Embedded newform 9.22.c.a.4.4

• Label: 9.22.c.a.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	9.4
Dual form			9.22.c.a.7.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-959.466 - 1661.84i) q^{2} +(-29737.5 + 97857.2i) q^{3} +(-792574. + 1.37278e6i) q^{4} +(-5.04427e6 + 8.73693e6i) q^{5} +(1.91156e8 - 4.44715e7i) q^{6} +(-7.19629e8 - 1.24643e9i) q^{7} -9.82502e8 q^{8} +(-8.69171e9 - 5.82006e9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 1023 q^{2} + 128841 q^{3} - 19922945 q^{4} + 32234853 q^{5} + 187115031 q^{6} - 189623959 q^{7} - 1296271662 q^{8} + 10117230687 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−959.466	−	1661.84i	−0.662544	−	1.14756i	−0.979945	−	0.199268i	\(-0.936144\pi\)
							0.317401	−	0.948291i	\(-0.397190\pi\)
\(3\)	−29737.5	+	97857.2i	−0.290758	+	0.956797i
							\(5\)	−5.04427e6	+	8.73693e6i	−0.231001	+	0.400105i	−0.958103	−	0.286424i	\(-0.907533\pi\)
0.727102	+	0.686529i	\(0.240867\pi\)
\(7\)	−7.19629e8	−	1.24643e9i	−0.962896	−	1.66778i	−0.715165	−	0.698956i	\(-0.753648\pi\)
							−0.247731	−	0.968829i	\(-0.579685\pi\)
\(11\)	8.94881e9	+	1.54998e10i	0.104026	+	0.180178i	0.913340	−	0.407198i	\(-0.133494\pi\)
							−0.809314	+	0.587376i	\(0.800161\pi\)
\(13\)	5.83955e9	−	1.01144e10i	0.0117483	−	0.0203486i	−0.860091	−	0.510140i	\(-0.829594\pi\)
							0.871840	+	0.489791i	\(0.162927\pi\)
\(17\)	−8.88490e11			−0.106890			−0.0534452	−	0.998571i	\(-0.517020\pi\)
							−0.0534452	+	0.998571i	\(0.517020\pi\)
\(19\)	−3.75758e13			−1.40603			−0.703016	−	0.711174i	\(-0.748164\pi\)
							−0.703016	+	0.711174i	\(0.748164\pi\)
\(23\)	1.29436e14	−	2.24189e14i	0.651497	−	1.12843i	−0.331263	−	0.943538i	\(-0.607475\pi\)
							0.982760	−	0.184887i	\(-0.0591919\pi\)
\(29\)	1.42399e15	+	2.46643e15i	0.628535	+	1.08865i	0.987846	+	0.155437i	\(0.0496785\pi\)
							−0.359311	+	0.933218i	\(0.616988\pi\)
\(31\)	−2.05944e15	+	3.56705e15i	−0.451284	+	0.781648i	−0.998466	−	0.0553663i	\(-0.982367\pi\)
							0.547182	+	0.837014i	\(0.315701\pi\)
\(37\)	4.25896e15			0.145608			0.0728040	−	0.997346i	\(-0.476805\pi\)
							0.0728040	+	0.997346i	\(0.476805\pi\)
\(41\)	6.26683e16	−	1.08545e17i	0.729150	−	1.26293i	−0.228092	−	0.973639i	\(-0.573249\pi\)
							0.957243	−	0.289286i	\(-0.0934178\pi\)
\(43\)	8.49316e16	+	1.47106e17i	0.599309	+	1.03803i	0.992923	+	0.118757i	\(0.0378911\pi\)
							−0.393615	+	0.919275i	\(0.628776\pi\)
\(47\)	2.78210e16	+	4.81874e16i	0.0771517	+	0.133631i	0.902020	−	0.431694i	\(-0.142084\pi\)
							−0.824868	+	0.565325i	\(0.808751\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.22.c.a.4.4	✓	40
3.2	odd	2		27.22.c.a.10.17		40
9.2	odd	6		27.22.c.a.19.17		40
9.4	even	3		81.22.a.c.1.17		20
9.5	odd	6		81.22.a.d.1.4		20
9.7	even	3	inner	9.22.c.a.7.4	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.22.c.a.4.4	✓	40	1.1	even	1	trivial
9.22.c.a.7.4	yes	40	9.7	even	3	inner
27.22.c.a.10.17		40	3.2	odd	2
27.22.c.a.19.17		40	9.2	odd	6
81.22.a.c.1.17		20	9.4	even	3
81.22.a.d.1.4		20	9.5	odd	6