Embedded newform 9.12.c.a.7.8

• Label: 9.12.c.a.7.8
• 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			7.8
Root			\(31.4671i\) of defining polynomial
Character	\(\chi\)	\(=\)	9.7
Dual form			9.12.c.a.4.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(25.7513 - 44.6025i) q^{2} +(-329.417 + 261.976i) q^{3} +(-302.258 - 523.527i) q^{4} +(4464.02 + 7731.91i) q^{5} +(3201.87 + 21439.1i) q^{6} +(-5876.40 + 10178.2i) q^{7} +74343.1 q^{8} +(39884.4 - 172599. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 33 q^{2} - 12 q^{3} - 9217 q^{4} - 7230 q^{5} + 20583 q^{6} + 8512 q^{7} - 29118 q^{8} + 135504 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	25.7513	−	44.6025i	0.569029	−	0.985586i	−0.427634	−	0.903952i	\(-0.640653\pi\)
							0.996662	−	0.0816343i	\(-0.0260140\pi\)
\(3\)	−329.417	+	261.976i	−0.782671	+	0.622435i
							\(5\)	4464.02	+	7731.91i	0.638839	+	1.10650i	0.985688	+	0.168580i	\(0.0539182\pi\)
−0.346849	+	0.937921i	\(0.612748\pi\)
\(7\)	−5876.40	+	10178.2i	−0.132151	+	0.228893i	−0.924506	−	0.381168i	\(-0.875522\pi\)
							0.792354	+	0.610061i	\(0.208855\pi\)
\(11\)	−342935.	+	593982.i	−0.642026	+	1.11202i	0.342954	+	0.939352i	\(0.388573\pi\)
							−0.984980	+	0.172669i	\(0.944761\pi\)
\(13\)	924126.	+	1.60063e6i	0.690308	+	1.19565i	0.971737	+	0.236066i	\(0.0758581\pi\)
							−0.281429	+	0.959582i	\(0.590809\pi\)
\(17\)	−2.62591e6			−0.448549			−0.224275	−	0.974526i	\(-0.572001\pi\)
							−0.224275	+	0.974526i	\(0.572001\pi\)
\(19\)	−3.45788e6			−0.320380			−0.160190	−	0.987086i	\(-0.551211\pi\)
							−0.160190	+	0.987086i	\(0.551211\pi\)
\(23\)	−1.51007e7	−	2.61551e7i	−0.489208	−	0.847333i	0.510715	−	0.859750i	\(-0.329381\pi\)
							−0.999923	+	0.0124174i	\(0.996047\pi\)
\(29\)	1.10182e8	−	1.90841e8i	0.997521	−	1.72776i	0.437820	−	0.899063i	\(-0.355751\pi\)
							0.559701	−	0.828695i	\(-0.310916\pi\)
\(31\)	7.20445e7	+	1.24785e8i	0.451972	+	0.782838i	0.998508	−	0.0545972i	\(-0.0173875\pi\)
							−0.546537	+	0.837435i	\(0.684054\pi\)
\(37\)	1.39068e8			0.329699			0.164849	−	0.986319i	\(-0.447286\pi\)
							0.164849	+	0.986319i	\(0.447286\pi\)
\(41\)	−3.21762e7	−	5.57308e7i	−0.0433734	−	0.0751249i	0.843524	−	0.537092i	\(-0.180477\pi\)
							−0.886897	+	0.461967i	\(0.847144\pi\)
\(43\)	8.71724e8	−	1.50987e9i	0.904279	−	1.56626i	0.0823973	−	0.996600i	\(-0.473742\pi\)
							0.821882	−	0.569658i	\(-0.192924\pi\)
\(47\)	−5.85483e8	+	1.01409e9i	−0.372371	+	0.644965i	−0.989930	−	0.141559i	\(-0.954788\pi\)
							0.617559	+	0.786525i	\(0.288122\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.12.c.a.7.8	yes	20
3.2	odd	2		27.12.c.a.19.3		20
9.2	odd	6		81.12.a.c.1.8		10
9.4	even	3	inner	9.12.c.a.4.8	✓	20
9.5	odd	6		27.12.c.a.10.3		20
9.7	even	3		81.12.a.e.1.3		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.12.c.a.4.8	✓	20	9.4	even	3	inner
9.12.c.a.7.8	yes	20	1.1	even	1	trivial
27.12.c.a.10.3		20	9.5	odd	6
27.12.c.a.19.3		20	3.2	odd	2
81.12.a.c.1.8		10	9.2	odd	6
81.12.a.e.1.3		10	9.7	even	3