. i) q^{6} +(-376400. - 651944. i) q^{7} -6.26837e6i q^{8} +(127396. - 4.78127e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 3 q^{2} - 2199 q^{3} + 98303 q^{4} - 107994 q^{5} + 24957 q^{6} - 146330 q^{7} - 5316525 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	209.734	−	121.090i	1.63855	−	0.946015i	0.657213	−	0.753705i	\(-0.271735\pi\)
							0.981334	−	0.192310i	\(-0.0615979\pi\)
\(3\)	−1566.90	+	1525.71i	−0.716462	+	0.697626i
							\(5\)	−69749.9	−	40270.1i	−0.892799	−	0.515458i	−0.0179418	−	0.999839i	\(-0.505711\pi\)
−0.874857	+	0.484381i	\(0.839045\pi\)
\(7\)	−376400.	−	651944.i	−0.457050	−	0.791633i	0.541754	−	0.840537i	\(-0.317761\pi\)
							−0.998803	+	0.0489039i	\(0.984427\pi\)
\(11\)	1.49749e7	−	8.64579e6i	0.768451	−	0.443666i	−0.0638706	−	0.997958i	\(-0.520344\pi\)
							0.832322	+	0.554293i	\(0.187011\pi\)
\(13\)	−2.09803e7	+	3.63390e7i	−0.334355	+	0.579121i	−0.983361	−	0.181663i	\(-0.941852\pi\)
							0.649005	+	0.760784i	\(0.275185\pi\)
\(17\)			4.35245e8i			1.06070i	0.847780	+	0.530349i	\(0.177939\pi\)
							−0.847780	+	0.530349i	\(0.822061\pi\)
\(19\)	9.97460e8			1.11589			0.557944	−	0.829879i	\(-0.311590\pi\)
							0.557944	+	0.829879i	\(0.311590\pi\)
\(23\)	−1.81438e9	−	1.04753e9i	−0.532885	−	0.307661i	0.209305	−	0.977850i	\(-0.432880\pi\)
							−0.742190	+	0.670189i	\(0.766213\pi\)
\(29\)	2.31452e10	−	1.33629e10i	1.34176	−	0.774667i	0.354696	−	0.934982i	\(-0.384584\pi\)
							0.987066	+	0.160315i	\(0.0512510\pi\)
\(31\)	1.04142e10	−	1.80379e10i	0.378524	−	0.655624i	−0.612323	−	0.790607i	\(-0.709765\pi\)
							0.990848	+	0.134984i	\(0.0430983\pi\)
\(37\)	5.64575e10			0.594716			0.297358	−	0.954766i	\(-0.403895\pi\)
							0.297358	+	0.954766i	\(0.403895\pi\)
\(41\)	−1.74088e11	−	1.00509e11i	−0.893883	−	0.516084i	−0.0186725	−	0.999826i	\(-0.505944\pi\)
							−0.875211	+	0.483742i	\(0.839277\pi\)
\(43\)	2.99784e10	+	5.19241e10i	0.110288	+	0.191025i	0.915886	−	0.401437i	\(-0.131489\pi\)
							−0.805598	+	0.592462i	\(0.798156\pi\)
\(47\)	−1.62621e11	+	9.38891e10i	−0.320989	+	0.185323i	−0.651834	−	0.758362i	\(-0.726000\pi\)
							0.330844	+	0.943685i	\(0.392667\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.15.d.a.5.13	yes	26
3.2	odd	2		27.15.d.a.17.1		26
9.2	odd	6	inner	9.15.d.a.2.13	✓	26
9.4	even	3		81.15.b.a.80.26		26
9.5	odd	6		81.15.b.a.80.1		26
9.7	even	3		27.15.d.a.8.1		26

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.15.d.a.2.13	✓	26	9.2	odd	6	inner
9.15.d.a.5.13	yes	26	1.1	even	1	trivial
27.15.d.a.8.1		26	9.7	even	3
27.15.d.a.17.1		26	3.2	odd	2
81.15.b.a.80.1		26	9.5	odd	6
81.15.b.a.80.26		26	9.4	even	3