Embedded newform 9.22.c.a.7.17

• Label: 9.22.c.a.7.17
• Level: $9$
• Weight: $22$
• Character: 9.7
• Analytic conductor: $25.153$
• Analytic rank: $0$
• Dimension: $40$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	9.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(25.1529609858\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			7.17
Character	\(\chi\)	\(=\)	9.7
Dual form			9.22.c.a.4.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(991.417 - 1717.18i) q^{2} +(-93811.7 + 40739.6i) q^{3} +(-917239. - 1.58870e6i) q^{4} +(5.12169e6 + 8.87103e6i) q^{5} +(-2.30492e7 + 2.01482e8i) q^{6} +(4.75645e8 - 8.23841e8i) q^{7} +5.20838e8 q^{8} +(7.14093e9 - 7.64370e9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 1023 q^{2} + 128841 q^{3} - 19922945 q^{4} + 32234853 q^{5} + 187115031 q^{6} - 189623959 q^{7} - 1296271662 q^{8} + 10117230687 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	991.417	−	1717.18i	0.684607	−	1.18577i	−0.288953	−	0.957343i	\(-0.593307\pi\)
							0.973560	−	0.228431i	\(-0.0733595\pi\)
\(3\)	−93811.7	+	40739.6i	−0.917242	+	0.398330i
							\(5\)	5.12169e6	+	8.87103e6i	0.234546	+	0.406246i	0.959141	−	0.282929i	\(-0.0913062\pi\)
−0.724594	+	0.689176i	\(0.757973\pi\)
\(7\)	4.75645e8	−	8.23841e8i	0.636434	−	1.10234i	−0.349775	−	0.936834i	\(-0.613742\pi\)
							0.986209	−	0.165503i	\(-0.0529246\pi\)
\(11\)	−4.33867e10	+	7.51480e10i	−0.504352	+	0.873563i	0.495636	+	0.868531i	\(0.334935\pi\)
							−0.999987	+	0.00503232i	\(0.998398\pi\)
\(13\)	−2.62920e11	−	4.55390e11i	−0.528954	−	0.916175i	−0.999430	−	0.0337620i	\(-0.989251\pi\)
							0.470476	−	0.882413i	\(-0.344082\pi\)
\(17\)	−9.08695e12			−1.09321			−0.546606	−	0.837390i	\(-0.684080\pi\)
							−0.546606	+	0.837390i	\(0.684080\pi\)
\(19\)	−8.79971e12			−0.329273			−0.164636	−	0.986354i	\(-0.552645\pi\)
							−0.164636	+	0.986354i	\(0.552645\pi\)
\(23\)	−1.93036e13	−	3.34348e13i	−0.0971620	−	0.168289i	0.813347	−	0.581779i	\(-0.197643\pi\)
							−0.910509	+	0.413490i	\(0.864310\pi\)
\(29\)	1.44890e14	−	2.50957e14i	0.0639528	−	0.110770i	−0.832276	−	0.554361i	\(-0.812963\pi\)
							0.896229	+	0.443592i	\(0.146296\pi\)
\(31\)	−3.02092e15	−	5.23239e15i	−0.661975	−	1.14657i	−0.980096	−	0.198525i	\(-0.936385\pi\)
							0.318121	−	0.948050i	\(-0.396948\pi\)
\(37\)	−4.41709e16			−1.51014			−0.755071	−	0.655642i	\(-0.772398\pi\)
							−0.755071	+	0.655642i	\(0.772398\pi\)
\(41\)	−4.43133e16	−	7.67528e16i	−0.515589	−	0.893026i	−0.999836	−	0.0180947i	\(-0.994240\pi\)
							0.484248	−	0.874931i	\(-0.339093\pi\)
\(43\)	−3.63458e16	+	6.29527e16i	−0.256469	+	0.444217i	−0.965294	−	0.261167i	\(-0.915893\pi\)
							0.708824	+	0.705385i	\(0.249226\pi\)
\(47\)	3.59772e17	−	6.23144e17i	0.997701	−	1.72807i	0.440157	−	0.897921i	\(-0.354923\pi\)
							0.557544	−	0.830147i	\(-0.311744\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.22.c.a.7.17	yes	40
3.2	odd	2		27.22.c.a.19.4		40
9.2	odd	6		81.22.a.d.1.17		20
9.4	even	3	inner	9.22.c.a.4.17	✓	40
9.5	odd	6		27.22.c.a.10.4		40
9.7	even	3		81.22.a.c.1.4		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.22.c.a.4.17	✓	40	9.4	even	3	inner
9.22.c.a.7.17	yes	40	1.1	even	1	trivial
27.22.c.a.10.4		40	9.5	odd	6
27.22.c.a.19.4		40	3.2	odd	2
81.22.a.c.1.4		20	9.7	even	3
81.22.a.d.1.17		20	9.2	odd	6