Embedded newform 9.12.c.a.7.5

• Label: 9.12.c.a.7.5
• 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			7.5
Root			\(-5.09438i\) of defining polynomial
Character	\(\chi\)	\(=\)	9.7
Dual form			9.12.c.a.4.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.91186 + 10.2396i) q^{2} +(-417.157 - 55.9222i) q^{3} +(954.100 + 1652.55i) q^{4} +(-2014.61 - 3489.40i) q^{5} +(3038.79 - 3940.93i) q^{6} +(34855.3 - 60371.1i) q^{7} -46777.0 q^{8} +(170892. + 46656.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 33 q^{2} - 12 q^{3} - 9217 q^{4} - 7230 q^{5} + 20583 q^{6} + 8512 q^{7} - 29118 q^{8} + 135504 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.91186	+	10.2396i	−0.130635	+	0.226266i	−0.923921	−	0.382582i	\(-0.875035\pi\)
							0.793287	+	0.608848i	\(0.208368\pi\)
\(3\)	−417.157	−	55.9222i	−0.991134	−	0.132867i
							\(5\)	−2014.61	−	3489.40i	−0.288307	−	0.499362i	0.685099	−	0.728450i	\(-0.259759\pi\)
−0.973406	+	0.229088i	\(0.926426\pi\)
\(7\)	34855.3	−	60371.1i	0.783843	−	1.35766i	−0.145845	−	0.989307i	\(-0.546590\pi\)
							0.929688	−	0.368348i	\(-0.120077\pi\)
\(11\)	418627.	−	725084.i	0.783732	−	1.35746i	−0.146021	−	0.989281i	\(-0.546647\pi\)
							0.929753	−	0.368183i	\(-0.120020\pi\)
\(13\)	−31447.4	−	54468.4i	−0.0234907	−	0.0406871i	0.854041	−	0.520206i	\(-0.174145\pi\)
							−0.877532	+	0.479518i	\(0.840811\pi\)
\(17\)	−6.17741e6			−1.05521			−0.527603	−	0.849491i	\(-0.676909\pi\)
							−0.527603	+	0.849491i	\(0.676909\pi\)
\(19\)	8.86431e6			0.821297			0.410648	−	0.911794i	\(-0.365302\pi\)
							0.410648	+	0.911794i	\(0.365302\pi\)
\(23\)	−1.43226e7	−	2.48074e7i	−0.463999	−	0.803670i	0.535157	−	0.844753i	\(-0.320253\pi\)
							−0.999156	+	0.0410827i	\(0.986919\pi\)
\(29\)	6.76828e7	−	1.17230e8i	0.612759	−	1.06133i	−0.378015	−	0.925800i	\(-0.623393\pi\)
							0.990773	−	0.135530i	\(-0.0432736\pi\)
\(31\)	6.39152e7	+	1.10704e8i	0.400973	+	0.694505i	0.993844	−	0.110793i	\(-0.0353389\pi\)
							−0.592871	+	0.805297i	\(0.702006\pi\)
\(37\)	7.85725e7			0.186278			0.0931389	−	0.995653i	\(-0.470310\pi\)
							0.0931389	+	0.995653i	\(0.470310\pi\)
\(41\)	−2.95238e8	−	5.11367e8i	−0.397979	−	0.689321i	0.595497	−	0.803357i	\(-0.296955\pi\)
							−0.993477	+	0.114037i	\(0.963622\pi\)
\(43\)	−3.13404e8	+	5.42831e8i	−0.325108	+	0.563104i	−0.981534	−	0.191287i	\(-0.938734\pi\)
							0.656426	+	0.754390i	\(0.272067\pi\)
\(47\)	−5.80615e8	+	1.00566e9i	−0.369275	+	0.639604i	−0.989452	−	0.144858i	\(-0.953727\pi\)
							0.620177	+	0.784462i	\(0.287061\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.12.c.a.7.5	yes	20
3.2	odd	2		27.12.c.a.19.6		20
9.2	odd	6		81.12.a.c.1.5		10
9.4	even	3	inner	9.12.c.a.4.5	✓	20
9.5	odd	6		27.12.c.a.10.6		20
9.7	even	3		81.12.a.e.1.6		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.12.c.a.4.5	✓	20	9.4	even	3	inner
9.12.c.a.7.5	yes	20	1.1	even	1	trivial
27.12.c.a.10.6		20	9.5	odd	6
27.12.c.a.19.6		20	3.2	odd	2
81.12.a.c.1.5		10	9.2	odd	6
81.12.a.e.1.6		10	9.7	even	3