Embedded newform 9.13.d.a.2.3

• Label: 9.13.d.a.2.3
• Level: $9$
• Weight: $13$
• Character: 9.2
• Analytic conductor: $8.226$
• Analytic rank: $0$
• Dimension: $22$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	9.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.22594435549\)
Analytic rank:	\(0\)
Dimension:	\(22\)
Relative dimension:	\(11\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2.3
Character	\(\chi\)	\(=\)	9.2
Dual form			9.13.d.a.5.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-64.6760 - 37.3407i) q^{2} +(721.758 + 102.503i) q^{3} +(740.658 + 1282.86i) q^{4} +(-13863.9 + 8004.33i) q^{5} +(-42852.9 - 33580.4i) q^{6} +(4312.89 - 7470.15i) q^{7} +195268. i q^{8} +(510427. + 147964. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 618 q^{3} + 20479 q^{4} - 15987 q^{5} + 67725 q^{6} + 34319 q^{7} - 315126 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−64.6760	−	37.3407i	−1.01056	−	0.583449i	−0.0992064	−	0.995067i	\(-0.531630\pi\)
							−0.911356	+	0.411618i	\(0.864964\pi\)
\(3\)	721.758	+	102.503i	0.990065	+	0.140607i
							\(5\)	−13863.9	+	8004.33i	−0.887290	+	0.512277i	−0.873055	−	0.487622i	\(-0.837865\pi\)
−0.0142348	+	0.999899i	\(0.504531\pi\)
\(7\)	4312.89	−	7470.15i	0.0366590	−	0.0634952i	−0.847114	−	0.531412i	\(-0.821662\pi\)
							0.883773	+	0.467916i	\(0.154995\pi\)
\(11\)	2.32788e6	+	1.34400e6i	1.31403	+	0.758655i	0.982761	−	0.184882i	\(-0.0591903\pi\)
							0.331268	+	0.943537i	\(0.392524\pi\)
\(13\)	87758.2	+	152002.i	0.0181814	+	0.0314911i	0.874973	−	0.484172i	\(-0.160879\pi\)
							−0.856792	+	0.515663i	\(0.827546\pi\)
\(17\)			3.43297e7i			1.42225i	0.703064	+	0.711126i	\(0.251815\pi\)
							−0.703064	+	0.711126i	\(0.748185\pi\)
\(19\)	4.52574e7			0.961984			0.480992	−	0.876725i	\(-0.340277\pi\)
							0.480992	+	0.876725i	\(0.340277\pi\)
\(23\)	−9.11261e7	+	5.26117e7i	−0.615568	+	0.355398i	−0.775141	−	0.631788i	\(-0.782321\pi\)
							0.159574	+	0.987186i	\(0.448988\pi\)
\(29\)	−5.00518e8	−	2.88974e8i	−0.841457	−	0.485816i	0.0163020	−	0.999867i	\(-0.494811\pi\)
							−0.857759	+	0.514051i	\(0.828144\pi\)
\(31\)	3.29226e8	+	5.70236e8i	0.370957	+	0.642517i	0.989713	−	0.143066i	\(-0.0456963\pi\)
							−0.618756	+	0.785583i	\(0.712363\pi\)
\(37\)	−3.60533e9			−1.40519			−0.702594	−	0.711591i	\(-0.747975\pi\)
							−0.702594	+	0.711591i	\(0.747975\pi\)
\(41\)	−5.05766e9	+	2.92004e9i	−1.06475	+	0.614732i	−0.926741	−	0.375700i	\(-0.877402\pi\)
							−0.138005	+	0.990432i	\(0.544069\pi\)
\(43\)	5.53613e9	−	9.58886e9i	0.875781	−	1.51690i	0.0198525	−	0.999803i	\(-0.493680\pi\)
							0.855928	−	0.517094i	\(-0.172986\pi\)
\(47\)	1.17967e10	+	6.81081e9i	1.09439	+	0.631847i	0.934742	−	0.355327i	\(-0.115630\pi\)
							0.159649	+	0.987174i	\(0.448964\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.13.d.a.2.3	✓	22
3.2	odd	2		27.13.d.a.8.9		22
9.2	odd	6		81.13.b.a.80.6		22
9.4	even	3		27.13.d.a.17.9		22
9.5	odd	6	inner	9.13.d.a.5.3	yes	22
9.7	even	3		81.13.b.a.80.17		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.13.d.a.2.3	✓	22	1.1	even	1	trivial
9.13.d.a.5.3	yes	22	9.5	odd	6	inner
27.13.d.a.8.9		22	3.2	odd	2
27.13.d.a.17.9		22	9.4	even	3
81.13.b.a.80.6		22	9.2	odd	6
81.13.b.a.80.17		22	9.7	even	3