Embedded newform 9.12.c.a.7.3

• Label: 9.12.c.a.7.3
• Level: $9$
• Weight: $12$
• Character: 9.7
• Analytic conductor: $6.915$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	9.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.91508862504\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial:	x^20 + 9863*x^18 + 40416552*x^16 + 89424581388*x^14 + 116167273852206*x^12 + 90155066123992770*x^10 + 40615438428348034476*x^8 + 9750032167348768665084*x^6 + 1016547159346572680888121*x^4 + 22772285013675084656406191*x^2 + 59348899944549565171293124
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{24}\cdot 3^{45} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			7.3
Root			\(-26.4646i\) of defining polynomial
Character	\(\chi\)	\(=\)	9.7
Dual form			9.12.c.a.4.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-24.4190 + 42.2949i) q^{2} +(129.541 + 400.457i) q^{3} +(-168.574 - 291.979i) q^{4} +(5966.72 + 10334.7i) q^{5} +(-20100.6 - 4299.85i) q^{6} +(19533.0 - 33832.1i) q^{7} -83554.5 q^{8} +(-143585. + 103751. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 33 q^{2} - 12 q^{3} - 9217 q^{4} - 7230 q^{5} + 20583 q^{6} + 8512 q^{7} - 29118 q^{8} + 135504 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−24.4190	+	42.2949i	−0.539589	+	0.934595i	0.459338	+	0.888262i	\(0.348087\pi\)
							−0.998926	+	0.0463329i	\(0.985246\pi\)
\(3\)	129.541	+	400.457i	0.307779	+	0.951458i
							\(5\)	5966.72	+	10334.7i	0.853887	+	1.47898i	0.877674	+	0.479259i	\(0.159095\pi\)
−0.0237863	+	0.999717i	\(0.507572\pi\)
\(7\)	19533.0	−	33832.1i	0.439268	−	0.760834i	−0.558365	−	0.829595i	\(-0.688571\pi\)
							0.997633	+	0.0687610i	\(0.0219046\pi\)
\(11\)	193586.	−	335301.i	0.362421	−	0.627732i	−0.625937	−	0.779873i	\(-0.715283\pi\)
							0.988359	+	0.152141i	\(0.0486168\pi\)
\(13\)	−84319.1	−	146045.i	−0.0629851	−	0.109093i	0.832813	−	0.553554i	\(-0.186729\pi\)
							−0.895798	+	0.444461i	\(0.853395\pi\)
\(17\)	1.08126e7			1.84698			0.923488	−	0.383628i	\(-0.125326\pi\)
							0.923488	+	0.383628i	\(0.125326\pi\)
\(19\)	−9.47464e6			−0.877845			−0.438923	−	0.898525i	\(-0.644640\pi\)
							−0.438923	+	0.898525i	\(0.644640\pi\)
\(23\)	2.90070e6	+	5.02416e6i	0.0939724	+	0.162765i	0.909179	−	0.416405i	\(-0.136710\pi\)
							−0.815207	+	0.579170i	\(0.803377\pi\)
\(29\)	−7.85142e6	+	1.35991e7i	−0.0710819	+	0.123118i	−0.899376	−	0.437176i	\(-0.855979\pi\)
							0.828294	+	0.560294i	\(0.189312\pi\)
\(31\)	−5.53679e6	−	9.59000e6i	−0.0347351	−	0.0601630i	0.848135	−	0.529780i	\(-0.177725\pi\)
							−0.882870	+	0.469617i	\(0.844392\pi\)
\(37\)	3.96041e8			0.938925			0.469462	−	0.882952i	\(-0.344448\pi\)
							0.469462	+	0.882952i	\(0.344448\pi\)
\(41\)	3.81251e8	+	6.60347e8i	0.513925	+	0.890145i	0.999869	+	0.0161550i	\(0.00514253\pi\)
							−0.485944	+	0.873990i	\(0.661524\pi\)
\(43\)	1.72564e8	−	2.98889e8i	0.179008	−	0.310051i	−0.762533	−	0.646949i	\(-0.776044\pi\)
							0.941541	+	0.336898i	\(0.109378\pi\)
\(47\)	−7.83082e8	+	1.35634e9i	−0.498045	+	0.862640i	−0.999997	−	0.00225561i	\(-0.999282\pi\)
							0.501952	+	0.864895i	\(0.332615\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.12.c.a.7.3	yes	20
3.2	odd	2		27.12.c.a.19.8		20
9.2	odd	6		81.12.a.c.1.3		10
9.4	even	3	inner	9.12.c.a.4.3	✓	20
9.5	odd	6		27.12.c.a.10.8		20
9.7	even	3		81.12.a.e.1.8		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.12.c.a.4.3	✓	20	9.4	even	3	inner
9.12.c.a.7.3	yes	20	1.1	even	1	trivial
27.12.c.a.10.8		20	9.5	odd	6
27.12.c.a.19.8		20	3.2	odd	2
81.12.a.c.1.3		10	9.2	odd	6
81.12.a.e.1.8		10	9.7	even	3