Embedded newform 9.22.c.a.4.20

• Label: 9.22.c.a.4.20
• 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			4.20
Character	\(\chi\)	\(=\)	9.4
Dual form			9.22.c.a.7.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1388.16 + 2404.36i) q^{2} +(50822.3 - 88755.0i) q^{3} +(-2.80539e6 + 4.85908e6i) q^{4} +(1.57654e7 - 2.73064e7i) q^{5} +(2.83948e8 - 1.01067e6i) q^{6} +(-2.72501e8 - 4.71986e8i) q^{7} -9.75494e9 q^{8} +(-5.29453e9 - 9.02147e9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 1023 q^{2} + 128841 q^{3} - 19922945 q^{4} + 32234853 q^{5} + 187115031 q^{6} - 189623959 q^{7} - 1296271662 q^{8} + 10117230687 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1388.16	+	2404.36i	0.958570	+	1.66029i	0.725978	+	0.687718i	\(0.241388\pi\)
							0.232593	+	0.972574i	\(0.425279\pi\)
\(3\)	50822.3	−	88755.0i	0.496914	−	0.867800i
							\(5\)	1.57654e7	−	2.73064e7i	0.721970	−	1.25049i	−0.238240	−	0.971206i	\(-0.576570\pi\)
0.960209	−	0.279282i	\(-0.0900963\pi\)
\(7\)	−2.72501e8	−	4.71986e8i	−0.364619	−	0.631538i	0.624096	−	0.781347i	\(-0.285467\pi\)
							−0.988715	+	0.149809i	\(0.952134\pi\)
\(11\)	−4.94006e10	−	8.55644e10i	−0.574261	−	0.994649i	−0.996121	−	0.0879885i	\(-0.971956\pi\)
							0.421860	−	0.906661i	\(-0.361377\pi\)
\(13\)	2.11169e11	−	3.65755e11i	0.424839	−	0.735843i	−0.571566	−	0.820556i	\(-0.693664\pi\)
							0.996405	+	0.0847129i	\(0.0269973\pi\)
\(17\)	8.22612e11			0.0989649			0.0494825	−	0.998775i	\(-0.484243\pi\)
							0.0494825	+	0.998775i	\(0.484243\pi\)
\(19\)	−2.94153e10			−0.00110068			−0.000550339	−	1.00000i	\(-0.500175\pi\)
							−0.000550339		1.00000i	\(0.500175\pi\)
\(23\)	5.19177e12	−	8.99242e12i	0.0261321	−	0.0452620i	−0.852664	−	0.522460i	\(-0.825014\pi\)
							0.878796	+	0.477198i	\(0.158348\pi\)
\(29\)	1.47961e15	+	2.56276e15i	0.653084	+	1.13117i	0.982371	+	0.186944i	\(0.0598583\pi\)
							−0.329287	+	0.944230i	\(0.606808\pi\)
\(31\)	−2.19286e15	+	3.79814e15i	−0.480521	+	0.832287i	−0.999750	−	0.0223478i	\(-0.992886\pi\)
							0.519229	+	0.854635i	\(0.326219\pi\)
\(37\)	4.72350e16			1.61490			0.807450	−	0.589936i	\(-0.200847\pi\)
							0.807450	+	0.589936i	\(0.200847\pi\)
\(41\)	−1.43202e15	+	2.48034e15i	−0.0166617	+	0.0288589i	−0.874236	−	0.485501i	\(-0.838637\pi\)
							0.857574	+	0.514360i	\(0.171971\pi\)
\(43\)	−4.07749e16	−	7.06241e16i	−0.287722	−	0.498350i	0.685543	−	0.728032i	\(-0.259565\pi\)
							−0.973266	+	0.229682i	\(0.926231\pi\)
\(47\)	8.34940e16	+	1.44616e17i	0.231541	+	0.401041i	0.958262	−	0.285892i	\(-0.0922899\pi\)
							−0.726721	+	0.686933i	\(0.758957\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.22.c.a.4.20	✓	40
3.2	odd	2		27.22.c.a.10.1		40
9.2	odd	6		27.22.c.a.19.1		40
9.4	even	3		81.22.a.c.1.1		20
9.5	odd	6		81.22.a.d.1.20		20
9.7	even	3	inner	9.22.c.a.7.20	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.22.c.a.4.20	✓	40	1.1	even	1	trivial
9.22.c.a.7.20	yes	40	9.7	even	3	inner
27.22.c.a.10.1		40	3.2	odd	2
27.22.c.a.19.1		40	9.2	odd	6
81.22.a.c.1.1		20	9.4	even	3
81.22.a.d.1.20		20	9.5	odd	6