Embedded newform 9.12.c.a.4.7

• Label: 9.12.c.a.4.7
• Level: $9$
• Weight: $12$
• Character: 9.4
• 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			4.7
Root			\(-15.2468i\) of defining polynomial
Character	\(\chi\)	\(=\)	9.4
Dual form			9.12.c.a.7.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.7041 + 20.2721i) q^{2} +(409.205 + 98.4789i) q^{3} +(750.028 - 1299.09i) q^{4} +(1606.12 - 2781.88i) q^{5} +(2793.01 + 9448.06i) q^{6} +(-185.973 - 322.114i) q^{7} +83053.6 q^{8} +(157751. + 80596.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 33 q^{2} - 12 q^{3} - 9217 q^{4} - 7230 q^{5} + 20583 q^{6} + 8512 q^{7} - 29118 q^{8} + 135504 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.7041	+	20.2721i	0.258627	+	0.447954i	0.965874	−	0.259011i	\(-0.0833967\pi\)
							−0.707248	+	0.706966i	\(0.750063\pi\)
\(3\)	409.205	+	98.4789i	0.972242	+	0.233979i
							\(5\)	1606.12	−	2781.88i	0.229849	−	0.398111i	−0.727914	−	0.685668i	\(-0.759510\pi\)
0.957763	+	0.287558i	\(0.0928433\pi\)
\(7\)	−185.973	−	322.114i	−0.00418225	−	0.00724387i	0.863927	−	0.503618i	\(-0.167998\pi\)
							−0.868109	+	0.496374i	\(0.834665\pi\)
\(11\)	190681.	+	330270.i	0.356984	+	0.618314i	0.987455	−	0.157898i	\(-0.0504719\pi\)
							−0.630472	+	0.776212i	\(0.717139\pi\)
\(13\)	−156096.	+	270366.i	−0.116601	+	0.201959i	−0.918419	−	0.395610i	\(-0.870533\pi\)
							0.801817	+	0.597569i	\(0.203867\pi\)
\(17\)	−3.99667e6			−0.682699			−0.341350	−	0.939936i	\(-0.610884\pi\)
							−0.341350	+	0.939936i	\(0.610884\pi\)
\(19\)	−1.78731e7			−1.65598			−0.827990	−	0.560742i	\(-0.810516\pi\)
							−0.827990	+	0.560742i	\(0.810516\pi\)
\(23\)	2.17979e7	−	3.77550e7i	0.706172	−	1.22313i	−0.260094	−	0.965583i	\(-0.583754\pi\)
							0.966267	−	0.257543i	\(-0.0829130\pi\)
\(29\)	−5.95443e7	−	1.03134e8i	−0.539077	−	0.933709i	−0.998954	−	0.0457264i	\(-0.985440\pi\)
							0.459877	−	0.887983i	\(-0.347894\pi\)
\(31\)	−1.04087e8	+	1.80284e8i	−0.652991	+	1.13101i	0.329403	+	0.944190i	\(0.393153\pi\)
							−0.982393	+	0.186824i	\(0.940181\pi\)
\(37\)	−7.64834e8			−1.81325			−0.906624	−	0.421939i	\(-0.861350\pi\)
							−0.906624	+	0.421939i	\(0.861350\pi\)
\(41\)	−3.33473e8	+	5.77592e8i	−0.449520	+	0.778592i	−0.998355	−	0.0573393i	\(-0.981738\pi\)
							0.548835	+	0.835931i	\(0.315072\pi\)
\(43\)	3.39684e8	+	5.88350e8i	0.352370	+	0.610322i	0.986664	−	0.162769i	\(-0.0520426\pi\)
							−0.634294	+	0.773092i	\(0.718709\pi\)
\(47\)	5.13602e8	+	8.89585e8i	0.326655	+	0.565782i	0.981846	−	0.189681i	\(-0.0607453\pi\)
							−0.655191	+	0.755463i	\(0.727412\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.12.c.a.4.7	✓	20
3.2	odd	2		27.12.c.a.10.4		20
9.2	odd	6		27.12.c.a.19.4		20
9.4	even	3		81.12.a.e.1.4		10
9.5	odd	6		81.12.a.c.1.7		10
9.7	even	3	inner	9.12.c.a.7.7	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.12.c.a.4.7	✓	20	1.1	even	1	trivial
9.12.c.a.7.7	yes	20	9.7	even	3	inner
27.12.c.a.10.4		20	3.2	odd	2
27.12.c.a.19.4		20	9.2	odd	6
81.12.a.c.1.7		10	9.5	odd	6
81.12.a.e.1.4		10	9.4	even	3