Embedded newform 9.22.c.a.4.17

• Label: 9.22.c.a.4.17
• Level: $9$
• Weight: $22$
• Character: 9.4
• Analytic conductor: $25.153$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• 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			4.17
Character	\(\chi\)	\(=\)	9.4
Dual form			9.22.c.a.7.17

$q$-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 \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

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