Embedded newform 9.76.c.a.4.9

• Label: 9.76.c.a.4.9
• Level: $9$
• Weight: $76$
• Character: 9.4
• Analytic conductor: $320.606$
• Analytic rank: $0$
• Dimension: $148$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 76 \)
Character orbit:	\([\chi]\)	\(=\)	9.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(320.605553540\)
Analytic rank:	\(0\)
Dimension:	\(148\)
Relative dimension:	\(74\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			4.9
Character	\(\chi\)	\(=\)	9.4
Dual form			9.76.c.a.7.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.61314e11 - 2.79404e11i) q^{2} +(1.21997e17 + 7.70314e17i) q^{3} +(-3.31548e22 + 5.74259e22i) q^{4} +(-3.63626e25 + 6.29819e25i) q^{5} +(1.95549e29 - 1.58349e29i) q^{6} +(2.74752e31 + 4.75884e31i) q^{7} +9.20481e33 q^{8} +(-5.78500e35 + 1.87952e35i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	148 * q - 137438953473 * q^2 + 785092363818710037 * q^3 - 1378931012997936402399233 * q^4 - 128851949435224459885424655 * q^5 + 538792206380571785448697304391 * q^6 + 2305169592807634547412733660793 * q^7 + 24179496392511120352908521695703106 * q^8 + 330787849704966093938607389501153829 * q^9

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.61314e11	−	2.79404e11i	−0.829940	−	1.43750i	−0.898084	−	0.439823i	\(-0.855041\pi\)
							0.0681442	−	0.997675i	\(-0.478292\pi\)
\(3\)	1.21997e17	+	7.70314e17i	0.156423	+	0.987690i
							\(5\)	−3.63626e25	+	6.29819e25i	−0.223501	+	0.387116i	−0.955869	−	0.293794i	\(-0.905082\pi\)
0.732367	+	0.680910i	\(0.238415\pi\)
\(7\)	2.74752e31	+	4.75884e31i	0.559454	+	0.969002i	0.997542	+	0.0700703i	\(0.0223224\pi\)
							−0.438088	+	0.898932i	\(0.644344\pi\)
\(11\)	8.65847e38	+	1.49969e39i	0.767743	+	1.32977i	0.938784	+	0.344506i	\(0.111953\pi\)
							−0.171042	+	0.985264i	\(0.554713\pi\)
\(13\)	−2.82431e41	+	4.89185e41i	−0.476472	+	0.825273i	−0.999637	−	0.0269583i	\(-0.991418\pi\)
							0.523165	+	0.852231i	\(0.324751\pi\)
\(17\)	1.58910e46			1.14635			0.573174	−	0.819434i	\(-0.305712\pi\)
							0.573174	+	0.819434i	\(0.305712\pi\)
\(19\)	−1.62968e48			−1.81486			−0.907429	−	0.420205i	\(-0.861958\pi\)
							−0.907429	+	0.420205i	\(0.861958\pi\)
\(23\)	4.55207e50	−	7.88441e50i	0.392116	−	0.679165i	−0.600612	−	0.799541i	\(-0.705076\pi\)
							0.992728	+	0.120375i	\(0.0384097\pi\)
\(29\)	6.04419e54	+	1.04688e55i	0.873803	+	1.51347i	0.858033	+	0.513595i	\(0.171687\pi\)
							0.0157701	+	0.999876i	\(0.494980\pi\)
\(31\)	−2.19853e55	+	3.80797e55i	−0.260657	+	0.451471i	−0.966417	−	0.256980i	\(-0.917273\pi\)
							0.705760	+	0.708451i	\(0.250606\pi\)
\(37\)	−5.28964e58			−0.823875			−0.411938	−	0.911212i	\(-0.635148\pi\)
							−0.411938	+	0.911212i	\(0.635148\pi\)
\(41\)	1.77432e60	−	3.07321e60i	0.588352	−	1.01906i	−0.406097	−	0.913830i	\(-0.633110\pi\)
							0.994448	−	0.105225i	\(-0.0335563\pi\)
\(43\)	−1.06813e61	−	1.85005e61i	−0.593684	−	1.02829i	−0.993731	−	0.111796i	\(-0.964340\pi\)
							0.400047	−	0.916495i	\(-0.368994\pi\)
\(47\)	−4.24357e62	−	7.35007e62i	−0.839578	−	1.45419i	−0.890248	−	0.455477i	\(-0.849469\pi\)
							0.0506691	−	0.998715i	\(-0.483865\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.76.c.a.4.9	✓	148
9.7	even	3	inner	9.76.c.a.7.9	yes	148

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.76.c.a.4.9	✓	148	1.1	even	1	trivial
9.76.c.a.7.9	yes	148	9.7	even	3	inner