Embedded newform 9.12.c.a.7.10

• Label: 9.12.c.a.7.10
• 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			7.10
Root			\(49.9602i\) of defining polynomial
Character	\(\chi\)	\(=\)	9.7
Dual form			9.12.c.a.4.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(41.7668 - 72.3422i) q^{2} +(296.011 + 299.207i) q^{3} +(-2464.93 - 4269.38i) q^{4} +(-2698.81 - 4674.47i) q^{5} +(34008.7 - 8917.18i) q^{6} +(23240.9 - 40254.5i) q^{7} -240732. q^{8} +(-1902.15 + 177137. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 33 q^{2} - 12 q^{3} - 9217 q^{4} - 7230 q^{5} + 20583 q^{6} + 8512 q^{7} - 29118 q^{8} + 135504 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	41.7668	−	72.3422i	0.922924	−	1.59855i	0.128058	−	0.991767i	\(-0.459126\pi\)
							0.794866	−	0.606785i	\(-0.207541\pi\)
\(3\)	296.011	+	299.207i	0.703300	+	0.710893i
							\(5\)	−2698.81	−	4674.47i	−0.386222	−	0.668956i	0.605716	−	0.795681i	\(-0.292887\pi\)
−0.991938	+	0.126725i	\(0.959553\pi\)
\(7\)	23240.9	−	40254.5i	0.522654	−	0.905263i	−0.476999	−	0.878904i	\(-0.658275\pi\)
							0.999653	−	0.0263591i	\(-0.00839134\pi\)
\(11\)	119491.	−	206965.i	0.223705	−	0.387469i	−0.732225	−	0.681063i	\(-0.761518\pi\)
							0.955930	+	0.293594i	\(0.0948514\pi\)
\(13\)	1.23615e6	+	2.14108e6i	0.923387	+	1.59935i	0.794134	+	0.607742i	\(0.207925\pi\)
							0.129253	+	0.991612i	\(0.458742\pi\)
\(17\)	3.32998e6			0.568818			0.284409	−	0.958703i	\(-0.408203\pi\)
							0.284409	+	0.958703i	\(0.408203\pi\)
\(19\)	3.63246e6			0.336555			0.168278	−	0.985740i	\(-0.446180\pi\)
							0.168278	+	0.985740i	\(0.446180\pi\)
\(23\)	−2.74059e6	−	4.74683e6i	−0.0887851	−	0.153780i	0.818213	−	0.574916i	\(-0.194965\pi\)
							−0.906998	+	0.421135i	\(0.861632\pi\)
\(29\)	−2.21997e7	+	3.84510e7i	−0.200983	+	0.348112i	−0.948845	−	0.315741i	\(-0.897747\pi\)
							0.747863	+	0.663853i	\(0.231080\pi\)
\(31\)	6.32869e7	+	1.09616e8i	0.397031	+	0.687678i	0.993358	−	0.115063i	\(-0.0367071\pi\)
							−0.596327	+	0.802742i	\(0.703374\pi\)
\(37\)	−5.73034e8			−1.35854			−0.679268	−	0.733891i	\(-0.737702\pi\)
							−0.679268	+	0.733891i	\(0.737702\pi\)
\(41\)	−5.96616e7	−	1.03337e8i	−0.0804236	−	0.139298i	0.823008	−	0.568029i	\(-0.192294\pi\)
							−0.903432	+	0.428732i	\(0.858961\pi\)
\(43\)	−6.48081e8	+	1.12251e9i	−0.672284	+	1.16443i	0.304971	+	0.952362i	\(0.401353\pi\)
							−0.977255	+	0.212068i	\(0.931980\pi\)
\(47\)	7.66251e8	−	1.32719e9i	0.487341	−	0.844099i	−0.512553	−	0.858655i	\(-0.671300\pi\)
							0.999894	+	0.0145566i	\(0.00463367\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.12.c.a.7.10	yes	20
3.2	odd	2		27.12.c.a.19.1		20
9.2	odd	6		81.12.a.c.1.10		10
9.4	even	3	inner	9.12.c.a.4.10	✓	20
9.5	odd	6		27.12.c.a.10.1		20
9.7	even	3		81.12.a.e.1.1		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.12.c.a.4.10	✓	20	9.4	even	3	inner
9.12.c.a.7.10	yes	20	1.1	even	1	trivial
27.12.c.a.10.1		20	9.5	odd	6
27.12.c.a.19.1		20	3.2	odd	2
81.12.a.c.1.10		10	9.2	odd	6
81.12.a.e.1.1		10	9.7	even	3